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Kohlberg and Mertens use Theorem 3 to show that each game has at least one compo nent of equilibria that does not vanish entirely when the payoffs of the game are slightly perturbed, a result that we will further discuss in Section 4. We now move on to show that the graph E is really simple as generically (i.e., except on a closed set of games with measure zero) the equilibrium correspondence consists of a finite (odd) number of differentiable functions. We proceed in the spirit of Harsanyi (1973a), but follow the more elegant elaboration of Ritzberger (1994). At the end of the subsection, we briefly discuss some related recent work that provides a more general perspective. Obviously, if s is a Nash equilibrium of g, then s is a solution to the following system of equations
s; (a;) [u; (s\a;)- u; (s) ] = 0
all i
E
/,
a; E A;.
(2. 1 5)
(The system (2. 1 5 ) also admits solutions that are not equilibria - for example, any pure strategy vector is a solution - but this fact need not bother us at present.) For each player i, one equation in (2. 15) is redundant; it is automatically satisfied if the others are. If we select, for each player i, one strategy a; E A; and delete the corresponding equation, we are left with m = L; [A; I I equations. Similarly we can delete the vari able s; (a;) for each i as it can be recovered from the constraint that probabilities add up to one. Hence, (2. 1 5 ) reduces to a system of m equations with m unknowns. Taking each pair (i, a) with i E I and a E A;\ {a; } as a coordinate, we can view S as a subset of JR.m and the left-hand side of (2. 1 5) as a mapping from S to JR.m , hence -
(2. 16) Write of(s) for the Jacobian matrix of partial derivates of f evaluated at s and [of(s)[ for its determinant. We say that s is a regular equilibrium of g if Iaf (s) I =/::. o, hence, if the Jacobian is nonsingular. The reader easily checks that for all i E I and a; E A;, if s; (a;) 0, then u; (s\a;) - u; (s) is an eigenvalue of of(s), hence, it follows that a regular equilibrium is necessarily quasi-strict. Furthermore, if s is a strict equilibrium, the above observation identifies m (hence, all) eigenvalues, so that any strict equilibrium is regular. A straightforward application of the implicit function theorem yields that, if s* is a regular equilibrium of a game g*, there exist neighborhoods U of g* in r and =
Ch. 41:
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V of s* in S and a continuous map s : U---+ V with s(g*) = s* and {s(g)} = E(g) n V for all g E U. Hence, if s* is a regular equilibrium of g*, then around (g*, s*) the equilibrium graph E looks like a continuous curve. By using Sard's theorem (in the manner initiated in Debreu ( 1 970)) Harsanyi showed that for almost all normal form games all equilibria are regular. Formally, the proof proceeds by constructing a subspace f of T and a polynomial map� cp : f X S ---+ T with the following properties (where g denotes the projection of g in T): (1) cp (g, s) = g if s E E(g); (2) 1acp(g, s) 1 = 0 if and only if laf(s) I = 0 . Hence, if s is an irregular equilibrium o f g, then g is a critical value o f cp and Sard's theorem guarantees that the set of such critical values has measure zero. (For further details we refer to Harsanyi (1 973a) and Van Damme ( 1 987a).) We summarize the above discussion in the following theorem. THEOREM 4 [Harsanyi ( 1 973a)] . Almost all normal form games are regular, that is, they have only regular equilibria. Around a regular game, the equilibrium correspon dence consists of a finite number of continuous functions. Any strict equilibrium is reg ular and any regular equilibrium is quasi-strict. Note that Theorem 4 may be of limited value for games given originally in extensive form. Any such nontrivial extensive form gives rise to a strategic form that is not in gen eral position, hence, that is not regular. We will return to generic properties associated with extensive form games in Section 4. We will now show that the finiteness mentioned in Theorem 4 can be strengthened to oddness. Again we trace the footsteps of Harsanyi ( 1 973a) with minor modifications as suggested by Ritzberger (1994), a paper that in tum builds on Dierker ( 1 972). Consider a regular game g and add to it a logarithmic penalty term so that the payoff to i resulting from s becomes
u f (s) = u; (s)+8
L ln s; (a; )
a;EA;
(i E I, s E S).
(2. 1 7)
Obviously, an equilibrium of this game has to be in completely mixed strategies. (Since the payoff function is not multilinear, (2. 10) and (2. 1 2) are no longer equivalent; by an equilibrium we mean a strategy vector satisfying (2. 1 2) with u; replaced by u f . It fol lows easily from Kakutani's theorem that an equilibrium exists.) Hence, the necessary and sufficient conditions for equilibrium are given by the first order conditions: (2. 1 8) Because of the regularity of g, g has finitely many equilibria, say s 1 , . . . , s K . The im plicit function theorem tells us that for small 8, system (2. 1 8) has at least K solutions {sk (8) }_f= 1 with s k (8) ---+ s k as 8 ---+ 0. In fact there must be exactly K solutions for
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E. van Damme
small £: because of regularity there cannot be two solution curves converging to the same s k , and if a solution curve remained bounded away from the set {s 1 , . . . , s K }, then it would have a cluster point and this would be an equilibrium of g. However, the latter is impossible since we have assumed g to be regular. Hence, if £ is small, J 8 has ex actly as many zero's as g has equilibria. An application of the Poincare-Hopf theorem for manifolds with boundary shows that each r has an odd number of zero's, hence, g has an odd number of equilibria. (To apply the Poincare-Hopf theorem, take a smooth approximation to the boundary of S, for example,
{
S(8) = s E S;
fl si (a ) ): 8 all i } .
aiEA i
i
(2. 19)
Then the Euler characteristic of S (8) is equal to 1 and, for fixed £ , if 8 is sufficiently small, J8 points outward at the boundary of S(8).) To summarize, we have shown: THEOREM 5 [Harsanyi ( 1973a), Wilson (1971), Rosenmtiller ( 1 97 1 )] .
gic fo rm games have an odd number o f equilibria.
Generic strate
Ritzberger notes that actually we can say a little more. Recall that the index of a zero s of f is defined as the sign of the determinant I B f (s) l . By the Poincare-Hopf theorem and the continuity of the determinant
L
sEE(g)
sgn J B f (s) J
=
1.
(2.20)
It is easily seen that the index of a pure equilibrium is + 1. Hence, if there are l pure equilibria, there must be at least l - 1 equilibria with index - 1 , and these must be mixed. This latter result was also established in Gul et al. ( 1993). In this paper, the authors construct a map g from the space of mixed strategies S into itself such that s is a fixed point of g if and only if s is a Nash equilibrium. They define an equilibrium s to be regular if it is quasi-strict and if det(I - g' (s)) -=/= 0. Using the result that the sum of the Lefschetz indices of the fixed points of a Lefschetz function is + 1 and the observation that a pure equilibrium has index + 1 , they obtain their result that a regular game that has k pure equilibria must have at least k - 1 mixed ones. The authors also show that almost all games have only regular equilibria. Recall that already Nash (195 1 ) worked with a function f of which the fixed points correspond with the equilibria of the game. (See also the remark immediately below Theorem 1 .) Nash's function is, however, different from that of Gul et al. ( 1993), and different from the function that we worked with in (2. 15). This raises the question of whether the choice of the function matters. In recent work, Govindan and Wilson (2000) show that the answer is no. These authors define a Nash map as a continuous func tion f : r X S -?> S that has the property that for each fixed game g the induced map fg : S _,. S has as its fixed points the set of Nash equilibria of g. Given such a Nash map,
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the index ind(C, f) of a component C of Nash equilibria of g is defined in the usual way [see Dold ( 1 972)] . The main result of Govindan and Wilson (2000) states that for any two Nash maps f, f' and any component C we have ind(C, f) = ind(C, f'). Fur thermore, if the degree of a component, deg(C) , is defined as the local degree of the projection map from the graph E of the equilibrium correspondence to the space of games (cf. Theorem 3), then ind(C, f) = deg(C) [see Govindan and Wilson ( 1 997)].
2.4. Computation of equilibria: The 2-person case The papers of Rosenmtiller and Wilson mentioned in the previous theorem proved the generic oddness of the number of equilibria of a strategic form game in a completely different way than we did. These papers generalized the Lemke and Howson (1964) al gorithm for the computation of equilibria in bimatrix games to n-person games. Lemke and Howson had already established the generic oddness of the number of equilibria for bimatrix games and the only difference between the 2-person case and the n-person case is that in the latter the pivotal steps involve nonlinear computations rather than the linear ones in the 2-person case. In this subsection we restrict ourselves to 2-person games and briefly outline the Lemke-Howson algorithm, thereby establishing another proof for Theorem 5 in the 2-person case. The discussion will be based upon Shapley ( 1974). Let g = (A , u) be a 2-person game. The nondegeneracy condition that we will use to guarantee that the game is regular is
I C (s ) l � I B (s ) l
for all s E S.
(2.21)
This condition is clearly satisfied for almost all bimatrix games and indeed ensures that all equilibria are regular. We write L (s;) for the set of "labels" associated with s; E S; (2.22) If m; = I Ai l , then, by (2.21), the number of labels if s; is at most m; . We will be interested in the set Nt of those s; that have exactly m; labels. This set is finite: the regularity condition (2. 2 1 ) guarantees that for each set L c A1 U Az with ILl = m1 there is at most one s; E s, such that L(st) = L. Hence, the labelling identifies the strategy, so that the word label is appropriate. If s; E N; \A; , then for each a; E L(s;) there exists (because of (2.21)) a unique ray in S; emanating at s; of points s; with L(s[) = L (s;)\{a; } , and moving in the direction of this ray we find a new point s;' E N; after a finite distance. A similar remark applies to s; E N; n A1 , except that in that case we cannot eliminate the label corresponding to Bj (s;). Consequently, we can construct a graph T; with node set N; that has m; edges (of points s; with IL(s[) l = m; - 1 ) originating from each node in N; \A; and that has m; 1 edges originating from each node in Nt n A; . We say that two nodes are adj acent if they are connected by an edge, hence, if they differ by one label. -
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E. van Damme
Now consider the "product graph" T in the product set S: the set of nodes is N = x N2 and two nodes s, s' are adjacent if for some i, s; = s; while for j =I= i we have that SJ and sj are adjacent in Nj . For s E S, write L(s) = L (si) U L (s2 ) . Obviously, we have that L(s) = A1 U A2 if and only if s is a Nash equilibrium of g. Hence, equilibria correspond to fully labelled strategy vectors and the set of such vectors will be denoted by E. The regularity assumption (2.2 1) implies that E c N, hence, E is a finite set. For a E A 1 U A2 write Na for the set of s E N that miss at most the label a. The observations made above imply the following fundamental lemma:
N1
LEMMA 1 . (i) If s E E, s; = a, then s is adjacent to no node in Na. (ii) If s E E, s; =I= a, then s is adjacent to exactly one node in Na. (iii) If s E Na\E, s; = a, then s is adjacent to exactly one node in Na. (iv) If s E Na\E, s; =I= a, then s is adjacent to exactly two nodes in Na. PROOF. (i) In this case s is a pure and strict equilibrium, hence, any move away from s eliminates labels other than a. (ii) If s i s a pure equilibrium, then the only move that eliminates only the label a is to increase the probability of a in T; . If s; is mixed, then (2.2 1 ) implies that si is mixed as well. We either have s; (a) 0 or a E B; (si ) . In the first case the only move that eliminates only label a is one in T; (increase the probability of a), in the second case it is the unique move in TJ away from the region where a is a best response. (iii) The only possibility that this case allows is s = (a, b) with b being the unique best response to a. Hence, if a' is the unique best response against b, the a' is the unique action that is labelled twice. The only possible move to an adjacent point in Na is to increase the probability of a' in T; . (iv) Let b be the unique action that is labelled by both SJ and s2 , hence {b} = L (s 1 ) n L (s2 ) . Note that s; is mixed. If s1 is mixed as well, then we can either drop b from L (si) in T; or drop b from L (s2 ) in TJ . This yields two different possibilities and these are the only ones. If si is pure, then b E A; and the same 0 argument applies. =
The lemma now implies that an equilibrium can be found by tracing a path of almost completely labelled strategy vectors in Na, i.e., vectors that miss at most a. Start at the pure strategy pair (a, b) where b is the best response to a. If a is also the best response to b, we are done. If not, then we are in case (iii) of the lemma and we can follow a unique edge in Na starting at (a, b). The next node s we encounter is one satisfying either condition (ii) of the lemma (and then we are done) or condition (iv). In the latter case, there are two edges of Na at s. We came in via one route, hence there is only one way to continue. Proceeding in similar fashion, we encounter distinct nodes of type (iv)
Ch. 41:
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1537
until we finally hit upon a node of type (ii). The latter must eventually happen since N a has finitely many nodes. The lemma also implies that the number of equilibria is odd. Consider an equilib rium s' different from the one found by the above construction. Condition (ii) from the lemma guarantees that this equilibrium is connected to exactly one node in N a as in condition (iv) of the lemma. We can now repeat the above constructive process until we end up at yet another equilibrium s". Hence, all equilibria, except the distinguished one constructed above, appear in pairs: the total number of equilibria is odd. Note that the algorithm described in this subsection offers no guarantee to find more than one equilibrium, let alone to find all equilibria. Shapley (1981) discusses a way of transforming the paths so as to get access to some of the previously inaccessible equilibria.
2.5. Purification of mixed strategy equilibria In Section 2.1 we noted that mixed strategies can be interpreted both as acts of deliberate randomization as well as representations of players' beliefs. The former interpretation seems intuitively somewhat problematic; it may be hard to accept the idea of making an important decision on the basis of a toss of a coin. Mixed strategy equilibria also seem unstable: to optimize his payoff a player does not need to randomize; any pure strategy in the support is equally as good as the equilibrium strategy itself. The only reason a player randomizes is to keep the other players in equilibrium, but why would a player want to do this? Hence, equilibria in mixed strategies seem difficult to interpret [Aumann and Maschler ( 1 972), Rubinstein ( 1 99 1 )] . Harsanyi ( 1973a) was the first to discuss the more convincing alternative interpreta tion of a mixed strategy of player i as a representation of the ignorance of the opponents as to what player i is actually going to do. Even though player i may follow a deter ministic rule, the opponents may not be able to predict i 's actions exactly, since i 's decision might depend on information that the opponents can only assess probabilisti cally. Harsanyi argues that each player always has a tiny bit of private information about his own payoffs and he modifies the game accordingly. Such a slightly perturbed game admits equilibria in pure strategies and the (regular) mixed equilibria of the original unperturbed game may be interpreted as the limiting beliefs associated with these pure equilibria of the perturbed games. In this subsection we give Harsanyi's construction and state and illustrate his main result. Let g = (A, u) be an /-person normal form game and, for each i E / , let X; be a random vector taking values in JR. A . Let X = (X; )i E / and assume that different compo nents of X are stochastically independent. Let F; be the distribution function of X; and assume that F; admits a continuously differentiable density f; that is strictly positive on some ball 8; around zero in JR A (and 0 outside that ball). For c: > 0, write g 8 (X) for the game described by the following rules: (i) nature draws x from X, (ii) each player i is informed about his component x; ,
1538
E. van Damme
(iii) simultaneously and independently each player i selects an action a; E A;, (iv) each player i receives the payoff u; (a) + ex; (a), where a i s the action combina tion resulting from (iii). Note that, if e is small, a player's payoff is close to the payoff from g with probability approximately 1 . What a player will do in g8 (X) depends on his observation and on his beliefs about what the opponents will do. Note that these beliefs are independent of his observation and that, no matter what the beliefs might be, the player will be indifferent between two pure actions with probability zero. Hence, we may assume that each player i restricts himself to a pure strategy in g 8 (X), i.e., to a map a; : G; --+ A; . (If a player is indifferent, he himself does not care what he does and his opponents do not care since they attach probability zero to this event.) Given a strategy vector a 8 in g 8 (X) and a; E A; write G�' (a 8 ) for the set of observations where a18 prescribes to play a; . If a player j =/= i believes i is playing a;" , then the probability that j assigns to i choosing a; is
sf (a;)
=
1
a
e, ' (rr8)
d F;
.
(2.23)
The mixed strategy vector s8 E S determined by (2.23) will be called the vector of beliefs associated with the strategy vector a 8 • Note that all opponents j of i have the same beliefs about player i since they base themselves on the same information. The strategy combination a 8 is an equilibrium of g8 (X) if, for each player i, it assigns an optimal action at each observation, hence (2.24) We can now state Harsanyi's theorem THEOREM 6 [Harsanyi ( l 973b)] . Let g be a regular normalform game and let the equi
libria be s 1 , . . . , s K . Then, for sufficiently small e, the game gt: (X) has exactly K equi librium belief vectors, say s 1 (e), . . . , s K (e), and these are such that lime--+osk (e) = sk for all k. Furthermore, the equilibrium ak (e) underlying the belief vector sk (e) can be taken to be pure. We will illustrate this theorem by means of a simple example, the game from Figure 1 . (The "t" stands for "tough", the " w " for "weak", the game is a variation of the battle of the sexes.) For analytical simplicity, we will perturb only one payoff for each player, as indicated in Figure 1 . The unperturbed game g ( e = 0 in Figure 1 ) has 3 equilibria, (t1 , w 2) , ( w 1 , t2) and a mixed equilibrium in which each player i chooses t; with probability s; = 1 - u i (i =/= j ) . The pure equilibria are strict, hence, it is easily seen that they can be approxi mated by equilibrium beliefs of the perturbed games in which the players have private information: if e is small, then (t; , w i ) is a strict equilibrium of gt: (x1 , x2) for a set of
Ch. 41:
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0, 0
Figure 1. A perturbed game g8 (XJ x2) (0 < u 1 , u 2 ,
<
1).
(x 1 , x2 ) -values with large probability. Let us show how the mixed equilibrium of g can be approximated. If player i assigns probability sj to j playing tj , then he prefers to play ti if and only if
1 - sj > Ui + 8Xi .
(2.25)
Writing Fi for the distribution of X; we have that the probability that j assigns to the event (2.25) is F; ((sj - u;)/8), hence, to have an equilibrium of the perturbed game we must have
sf = F; ((1 - sj - u;)/8) , i, j E { 1 , 2} , i # J . Writing
(2 . 26)
G; for the inverse of F; , we obtain the equivalent conditions
1 - sj - u; - 8G; (sf ) = O , i, j E { 1 , 2}, i # J .
(2.27)
For 8 = 0, the system of equations has the regular, completely mixed equilibrium of g as a solution, hence, the implicit function theorem implies that, for 8 sufficiently small, there is exactly one solution (sf , sD of (2.27) with sf --+ 1 - u j as 8 --+ 0. These beliefs are the ones mentioned in Theorem 6. A corresponding pure equilibrium strategy for each player i is: play w; if x; :::;; (1 - sj - u;) /8 and play b; otherwise. For more results on purification of mixed strategy equilibria, we refer to Aumann et al. ( 1 983), Milgram and Weber ( 1 985) and Radner and Rosenthal ( 1 982). These papers consider the case where the private signals that players receive do not influence the payoffs and they address the question of how much randomness there should be in the environment in order to enable purification. In Section 5 we will show that completely different results are obtained if players make common noisy observations on the entire game: in this case even some strict equilibria cannot be approximated.
3. Backward induction equilibria in extensive form games
Selten ( 1 965) pointed out that, in extensive form games, not every Nash equilibrium can be considered self-enforcing. Selten's basic example is similar to the game g from Figure 2, which has (/1 , h) and (rt , r2 ) as its two pure Nash equilibria. The equilibrium (!1 , l2 ) is not self-enforcing. Since the game is noncooperative, player 2 has no ability
1540
E. van Damme (3 , 1 )
(0, 0)
( 1 , 3) 2
Figure 2. A Nash equilibrium that is not self-enforcing.
to commit himself to !2 . If he is actually called upon to move, player 2 strictly prefers to play r2, hence, being rational, he will indeed play r2 in that case. Player 1 can foresee that player 2 will deviate to r2 if he himself deviates to r 1 , hence, it is in the interest of player 1 to deviate from an agreement on (l1 , h). Only an agreement on (r 1 , r2 ) is self-enforcing. Being a Nash equilibrium, (lJ , l2 ) has the property that no player has an incentive to deviate from it if he expects the opponent to stick to this strategy pair. The example, however, shows that player 1 's expectation that player 2 will abide by an agreement on (! 1 , !2 ) is nonsensical. For a self-enforcing agreement we should not only require that no player can profitably deviate if nobody else deviates, we should also require that the expectation that nobody deviates be rational. In this section we discuss several solution concepts, refinements of Nash equilibrium, that have been proposed as formalizations of this requirement. In particular, attention is focussed on sequential equilibria [Kreps and Wilson ( 1982a)] and on perfect equilibria [Selten (1975)]. Along the way we will also discuss Myerson's ( 1 978) notion of proper equilibrium. First, however, we introduce some basic concepts and notation related to extensive form games. 3. 1.
Extensive form and related normalforms
Throughout, attention will be confined tofinite extensive form games with perfect recall. Such a game g is given by (i) a collection 1 of players, (ii) a game tree K specifying the physical order of play, (iii) for each player i a collection H; of information sets specifying the information a player has when he has to move. Hence H; is a partition of the set of decision points of player i in the game and if two nodes x and y are in the same element h of the partition H; , then i cannot distinguish between x and y , (iv) for each information set h, a specification of the set of choices Ch that are fea sible at that set, (v) a specification of the probabilities associated with chance moves, and (vi) for each end point z of the tree and each player i a payoff u; (z) that player i receives when z is reached.
Ch. 41:
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1541
For formal definitions, we refer to Selten (1975), Kreps and Wilson (1982a) or Hart ( 1 992). For an extensive form game g we write g = (r, u) where r specifies the struc tural characteristics of the game and u gives the payoffs. r is called a game form. The set of all games with game form r can be identified with an I I I x I Z I Euclidean space, where I is the player set and Z the set of end points. The assumption of perfect recall, saying that no player ever forgets what he has known or what he has done, implies that each H; is a partially ordered set. A local strategy Si h of player i at h E H; is a probability distribution on the set ch of choices at this information set h. It is interpreted as a plan for what i will do at h or as the beliefs of the opponents of what i will do at that information set. Note that the latter interpretation assumes that different players hold the same beliefs about what i will do at h and that these beliefs do not change throughout the game. A behavior strategy s; of player i assigns a local strategy s;h to each h E H; . We write S;h for the set of local strategies at h and S; for the set of all behavior strategies of player i . A behavior strategy s; is called pure if it associates a pure action at each h E H; and the set of all these strategies is denoted A; . A behavior strategy combination s = (s 1 , . . . , s1) specifies a behavior strategy for each player i . The probability distribution p5 that s induces on Z is called the outcome of s. Two strategies s; and s[' of player i are said to be realization equivalent if ps \s; =
p5 \5i" for each strategy combination s , i.e., if they induce the same outcomes against any strategy profile of the opponents. Player i 's expected payoff associated with s is u; (s) = Lz p5 (z)u; (z). If x is a node of the game tree, then p� denotes the probability distribution that results on Z when the game is started at x with strategies s and U ix (s) denotes the associated expectation of u; . If every information set h of g that contains a node y after x actually has all its nodes after x , then that part of the tree of g that comes after x is a game of its own. It is called the subgame of g starting at x . The normal form associated with g is the normal form game ( A , u) which has the same player set, the same sets of pure strategies and the same payoff functions as g has. A mixed strategy from the normal form induces a behavioral strategy in the ex tensive form and Kuhn's (1953) theorem for games with perfect recall guarantees that, conversely, for every mixed strategy, there exists a behavior strategy that is realization equivalent to it. [See Hart (1 992) for more details.] Note that the normal form frequently contains many realization equivalent pure strategies for each player: if the information set h E H; is excluded by player i 's own strategy, then it is "irrelevant" what the strat egy prescribes at h. The game that results from the normal form if we replace each equivalence class (of realization equivalent) pure strategies by a representative from that class, will be called the semi-reduced normalform. Working with the semi-reduced normal form implies that we do not specify playerj 's beliefs about what i will do at an information set h E H; that is excluded by i 's own strategy. The agent normal form associated with g is the normal form game ( C, u) that has a player i h associated with every information set h of each player i in g. This player i h has the set Ch of feasible actions as his pure strategy set and his payoff function is the payoff of the player i to whom he belongs. Hence, if Ci h E ch for each h E U ; H;,
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E. van Damme
then s = (c; h ) ih is a (pure) strategy combination in g and we define Uih (s) = u; (s) for h E Hi . The agent normal form was first introduced in Selten ( 1 975). It provides a lo cal perspective, it decentralizes the strategy decision of player i into a number of local decisions. When planning his decision for h, the player does not necessarily assume that he is in full control of the decision at an information set h' E Hi that comes af ter h, but he is sure that the player/agent making the decision at that stage has the same objectives as he has. Hence, a player is replaced by a team of identically motivated agents. Note that a pure strategy combination is a Nash equilibrium of the agent normal form if and only if it is a Nash equilibrium of the normal form. Because of perfect re call, a similar remark applies to equilibria that involve randomization, provided that we identify strategies that are realization equivalent. Hence, we may define a Nash equi librium of the extensive form as a Nash equilibrium of the associated (agent) normal form and obtain (2. 1 2) as the defining equations for such an equilibrium. It follows from Theorem 1 that each extensive form game has at least one Nash equilibrium. Theorems 2 and 3 give information about the structure of the set of Nash equilibria of extensive form games. Kreps and Wilson proved a partial generalization of Theo rem 4: THEOREM 7 [Kreps and Wilson ( 1982a)] . Let T be any game form. Then, for almost all u, the extensive form game (T, u} has.finitely many Nash equilibrium outcomes (i.e., the set {ps (u): s is a Nash equilibrium of (T, u}} is finite) and these outcomes depend continuously on u. Note that in this theorem, finiteness cannot be strengthened to oddness: any extensive form game with the same structure as in Figure 2 and with payoffs close to those in Figure 2 has lr and (rr , rz) as Nash equilibrium outcomes. Hence, Theorem 5 does not hold for extensive form games. Little is known about whether Theorem 6 can be extended to classes of extensive form games. However, see Fudenberg et al. ( 1988) for results concerning various forms of payoff uncertainty in extensive form games. Before moving on to discuss some refinements in the next subsections, we briefly mention some coarsenings of the Nash concept that have been proposed for extensive form games. Pearce (1984), Battigalli ( 1997) and Bi:irgers ( 1991) propose concepts of extensive form rationalizability. Some of these also aim to capture some aspects of for ward induction (see Section 4). Fudenberg and Levine ( 1 993a, 1 993b) and Rubinstein and Wolinsky (1994) introduce, respectively, the concepts of "self-confirming equilib ria" and of "rationalizable conjectural equilibria" that impose restrictions that are in between those of Nash equilibrium and rationalizability. These concepts require play ers to hold identical and correct beliefs about actions taken at information sets that are on the equilibrium path, but allow players to have different beliefs about oppo nents' play at information sets that are not reached. Hence, in such an equilibrium, if players only observe outcomes, no player will observe play that contradicts his predic tions.
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3.2. Subgame perfect equilibria The Nash equilibrium condition (2. 1 2) requires that each player's strategy be optimal from the ex ante point of view. Ex ante optimality implies that the strategy is also ex post optimal at each information set that is reached with positive probability in equilib rium, but, as the game of Figure 2 illustrates, such ex post optimality need not hold at the unreached information sets. The example suggests imposing ex post optimality as a nec essary requirement for self-enforcingness but, of course, this requirement is meaningful only when conditional expected payoffs are well-defined, i.e., when the information set is a singleton. In particular, the suggestion is feasible for games with perfect informa tion, i.e., games in which all information sets are singletons, and in this case one may require as a condition for s* to be self-enforcing that it satisfies (3. 1) Condition (3 . 1 ) states that at no decision point h can a player gain by deviating from s* if after h no other player deviates from s * . Obviously, equilibria satisfying (3 . 1 ) can be found by rolling back the game tree in a dynamic programming fashion, a procedure already employed in Zermelo ( 1 9 1 2). It is, however, also worthwhile to remark that al ready in von Neumann and Morgenstern ( 1 944) it was argued that this backward induc tion procedure was not necessarily justified as it incorporates a very strong assumption of "persistent" rationality. Recently, Hart (1999) has shown that the procedure may be justified in an evolutionary setting. Adopting Zermelo's procedure one sees that, for per fect information games, there exists at least one Nash equilibrium satisfying (3 . 1 ) and that, for generic perfect information games, (3 . 1 ) selects exactly one equilibrium. Fur thermore, in the latter case, the outcome of this equilibrium is the unique outcome that survives iterated elimination of weakly dominated strategies in the normal form of the game. (Each elimination order leaves at least this outcome and there exists a sequence of eliminations that leaves nothing but this outcome; cf. Moulin (1979).) Selten ( 1 978) was the first paper to show that the solution determined by (3 . 1 ) may be hard to accept as a guide to practical behavior. (Of course, it was already known for a long time that in some games, such as chess, playing as (3 . 1 ) dictates may be infea sible since the solution s* cannot be computed.) Selten considered the finite repetition of the game from Figure 2, with one player 2 playing the game against a sequence of different players in each round and with players always being perfectly informed about the outcomes in previous rounds. In the story that Selten associates with this game, player 2 is the owner of a chain store who is threatened by entry in each of finitely many towns. When entry takes place (rr is chosen), the chain store owner either acquiesces (chooses r2 ) or fights entry (chooses h). The backward induction solution has players play (rr , r2 ) in each round, but intuitively, we expect player 2 to behave aggressively (choose h) at the beginning of the game with the aim of inducing later entrants to stay out. The chain store paradox is the paradox that even people who accept the logical va lidity of the backward induction reasoning somehow remain unconvinced by it and do
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not act in the manner that it prescribes, but rather act according to the intuitive solution. Hence, there is an inconsistency between plausible human behavior and game-theoretic reasoning. Selten's conclusion from the paradox is that a theory of perfect rationality may be of limited relevance for actual human behavior and he proposes a theory of lim ited rationality to resolve the paradox. Other researchers have argued that the paradox may be caused more by the inadequacy of the model than by the solution concept that is applied to it. Our intuition for the chain store game may derive from a richer game in which the deterrence equilibrium indeed is a rational solution. Such richer models have been constructed in Kreps and Wilson ( 1 982b), Milgram and Roberts ( 1 982) and Au mann ( 1 992). These papers change the game by allowing a tiny probability that player 2 may actually find it optimal to fight entry, which has the consequence that, when the game still lasts for a long time, player 2 will always play as if it were optimal to fight entry which forces player 1 to stay out. The cause of the chain store paradox is the assumption of persistent rationality that underlies (3 . 1 ) , i.e., players are forced to believe that even at information sets h that can be reached only by many deviations from s*, behavior will be in accordance with s*. This assumption that forces a player to believe that an opponent i s rational even after he has seen the opponent make irrational moves has been extensively discussed and criticized in the literature, with many contributions being critical [see, for example, Basu ( 1 988, 1 990), Ben-Porath ( 1993), Binmore ( 1 987), Reny ( 1992a, 1 992b, 1 993) and Rosenthal (198 1)]. Binmore argues that human rationality may differ in systematic ways from the perfect rationality that game theory assumes, and he urges theorists to build richer models that incorporate explicit human thinking processes and that take these systematic deviations into account. Reny argues that (3 . 1) assumes that there is common knowledge of rationality throughout the game, but that this assumption is self contradicting: once a player has "shown" that he is irrational (for example, by playing a strictly dominated move), rationality can no longer be common knowledge and solution concepts that build on this assumption are no longer appropriate. Aumann and Bran denburger (1995) however argue that Nash equilibrium does not build on this common knowledge assumption. Reny ( 1 993), on the other hand, concludes from the above that a theory of rational behavior cannot be developed in a context that does not allow for irrational behavior, a conclusion similar to the one also reached in Selten ( 1 975) and Aumann ( 1987). Aumann ( 1 995), however, disagrees with the view that the assumption of common knowledge of rationality is impossible to maintain in extensive form games with perfect information. As he writes, "The aim of this paper is to present a coherent formulation and proof of the principle that in PI games, common knowledge of ratio nality implies backward induction" (p. 7) (see also Aumann (1998) for an application to Rosenthal's centipede game; the references in that paper provide further information, also on other points of view). We now leave this discussion on backward induction in games with perfect informa tion and move on to discuss more general games. Selten ( 1965) notes that the argument leading to (3 . 1) can be extended beyond the class of games with perfect information. If the game g admits a subgame y , then the expected payoffs of s* in y depend only on
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what s* prescribes in y . Denote this restriction of s* to y by s; . Once the subgame y is reached, all other parts of the game have become strategically irrelevant, hence, Selten argues that, for s* to be self-enforcing, it is necessary that s ; be self-enforcing for every subgame y . Selten defined a sub game perfect equilibrium as an equilibrium s* of g that induces a Nash equilibrium s; in each subgame y of g and he proposed subgame perfection as a necessary requirement for self-enforcingness. Since every equilibrium of a subgame of a finite game can be "extended" to an equilibrium of the overall game, it follows that every finite extensive form game has at least one subgame perfect equi librium. Existence is, however, not as easily established for games in which the strategy spaces are continuous. In that case, not every subgame equilibrium is part of an overall equilib rium: players moving later in the game may be forced to break ties in a certain way, in order to guarantee that players who moved earlier indeed played optimally. (As a simple example, let player 1 first choose x E [0, 1 ] and let then player 2, knowing x, choose y E [0, 1 ] . Payoffs are give by u 1 (x, y) = xy and u z (x , y) = (1 - x)y. In the unique subgame perfect equilibrium both players choose 1 even though player 2 is completely indifferent when player 1 chooses x = 1 .) Indeed, well-behaved continuous extensive form games need not have a subgame perfect equilibrium, as Harris et al. (1995) have shown. However, these authors also show that, for games with almost perfect infor mation ("stage" games), existence can be restored if players can observe a common random signal before each new stage of the game which allows them to correlate their actions. For the special case where information is perfect, i.e., information sets are sin gletons, Harris (1985) shows that a subgame perfect equilibrium does exist even when correlation is not possible [see also Hellwig et al. ( 1 990)] . Other chapters of this Handbook contain ample illustrations of the concept of sub game perfect equilibrium, hence, we will not give further examples. It suffices to remark here that subgame perfection is not sufficient for self-enforcingness, as is illustrated by the game from Figure 3 . The left-hand side of Figure 3 illustrates a game where player 1 first chooses whether or not to play a 2 x 2 game. If player 1 chooses r 1 , both players are informed that r 1 has been chosen and that they have to play the 2 x 2 game. This 2 x 2 game is a subgame of the overall game and it has (t, [z) as its unique equilibrium. Consequently,
( 2, 2 ) ll
b
12
r2
3, 1
1,0
0, 1
O, x
rl
ll
2, 2
r1t
3, 1
1, 0
r1b 0, 1
O, x
2, 2
Figure 3. Not all subgame perfect equilibria are self-enforcing.
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(TI t, h) is the unique subgame perfect equilibrium. The game on the right is the (semi reduced) normal form of the game on the left. The only difference between the games is tha� in the normal form, player 1 chooses simultaneously between / 1 , rt t and T] b and that player 2 does not get to hear that player 1 has not chosen f t . However, these changes appear inessential since player 2 is indifferent between h and r2 when player 1 chooses / 1 . Hence, it would appear that an equilibrium is self-enforcing in one game only if it is self-enforcing in the other. However, the sets of subgame perfect equilibria of these games differ. The game on the right does not admit any proper subgames so that the Nash equilibrium (1 1 , r2 ) is trivially subgame perfect. 3.3.
Perfect equilibria
We have seen that Nash equilibria may prescribe irrational, non-maximizing behavior at unreached information sets. Selten ( 197 5) proposes to eliminate such non-self-enforcing equilibria by eliminating the possibility of unreached information sets. He proposes to look at complete rationality as a limiting case of incomplete rationality, i.e., to assume that players make mistakes with small vanishing probability and to restrict attention to the limits of the corresponding equilibria. Such equilibria are called (trembling hand) perfect equilibria. Formally, for an extensive form game g, Selten (1975) assumes that at each informa tion set h E H; player i will, with a small probability t:h > 0, suffer from "momentary insanity" and make a mistake. Note that t:h is assumed not to depend on the intended action at h . If such a mistake occurs, player i 's behavior is assumed to be governed by some unspecified psychological mechanism which results in each choice c at h occur ring with a strictly positive probability ah (c). Selten assumes each of these probabilities t:h and ah(c) (h E H;, c E Ch) to be independent of each other and also to be indepen dent of the corresponding probabilities of the other players. As a consequence of these assumptions, if a player i intends to play the behavior strategy s; , he will actually play the behavior strategy s:·" given by (3 .2) Obviously, given these mistakes all information sets are reached with positive probabil ity. Furthermore, if players intend to play s, then, given the mistake technology specified by (t:, a), each player i will at each information set h intend to choose a local strategy Si h that satisfies (3 .3) If (3.3) is satisfied by s;h = s;h at each h E U ; H; (i.e., if the intended action optimizes the payoff taking the constraints into account), then s is said to be an equilibrium of the perturbed game g 8 ·" . Hence, (3 .3) incorporates the assumption of persistent rationality. Players try to maximize whenever they have to move, but each time they fall short of the
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ideal. Note that the definitions have been chosen to guarantee that s is an equilibrium of 8 (J g · if and only if s is an equilibrium of the corresponding perturbation of the agent nor mal form of g . A straightforward application of Kakutani 's fixed point theorem yields that each perturbed game has at least one equilibrium. Selten ( 1 975) then defines s to be a peifect equilibrium of g if there exist sequences ck, uk of mistake probabilities (ck > 0, ck -+ 0) and mistake vectors a-A (c) > 0 and an associated sequence sk with sk k
k
being an equilibrium of the perturbed game g 8 , (J such that sk -+ s as k -+ oo . Since the set of strategy vectors is compact, it follows that each game has at least one perfect equilibrium. It may also be verified that s is a perfect equilibrium of g if and only if there exists a sequence sk of completely mixed behavior strategies (s� (c ) > 0 for all i , h, c , k ) that converges to s as k -+ oo , such that su, i s a local best reply against any element in the sequence, i.e., (3.4) Note that for s to be perfect, it is sufficient that s can be rationalized by some sequence of vanishing trembles, it is not necessary that s be robust against all possible trembles. In the next section we will discuss concepts that insist on such stronger stability. We will also encounter concepts that require robustness with respect to specific sequences of trembles. For example, Harsanyi and Selten's ( 1 988) concept of uniformly perfect equilibria is based on the assumption that all mistakes are equally likely. In contrast, Myerson's ( 1 978) properness concept builds on the assumption that mistakes that are more costly are much less likely. It is easily verified that each perfect equilibrium is subgame perfect. The converse is not true: in the game on the right of Figure 3 with x � 1 , player 2 strictly prefers to play l2 if player 1 chooses r1 t and r1 b by mistake, hence, only (r 1 t, l2 ) is perfect. However, since there are no subgames, (l 1 , r2 ) is subgame perfect. By definition, the perfect equilibria of the extensive form game g are the perfect equilibria of the agent normal form of g. However, they need not coincide with the perfect equilibria of the associated normal form. Applying the above definitions to the normal form shows that s is a perfect equilibrium of a normal form game g = (A, u) if there exists a sequence of completely mixed strategy profiles sk with sk -+ s such that s E B(sk) for all k, i.e., (3.5) Hence, we claim that the global conditions (3.5) may determine a different set of solu tions than the local conditions (3.4). As a first example, consider the game from Fig ure 4. In the extensive form, player 1 is justified to choose L if he expects himself, at his second decision node, to make mistakes with a larger probability than player 2 does. Hence, the outcome ( 1 , 2) is perfect in the extensive form. In the normal form, how ever, Rlt is a strategy that guarantees player 1 the payoff 1 . This strategy dominates all
1548
E. van Damme (1, 2 ) l2
(0, 0) r2
2 L
( 1, 1 )
(0, 0)
ll
rl
R
L
1, 2
0, 0
Rl 1
1, 1
1, 1
Rr1
0, 0 0, 0
Figure 4. A perfect equilibrium of the extensive form need not be perfect in the normal form.
others, so that perfectness forces player 1 to play it, hence, only the outcome ( 1 , 1) is perfect in the normal form. Motivated by the consideration that a player may be more concerned with mistakes of others than with his own, Van Damme ( 1984) introduces the concept of a quasi-perfect equilibrium. Here each player follows a strategy that at each node specifies an action that is optimal against mistakes of other players, keeping the player's own strategy fixed throughout the game. Mertens (1992) has argued that this concept of "quasi-perfect equilibria" is to be preferred above "extensive form perfect equilibria". (We will return to the concept below.) Conversely, we have that a perfect equilibrium of the normal form need not even be subgame perfect in the extensive form. The game from Figure 3 with x > 1 provides an example. Only the outcome (3 , 1 ) is subgame perfect in the extensive form. In the normal form, player 2 is justified in playing r2 if he expects that player 1 is (much) more likely to make the mistake r 1 b than to make the mistake r 1 t. Hence, (! 1 r2 ) is a perfect equilibrium in the normal form. Note that in both examples there is at least one equilibrium that is perfect in both the extensive and the normal form. Mertens (1 992) discusses an example in which the sets of perfect equilibria of these game forms are disjoint: the normal form game has a dominant strategy equilibrium, but this equilibrium is not perfect in the extensive form of the game. It follows from (3.5) that a perfect equilibrium strategy of a normal form game cannot be weakly dominated. (Strategy s; is said to be weakly dominated by s;' if ui (s\s;') � ui (s\s;) for all s and Ui (s\s;') > ui (s\s;) for some s.) Equilibria in un dominated strategies are not necessarily perfect, but an application of the separating hyperplane theorem shows that the two concepts coincide in the 2-person case [Van Damme ( 1 983)] . (In the general case a strategy Si is not weakly dominated if and only if it is a best reply against a completely mixed correlated strategy of the oppo nents.) Before summarizing the discussion from this section in a theorem we note that games in which the strategy spaces are continua and payoffs are continuous need not have equilibria in undominated strategies. Consider the 2-player game in which each player i chooses Xi from [0, 1 /2] and in which ui (x) = Xi if Xi � xJ /2 and ui (x) = Xj ( 1 - Xi) / 2 x ; otherwise. Then the unique equilibrium is x = 0, but this is in dominated strate gies. We refer to Simon and Stinchcombe (1995) for definitions of perfectness concepts for continuous games. ,
-
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THEOREM 8 [Selten ( 1 975)] . Every game has at least one peifect equilibrium. Every extensive form perfect equilibrium is a subgame peifect equilibrium, hence, a Nash equilibrium. An equilibrium of an extensive form game is peifect if and only if it is peifect in the associated agent normal form. A peifect equilibrium of the normal form need not be peifect in the extensive form and also the converse need not be true. Every peifect equilibrium of a strategic form game is in undominated strategies and, in 2-per son normalform games, every undominated equilibrium is peifect. 3.4.
Sequential equilibria
Kreps and Wilson ( 1982a) propose to eliminate irrational behavior at unreached infor mation sets in a somewhat different way than Selten does. They propose to extend the applicability of (3 . 1 ) by explicitly specifying beliefs (i.e., conditional probabilities) at each information set so that posterior expected payoffs can always be computed. Hence, whenever a player reaches an information set, he should, in conformity with Bayesian decision theory, be able to produce a probability distribution on the nodes in that set that represents his uncertainty. Of course, players' beliefs should be consistent with the strategies actually played (i.e., beliefs should be computed from Bayes' rule whenever possible) and they should respect the structure of the game (i.e., if a player has es sentially the same information at h as at h', his beliefs at these sets should coincide). Kreps and Wilson ensure that these two conditions are satisfied by deriving the beliefs from a sequence of completely mixed strategies that converges to the strategy profile in question. Formally, a system of beliefs fL is defined as a map that assigns to each information set h E U ; H; a probability distribution /Lh on the nodes in that set. The interpretation is that, when h E H; is reached, player i assigns a probability /Lh (x) to each node x in h. The system of beliefs fL i s said to be consistent with the strategy profile s if there exists k a sequence s of completely mixed behavior strategies (s�h (c) > 0 for all i, h, k, c) with k s --+ s as k --+ oo such that fL
k p5 (x fh) h (x ) = khm --+ oo •
for all h,
x,
(3.6)
k
where p5 (x fh) denotes the (well-defined) conditional probability that x is reached k given that h is reached and s is played. Write u 0, (s) for player i 's expected payoff at h associated with s and fL, hence u i17 (s) = LxEh fLh (x ) u;x (s ) , where u;x is as de fined in Section 3 . 1 . The profile s is said to be sequentially rational given fL if
u 0, (s) � u �,{s\s;)
all i,
h, s; .
(3.7)
An assessment (s, tL) is said to be a sequential equilibrium if fL is consistent with s and if s is sequentially rational given fL. Hence, the difference between perfect equi libria and sequential equilibria is that the former concept requires ex post optimality
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approaching the limit, while the latter requires this only at the limit. Roughly speaking, perfectness amounts to sequentiality plus admissibility (i.e., the prescribed actions are not locally dominated). Hence, if s is perfect, then there exists some fJ such that (s, fJ) is a sequential equilibrium, but the converse does not hold: in a normal form game every Nash equilibrium is sequential, but not every Nash equilibrium is perfect. The differ ence between the concepts is only marginal: for almost all games the concepts yield the same outcomes. The main innovation of the concept of sequential equilibrium is the explicit incorporation of the system of beliefs sustaining the strategies as part of the definition of equilibrium. In this, it provides a language for discussing the relative plau sibility of various systems of beliefs and the associated equilibria sustained by them. This language has proved very effective in the discussion of equilibrium refinements in games with incomplete information [see, for example, Kreps and Sobel (1 994)] . We summarize the above remarks in the following theorem. (In it, we abuse the language somewhat: s E S is said to be a sequential equilibrium if there exists some fJ such that (s, JJ) is sequential.)
Every perfect equi librium is sequential and every sequential equilibrium is subgame perfect. For any game structure r we have thatfor almost all games (T, u) with that structure the sets ofper fect and sequential equilibria coincide. For such generic payoffs u, the set of perfect equilibria depends continuously on u. THEOREM 9 [Kreps and Wilson (1 982a), Blume and Zame (1994)].
Let us note that, if the action spaces are continua, and payoffs are continuous, a sequential equilibrium need not exist. A simple example is the following signalling game [Van Damme (1 987b)]. Nature first selects the type t of player 1 , t E {0, 2} with both possibilities being equally likely. Next, player 1 chooses x E [0, 2] and thereafter player 2, knowing x but not knowing t, chooses y E [0, 2] . Payoffs are u 1 (t, x , y) = (x - t)(y - t) and u 2 (t, x , y) = (1 - x)y. If player 2 does not choose y = 2 - t at x = 1 , then type t of player 1 does not have a best response. Hence, there is at least one type that does not have a best response, and a sequential equilibrium does not exist. In the literature one finds a variety of solution concepts that are related to the se quential equilibrium notion. In applications it might be difficult to construct an approx imating sequence as in (3 .6), hence, one may want to work with a more liberal con cept that incorporates just the requirement that beliefs are consistent with s whenever possible, hence !Jh(x) = ps (s [h) whenever ps (h) > 0. Combining this condition with the sequential rationality requirement (3.7) we obtain the concept of perfect Bayesian equilibrium which has frequently been applied in dynamic games with incomplete in formation. Some authors have argued that in the context of an incomplete information game, one should impose a support restriction on the beliefs: once a certain type of a player is assigned probability zero, the probability of this type should remain at zero for the remainder of the game. Obviously, this restriction comes in handy when doing back ward induction. However, the restriction is not compelling and there may exist no Nash
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equilibria satisfying it [see Madrigal et al. ( 1 987), Neyman and Van Damme ( 1990)]. For further discussions on variations of the concept of perfect Bayesian equilibrium, the reader is referred to Fudenberg and Tirole ( 1 99 1 ) . Since the sequential rationality requirement (3.7) has already been discussed exten sively in Section 3 .2, there is no need to go into detail here. Rather let us focus on the consistency requirement (3.6). When motivating this requirement, Kreps and Wilson re fer to the intuitive idea that when a player reaches an information set h with ps (h) = 0, ' he reassesses the game, comes up with an alternative hypothesis s ' (with p s (h) > 0) about how the game is played and then constructs his beliefs at h from s ' . A system of beliefs is called structurally consistent if it can be constructed in this way. Kreps and Wilson claimed that consistency, as in (3.6), implies structural consistency, but this claim was shown to be incorrect in Kreps and Ramey (1987): there may not exist an equilibrium that can be sustained by beliefs that are both consistent and structurally consistent. At first sight this appears to be a serious blow to the concept of sequential equilibrium, or at least to its motivation. However, the problem may be seen to lie in the idea of reassessing the game, which is not intuitive at all. First of all, it goes counter to the idea of rational players who can foresee the play in advance: they would have to reassess at the start. Secondly, interpreting strategy vectors as beliefs about how the game will be played implies there is no reassessment: all agents have the same beliefs about the behavior of each agent. Thirdly, the combination of structural consistency with the sequential rationality requirement (3 .7) is problematic: if player i believes at h that s ' is played, shouldn't he then optimize against s ' rather then against s? Of course, rejecting structural consistency leaves us with the question of whether an al ternative justification for (3 . 6) can be given. Kohlberg and Reny ( 1997) provide such a natural interpretation of consistency by relying on the idea of consistent probability systems. 3. 5.
Proper equilibria
In Section 3 . 1 we have seen that perfectness in the normal form is not sufficient to guar antee (subgame) perfectness in the extensive form. This observation raises the question of whether backward induction equilibria (say sequential equilibria) from the extensive form can already be detected in the normal form of the game. This question is important since it might be argued that, since a game is nothing but a collection of simultaneous individual decision problems, all information that is needed to solve these problems is already contained in the normal form of the game. The criteria for self-enforcingness in the normal form are no different from those in the extensive form: if the opponents of player i stick to s, then the essential information for i 's decision problem is contained in this normal form: if i decides to deviate from s at a certain information set h, he can already plan that deviation beforehand, hence, he can deviate in the normal form. It turns out that the answer to the opening question is yes: an equilibrium that is proper in the normal form induces a sequential equilibrium outcome in every extensive form with that normal form.
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Proper equilibria were introduced in Myerson ( 1 978) with the aim of eliminating certain deficiencies in Selten's perfectness concept. One such deficiency is that adding strictly dominated strategies may enlarge the set of perfect equilibria. As an example, consider the game from the right-hand side of Figure 3 with the strategy r 1 b elimi nated. In this 2 x 2 game only (r 1 t, b) is perfect. If we then add the strictly dominated strategy q b, the equilibrium (!1 , r2) becomes perfect. But, of course, strictly dominated strategies should be irrelevant; they cannot determine whether or not an outcome is self enforcing. Myerson argues that, in Figure 3, player 2 should not believe that the mistake q b is more likely than r1 t. On the contrary, since r1 t dominates r1 b, the mistake r1 b is more severe than the mistake r 1 t; player 1 may be expected to spend more effort at preventing it and as a consequence it will occur with smaller probability. In fact, Myer son's concept of proper equilibrium assumes such a more costly mistake to occur with a probability that is of smaller order. Formally, for a normal form game {A, u) and some £ > 0, a strategy vector s 8 E S is said to be an £-proper equilibrium if it is completely mixed (i.e., sf (a;) > 0 for all i , all a; E A;) and satisfies (3.8) A strategy vector s E S is a proper equilibrium if it is a limit, as £ --+ 0, of a sequence 5 of £-proper equilibria. Myerson ( 1 978) shows that each strategic form game has at least one proper equilib rium and it is easily seen that any such equilibrium is perfect. Now, let g be an extensive form game with semi-reduced normal form n(g) and, for £ --+ 0, let s 5 be an 8-proper equilibrium of n (g) with s 8 --+ s as 8 --+ 0. Since s c is completely mixed, it induces a completely mixed behavior strategy s8 in g. Let s = limc -+ 0 s8 • Then s is a behavior strategy vector that induces the same outcome as s does, p ' = ps . (Note that s need not induce a full behavior strategy vector; as s was defined in the semi-reduced nor mal form, it does not necessarily specify a unique action at information sets that are excluded by the players themselves.) Condition (3.8) now implies that at each informa tion set h, s; assigns positive probability only to the pure actions at h that maximize the local payoff at h against s8 . Namely, if c is a best response at h and c' is not, then for each pure strategy in the normal form that prescribes to play c' there exists a pure strategy that prescribes to play c and that performs strictly better against s 8 • (Take strategies that differ only at h.) Condition (3 . 8) then implies that in the normal form the total probability of the set of strategies choosing c' is of smaller order than the total probability of choosing c, hence, the limiting behavior strategy assigns probability 0 to c ' . Hence, we have shown that each player always maximizes his local payoff, taking the mistakes of opponents into account. In other words, using the terminology of Van Damme ( 1 984), the profile s is a quasi-perfect equilibrium. By the same argument, s is a sequential equilibrium. Formally, let f.Lc be the system of beliefs associated with s8 and let f.L = Iims-+O f.L 8 • Then the assessment (s, f.L) satisfies (3.6) and (3.7), hence, it is a sequential equilibrium of g. The following theorem summarizes the above discus sion. s
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THEOREM 1 0 . (i) [Myerson ( 1 978)]. (ii)
Every strategic form game has at least one proper equilib rium. Every proper equilibrium is peifect. [Van Damme (1 984), Kohlberg and Mertens (1986)] . Let g be an extensive form game with semi-reduced normal form n(g). If s is a proper equilibrium ofn(g), then ps is a quasi-peifect and a sequential equilibrium outcome in g.
Mailath et aL ( 1 997) have shown that sorts of converses to Theorem 10(ii) hold as welL Let {s8} be a converging sequence of completely mixed strategies in a semi reduced normal form game n(g). This sequence induces a quasi-perfect equilibrium in every extensive form game with semi-reduced normal form n(g) if and only if the limit of {s8} is a proper equilibrium that is supported by the sequence. It is important that the same sequence be used: Hillas ( 1 996) gives an example of a strategy profile that is not proper and yet is quasi-perfect in every associated extensive form. Secondly, Mailath et aL ( 1 997) define a concept of normal form sequential equilibrium and they show that an equilibrium is normal form sequential if and only if it is sequential in every extensive form game with that semi-reduced normal form. Theorem l O(ii) appears to be the main application of proper equilibrium. One other application deserves to be mentioned: in 2-person zero-sum games, there is essentially one proper equilibrium and it is found by the procedure of cautious exploitation of the mistakes of the opponent that was proposed by Dresher (196 1 ) [see Van Damme (1983, Section 3.5)] . 4. Forward induction and stable sets of equilibria
Unfortunately, as the game of Figure 5 (a modification of a game discussed by Kohlberg (198 1 )) shows, none of the concepts discussed thus far provides sufficient conditions for self-enforcingness. In this game player 1 first chooses between taking up an outside option that yields him 2 (and the opponent 0) and playing a battle-of-the-sexes game. Player 2 only has to move when player 1 chooses to play the subgame. In this game player 1 taking up his option and players continuing with ( w 1 s2) in the subgame con stitutes a subgame perfect equilibrium. The equilibrium is even perfect: player 2 can ,
2, 0
S]
3, 1
0, 0
W]
0, 0
1, 3
p
Figure 5. Battle of the sexes with an outside option.
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van Damme
argue that player 1 must have suffered from a sudden stroke of irrationality at his first move, but that he (player 1 ) will come back to his senses before his second move and continue with the plan (i.e., play W J ) as if nothing had happened. In fact, the equilib rium (t, sz) is even proper in the normal form of the game: properness allows player 2 to conclude that the mistake PW J is more likely than the mistake ps 1 since PW J is better than ps 1 when player 2 plays sz. However, the outcome where player 1 takes up his option does not seem self enforcing. If player 1 deviates and decides to play the battle-of-the-sexes game, player 2 should not rush to conclude that player 1 must have made a mistake; rather he might first investigate whether he can give a rational interpretation of this deviation. In the case at hand, such an explanation can indeed be given. For a rational player 1 it does not make sense to play W J in the subgame since the plan pw 1 is strictly dominated by the outside option. Hence, combining the rationality of player 1 with the fact that this player chose to play the subgame, player 2 should come to the conclusion that player 1 intends to play SJ in the subgame, i.e., that player 1 bets on getting more than his option and that player 2 is sufficiently intelligent to understand this. Consequently, player 2 should respond by wz, a move that makes the deviation of player 1 profitable, hence, the equilibrium is not self-enforcing. Essentially what is involved here is an argument of forward induction: players' de ductions about other players should be consistent with the assumption that these players are pursuing strategies that constitute rational plans for the overall game. The backward induction requirements discussed before were local requirements only taking into ac count rational behavior in the future. Forward induction requires that players' deduc tions be based on overall rational behavior whenever possible and forces players to take a global perspective. Hence, one is led to an analysis by means of the normal form. In this section we take such a normal form perspective and ask how forward induction can be formulated. The discussion will be based on the seminal work of Elon Kohlberg and Jean-Franc;ois Mertens [Kohlberg and Mertens (1986), Kohlberg ( 1 989), Mertens (1987, 1989a, 1989b, 199 1)]. At this stage the reader may wonder whether there is no loss of information in moving to the normal form, i.e., whether the concepts that were discussed before can be recovered in the normal form. Theorem l O(ii) already provides part of the answer as it shows that sequential equilibria can be recovered. Mailath et al. ( 1 993) discuss the question in detail and they show that also subgames and subgame perfect equilibria can be recovered in the normal form. 4. 1.
Set-valuedness as a consequence of desirable properties
Kohlberg and Mertens (1986) contains a first and partial axiomatic approach to the problem of what constitutes a self-enforcing agreement. (It should, however, be noted that the authors stress that their requirements should not be viewed as axioms since some of them are phrased in terms that are outside of decision theory.) Kohlberg and Mertens argue that a solution of a game should: (i) always exist,
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(ii) be consistent with standard one-person decision theory, (iii) be independent of irrelevant alternatives, and (iv) be consistent with backward induction. (The third requirement states that strategies which certainly will not be used by ratio nal players can have no influence on whether a solution is self-enforcing; it is the for malization of the forward induction requirement that was informally discussed above; it will be given a more precise meaning below.) In this subsection we will discuss these requirements (except for (iv), which was extensively discussed in the previous section), and show that they imply that a solution cannot just be a single strategy profile but rather has to be a set of strategy profiles. In the next subsection, we give formalized versions of these basic requirements. The existence requirement is fundamental and need not be discussed further. It guar antees that, if our necessary conditions for self-enforcingness leave only one candidate solution, that solution is indeed self-enforcing. Without having an existence theorem, we would run the risk of working with requirements that are incompatible, hence, of proving vacuous theorems. The second requirement from the above list follows from the observation that a game is nothing but a simultaneous collection of one-person decision problems. In particular, it implies that the solution of a game can depend only on the normal form of that game. As a matter of fact, Kohlberg and Mertens argue that even less information than is con tained in the normal form should be sufficient to decide on self-enforcingness. Namely, they take mixed strategies seriously as actions, and argue that a player is always able to add strategies that are just mixtures of strategies that are already explicitly given to them. Hence, they conclude that adding or deleting such strategies can have no influ ence on self-enforcingness. Formally, define the reduced normal form of a game as the game that results when all pure strategies that are equivalent to mixtures of other pure strategies have been deleted. (Hence, strategy ai E A i is deleted if there exists s; E St with s;(at) = 0 such that Uj (s\s;) = UJ (s\at ) for all j . The reader may ask whether the reduced normal form is well-defined. We return to this issue in the next subsec tion.) As a first consequence of consistency with one-person decision theory, Kohlberg and Mertens insist that two games with the same reduced normal form be considered equivalent and, hence, as having the same solutions. Kohlberg and Mertens accept as a basic postulate from standard decision theory that a rational agent will only choose undominated strategies, i.e., that he will not choose a strategy that is weakly dominated. Hence, a second consequence of (ii) is that game solutions should be undominated (admissible) as well. Furthermore, if players do not choose undominated strategies, such strategies are actually irrelevant alternatives, hence, (iii) requires that they can be deleted without changing the self-enforcingness of the solution. Hence, the combination of (ii) and (iii) implies that self-enforcing solu tions should survive iterated elimination of weakly dominated strategies. Note that the requirement of independence of dominated strategies is a "global" requirement that is applicable independent of the specific game solution that is considered. Once one has a specific candidate solution, one can argue that, if the solution is self-enforcing, no
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player will use a strategy that is not a best response against the solution, and, hence, that such inferior strategies should be irrelevant for the study of the self-enforcingness of the solution. Consequently, Kohlberg and Mertens require as part of (iii) that a self enforcing solution remain self-enforcing when a strategy that is not a best response to this solution is eliminated. Note that "axioms" (ii) and (iii) force the conclusion that only (3 , 1 ) can be self enforcing in the game of Figure 5: only this outcome survives iterated elimination of weakly dominated strategies. The same conclusion can also be obtained without using such iterative elimination: it follows from backward induction together with the require ment that the solution should depend only on the reduced normal form. Namely, add to the normal form of the game of Figure 5 the mixed strategy m = At + (1 - A)S J with 1/2 < A < 1 as an explicit pure strategy. The resulting game can be viewed as the normal form associated with the extensive form game in which first player 1 decides between the outside option t and playing a subgame with strategy sets {s1 , WJ , m } and {s2 , w2 } . This extensive form game is equivalent to the extensive form from Figure 5, hence, they should have the same solutions. However, the newly constructed game only has (3, 1 ) as a subgame perfect equilibrium outcome. (In the subgame w 1 is strictly dominated by m , hence player 2 i s forced to play w2 .) We will now show why (subsets of ) the above "axioms" can only be satisfied by set valued solution concepts. Consider the trivial game from Figure 6. Obviously, player 1 choosing his outside option is self-enforcing. The question is whether a solution should contain a unique recommendation for player 2. Note that the unique subgame perfect equilibrium of the game requires player 2 to choose �h + �r2, hence, according to (i) and (iv) this strategy should be the solution for player 2. However, according to (i), (ii), and (iii), the solution should be h (eliminate the strategies in the order pl 1 , rz), while according to these same axioms the solution should also be r2 (take the elimi nation order pr 1 , l2 ). Hence, we see that, to guarantee existence, we have to allow for set-valued solution concepts. Furthermore, we see that, even with set-valued concepts, only weak versions of the axioms - that just require set inclusion - can be satisfied. Ac tually, these weak versions of the axioms imply that in this game all equilibria should belong to the solution. Namely, add to the normal form of the game of Figure 6 the mixed strategy At + ( 1 - A)plJ with 0 < A < 1 /2 as a pure strategy. Then the resulting
2, 0
11
1, 0
0, 1
'1
0, 1
1, 0
p
Figure 6. All Nash equilibria are equally good.
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game is the normal form of an extensive form game that has JLh + (1 - JL)r2 with JL = ( 1 - 2A ) j (2 - A) as the unique subgame perfect equilibrium strategy for player 2. Hence, as A moves through (0, 1 /2) , we trace half of the equilibrium set of player 2, viz. the set {JLl2 + (1 - JL)r2 : 0 < JL < 1/2}. The other half can be traced by adding the mixed strategy At + (1 - A) prr in a similar way. Hence, axioms (i), (ii) and (iv) imply that all equilibrium strategies of player 2 belong to the solution. The game of Figure 6 suggests that we should broaden our concept of a solution, that we should not always aim to give a unique strategy recommendation for a player at an information set that can be reached only by irrational moves of other players. In the game of Figure 6, it is unnecessary to give a specific recommendation to player 2 and any such recommendation is somewhat artificial. Player 2 is dependent upon player 1 so that his optimal choice seems to depend on the exact way in which player 1 is ir rational. However, our analysis has assumed rational players, and since no model of irrationality has been provided, the theorist could be content to remain silent. Hence, a self-enforcing solution should not necessarily pin down completely the behavior of players at unreached points of the game tree. We may be satisfied if we can recommend what players do in those circumstances that are consistent with players being rational, i.e., as long as the play is according to the self-enforcing solution. Note that by extending our solution notion to allow for multiple beliefs and actions after irrational moves we can also get rid of the unattractive assumption of persistent rationality that was discussed in Section 3.2 and that corresponds to a narrow reading of axiom (iv). We might just insist that a solution contains a backward induction equi librium, not that it consists exclusively of backward induction equilibria. We should not fully exclude the possibility that a player just made a one-time mistake and will continue to optimize, but we should not force this assumption. In fact, the axioms imply that the solution of a perfect information game frequently cannot just consist of the sub game perfect equilibrium. Namely, consider the game TOL(3) represented in Figure 7, which is a variation of a game discussed in Reny ( 1993). (TOL(n) stands for "Take it or leave it with n rounds".) The game starts with $ 1 on the table in round 1 and each time the (4 , 0)
(3, 3)
(0 , 2)
( 1 , 0) T1
Tz
Lz
2 L1
Figure 7. The game TOL(3).
T1
1, 0
1,0
L1 t
0, 2 4 0
L1 l
0, 2 3, 3
,
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game moves to a next round, the amount of money doubles. In round t, the player i with i ( mod 2) = t ( mod 2) has to move. The game ends as soon as a player takes the money; if the game continues till the end, each player receives $ 2n - 1 1 . (In the unique backwards induction equilibrium, player 1 takes the first dollar.) The unique subgame perfect equilibrium ofTOL(3) is (T1 t, T2 ) , which corresponds to (TJ , T2) in the semi-reduced normal form. If the solution of the game were just (TJ , T2) , then L 1 t would not be a best reply against the solution and according to "axiom" (iii), (T1 , T2) should remain a solution when L 1 t is eliminated. However, in the resulting 2 x 2 game, the unique perfect equilibrium is (Lt l , L2) so that the axioms force this outcome to be the solution. Hence, the axioms imply that the strategy � L2 + � T2 of player 2 has to be part of the solution (in order to make L 1 t a best response against the solution): player 1 cannot believe that, after player 2 has seen player 1 making the move L 1 , player 2 believes player 1 to be rational. Intuitively, stable sets have to be large since they must incorporate the possibility of irrational play. Once we start eliminating dominated and/or inferior strategies, we attribute more rationality to players, make them more predictable and hence can make do with smaller stable sets. In formalizations of iterated elimination, we naturally have set inclusion. The question remains of what type of mathematical objects are candidates for so lutions of games now that we know that single strategy profiles do not qualify. In the above examples, the set of all equilibria was suggested, but the examples were special since there was only one connected component of Nash equilibria. More generally, one might consider connected components as solution candidates; however, this might be too coarse. For example, if, in Figure 6, we were to change player 2's payoffs in the subgame in such a way as to make h strictly dominant, we would certainly recommend player 2 to play /2 even if player 1 has made the irrational move. Hence, the answer to the question appears unclear. Motivated by constructions like the above, and by the interpretation of stable sets as patterns in which equilibria vary smoothly with beliefs or presentation effects, Kohlberg and Mertens suggest connected subsets of equilibria as solution candidates. Hence, a solution is a subset of a component of the equilibrium set (cf. Theorem 2). Note that since a generic game has only finitely many Nash equilib rium outcomes (Theorem 7), all equilibria in the same connected component yield the same outcome (since outcomes depend continuously on strategies); hence, for generic games each Kohlberg and Mertens solution indeed generates a unique outcome. (See also Section 4.3.) �
4.2.
Desirable properties for strategic stability
In this subsection we rephrase and formalize (some consequences of) the requirements (i)-(iv) from the previous subsection, taking the discussion from that subsection into account. Let r be the set of all finite games. A solution concept is a map S that assigns to each game g E r a collection of non-empty subsets of mixed strategy profiles for the game. A solution T of g is a subset T of the set of mixed strategies of (the normal form
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Ch. 41:
of) g with T E S(g) , hence, it is a set of profiles that S allows. The first fundamental requirement that we encountered in the previous subsection was: (E)
Existence: S (g) i= 0 .
(We adopt the convention that, whenever a quantifier is missing, it should be read as "for all", hence (E) requires existence of at least one solution for each game.) Secondly, we will accept Nash equilibrium as a necessary requirement for self-enforcingness: (NE)
Equilibrium: If T E S (g), then T c E (g) .
A third requirement discussed above was: (C)
Connectedness: If T E S (g), then T is connected.
As discussed in the previous subsection, Kohlberg and Mertens insist that rational play ers only play admissible strategies. One formalization of admissibility is the restriction to undominated strategies, i.e., strategies that are best responses to correlated strate gies of the opponents with full support. If players make their choices independently, a stronger admissibility suggests itself, viz. each player chooses a best response against a completely mixed strategy combination of the opponents. Formally, say that s; is an ad missible best reply against s if there exists a sequence s k of completely mixed strategy vectors converging to s such that s; is a best response against any element in the se quence. Write Bf (s) for the set of all such admissible best replies, Bf (s) = Bf (s) n A i , and let Ba(s) = XiBf (s). For any subset S' of S write Ba (S') = UsES' Ba (s) . We can now write the admissibility requirement as: (A)
Admissibility: If T E S (g), then T c Ba (S).
Note that the combination of (NE) and (A) is almost equivalent to requiring perfec tion. The difference is that, as (3.5) shows, perfectness requires the approximating se k quence s to be the same for each player. Accepting that players only use admissible best responses implies that a strategy that is not an admissible best response against the solution is certain not to be played and, hence, can be eliminated. Consequently, we can write the independence of irrelevant alternatives requirement as: (IIA)
Independence of irrelevant alternatives: If a ¢. Bf (T), then T contains a solution of the game in which a has been eliminated.
Note that Bf (T) C Bi (T) n Bf (S), hence, (IIA) implies the requirements that strategies that are not best responses against the solution can be eliminated and that strategies that are not admissible can be eliminated. It is also a fundamental requirement that irrelevant players should have no influ ence on the solutions. Formally, following Mertens (1990), say that a subset 1 of the player set constitutes a small world if their payoffs do not depend on the actions of the players not in 1 , i.e., if SJ
=
sj for all j E 1,
then u 1 (s)
=
u j (s ' ) for all j E 1.
(4. 1)
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A solution has the small worlds property if the players outside the small world have no influence on the solutions inside the small world. Formally, if we write g 1 for the game played by the insiders, then (SMW)
Small worlds property: If J is a small world in g, then T; is a solution in g; if and only if it is a projection of a solution T in g.
Closely related to the small worlds property is the decomposition property: if two dis joint player sets play different games in different rooms, it does not matter whether one analyzes the games separately or jointly. Formally, say that g decomposes at J if both J and J = 1\f are small worlds in g. (D)
Decomposition: If g decomposes at J, then T E S(g) if and only if T with Tk E S(gk) (k E {J, J}).
=
T; x TJ
We now discuss the "player splitting property" which deals with another form of decom position. Suppose g is an extensive form game and assume that there exists a partition Pi of Hi (the set of information sets of player i ) such that, if h, h' belong to differ ent elements of Pi , there is no path in the tree that cuts both h and h'. In such a case, the player can plan his actions at h without having to take into consideration his plans at h'. More generally, plans at one element of the partition can be made independently of plans at the other part and we do not limit the freedom of action of player i if we replace this player by a collection of agents, one agent for each element of Pi . Consequently, we should require the two games to have the same self-enforcing solutions: (PSP)
Player splitting property: If g' is obtained from g by splitting some player i into a collection of independent agents, then S (g) = S (g1) .
Note that for a solution concept having this property it does not matter whether a sig nalling game [Kreps and Sobel ( 1 994)] is analyzed in normal form (also called the Harsanyi-form in this case) or in agent normal form (also called the Selten-form). Also note that in (PSP) the restriction to independent agents is essential: in the agent normal form of the game from Figure 5, the first agent of player 1 taking up his outside option is a perfectly sensible outcome: once the decisions are decoupled, the first action cannot signal anything about the second action. We will return to this in Section 5. We will now formalize the requirement that the solution of a game depends only on those aspects of the problem that are relevant for the players' individual decision problems, i.e., that the solution is ordinal [cf. Mertens ( 1 987)]. As already discussed above, Mertens argues that rational players will only play admissible best responses. A natural invariance requirement thus is that the solutions depend only on the admissible best-reply correspondence, formally (BRI)
Best reply invariance: If B� = B�,, then S(g) = S(g').
Note that the application of (BRI) is restricted to games with the same player sets and the same strategy spaces, hence, this requirement should be supplemented with require ments that the names of the players and the strategies do not matter, etc.
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In the previous subsection we also argued that games with the same reduced nor mal form should be considered equivalent. In order to be able to properly formalize this invariance requirement it turns out to be necessary to extend the domain of games somewhat: after one has eliminated all equivalent strategies of a player, this player's strategy set need no longer be a full simplex. To deal with such possibilities, define an I -person strategic form game as a tuple (S, u) where S = Xi Si is a product of compact polyhedral sets and u is a multilinear map on S. Note that each such strategic form game has at least one equilibrium, and that the equilibrium set consists of finitely many con nected components. Furthermore, all the requirements introduced above are meaningful for strategic form games. Say that an I -person strategic form game g' = (S', u') is a re duction of the I -person normal form game g = ( A , u) if there exists a map f = Cfi ) iEI with fi : si ---+ s; being linear and surjective, such that u = u' 0 f, hence, f preserves payoffs. Call such a map f an isomorphism from g onto g'. The requirement that the solution depends only on the reduced normal form may now be formalized as:
Invariance: If f is an isomorphism from g onto g', then S(g') = {f(T) : T E S(g)} and f - 1 (T') = U { T E S(g): f(T) = T ' } for all T ' E S(g') . It should be stressed here that in Mertens ( 1987) the requirements (BRI) and (I) are (I)
derived from more abstract requirements of ordinality. The final requirement that was discussed in the previous subsection was the back wards induction requirement, which, in view of Theorem 1 0, can be formalized as: (BI) 4.3.
Backwards induction: If T E S(g), then T contains a proper equilibrium of g.
Stable sets of equilibria
In Kohlberg and Mertens ( 1 986), three set-valued solution concepts are introduced that aim to capture self-enforcingness. Unfortunately, each of these fails to satisfy at least one of the above requirements so that that seminal paper does not come up with a def inite answer as to what constitutes a self-enforcing outcome. The definitions of these concepts build on Theorem 3 that describes the structure of the Nash equilibrium corre spondence. The idea is to look at components of Nash equilibria that are robust to slight perturbations in the data of the game. The structure theorem implies that at least one such component exists. By varying the class of perturbations that are allowed, different concepts are obtained. Formally define (i) T is a stable set of equilibria of g if it is minimal among all the closed sets of equilibria T ' that have the property that each perturbed game g 8 ·a with s close to zero has an equilibrium close to T ' . (ii) T is afully stable set of equilibria of g if it is minimal among all the closed sets of equilibria T ' that have the property that each game (S', u) with S[ a polyhedral set in the interior of s; (for each i) that is close to g has an equilibrium close to T ' . (iii) T is a hyperstable set of equilibria of g if it is minimal among all the closed sets of equilibria T' that have the property that for each game g' = (A', u') that is
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equivalent to g and for each small payoff perturbation (A', u 8 ) of g ' there exists an equilibrium close to T'. Kohlberg and Mertens (1 986) show that every hyperstable set contains a set that is fully stable and that every fully stable set contains a stable set. Furthermore, from The orem 3 they show that every game has a hyperstable set that is contained in a single connected component of Nash equilibria and, hence, that the same property holds for fully stable sets and stable sets. They, however, reject the (preliminary) concepts of hy perstability and full stability because these do not satisfy the admissibility requirement. Kohlberg and Mertens write that stability seems to be the "right" concept but they are forced to reject it since it violates (C) and (BI). (This concept does satisfy (E), (NE), (A), (IIA), (BRI), and (1).) Kohlberg and Mertens conclude with "we hope that in the future some appropriately modified definition of stability will, in addition, imply connected ness and backwards induction". Mertens (1989a, 199 1 ) gives such a modification. We will consider it below. An example of a game in which every fully stable set contains an inadmissible equi librium (and hence in which every hyperstable set contains such an equilibrium) is ob tained by changing the payoff vector (0, 2) in TOL(3) (Figure 7) to (5, 5). The unique admissible equilibrium then is (L 1 t, T2) but every fully stable set has to contain the strategy (L 1 l) of player 1 . Namely, if (in the normal form) player 1 trembles with a larger probability to T1 when playing L 1 t than when playing L 1 l, we obtain a perturbed game in which only (L 1 l, T2) is an equilibrium. We now describe a 3-person game (attributed to Faruk Gul in Kohlberg and Mertens (1986)) that shows that stable sets may contain elements from different equilibrium components and need not contain a subgame perfect equilibrium. Player 3 starts the game by choosing between an outside option T (which yields payoffs (0, 0, 2)) or play ing a simultaneous move subgame with players 1 and 2 in which each of the three players has strategy set {a, b} and in which the payoffs are as in the matrix from the left-hand side of Figure 8 (x, y E {a, b}, x f. y) . Hence, players 1 and 2 have identical payoffs and they want to make the same choice as player 3 . Player 3 prefers these play-
X y
!±i
y
X
1, 1, 5
3, 3, 0
1 , 1 , 5 0, 0, 1 X
ai
\
a
a 3a3, 3a3
� ai
b
1, 1
\
Figure 8. Stable sets need not contain a subgame perfect equilibrium.
b
1, 1
3b3 , 3b3
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ers to make different choices, but, if they make the same choice, he wants his choice to be different from theirs. The game g described in the above story has a unique subgame perfect equilibrium: player 3 chooses to play the subgame and each player chooses � a + � b in this subgame. This strategy vector constitutes a singleton component of the set of Nash equilibria. In addition, there are two components in which player 3 takes up his option T. Writing a; (resp. b; ) for the probability with which player i (i = 1 , 2) chooses a (resp. b), the strategies of players 1 and 2 in this component are the solutions to the pair of inequalities (4.2) Note that the solution set of (4.2) indeed consists of two connected components, one around (a , a) (i.e., a 1 = a2 = 1) and one around (b , b) . Now, let us look at perturba tions of (the normal form of) g. If player 3 chooses to play the subgame with positive stability 8 and if, conditional on such a mistake, he chooses a (resp. b) with probabil ity a3 (resp. b 3 ), players 1 and 2 face the game from the right-hand side of Figure 8. The equilibria of this game are given by if a3 < 1/3, if 1/3 < a3 < 2j3, if 2/3 < a3 , hence, restricted to players 1 and 2, each perturbed game has (a , a) or (b, b) (or both) as a strict equilibrium. If players 1 and 2 coordinate on any of these strict equilibria, player 3 strictly prefers to play T, hence, { (a J , a2 , T), (h i , b2 , T)} is a stable set of g . Obviously, this set does not contain the subgame perfect equilibrium, and even yields a different outcome. A closer investigation may reveal the source of the difficulty and suggest a resolu tion of the problem. Since problematic zero-probability events arise only from player 3 choosing T, let us insist that he chooses to play the subgame with probability 8 but, for simplicity, let us not perturb the strategies of players 1 and 2. Formally, consider a perturbed game g c:, a with 8! = 82 = 0, 83 = 8 > 0 and o-3 = (0, a , 1 - a ) , hence a is the probability that player 3 chooses a if he makes a mistake. The middle panel in Figure 8 displays, for any small 8 > 0, the equilibrium correspondence as a function of a. (The horizontal axis corresponds to a, the vertical one to a; .) Each perturbed game has an equilibrium close to the subgame perfect equilibrium of g . This equilibrium is repre sented by the horizontal line at a; = 1/2. The inverted z-shaped figure corresponds to the solutions of (4.3). If players 1 and 2 play such a solution that is sufficiently close to a pure strategy, then T is the unique best response of player 3, hence, in that case we have an equilibrium of the perturbed game with a3 = a. If players 1 and 2 play a solution of (4.3) that is sufficiently close to a; = 1/2 (i.e., they choose a; E (a; , f!; ) corresponding to the dashed part of the z-curve ), then we do not have an equilibrium unless a 1 = a2 = a 3 = 1/2. (If a; > 1/2, then the unique best response of player 3 is to
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play b, hence a 3 = w < 1/3 so that by (4.3) we should have a; = 0.) The points !! and a where the solid z-curve changes into the dashed z-curve are somewhat special. Writing f!:.; = 2 - 3!! we have that if each player i (i = 1, 2) chooses a with probability f!:.; , then player 3's best responses are T and b. Consequently, by playing b voluntarily with the appropriate probability, player 3 can enforce any a3 E (c:a, a), hence, if c: is sufficiently small and a > !!, player 3 can enforce a3 = !!· We see that for each a � !!, the perturbed game has an equilibrium with a; = f!:.; . In the diagram, this branch is represented by the horizontal line at f!:.; . Of course, there is a similar branch at a; . Since the above search was exhaustive, the middle panel in Figure 8 contains a complete description of the equilibrium graph, or at least of its projection on the (a, a;) -space. The critical difference between the "middle" branch of the equilibrium correspon dence and each of the other two branches is that in the latter cases it is possible to con tinuously deform the graph, leaving the part over the extreme perturbations (a E {0, 1 }) intact, in such a way that the interior is no longer covered, i.e., such that there are no longer "equilibria" above the positive perturbations. Hence, although the projection from the union of the top and bottom branches to the perturbations is surjective (as re quired by stability), this projection is homologically trivial, i.e., it is homologous to the identity map of the boundary of the space of perturbations. Building on this observa tion, and on the topological structure of the equilibrium correspondence more generally, Mertens (1989a, 199 1) proposes a refinement of stability (to be called M-stability) that essentially requires that the projection from a neighborhood of the set to a neighborhood of the game should be homologically nontrivial. As the formal definition is somewhat involved we will not give it here but confine ourselves to stating its main properties. Let us, however, note that Mertens does not insist on minimality; he shows that this conflicts with the ordinality requirement (cf. Section 4.5). THEOREM 1 1 [Mertens (1989a, 1990, 1991)]. M-stable sets are closed sets of normal form peifect equilibria that satisfy all properties listed in the previous subsection. We close this subsection with a remark and with some references to recent literature. First of all, we note that also in Hillas (1990) a concept of stability is defined that satis fies all properties from the list of the previous subsection. (We will refer to this concept as H-stability. To avoid confusion, we will refer to the stability concept that was defined in Kohlberg and Mertens as KM-stability.) T is an H-stable set of equilibria of g if it is minimal among all the closed sets of equilibria T ' that have the following property: each upper-hemicontinuous compact convex-valued correspondence that is pointwise close to the best-reply correspondence of a game that is equivalent to g has a fixed point close to T ' . The solution concept of H-stable sets satisfies the requirements (E), (NE), (C), (A), (IIA), (BRI), (I) and (BI), but it does not satisfy the other requirement from Section 4.2. (The minimality requirement forces H-stable sets to be connected, hence, in the game of Figure 8 only the subgame perfect equilibrium outcome is H-stable.) In Hillas et al. (1999) it is shown that each M-stable set contains an H-stable set. That paper discusses a couple of other related concepts as well.
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I conclude this section by referring to some other recent work. Wilson (1 997) dis cusses the role of admissibility in identifying self-enforcing outcomes. He argues that admissibility criteria should be deleted when selecting among equilibrium components, but that they may be used in selecting equilibria from a component, hence, Wilson ar gues in favor of perfect equilibria in essential components, i.e., components for which the degree (cf. Section 2.3) is non-zero. Govindan and Wilson (1999) show that, in 2-player games, maximal M-stable sets are connected components of perfect equilibria, hence, such sets are relatively easy to compute and their number is finite. (On finite ness, see Hillas et al. ( 1 997).) The result implies that an essential component contains a stable set; however, as Govindan and Wilson illustrate by means of several examples, inessential components may contain stable sets as well. 4.4.
Applications ofstability criteria
Concepts related to strategic stability have been frequently used to narrow down the number of equilibrium outcomes in games arising in economic contexts. (Recall that in generic extensive games all equilibria in the same component have the same outcome so that we can speak of stable and unstable outcomes.) Especially in the context of signalling games many refinements have been proposed that were inspired by stability or by its properties [cf. Cho and Kreps ( 1 987), Banks and Sobel (1 987) and Cho and Sobel (1990)] . As this literature is surveyed in the chapter by Kreps and Sobel ( 1 994), there is no need to discuss these applications here [see Van Damme ( 1 992)] . I shall confine myself here to some easy applications and to some remarks on examples where the fine details of the definitions make the difference. It is frequently argued that the Folk theorem, i.e., the fact that repeated games have a plethora of equilibrium outcomes (see Chapter 4 in this Handbook) shows a funda mental weakness of game theory. However, in a repeated game only few outcomes may actually be strategically stable. (General results, however, are not yet available.) To il lustrate, consider the twice-repeated battle-of-the-sexes game, where the stage game payoffs are as in (the subgame occurring in) Figure 5 and that is played according to the standard information conditions. The path ((s 1 , w2 ), (s 1 , w2 )) in which player 1 's most preferred stage equilibrium is played twice is not stable. Namely, the strategy s2 w2 (i.e., deviate to s2 and then play w2 ) is not a best response against any equilibrium that supports this path, hence, if the path were stable, then according to (IIA) it should be possible to delete this strategy. However, the resulting game does not have an admissible equilibrium with payoff (6, 2) so that the path cannot be stable. (Admissibility forces player 1 to respond with WJ after 2 has played s2 ; hence, the deviation s2 s2 is profitable for player 2.) For further results on stability in repeated games, the reader is referred to Balkenborg ( 1 993), Osborne ( 1990), Ponssard (199 1 ) and Van Damme ( 1 989a). Stability implies that the possibility to inflict damage on oneself confers power. Sup pose that before playing the one-shot battle-of-the-sexes game, player 1 has the opportu nity to burn 1 unit of utility in a way that is observable to player 2. Then the only stable outcome is the one in which player 1 does not burn utility and players play (s 1 , w 1 ),
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hence, player 1 gets his most preferred outcome. The argument is simply that the game can be reduced to this outcome by using (IIA). If both players can throw away util ity, then stability forces utility to be thrown away with positive probability: any other outcome can be upset by (IIA). [See Van Damme ( 1989a) for further details and Ben Porath and Dekel (1 992), Bagwell and Ramey ( 1 996), Glazer and Weiss (1990) for applications.] Most applications of stability in economics use the requirements from Section 4.2 to limit the set of solution candidates to one and they then rely on the existence theorem to conclude that the remaining solution must be stable. Direct verification of stability may be difficult; one may have to enumerate all perturbed games and investigate how the equilibrium graph hangs together [see Mertens ( 1 987, 1 989a, 1 989b, 1991) for var ious illustrations of this procedure and for arguments as to why certain shortcuts may not work] . Recently, Wilson (1992) has constructed an algorithm to compute a simply stable component of equilibria in bimatrix games. Simply stable sets are robust against a restricted set of perturbations, viz. one perturbs only one strategy (either its probabil ity or its payoff). Wilson amends the Lemke and Howson algorithm from Section 2.3 to make it applicable to nongeneric bimatrices and he adds a second stage to it to en sure that it can only terminate at a simply stable set. Whenever the Lemke and Howson algorithm terminates with an equilibrium that is not strict, Wilson uses a perturbation to transit onto another path. The algorithm terminates only when all perturbations have been covered by some vertex in the same component. Unfortunately, Wilson cannot guarantee that a simply stable component is actually stable. In Van Damme ( 1 989a) it was argued that stable sets (as originally defined by Kohlberg and Mertens) may not fully capture the logic of forward induction. Following an idea originally discussed in McLennan ( 1 985) it was argued that if an information set h E H; can be reached only by one equilibrium s * , and if s * is self-enforcing, player i should indeed believe that s * is played if h is reached and, hence, only s;h should be allowed at h. A 2-person example in Van Damme (1989a) showed that stable equilibria need not satisfy this forward-induction requirement. (Actually Gul's example (Figure 8) already shows this.) Hauk and Hurkens ( 1999) have recently shown that this forward induction property is satisfied by none of the stability concepts discussed above. On the other hand they show that this property is satisfied by some evolutionary equilibrium concepts that are related to those discussed in Section 4.5 below. Gul and Pearce ( 1996) argue that forward induction loses much of its power when public randomization is allowed; however, Govindan and Robson ( 1 998) show that the Gul and Pearce argument depends essentially on the use of inadmissible strategies. Mertens (1 992) describes a game in which each player has a unique dominant strat egy, yet the pair of these dominant strategies is not perfect in the agent normal form. Hence, the M-stable sets of the normal form and those of the agent normal form may be disjoint. That same paper also contains an example of a nongeneric perfect information game (where ties are not noticed when doing the backwards induction where the unique M-stable set contains other outcomes besides the backwards induction outcome). [See also Van Damme (1987b, pp. 32, 33).]
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Govindan ( 1 995) has applied the concept of M-stability to the Kreps and Wilson ( 1 982b) chain store game with incomplete information. He shows that only the outcome that was already identified in Kreps and Wilson (1 982b) as the unique "reasonable" one, is indeed the unique M-stable outcome. Govindan's approach is to be preferred to Kreps and Wilson's since it does not rely on ad hoc methods. It is worth remarking that Govindan is able to reach his conclusion just by using the properties of M-stable equilibria (as mentioned in Theorem 1 1) and that the connectedness requirement plays an important role in the proof. 4.5.
Robustness and persistent equilibria
Many game theorists are not convinced that equilibria in mixed strategies should be treated on an equal footing with pure, strict equilibria; they express a clear preference for pure equilibria. For example, Harsanyi and Selten ( 1988, p. 198) write: "Games that arise in the context of economic theory often have many strict equilibrium points. Obviously in such cases it is more natural to select a strict equilibrium point rather than a weak one. Of course, strict equilibrium points are not always available ( . . . ) but it is still possible to look for a principle that helps us to avoid those weak equilibrium points that are especially unstable." (They use the term "strong" where I write "strict".) In this subsection we discuss such principles. Harsanyi and Selten discuss two forms of instability associated with mixed strategy equilibria. The first, weak form of instability results from the fact that even though a player might have no incentive to deviate from a mixed equilibrium, he has no positive incentive to play the equilibrium strategy either: any pure strategy that is used with positive probability is equally good. As we have seen in Section 2.5, the reinterpretation of mixed equilibria as equilibria in beliefs provides an adequate response to the criticism that is based on this form of instability. The second, strong form of instability is more serious and cannot be countered so easily. This form of instability results from the fact that, in a mixed equilibrium, if a player's beliefs differ even slightly from the equilibrium beliefs, optimizing behavior will typically force the player to deviate from the mixed equilibrium strategy. In contrast, if an equilibrium is strict, a player is forced to play his equilibrium strategy as long as he assigns a sufficiently high probability to the opponents playing this equilibrium. For example, in the battle-of-the-sexes game (that occurs as the subgame in Figure 5), each player is willing to follow the recommendation to play a pure equilibrium as long as he believes that the opponent follows the recommendation with a probability of at least 2/3. In contrast, player i is indifferent between Si and Wi only if he assigns a probability of exactly 1 /3 to the opponent playing wJ . Hence, it seems that strict equilibria possess a type of robustness property that the mixed equilibrium lacks. However, this difference is not picked up by any of the stability concepts that have been discussed above: the mixed strategy equilibrium of the battle-of-the-sexes game constitutes a singleton stable set according to each of the above stability definitions. In this subsection, we will discuss some set-valued generalizations of strict equilibria that do pick up the difference. They all aim at capturing the idea that equilibria should
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be robust to small trembles in the equilibrium beliefs, hence, they address the question of what outcome an outsider would predict who is quite sure, but not completely sure, about the players' beliefs. The discussion that follows is inspired by Balkenborg (1992). If s is a strict equilibrium of g = (A, u), then s is the unique best response against s, hence {s} = B(s). We have already encountered a set-valued analogue of this unique ness requirement in Section 2.2, viz. the concept of a minimal curb set. Recall that C C A is a curb set of g if
B(C) c C ,
(4.3)
i.e., if every best reply against beliefs that are concentrated on C again belongs to C. Obviously, a singleton set C satisfies (4.4) only if i t i s a strict equilibrium. Nonsingleton curb sets may be very large (for example, the set A of all strategy profiles trivially satisfies (4.4)), hence in order to obtain more definite predictions, one can investigate minimal sets with the property (4.4). In Section 2.2 we showed that such minimal curb sets exist, that they are tight, i.e., B(C) = C, and that distinct minimal curb sets are disjoint. Furthermore, curb sets possess the same neighborhood stability property as strict equilibria, viz. if C satisfies (4.4), then there exists a neighborhood U of X; L1 (C;) in S such that
B(U) c C.
(4.4)
Despite all these nice properties, minimal curb sets do not seem to be the appropriate generalization of strict equilibria. First, if a player i has payoff equivalent strategies, then ( 4.4) requires all of these to be present as soon as one is present in the set, but optimizing behavior certainly does not force this conclusion: it is sufficient to have at least one member of the equivalence class in the curb set. (Formally, define the strategies s[ and sf' of player i to be i-equivalent if u; (s\s;) = u; (s\s;') for all s E S, and write s[ � ; s;' if sf and s[' are i -equivalent.) Secondly, requirement (4.4) does not differentiate among best responses; it might be preferable to work with the narrower set of admissible best responses. As a consequence of these two observations, curb sets may include too many strategies and minimal curb sets do not provide a useful generalization of the strict equilibrium concept. Kalai and Samet's (1984) concept of persistent retracts does not suffer from the two drawbacks mentioned above. Roughly, this concept results when requirement (4.5) is weakened to "B(s) n C -=f. 0 for any s E U". Formally, define a retract R as a Cartesian product R = X; R; where each R; is a nonempty, closed, convex subset of S; . A retract is said to be absorbing if
B(s) n R -=f. 0
for all s in a neighborhood U of R,
(4.5)
that is, if against any small perturbation of strategy profile in R there exists a best re sponse that is in R. A retract is defined to be persistent if it is a minimal absorbing
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retract. Zorn's lemma implies that persistent retracts exist; an elementary proof is indi cated below. Kakutani's fixed point theorem implies that each absorbing retract contains a Nash equilibrium. A Nash equilibrium that belongs to a persistent retract is called a persistent equilibrium. A slight modification of Myerson's proof for the existence of proper equilibrium actually shows that each absorbing retract contains a proper equi librium. Hence, each game has an equilibrium that is both proper and persistent. Below we give examples to show that a proper equilibrium need not be persistent and that a persistent equilibrium need not be proper. Note that each strict equilibrium is a singleton persistent retract. The reader can easily verify that in the battle-of-the-sexes game only the pure equilibria are persistent and that (in the normal form of ) the overall game in Figure 5 only the equilibrium (ps 1 , w2) is persistent, hence, in this example, persistency selects the forward induction outcome. As a side remark, note that s is a Nash equilibrium if and only if {s} = R is a minimal retract with the property "B(s) n R =1- 0 for all s E R", hence, persistency corresponds to adding neighborhood robustness to the Nash equilibrium requirement. Kalai and Samet ( 1 984) show that persistent retracts have a very simple structure, viz. they contain at most one representative from each i -equivalence class of strategies for each player i . To establish this result, Kalai and Samet first note that two strategies s; and s;' are i -equivalent if and only if there exists an open set U in S such that, against any strategy in U, s; and s;' are equally good. Hence, it follows that, up to equivalence, the best response of a player is unique (and pure) on an open and dense subset of S. Note that, to a certain extent, a strategy that is not a best response against an open set of beliefs is superfluous, i.e., a player always has a best response that is also a best response to an open set in the neighborhood. Let us call sf a robust best response against s if there exists an open set U E S with s in its closure such that s; is a best response against all elements in U. (Balkenborg ( 1 992) uses the term semi-robust best response, in order to avoid confusion with Okada's (1 983) concept.) Write B[ (s) for all robust best responses of player i against s and sr (s) = Xi B[ (s). Note that W (s) c Ba (s) c B (s) for all s . Also note that a mixed strategy is a robust best response only if it is a mixture of equivalent pure robust best responses. Hence, up to equivalence, robustness restricts players to using pure strategies. Finally, note that an outside observer, who is somewhat uncertain about the players' beliefs and who represents this uncertainty by continuous distributions on S, will assign positive probability only to players playing robust best responses. The reader can easily verify that (4.6) is equivalent to if s E R and a E
B[(s)
then a
�; s; for some s; E R;
(all i, s).
(4.6)
Hence, up to equivalence, all robust best responses against the retract must belong to the absorbing retract. Minimality thus implies that a persistent retract contains at most one representative from each equivalence class of robust best responses. From this observa tion it follows that there exists an absorbing retract that is spanned by pure strategies and that there exists at least one persistent retract. (Consider the set of all retracts that are
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spanned by pure strategies. The set is finite, partially ordered and the maximal element (R = S) is absorbing, hence, there exists a minimal element.) Of course, for generic strategic form games, no two pure strategies are equivalent and any pure best response is a robust best response. For such games it thus follows that R is a persistent retract if and only if there exists a minimal curb set C such that R; = .1 ( C; ) for each player i . We will now investigate which properties from Section 4.2 are satisfied by persistent retracts. We have already seen that persistent retracts exist; they are connected and con tain a proper equilibrium. Hence, the properties (E), (C), and (BI) hold. Also (IIA) is satisfied, as follows easily from (4.7) and the fact that Br(s) c Ba(s). Also (BRI) fol lows easily from ( 4. 7). However, persistent retracts do not satisfy (NE). For example, in the matching pennies game the entire set of strategies is the unique persistent retract. Of course, persistency satisfies a weak form of (NE): any persistent retract contains a Nash equilibrium. In fact, it can be shown that each persistent retract contains a stable set of equilibria. (This is easily seen for stability as defined by Kohlberg and Mertens, Mertens ( 1 990) proves it for M-stability and Balkenborg ( 1 992) proves the property for H-stable sets.) Similarly, persistency satisfies a weak form of (A) : (4. 7) implies that if R is a per sistent retract and s; is an extreme point of R; , then s; is a robust best response, hence, s; is admissible. Consequently, property (A) holds for the extreme points of R, and each element in R only assigns positive probability to admissible pure strategies. This, how ever, does not imply that the elements of R are themselves admissible. For example, in the game of Figure 9, the only persistent retract is the entire game, but the strategy (�, �' 0) of player 1 is dominated. In particular the equilibrium ( ( � , � , 0) , (0, 0, 1)} is persistent but not perfect. Persistent retracts are not invariant. In Figure 9, replace the payoff "2" by "�" so that the third strategy becomes a duplicate of the mixture ( � ,
� , 0) . The unique persistent re
tract contains the mixed strategy ( ! , � , 0) , but it does not contain the equivalent strategy (0, 0, 1 ) . Hence, the invariance requirement (I) is violated. Balkenborg ( 1 992), however, shows that the extreme points of a persistent retract satisfy (I). He also shows that this set of extreme points satisfies the small worlds property (SWP) and the decomposition property (D). A serious drawback of persistency is that it does not satisfy the player splitting prop erty: the agent normal form and the normal form of an incomplete information game can have different persistent retracts. The reason is that the normal form forces differ ent types to have the same beliefs about the opponent, whereas the Selten form (i.e., the agent normal form) allows different types to have different conjectures. (Cf. our
Figure 9.
3, 0
0, 3
0, 2
0, 3
3, 0
0, 2
2, 0
2, 0
0, 0
A persistent equilibrium need not be perfect.
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discussion in Section 2.2.) Perhaps it is even more serious that also other completely inessential changes in the game may induce changes in the persistent retracts and may make equilibria persistent that were not persistent before. As an example, consider the game from Figure 5 in which only the outcome (3 , 1 ) is persistent. Now change the game such that, when (pw 1 , w2 ) is played, the players do not receive zero right away, but are rather forced to play a matching pennies game. Assume players simultaneously choose "heads" or "tails", that player 1 receives 4 units from player 2 if choices match and that he has to pay 4 units if choices differ. The change is completely inessential (the game that was added has unique optimal strategies and value zero), but it has the conse quence that in the normal form, only the entire strategy space is persistent. In particular, player 1 taking up his outside option is a persistent and proper equilibrium outcome of the modified game. For applications of persistent equilibria the reader is referred to Kalai and Samet (1985), Hurkens ( 1996), Van Damme and Hurkens (1996), Blume ( 1 994, 1 996), and Balkenborg (1993). Kalai and Samet consider "repeated" unanimity games. In each of finitely many periods, players simultaneously announce an outcome. The game stops as soon as players announce that same outcome, and then that outcome is implemented. Kalai and Samet show that if there are at least as many rounds as there are outcomes, players will agree on an efficient outcome in a (symmetric) persistent equilibrium. Hurkens (1996) analyzes situations in which some players can publicly bum utility be fore the play of a game. He shows that if the players who have this option have common interests [Aumann and Sorin (1989)], then only the outcome that these players prefer most is persistent. Van Damme and Hurkens (1996) study games in which players have common interests and in which the timing of the moves is endogenous. They show that persistency forces players to coordinate on the efficient equilibrium. Blume ( 1 994, 1 996) applies persistency to a class of signalling games and he also obtains that persis tent equilibria have to be efficient. Balkenborg ( 1 993) studies finitely repeated common interest games. He shows that persistent equilibria are almost efficient. The picture that emerges from these applications (as well as from some theoretical considerations not discussed here, see Van Damme ( 1 992)) is that persistency might be more relevant in an evolutionary and/or learning context, rather than in the pure deductive context we have assumed in this chapter. Indeed, Hurkens (1994) discusses an explicit learning model in which play eventually settles down in a persistent re tract. The following proposition summarizes the main elements from the discussion in this section: THEOREM 1 2 . (i) [Kalai and Samet ( 1985)].
(ii)
Every game has a persistent retract. Each persistent retract contains a proper equilibrium. Each strategy in a persistent retract as signs positive probability only to robust best replies. [Balkenborg ( 1 992)]. For generic strategic form games, persistent retracts cor respond to minimal curb sets.
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(iii) [Balkenborg ( 1 992)]. Persistent retracts satisfy the properties (E), (C), (IIA), (BRI) and (BI) from Section 4.2, but violate the other properties. The set of extreme points ofpersistent retracts satisfies (SWP), (D) and (1). (iv) [Mertens ( 1 990), Balkenborg ( 1 992)] . Each persistent retract contains an M
stable set. It also contains an H-stable set as well as a KM-stable set.
5. Equilibrium selection
Up to now this paper has been concerned just with the first and basic question of non cooperative game theory: which outcomes are self-enforcing? The starting point of our investigations was that being a Nash equilibrium is necessary but not sufficient for self enforcingness, and we have reviewed several other necessary requirements that have been proposed. We have seen that frequently even the most stringent refinements of the Nash concept allow multiple outcomes. For example, many games admit multiple strict equilibria and any such equilibrium passes every test of self-enforcingness that has been proposed up to now. In the introduction, however, we already argued that the "theory" rationale of Nash equilibrium relies essentially on the assumption that players can co ordinate on a single outcome. Hence, we have to address the questions of when, why and how players can reach such a coordinated outcome. One way in which such coordi nation might be achieved is if there exists a convincing theory of rationality that selects a unique outcome in every game and if this theory is common knowledge among the players. One such theory of equilibrium selection has been proposed in Harsanyi and Selten ( 1 988). In this section we will review the main building blocks of that theory. The theory from Harsanyi and Selten may be seen as derived from three basic postu lates, viz. that a theory of rationality should make a recommendation that is (i) a unique strategy profile, (ii) self-enforcing, and (iii) universally applicable. The latter require ment says that no matter the context in which the game arises, the theory should apply. It is a strong form of history-independence. Harsanyi and Selten ( 1 988, pp. 342, 343) re fer to it as the assumption of endogenous expectations: the solution of the game should depend only on the mathematical structure of the game itself, no matter the context in which this structure arises. The combination of these postulates is very powerful; for ex ample, one implication is that the solution of a symmetric game should be symmetric. The postulates also force an agent normal form perspective: once a subgame is reached, only the structure of the subgame is relevant, hence, the solution of a game has to project onto the solution of the subgame. Harsanyi and Selten refer to this requirement as "sub game consistency". It is a strong form of the requirement of "persistent rationality" that was extensively discussed in Section 3 . Of course, subgame consistency is naturally ac companied by the axiom of truncation consistency: to find the overall solution of the game it should be possible to replace a subgame by its solution. Indeed, Harsanyi and Selten insist on truncation consistency as well. It should now be obvious that the require ments that Harsanyi and Selten impose are very different from the requirements that we discussed in Section 4.2. Indeed the requirements are incompatible. For example, the
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Harsanyi and Selten requirements imply that the solution of the game from Figure 5 is (tm 1 , m 2 ) where m i = �Si + � w; . Symmetry requires the solution of the subgame to be (m 1 , m 2) and the axioms of sub game and truncation consistency prevent player 1 from signalling anything. If one accepts the Harsanyi and Selten postulates, then it is common knowledge that the battle-of-the-sexes subgame has to be played according to the mixed equilibrium, hence, if he has to play, player 2 must conclude that player 1 has made a mistake. Note that uniqueness of the solution is already incompatible with the pair (1), (BI) from Section 4.2. We showed that (I) and (BI) leave only the payoff (3 , 1 ) in the game of Figure 5, hence, uniqueness forces (3 , 1) as the unique solution of the "battle of the sexes". However, if we would have given the outside option to player 2 rather than to player 1, we would have obtained ( 1 , 3) as the unique solution. Hence, to guarantee existence, the approach from Section 4 must give up uniqueness, i.e., it has to allow multiple solutions. Both (3, 1) and ( 1 ,3) have to be admitted as so lutions of the battle-of-the-sexes game, in order to allow the context in which the game is played to determine which of these equilibria will be selected. The approach to be discussed in this section, which requires context independence, is in sharp conflict with that from the previous section. However, let us note that, although the two approaches are incompatible, each of the approaches corresponds to a coherent point of view. We confine ourselves to presenting both points of view, to allow the reader to make up his own mind. 5. 1.
Overview of the Harsanyi and Selten solution procedure
The procedure proposed by Harsanyi and Selten to find the solution of a given game generates a number of "smaller" games which have to be solved by the same procedure. The process of reduction and elimination should continue until finally a basic game is reached which cannot be scaled down any further. The solution of such a basic game can be determined by applying the tracing procedure to which we will return below. Hence, the theory consists of a process of reducing a game to a collection of basic games, a rule for solving each basic game, and a procedure for aggregating these basic solutions to a solution of the overall game. The solution process may be said to consist of five main steps, viz. (i) initialization, (ii) decomposition, (iii) reduction, (iv) formation splitting, and (v) solution using dominance criteria. To describe these steps in somewhat greater detail, we first introduce some terminology. The Harsanyi and Selten theory makes use of the so-called standard form of a game, a form that is in between the extensive form and the normal form. Formally, the standard form consists of the agent normal form together with information about which agents belong to the same player. Write I for the set of players in the game and for each i E I, let Hi = {ij : j E 1; } b e the set of agents o f player i . Writing H = U ; H; for the set o f all agents in the game, a game in standardform is a tuple g = (A , u) H where A = Xij Aij with A ;J being the action set of agent ij, and u; : A � � for each player i . Harsanyi and Selten work with this form since on the one hand they want to guarantee perfectness in the extensive form, while on the other hand they want different agents of the same player to have the same expectations about the opponents.
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Given a game in extensive form, the Harsanyi and Selten theory should not be di rectly applied to its associated standard form g; rather, for each £ > 0 that is sufficiently small, the theory should be applied to the uniform £-perturbation g" of the game. The solution of g is obtained by taking the limit, as £ tends to zero, of the solution of g " . The question of whether the limit exists is not treated in Harsanyi and Selten ( 1988); the authors refer to the unpublished working paper Harsanyi and Selten (1977) in which it is suggested that there should be no difficulties. Formally, g" is defined as follows. For each agent ij let O'ij be the centroid of Aij , i.e., the strategy that chooses all pure actions in Aij with probability IAij 1 - 1 . For £ > 0 sufficiently small, write Eij = £ 1 Aij I and let 8 = (Eij)ij E H · Recall, from Equation (3 .2) that i·a denotes the strategy vec tor that results when each player intends to play s, but players make mistakes with probabilities determined by 8 and mistakes are given by 0' . The uniformly perturbed game g8 is the standard form game (A, u") H where the payoff function u" is defined by uf(s) = u; (s 8 ·a). Hence, in g8 each agent ij mistakenly chooses each action with probability £ and the total probability that agent ij makes a mistake is IAij 1 £ . Let C b e a collection of agents in a standard form game g and denote the complement of C by C. Given a strategy vector t for the agents in C, write g� = (A, u 1 )c for the reduced game faced by the agents in C when the agents in C play t, hence u i/s) =
for ij E C. Write g� = gc and u 1 = u c in the special case where t is the centroid strategy for each agent in C. The set C is called cell in g if for each t and each player i with an agent in C there exist constants a; (t) > 0 and {3; (t) E lR such that
Uij (s, t)
(5 . 1 ) Hence, if C i s a cell, then up to positive linear transformations, the payoffs to agents in C are completely determined by the agents in C . Since the intersection of two cells is again a cell whenever this intersection is nonempty, there exist minimal cells. Such cells are called elementary cells. Two elementary cells have an empty intersection. Note that for the special case of a normal form game (each player has only one agent), each cell is a small world. Also note that a transformation as in (5 . 1 ) leaves the best-reply structure unchanged. Hence, if we had defined a small world as a set of players whose (admissible) best responses are not influenced by outsiders, then each small world would have been a cell. A solution concept that assigns to each standard form game g a unique strategy vector f (g) is said to satisfy cell and truncation consistency if for each C that is a cell in g we have if ij E C, if ij rf_ C .
(5.2)
The reader may check that a subgame of a uniformly perturbed extensive form game induces a cell in the associated perturbed standard form; hence, the axiom of cell and truncation consistency formalizes the idea that the solution is determined by backward induction in the extensive form.
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If g is a standard form game and BiJ is a nonempty set of actions for each agent ij, then B = XiJ BiJ is called a formation if for each agent ij, each best response against any correlated strategy that only puts probability on actions in B belongs to BiJ . Hence, in normal form games, formations are just like curb sets (cf. Section 2.2), the only dif ference being that formations allow for correlated beliefs. As the intersection of two formations is again a formation, we can speak about primitive formations, i.e., forma tions that do not contain a proper subformation. An action a of an agent ij is said to be inferior if there exists another action b of this agent that is a best reply against a strictly larger set of (possibly) correlated beliefs of the agents. Hence, noninferiority corresponds to the concept of robustness that we encountered (for the case of independent beliefs) in Section 4.5. Any strategy that is weakly dominated is inferior, but the converse need not hold. Using the concepts introduced above, we can now describe the main steps employed in the Harsanyi and Selten solution procedure: 1 . Initialization: Form the standard normal form g of the game, and, for each E > 0 that is sufficiently small, compute the uniformly perturbed game g 8 ; com pute the solution f (gE:) according to the steps described below and put f (g) = limc+of(g 8 ) . 2. Decomposition: Decompose the game into its elementary cells; compute the solu tion of an indecomposable game according to the steps described below and form the solution of the overall game by using cell and truncation consistency. 3. Reduction: Reduce the game by using the next three operations: (i) Eliminate all inferior actions of all agents. (ii) Replace each set of equivalent actions of each agent ij (i.e., actions among which all players are indifferent) by the centroid of that set. (iii) Replace, for each agent ij , each set of ij -equivalent actions (i.e., actions among which ij is indifferent no matter what the others do) by the centroid strategy of that set. By applying these steps, an irreducible game results. The solution of such a game is by means of Step 4. 4. Solution: (i) Initialization: Split the game into its primitive formations and determine the solution of each basic game associated with each primitive formation by ap plying the tracing procedure to the centroid of that formation. The set of all these solutions constitutes the first candidate set f2 1 . (ii) Candidate elimination and substitution: Given a candidate set f2 , determine the set M ( f2) of maximally stable elements in f2 . These are those equilibria in Q that are least dominated in Q . Dominance involves both payoff domi nance and risk dominance and payoff dominance ranks more important than risk dominance. The latter is defined by means of the tracing procedure (see below) and need not be transitive. Form the chain Q = !2 1 , gt+l = M(Q 1 ) until Q T + 1 = Q T . If I Q T I = 1 , then Q T is the solution, otherwise replace
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Q T by the trace, t (Q T ), of its centroid and repeat the process with the new candidate set Q = Q T- 1 \Q T U {t (D T )}.
It should be noted that i t may b e necessary to go through these steps repeatedly. Furthermore, the steps are hierarchically ordered, i.e., if the application of step 3(i) (i.e., the elimination of inferior actions) results in a decomposable game, one should first return to step 2. The reader is referred to the flow chart on p. 1 27 of Harsanyi and Selten ( 1988) for further details. The next two sections of the present paper are devoted to step 4, the core of the solution procedure. We conclude this subsection with some remarks on the other steps. We already discussed step 2, as well as the reliance on the agent normal form in the previous subsection. Deriving the solution of an unperturbed game as a limit of solu tions of uniformly perturbed games has several consequences that might be considered undesirable. For one, duplicating strategies in the unperturbed game may have an effect on the outcome. Consider the normal form of the game from Figure 6. If we duplicate the strategy pl 1 of player 1 , the limit solution prescribes r2 for player 2 (since the mis take pl 1 is more likely than the mistake pr 1 ), but if we duplicate pr 1 then the solution prescribes player 2 to choose l2 . Hence, the Harsanyi-Selten solution does not satisfy the invariance requirement (I) from Section 4.2, nor does it satisfy (IIA). Secondly, an action that is dominated in the unperturbed game need no longer be dominated in the £ perturbed version of the game and, consequently, it is possible to construct an example in which the Harsanyi-Selten solution is an equilibrium that uses dominated strategies [Van Damme (1990)]. Hence, the Harsanyi-Selten solution violates (A). Turning now to the reduction step, we note that the elimination procedure implies that invariance is violated. (Cf. the discussion on persistency in Section 4.5; note that any pure strategy that is a mixture of non-equivalent pure strategies is inferior.) Let us also remark that stable sets need not survive when an inferior strategy is eliminated. [See Van Damme (1 987a, Figure 10.3. 1) for an example.] Finally, we note that since the Harsanyi-Selten theory makes use of payoff comparisons of equilibria, the solution of that theory is not best reply invariant. We return to this below.
5.2. Risk dominance in 2 2 games x
The core of the Harsanyi-Selten theory of equilibrium selection consists of a procedure that selects, in each situation in which it is common knowledge among the players that there are only two viable solution candidates, one of these candidates as the actual solu tion for that situation. A simple example of a game with two obvious solution candidates (viz. the strict equilibria (a, a) and (a, a)) is the stag-hunt game of the left-hand panel of Figure 10, which is a slight modification of a game first discussed in Aumann ( 1 990). (The only reason to discuss this variant is to be able to draw simpler pictures.) The stag hunt from the left-hand panel is a symmetric game with common interests [Aumann and Sorin ( 1989)], i.e., it has (a, a) as the unique Pareto-efficient outcome. Playing a, however, is quite risky: if the opponent plays his alternative equilibrium strat egy a , the payoff is only zero. Playing a is much safer: one is guaranteed the equilib rium payoff and, if the opponent deviates, the payoff is even higher. Harsanyi and Selten
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a
ii
a
4, 4 0, 3
a
3, 0
2, 2
a
a
a
4, 4
O, x + 1
a
x + l, O
x, x
0 Figure 10. The stag hunt.
discuss a variant of this game extensively since it is a case where the two selection cri teria that are used in their theory (viz. those of payoff dominance and risk dominance) point in opposite directions. [See Harsanyi and Selten ( 1 988, pp. 88, 89, and 358, 359).] Obviously, if each player could trust the other to play a, he would also play a, and players clearly prefer such mutual trust to exist. The question, however, is under which conditions such trust exists and how it can be created if it does not exist. As Aumann ( 1990) has argued, preplay communication cannot create trust where it does not exist initially. In the end, Harsanyi and Selten decide to give precedence to the payoff domi nance criterion, i.e., they assume that rational players can rely on collective rationality and they select (a, a) in the game of Figure 10. However, the arguments given are not fully convincing. We will use the game of Figure 10 to illustrate the concept of risk dominance, which is based on strictly individualistic rationality considerations. Intuitively, the equilibrium s risk dominates the equilibrium s if, when players are in a state of mind where they think that either s or s should be played, they eventually come to the conclusion that s is too risky and, hence, they should play s . For general games, risk dominance is defined by means of the tracing procedure. For the special case of 2-player 2 x 2 normal form games with two strict equilibria, the concept is also given an axiomatic foundation. Before discussing this axiomatization, we first illustrate how riskiness of an equilibrium can be measured in 2 x 2 games. Let G(a, a) be the set of all 2-player normal form games in which each player i has the strategy set {a, a} available and in which (a, a) and (a, a) are strict Nash equilibria. For g E G(a, a), we identify a mixed strategy of player i with the probability ai that this strategy assigns to a and we write ai = 1 ai . We also write di (a) for the loss that player i incurs when he unilaterally deviates from (a, a) (hence, d1 (a) = u 1 (a, a) u 1 (a, a)) and we define di (a) similarly. Note that when player j plays a with probability aj given by -
-
aj = di (a)/(di (a) + di (a) ) ,
(5 .3)
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E. van Damme
player i is indifferent between a and a . Hence, the probability aj as in (5 .3) represents the risk that i is willing to take at (a, a)) before he finds it optimal to switch to a. In a symmetric game (such as that of Figure 10) aj = a{, hence aj (resp. aj) is a natural measure of the riskiness of the equilibrium (a, a) (resp. (a, a)) and (a , a) is more risky if aj < aj, that is, if aj < 1 /2. In the game of Figure 10, we have that aj = 2j3 , hence (a, a) is more risky than (a, a). More generally, let us measure the riskiness of an equilibrium as the sum of the players' risks. Formally, say that (a, a) risk dominates (a, a) in g (abbreviated a >- g a) if
(5.4) say that (a, a) risk dominates (a, a) (written a >-g a) if the reverse strict inequality holds, and say that there is no dominance relationship between (a, a) and (a , a)) (writ ten a �g a) if (5 . 4) holds with equality. In the game of Figure 10, we have that (a, a) risk dominates (a, a). To show that these definitions are not "ad hoc", w e now give an axiomatization of risk-dominance. On the class G(a, a), Harsanyi and Selten (1988, Section 3.9) charac terize this relation by the following axioms. 1. (Asymmetry and completeness): For each g exactly one of the following holds: a >-g a or a >-g a or a �g a. 2. (Symmetry): I f g i s symmetric and player i prefers (a, a) while player j ( j i= i) prefers (a, a), then a �g ii . 3 . (Best-reply invariance): If g and g ' have the same best-reply correspondence, then a >-g a , if and only if a >-g' a . ' 4 . (Payoff monotonicity): If g results from g by making (a, a ) more attractive for some player i while keeping all other payoffs the same, then a >-g' a whenever a >-g a or a �g a. The proof i s simple and follows from the observations that (i) games are best-reply-equivalent if and only if they have the same (aj, a{), (ii) symmetric games with conflicting interests satisfy (5 .4) with equality, and (iii) increasing u; (a, a) decreases aj. Harsanyi and Selten also give an alternative characterization of risk-dominance. Con dition (5.4) is equivalent to the (Nash) product of players' deviation losses at (a, a) being larger than the corresponding Nash product at (a, ii), hence
(5.5) and, in fact, the original definition is by means of this inequality. Yet another equivalent characterization is that the area of the stability region of (a, a) (i.e., the set of mixed strategies against which a is a best response for each player) is larger than the area of the stability region of (a, a). (Obviously, the first area is ajar, the second is ajar.> For the stag hunt game, the stability regions have been displayed in the middle panel of Figure 10. (The diagonal represents the line a 1 + a2 = 1 ; the upper left corner of the
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1579
diagram is the point a 1 = 1, a2 = 1, it corresponds to the upper left comer of the matrix, and similarly for other points.) In Carlsson and Van Damme ( 1 993a), equilibrium selection according to the risk dominance criterion is derived from considerations related to uncertainty concerning the payoffs of the game. These authors assume that players can observe the payoffs in a game only with some noise. In contrast to Harsanyi's model that was discussed in Section 2.5, Carlsson and Van Damme assume that each player is uncertain about both players' payoffs. Because of the noise, the actual best-reply structure will not be common knowledge and as a consequence of this lack of common knowledge, players' behavior at each observation may be governed by the behavior at some remote observa tion [also cf. Rubinstein ( 1989)] . In the noisy version of the stag hunt game of Figure 8, even though players may know to a very high degree that (a, a) is the Pareto-dominant equilibrium, they might be unwilling to play it since each player i might think that j will play a since i will think that j will think. . . that a is a dominant action. Hence, even though this model superficially resembles that of Harsanyi ( 1 973a), it leads to completely different results. As a simple and concrete illustration of the model, suppose that it is common knowl edge among the players that payoffs are related to actions as in the right panel g(x) of Figure 10. A priori, players consider all values x E [ - 1 , 4] to be possible and they consider all such values to be equally likely. (Carlsson and Van Damme (1 993a) show that the conclusion is robust with respect to such distributional assumptions, as well as with respect to assumptions on the structure of the noise.) Note that g(x) E G(a, a) for X E (0, 3) , that a is a dominant strategy if X < 0 and that a is dominant if X > 3. Suppose now that players can observe the actual value of x that prevails only with some slight noise. Specifically, assume player i observes x; = x + 8e; where x , e 1 , e2 are independent and e; is uniformly distributed on [ - 1 , 1 ] . Obviously, if x; < -8 (resp. x; > 3 + 8), player i will play a (resp. a) since he knows that that action is dominant at each actual value of x that corresponds to such an observation. Forcing players to play their dominant actions at these observations will make a and a dominant at a larger set of observations and the process can be continued iteratively. Let !. (resp. i) be the supremum (resp. infimum) of the set of observations y for which each player i has a (resp. a) as an iteratively dominant action for each x; < y (resp. x; > y). Then there must be a player i who is indifferent between a and a when he observes !. (resp. i ) . Writing ai (x;) for the probability that i assigns to j playing a when he observes x; , we can write the indifference condition of player i at x; (approximately) as (5.6) Now, at x; = !_, we have that ai (x;) is at least 1 /2 because of our symmetry assumptions and since j has a as an iteratively dominant strategy for each x j < !.· Consequently, !. ;) 3 j2. A symmetric argument establishes that i ::( 3 j2, hence !. = i = 3/2, and each player i should choose a if he observes x; < 3 j2 while he should choose a if x; > 3 j2. Hence, in the noisy version of the game, each player should always play the risk-dominant equilibrium of the game that corresponds to his observation.
E. van Damme
1580
To conclude this subsection, we remark that the concept of risk dominance also plays an important role in the literature that derives Nash equilibrium as a stationary state of processes of learning or evolution. Even though each Nash equilibrium may be a stationary state of such a process, occasional experimentation or mutation may result in only the risk-dominant equilibrium surviving in the long run: this equilibrium has a larger stability region, hence, a larger basin of attraction, so that the process is more easily trapped there and mutations have more difficulty in upsetting it [see Kandori et al. ( 1 993), Young (1993a, 1993b), Ellison (1993)]. 5.3.
Risk dominance and the tracing procedure
Let us now consider a more general normal form game g = (A, u) where the players are uncertain which of two equilibria, s or s, should be played. Risk dominance tries to capture the idea that in this state of confusion the players enter a process of expectation formation that converges on that equilibrium which is the least risky of the two. (Note that a player i with s; = s; is not confused at all. Harsanyi and Selten first eliminate all such players before making risk comparisons. For the remaining players they similarly delete strategies not in the formation spanned by s and s since these are never best responses, no matter what expectations the players have. To the smaller game that results in this way, one should then first apply the decomposition and reduction steps from Section 5.2. We shall assume that all these transformations have been made and we will denote the resulting game again by g.) Harsanyi and Selten view the rational formation of expectations as a two-stage process. In the first stage, players form preliminary expectations which are based on the structure of the game. These preliminary expectations take the form of a mixed strategy vector s 0 for the game. On the basis of s 0 , players can already form plans about how to play the game. A naive plan would be for each player to play the best response against s 0 , but, of course, these plans are not necessarily consistent with the prelimi nary expectations. The second stage of the expectation formation process then consists of a procedure that gradually adjusts plans and expectations until they are consistent and yield an equilibrium of the game g. Harsanyi and Selten actually make use of two adjustment processes, the linear tracing procedure T and the logarithmic tracing pro cedure T . Formally, each of these is a map that assigns to a mixed strategy vector s 0 exactly one equilibrium of g. The linear tracing procedure is easier to work with, but it is not always well-defined. The logarithmic tracing procedure is well-defined and yields the same outcome as the linear one whenever the latter is well-defined. We now first dis cuss these tracing procedures. Thereafter, we return to the question of how to form the preliminary expectations and how to define risk dominance for general games. Let g = (A, u) be a normal form game and let p be a vector of mixed strategies for g, interpreted as the players' prior expectations. For t E [0, 1 ] define the game gi·P = (A, u 1 · P) by
u ; · P = tu; (s) + ( 1 - t)u; (p\s; ) .
(5.7)
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1581
Hence, for t = 1 the game coincides with g, while g0·P is a trivial game in which each player's payoff depends only on this player's prior expectations, not on what the oppo nents are actually doing. Write r (p) for the graph of the equilibrium correspondence, hence
r(p) = { (t, s) E [0, 1]
x
S:
s is an equilibrium of g1·P } .
(5.8)
In nondegenerate cases, g0·P will have exactly one (and strict) equilibrium s(O, p) and this equilibrium will remain an equilibrium for sufficiently small t. Let us denote it by s(t, p). The linear tracing procedure now consists in following the curve s (t, p) until, at its endpoint T(p) = s(l, p), an equilibrium of g is reached. Hence, as the tracing procedure progresses, plans and expectations are continuously adjusted until an equilibrium is reached. The parameter t may be interpreted as the degree of confidence players have in the solution s (t, p). Formally, the linear tracing procedure with prior p is well-defined if the graph r(p) contains a unique connected curve that contains endpoints both at t = 0 and t = 1 . In this case, the endpoint T (p) at t = 1 is called the linear trace of p. (Note the requirement that there be a unique connecting curve. Herings (2000) shows that there will always be at least one such curve, hence, the procedure is feasible in principle.) We can illustrate the procedure by means of the stag hunt game from Figure 10. Write p; for the prior probability that i plays a. If Pi > 2/3 for i = 1, 2, then g0·P has (a, a) as its unique equilibrium and this strategy pair remains an equilibrium for all t. Furthermore, for any t E [0, 1], (a, a) is disconnected in r(p) from any other equilibrium of l·P. Hence, in this case the linear tracing procedure is well-defined and we have T(p) = (a, a). Similarly, T(p) = (a, a) if Pi < 2/3 for i = 1 , 2. Next, assume P i < 2/3 and P2 > 2/3 so that s(O, p) = (a, a). In this case the initial plans do not constitute an equilibrium of the final game so that adjustments have to take place along the path. The strategy pair (a, a) remains an equilibrium of g1·P as long as
4(1 - t)p2 � 2t + (1 - t)(2 + P2 )
(5.9)
and
(1 - t)(2 + PI) + 3t � 4pl (l - t) + 4t. Hence, provided that no player switches before t, player given by
t 1-t
(5. 10) 1 has to switch at the value of t (5. 1 1)
while player 2 has to switch when
t = 2 - 3P 1 · 1 -t
--
(5. 1 2)
1582
E. van Damme P2
0
I
3
(a)
(b)
Figure l l . (a) In the interior of the shaded area T (p) = (a, a). In the interior of the complement T(p) = (a, a). (b) A case where the linear tracing procedure is not well-defined.
Assume PI + P2 /2 < 1 so that the t-value determined by (5. 1 1) is smaller than the value determined by (5. 1 2). Hence, player 1 has to switch first and, following the branch (a , a), the linear tracing procedure continues with a branch (a, a). Since (a, a) is a strict equilibrium of g, this branch continues until t = 1 , hence T (p) = (a, a) in this case. Similarly, T (p) = (a , a) if P I < 2/3, P2 > 2/3 and PI + P2 /2 > 1 . In the case where P I > 2/3 and P2 < 2/3, the linear trace of p follows by symmetry. The results of our computations are summarized in the left-hand panel of Figure 1 1. If P I < 2/3, P2 > 2/3 and Pl + P2 /2 = 1 , then the equations (5. 1 1)-(5.12) deter mine the same t-value, hence, both players want to switch at the same time f. In this case, the game g i· P is degenerate with equilibria both at (a , a ) and at (a, a). Now there exists a path in r that connects (a , a) with (a , a ) as well as a path that connects (a , a) with (a, a). In fact, all three equilibria of g (including the mixed one) are connected to the equilibrium of g 0· P , hence, the linear tracing procedure is not well-defined in this case. Figure 1 1 (b) gives a graphical display of this case. (The picture is drawn for the case where PI = 1 /2, P2 = 1 and displays the probability of 1 choosing a . ) The logarithmic tracing procedure has been designed to resolve ambiguities such as those in Figure l l (b). For c E (0, 1], t E [0, 1) and p E S, define the game gO ,p by means of
u �, t ,p (s) = u � · P (s) + c ( l t)ai 2 )n si (a ) ,
(5. 1 3)
-
a
where ai is a constant defined by
(5.14) Hence, u : · t ,p (s) results from adding a logarithmic penalty term to u ; P (s). This term ensures that all equilibria are completely mixed and that there is a unique equilibrium ·
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s(&, 0, p) if t = 0. Write T (p) for the graph of the equilibrium correspondence T (p) = { (&, t, s) E (0, 1] x [0, 1) x S: s is an equilibrium of g 8· 1 · P } .
(5. 15)
T (p) is the zero set of a polynomial and, hence, is an algebraic set. Loosely speaking, the logarithmic tracing procedure consists of following, for each & > 0, the analytic continuation s (t:, t, p) of s (&, 0, p) till t = 1 and then taking the limit, as & -7 0, of the end points. Harsanyi and Selten (1988) and Harsanyi (1975) claim that this construction can indeed be carried out, but Schanuel et al. (1991) pointed to some difficulties in this construction: the analytic continuation need not be a curve and there is no reason for the limit to exist. Fortunately, these authors also showed that, apart from a finite set E of &-values, the construction proposed by Harsanyi and Selten is indeed feasible. Specifically, if & ¢ E, then there exists a unique analytic curve in f (p) that contains s (& , 0, p). If we write s (& , t, p) for the strategy component of this curve, then T (p) = lim8 �o limt-> I S (&, t, p) exists. T (p) is called the logarithmic trace of p. Hence, the logarithmic tracing procedure is well-defined. Furthermore, Schanuel et al. ( 1991) show that there exists a connected curve in r (p) connecting T (p) to an equilibrium in g 0 • P implying that T (p)
=
T (p) whenever the latter is well-defined. Hence, we have
1 3 [Harsanyi (1975), Schanuel et al. (1991)]. The logarithmic tracing pro cedure T is well-defined. The linear tracing procedure T is well-definedfor almost all priors and T (p) = T (p) whenever the latter is well-defined. THEOREM
The logarithmic penalty term occurring in (5.13) gives players an incentive to use completely mixed strategies. It has the consequence that in Figure 1 1 (b) the interior mixed strategy path is approximated as & --+ 0. Hence, if p is on the south-east boundary of the shaded region in Figure l l (a), then T (p) is the mixed strategy equilibrium of the game g. We finally come to the construction o f the prior probability distribution p used in the risk dominance comparison between s and s. According to Harsanyi and Selten, each player i will initially assume that his opponents already know whether s or s is the solution. Player i will assign a subjective probability z; to the solution being s and a probability Zi = 1 z; to the solution being s. Given his beliefs Z; player i will then choose a best response b�; to the correlated strategy z;s_; + z; s-; of his opponents. (In case of multiple best responses, i chooses all of them with the same probability.) An opponent j of player i is assumed not to know i 's subjective probability z; ; how ever, j knows that i is following the above reasoning process. Applying the principle of insufficient reasoning, Harsanyi and Selten assume that j considers all values of z; to be equally likely, hence, j considers z; to be uniformly distributed on [0, 1]. Conse quently, j believes that i will play a; E A; with a probability given by -
p; (a;) =
f b; (a;) dz; . Zi
(5. 16)
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1584
Equation (5. 16) determines the players' prior expectations p to be used for risk dominance comparison between s and s. If T(p) = s (resp. T (p) = s) then s is said to risk dominate s (resp. s risk dominates s). If T (p) tJ_ {s , S}, neither equilibrium risk dominates the other. The reader may verify that for 2 x 2 games this definition of risk dominance is in agreement with the one given in the previous section. For example, in the stag hunt game from Figure 8 we have that b:; (a) = 1 if z; > 2/3 and b�; (a) = 0 if z; < 2/3, hence p; (a) = 1/3. Consequently, p lies in the non-shaded region in Fig ure l l (a) and T (p) = (a, a), hence, (a, a) risk dominates (a, a). Unfortunately, for games larger than 2 x 2, the risk dominance relation need not be transitive [see Harsanyi and Selten (1988, Figure 3.25) for an example] and selection on the basis of this criterion need not be in agreement with selection on the basis of stability with respect to payoff perturbations [Carlsson and Van Damme (1993b)] . To illustrate the latter, consider the n-player stag hunt game in which each player i has the strategy set {a , a}. A player choosing a gets the payoff 1 if all players choose a, and 0 otherwise. A player choosing a gets the payoff X E (0, 1) irrespective of what the others do. There are two strict Nash equilibria, viz. "all a" and "all a". If player i assigns prior probability z to his opponents playing the former, then he will play a if z > x , hence, p; (a) = 1 - x according to (5. 16). Consequently, the risk-dominant solution is "all a" if
(1 - xt - 1 > X
(5. 1 7)
and it is "all a" if the reverse strict inequality is satisfied. On the other hand, Carls son and Van Damme ( 1 993b) derive that, whenever there is slight payoff uncertainty, a player should play a if 1/ n > x . It is interesting to note that this n-person stag hunt game has a potential (cf. Section 2.3) and that the solution identified by Carlsson and Van Damme maximizes the potential. More generally, suppose that, when there are k players choosing a, the payoff to a player choosing a equals f(k) (with f(O) = 0, f (n) = 1 ) and that the payoff to a player choosing a equals X E (0, 1 ) . Then the func tion p that assigns to each outcome in which exactly k players cooperate the value k
p(k) = �]f(l) - X]
1= 1
(5 . 1 8 )
is an exact potential for the game. "All a" maximizes the potential if and only if L�= 1 f(l)/n > x and this condition is identical to the one that Carlsson and Van Damme derive for a to be optimal in their model. To conclude this subsection, we remark that, in order to derive (5. 16), it was assumed that player i 's uncertainty can be represented by a correlated strategy of the opponents. Gtith (1985) argues that such correlated beliefs may reflect the strategic aspects rather poorly and he gives an example to show that such a correlated belief may lead to coun terintuitive results. Gtith suggests computing the prior as above, save by starting from
Ch. 41:
Strategic Equilibrium
the assumption that i believes j =/= i to play different z's being independent. 5.4.
1585
z1s1 + z/ si
with ZJ uniform on [0, 1] and
Risk dominance and payoff dominance
We already encountered the fundamental conflict between risk dominance and payoff dominance when discussing the stag hunt game in Section 5.2 (Figure 10). In that game, the equilibrium (a, a) Pareto dominates the equilibrium (a, a), but the latter is risk dom inant. In cases of such conflict, Harsanyi and Selten have given precedence to the payoff dominance criterion, but their arguments for doing so are not compelling, as they indeed admit in the postscript of their book, when they discuss Aumann's argument (also al ready mentioned in Section 5.2) that pre-play communication cannot make a difference in this game. After all, no matter what a player intends to play he will always attempt to induce the other to play a as he always benefits from this. Knowing this, the opponent cannot attach specific meaning to the proposal to play (a, a), communication cannot change a player's beliefs about what the opponent will do and, hence, communication can make no difference to the outcome of the game [Aumann ( 1990)]. As Harsanyi and Selten ( 1 988, p. 359) write, "This shows that in general we cannot expect the players to implement payoff dominance unless, from the very beginning, payoff dominance is part of the rationality concept they are using. Free communication among the players in it self might not help. Thus if one feels that payoff dominance is an essential aspect of game-theoretic rationality, then one must explicitly incorporate it into one's concept of rationality". Several equilibrium concepts exist that explicitly incorporate such considerations. The most demanding concept is Aumann's ( 1 959) notion of a strong equilibrium: it requires that no coalition can deviate in a way that makes all its members better off. Already in simple examples such as the prisoners' dilemma, this concept generates an empty set of outcomes. (In fact, generically all Nash equilibria are inefficient [see Dubey ( 1 986)] .) Less demanding is the idea that the grand coalition not be able to renegotiate to a more attractive stable outcome. This idea underlies the concept of renegotiation proof equilibrium from the literature on repeated games [see Bernheim and Ray ( 1989), Farrell and Maskin ( 1989) and Van Damme ( 1 988, 1 989a)]. Bernheim et al. (1987) have proposed the interesting concept of coalition-proofNash equilibrium as a formalization of the requirement that no subcoalition should be able to profitably deviate to a strat egy vector that is stable with respect to further renegotiation. The concept is defined for all normal form games and the formal definition is by induction on the number of players. For a one-person game any payoff-maximizing action is defined to be coalition proof. For an I -person game, a strategy profile s is said to be weakly coalition-proof, if, for any proper subcoalition coalition C of I, the strategy profile sc is coalition proof in the reduced game in which the complement C is restricted to play sz;, and s is said to be coalition-proof if there is no other weakly coalition-proof profile s' that strictly Pareto dominates it. For 2-player games, coalition-proof equilibria exist, but ex istence for larger games is not guaranteed. Furthermore, coalition-proof equilibria may be Pareto dominated by other equilibria.
E. van Damme
1586 a
a
a
2, 2, 2
0, 0, 0
a
o, o, o
3, 3, 0
Figure 12. Renegotiation as a constraint.
The tension between "global" payoff dominance and "local" efficiency was already pointed out in Harsanyi and Selten ( 1 988): an agreement on a Pareto-efficient equi librium may not be self-enforcing since, with the agreement in place, and accepting the logic of the concept, a subcoalition may deviate to an even more profitable agree ment. The following provides a simple example. Consider the 3-player game g in which player 3 first decides whether to take up an outside option T (which yields all players the payoff 1 ) or to let players 1 and 2 play a subgame in which the payoffs are as in Figure 12. The game g from Figure 1 2 has two Nash equilibrium outcomes. In the first, player 3 chooses T (in the belief that 1 and 2 will choose a with sufficiently high probabil ity); in the second, player 3 chooses p, i.e., he gives the move to players 1 and 2, who play (a, a). Both outcomes are subgame perfect (even stable) and the equilibrium (a, a, p) Pareto dominates the equilibrium T. At the beginning of the game it seems in the interest of all players to play (a, a, p). However, once player 3 has made his move, his interests have become strategically irrelevant and it is in the interest of players 1 and 2 to renegotiate to (a , a). Although the above argument was couched i n terms o f the extensive form of the game, it is equally relevant for the case in which the game is given in strategic form, i.e., when players have to move simultaneously. After agreeing to play (a, a, p), players 1 and 2 could secretly get together and arrange a joint deviation to (a, a). This deviation is in their interest and it is stable since no further deviations by subgroups are profitable. Hence, the profile (a, a, p) is not coalition-proof. The reader may argue that these "cooperative refinements" in which coalitions of players are allowed to deviate jointly have no place in the theory of strategic equilib rium, and that, as suggested in Nash (1 953), it is preferable to stay squarely within the non-cooperative framework and to fully incorporate possibilities for communication and cooperation in the game rather than in the solution concept. The present author agrees with that view. The above discussion has been included to show that, while it is tempting to argue that equilibria that are Pareto-inferior should be discarded, this view encoun ters difficulties and may not stand up to closer scrutinity. Nevertheless, the shortcut may sometimes yield valuable insights. The interested reader is referred to Bernheim and Whinston (1987) for some applications using the shortcut of coalition-proofness. 5.5.
Applications and variations
Nash ( 1 953) already noted the need for a theory of equilibrium selection for the study of bargaining. He wrote: "Thus the equilibrium points do not lead us immediately to
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a solution of the game. But if we discriminate between them by studying their relative stabilities we can escape from this troublesome nonuniqueness" [Nash ( 1 953, pp. 1 3 1 , 1 32)]. Nash studied 2-person bargaining games in which the players simultaneously make payoff demands, and in which each player receives his demand if and only if the pair of demands is feasible. Since each pair that is just compatible (i.e., is Pareto optimal) is a strict equilibrium, there are multiple equilibria. Using a perturbation argu ment, Nash suggested taking that equilibrium in which the product of the utility gains is largest as the solution of the game. The desire to have a solution with this "Nash prod uct property" has been an important guiding principle for Harsanyi and Selten when developing their theory (cf. (5.5)). One of the first applications of that theory was to unanimity games, i.e., games in which each player's payoff is zero unless all players si multaneously choose the same alternative. As the reader can easily verify, the Harsanyi and Selten solution of such a game is indeed the outcome in which the product of the payoffs is largest, provided that there is such a unique maximizing outcome. Another early application of the theory was to market entry games [Selten and Giith (1982)]. In such a game there are I players who simultaneously decide whether to enter a market or not If k players enter, the payoff to a player i that enters is rr(k) - c; , while his payoff is zero otherwise (rr is a decreasing function). The Harsanyi and Selten solution prescribes entry of the players with the lowest entry costs up to the point where entry becomes unprofitable. The Harsanyi and Selten theory has been extensively applied to bargaining prob lems [cf. Harsanyi and Selten (1988, Chapters 6-9), Harsanyi ( 1 980, 1982), Leopold Wildburger ( 1 985), Selten and Giith ( 1 99 1 ), Selten and Leopold ( 1 983)] . Such problems are modelled as unanimity games, i.e., a set of possible agreements is specified, play ers simultaneously choose an agreement and an agreement is implemented if and only if it is chosen by all players. In case there is no agreement, trade does not take place. For example, consider bargaining between two risk-neutral players about how to di vide one dollar and suppose that one of the players, say player 1, has an outside option of a. The Harsanyi-Selten solution allocates max(y(¥, 1 /2) to player I and the rest to player 2. Hence, the outside option influences the outcome only if it is sufficiently high [Harsanyi and Selten ( 1 988, Chapter 6)] . As another example, consider bargaining be tween one seller and n identical buyers about the sale of an indivisible object If the seller's value is 0 and each buyer's value is I , the Harsanyi-Selten solution is that each player proposes a sale at the price p (n) = (2n - 1) j (2n - 1 + n). Harsanyi and Selten ( 1 988, Chapters 8 and 9) apply the theory to simple bargaining games with incomplete information. Players bargain about how to divide one dollar; if there is disagreement, a player receives his conflict payoff, which may be either 0 or x (both with probability 1 /2) and which is private information. In the case of one-sided incomplete information (it is common knowledge that player 1 's conflict payoff is zero), player 1 proposes that he get a share x (a ) of the cake, where x (a ) is some decreasing square root function of a with x (0) = 50. The weak type of player 2 (i.e., the one with conflict payoff 0) proposes that player 1 get x (a ) , while the strong type proposes x (a ) if a < a* (� 8 1 ) and 0 in case a > a * . Hence, the bargaining outcome may be ex
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post inefficient. Gtith and Selten (1991) consider a simple version of Akerlof's lemons problem [Akerlof ( 1 970)]. A seller and a buyer are bargaining about the price of an object of art, which may be either worth 0 to each of them (it is a forgery) or which may be worth 1 to the seller and v > 1 to the buyer. The seller knows whether the object is original or fake, but the buyer only knows that both possibilities have positive probability. The solution either is disagreement, or exploitation of the buyer by the seller (i.e., the price equals the buyer's expected value), or some compromise in which the buyer bears a greater part of the fake risk than the seller does. At some parameter values, the solution (the price) changes discontinuously, and Gtith and Selten admit that they cannot give plausible intuitive interpretations for these jumps. Van Damme and Gtith ( 1 99 l a, 199 l b) apply the Harsanyi-Selten theory to signalling games. In Van Damme and Gtith ( 1 99 1 a) the most simple version of the Spence ( 1 973) signalling game is considered. There are two types of workers, one productive, the other unproductive, who differ in their education costs and who can use the education level to signal their type to uninformed employers who compete in prices a la Bertrand. It turns out that the Harsanyi and Selten solution coincides with the £2-equilibrium that was proposed in Wilson (1977). Hence, the solution is the sequential equilibrium that is most preferred by the high quality worker, and this worker signals his type if and only if signalling yields higher utility than pooling with the unproductive worker does. It is worth remarking that this solution is obtained without invoking payoff dominance. Note that the solution is again discontinuous in the parameter of the problem, i.e., in the ex ante probability that the worker is productive. The discontinuity arises at points where a different element of the Harsanyi-Selten solution procedure has to be invoked. Specif ically, if the probability of the worker being unproductive is small, then there is only one primitive formation and this contains only the Pareto-optimal pooling equilibrium. As soon as this probability exceeds a certain threshold, however, also the formation spanned by the Pareto-optimal separating equilibrium is primitive, and, since the sepa rating equilibrium risk dominates the pooling equilibrium, the solution is separating in this case. We conclude this subsection by mentioning some variations of the Harsanyi-Selten theory that have recently been proposed. Gtith and Kalkofen ( 1 989) propose the ESE ORA theory, whose main difference to the Harsanyi-Selten theory is that the (intran sitive) risk dominance relation is replaced by the transitive relation of resistance dom inance. The latter takes the intensity of the dominance relation into account. Formally, given two equilibria s and s ' , define player i 's resistance at s against s ' as the largest probability z such that, when each player j =!= i plays (1 z)s j + zsj , player i still prefers Si to s� . Gtith and Kalkofen propose ways to aggregate these individual resis tances into a resistance of s against s ' which can be measured by a number r (s, s ' ) . The resistance against s ' can then be represented by the vector R (s ' ) = (r(s, s ' ) ) s and Gtith and Kalkofen propose to select that equilibrium s ' for which the vector R (s ' ) , written in nonincreasing order, is lexicographically minimal. At present the ESBORA theory is still incomplete: the individual resistances can be aggregated in various ways and the solution may depend in an essential way on which aggregation procedure is adopted, as -
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examples in Gtith and Kalkofen ( 1 989) show [see also Gtith (1992) for different aggre gation procedures] . For a restricted class of games (specifically, bipolar games with linear incentives), Selten (1995) proposes a set of axioms that determine a unique rule to aggregate the players individual resistances into an overall measure of resistance (or risk) dominance. For 2 x 2 games, selection on the basis of this measure is in agreement with selec tion as in Section 5.2, but for larger games, this need no longer be true. In fact, for 2-player games with incomplete information, selection according to the measure pro posed in Selten (1995) has close relations with selection according to the "generalized Nash product" as in Harsanyi and Selten ( 1 972). Finally, we mention that Harsanyi (1995) proposes to replace the bilateral risk com parisons between pairs of equilibria by a multilateral comparison involving all equilibria that directly identifies the least risky of all of them. He also proposes not to make use of payoff comparisons, a suggestion that brings us back the to fundamental conflict between payoff dominance and risk dominance that was discussed in Section 5.4. 5. 6.
Final remark
We end this section and chapter by mentioning a result from Norde et al. ( 1 996) that puts all the attempts to select a unique equilibrium in a different perspective. Recall that in Section 2 we discussed the axiomatization of Nash equilibrium using the concept of consistency, i.e., the idea that a solution of a game should induce a solution of any reduced game in which some players are committed to playing the solution. Norde et al. (1 996) show that if s is a Nash equilibrium of a game g, g can be embedded in a larger game that only has s as an equilibrium; consequently consistency is incompatible with equilibrium selection. More precisely, Norde et al. (1996) show that the only solution concept that satisfies consistency, nonemptiness and one-person rationality is the Nash concept itself, so that not only equilibrium selection, but even the attempt to refine the Nash concept is frustrated if one insists on consistency. References Akerlof, G. ( 1970), "The market for lemons", Quarterly Journal of Economics 84:488-500. Aumann, R.J. ( 1959), "Acceptable points in general cooperative n-person games", in: R.D. Luce and A.W. Tucker, eds., Contributions to the Theory of Games, IV, Annals of Mathematics Studies, Vol. 40 (Princeton, NJ) 287-324. Aumann, R.J. (1974), "Subjectivity and correlation in randomized strategies", Journal of Mathematical Eco nomics 1 :67-96. Aumann, R.J. (1985), "What is game theory trying to accomplish?", in: K. Arrow and S. Honkapohja, eds., Frontiers of Economics (Basil Blackwell, Oxford) 28-76. Aumann, R. J . ( 1987), "Game theory", in: J. Eatwell, M. Milgate and P. Newman, eds., The New Palgrave Dictionary of Economics (Macmillan, London) 460-482. Aumann, R.J. ( 1990), "Nash equilibria are not self-enforcing", in: J.J. Gabszewicz, J.-F. Richard and L.A. Wolsey, eds., Economic Decision-Making: Games, Econometrics and Optimisation (Elsevier, Am sterdam) 201-206.
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Jansen, M.J.M., A.P. Jurg and P.E.M. Bonn (1990), "On the finiteness of stable sets", mimeo (University of Nijmegen). Kalai, E., and D. Samet ( 1984), "Persistent equilibria", International Journal of Game Theory 1 3 : 1 29-141 . Kalai, E . , and D. Samet (1985), "Unanimity games and Pareto optimality", International Journal of Game Theory 14:41-50. Kandori, M., G.J. Mailath and R. Rob (1993), "Learning, mutation and long-run equilibria in games", Econo metrica 61:29-56. Kohlberg, E. (198 1 ), "Some problems with the concept of perfect equilibrium", NBER Conference on Theory of General Economic Equilibrium, University of California, Berkeley. Kohlberg, E. ( 1989), "Refinement of Nash equilibrium: The main ideas", mimeo (Harvard University). Kohlberg, E., and J.-F. Mertens ( 1986), "On the strategic stability of equilibria", Econometrica 54: 1 003-1037. Kohlberg, E., and P. Reny (1997), "Independence on relative probability spaces and consistent assessments in game trees", Journal of Economic Theory 75:280-3 1 3 . Kreps, D . , and G . Ramey ( 1987), "Structural consistency, consistency, and sequential rationality", Economet rica 55: 1 33 1-1348. Kreps, D., and J. Sobel ( 1994), "Signalling", in: R.J. Aumann and S. Hart, eds., Handbook of Game Theory, Vol. 2 (North-Holland, Amsterdam) Chapter 25, 849-868. Kreps, D., and R. Wilson (1 982a), "Sequential equilibria", Econometrica 50:863-894. Kreps, D., and R. Wilson ( 1982b), "Reputation and imperfect information", Journal of Economic Theory 27:253-279.
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Chapter 42 FOUNDATIONS OF STRATEGIC EQUILIBRIUM JOHN HILLAS
Department ofEconomics, University ofAuckland, Auckland, New Zealand ELON KOHLBERG
Harvard Business School, Boston, MA, USA Contents
1. 2.
Introduction Pre-equilibrium ideas 2. 1 . Iterated dominance and rationalizability 2.2. Strengthening rationalizability
3.
The idea o f equilibrium 3 . 1 . Self-enforcing plans 3.2. Self-enforcing assessments
4. 5. 6. 7. 8.
The mixed extension of a game The existence of equilibrium Correlated equilibrium The extensive form Refinement o f equilibrium 8 . 1 . The problem of perfection 8.2. Equilibrium refinement versus equilibrium selection
9. 10.
Admissibility and iterated dominance Backward induction 10. 1. The idea of backward induction 10.2. Subgame perfection 10.3. Sequential equilibrium 10.4. Perfect equilibrium 10.5. Perfect equilibrium and proper equilibrium 10.6. Uncertainties about the game
1 1. 12.
Forward induction Ordinality and other invariances 12. 1 . Ordinality 12.2. Changes in the player set
13.
Strategic stability
Handbook of Game Theory, Volume 3, Edited by R.I. Aumann and S. Hart © 2002 Elsevier Science B. V. All rights reserved
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1598 1 3 . 1 . The requirements for strategic stability 13.2. Comments on sets of equilibria as solutions to non-cooperative games 13.3. Forward induction 13 .4. The definition of strategic stability 13.5. Strengthening forward induction 13.6. Forward induction and backward induction
14. 15.
An assessment of the solutions Epistemic conditions for equilibrium References
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Abstract
This chapter examines the conceptual foundations of the concept of strategic equilib rium and its various variants and refinements. The emphasis is very much on the under lying ideas rather than on any technical details. After an examination of some pre-equilibrium ideas, in particular the concept of ra tionalizability, the concept of strategic (or Nash) equilibrium is introduced. Various in terpretations of this concept are discussed and a proof of the existence of such equilibria is sketched. Next, the concept of correlated equilibrium is introduced. This concept can be thought of as retaining the self-enforcing aspect of the idea of equilibrium while relaxing the independence assumption. Most of the remainder of the chapter is concemed with the ideas underlying the re finement of equilibrium: admissibility and iterated dominance; backward induction; for ward induction; and ordinality and various invariances to changes in the player set. This leads to a consideration of the concept of strategic stability, a strong refinement satisfy ing these various ideas. Finally there is a brief examination of the epistemic approach to equilibrium and the relation between strategic equilibrium and correlated equilibrium.
Keywords
Nash equilibrium, strategic equilibrium, correlated equilibrium, equilibrium refinement, strategic stability
JEL classification: C72
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1. Introduction
The central concept of noncooperative game theory is that of the strategic equilibrium (or Nash equilibrium, or noncooperative equilibrium). A strategic equilibrium is a pro file of strategies or plans, one for each player, such that each player's strategy is optimal for him, given the strategies of the others. In most of the early literature the idea of equilibrium was that it said something about how players would play the game or about how a game theorist might recommend that they play the game. More recently, led by Harsanyi ( 1973) and Aumann (1987a), there has been a shift to thinking of equilibria as representing not recommendations to players of how to play the game but rather the expectations of the others as to how a player will play. Further, if the players all have the same expectations about the play of the other players we could as well think of an outside observer having the same information about the players as they have about each other. While we shall at times make reference to the earlier approach we shall basically follow the approach of Harsanyi and Aumann or the approach of considering an outside observer, which seem to us to avoid some of the deficiencies and puzzles of the earlier approach. Let us consider the example of Figure 1 . In this example there is a unique equilibrium. It involves Player 1 playing T with probability 1/2 and B with probability 1/2 and Player 2 playing L with probability 1/3 and R with probability 2/3 . There is some discomfort in applying the first interpretation to this example. In the equilibrium each player obtains the same expected payoff from each of his strategies. Thus the equilibrium gives the game theorist absolutely no reason to recommend a particular strategy and the player no reason to follow any recommendation the theorist might make. Moreover one often hears comments that one does not, in the real world, see players actively randomizing. If, however, we think of the strategy of Player 1 as representing the uncertainty of Player 2 about what Player 1 will do, we have no such problem. Any assessment other than the equilibrium of the uncertainty leads to some contradiction. Moreover, if we assume that the uncertainty in the players' minds is the objective uncertainty then we also have tied down exactly the distribution on the strategy profiles, and consequently the expected payoff to each player, for example 2/3 to Player 1 . This idea of strategic equilibrium, while formalized for games by Nash (1950, 195 1), goes back at least to Coumot ( 1 838). 1t is a simple, beautiful, and powerful concept. It seems to be the natural implementation of the idea that players do as well as they can, taking the behavior of others as given. Aumann (1974) pointed out that there is someL
R
Figure 1 .
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thing more involved in the definition given by Nash, namely the independence of the strategies, and showed that it is possible to define an equilibrium concept that retains the idea that players do as well as they can, taking the behavior of others as given while dropping the independence assumption. He called such a concept correlated equilib rium. Any strategic equilibrium "is" a correlated equilibrium, but for many games there are correlated equilibria which are quite different from any of the strategic equilibria. In another sense the requirements for strategic equilibrium have been seen to be too weak. Selten ( 1 965, 1 975) pointed out that irrational behavior by each of two different players might make the behavior of the other look rational, and proposed additional requirements, beyond those defining strategic equilibrium, to eliminate such cases. In doing so Selten initiated a large literature on the refinement of equilibrium. Since then many more requirements have been proposed. The question naturally arises as to whether it is possible to simultaneously satisfy all, or even a large subset of such re quirements. The program to define strategically stable equilibria, initiated by Kohlberg and Mertens ( 1 986) and brought to fruition by Mertens ( 1987, 1989, 199lb, 1 992) and Govindan and Mertens ( 1993), answers this question in the affirmative. This chapter is rather informal. Not everything is defined precisely and there is little use, except in examples, of symbols. We hope that this will not give our readers any problem. Readers who want formal definitions of the concepts we discuss here could consult the chapters by Hart (1 992) and van Damme (2002) in this Handbook.
2. Pre-equilibrium ideas
Before discussing the idea of equilibrium in any detail we shall look at some weaker conditions. We might think of these conditions as necessary implications of assuming that the game and the rationality of the players are common knowledge, in the sense of Aumann ( 1 976). 2.1.
Iterated dominance and rationalizability
Consider the problem of a player in some game. Except in the most trivial cases the set of strategies that he will be prepared to play will depend on his assessment of what the other players will do. However it is possible to say a little. If some strategy was strictly preferred by him to another strategy s whatever he thought the other players would do, then he surely would not play s. And this remains true if it was some lottery over his strategies that was strictly preferred to s . We call a strategy such as s a strictly
dominated strategy.
Perhaps we could say a little more. A strategy s would surely not be played unless there was some assessment of the manner in which the others might play that would lead s to be (one of ) the best. This is clearly at least as restrictive as the first requirement. (If s is best for some assessment it cannot be strictly worse than some other strategy for all assessments.) In fact, if the set of assessments of what the others might do is convex
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(as a set of probabilities on the profiles of pure strategies of the others) then the two requirements are equivalent. This will be true if there is only one other player, or if a player's assessment of what the others might do permits correlation. However, the set of product distributions over the product of two or more players' pure strategy sets is not convex. Thus we have two cases: one in which we eliminate strategies that are strictly domi nated, or equivalently never best against some distribution on the vectors of pure strate gies of the others ; and one in which we eliminate strategies that are never best against some product of distributions on the pure strategy sets of the others. In either case we have identified a set of strategies that we argue a rational player would not play. But since everything about the game, including the rationality of the players, is assumed to be common knowledge no player should put positive weight, in his assessment of what the other players might do, on such a strategy. And we can again ask: are there any strategies that are strictly dominated when we restrict attention to the assessments that put weight only on those strategies of the others that are not strictly dominated? If so, a rational player who knew the rationality of the others would surely not play such a strategy. And similarly for strategies that were not best responses against some assess ment putting weight only on those strategies of the others that are best responses against some assessment by the others. And we can continue for an arbitrary number of rounds. If there is ever a round in which we don't find any new strategies that will not be played by rational players com monly knowing the rationality of the others, we would never again "eliminate" a strat egy. Thus, since we start with a finite number of strategies, the process must eventually terminate. We call the strategies that remain iteratively undominated or correlatedly rationalizable in the first case; and rationalizable in the second case. The term rational izable strategy and the concept were introduced by Bernheim ( 1984) and Pearce ( 1 984). The term correlatedly rationalizable strategy and the concept were explicitly introduced by Brandenburger and Dekel ( 1 987), who also show the equivalence of this concept to what we are calling iteratively undominated strategies, though both the concept and this equivalence are alluded to by Pearce ( 1 984). The issue of whether or not the assessments of one player of the strategies that will be used by the others should permit correlation has been the topic of some discussion in the literature. Aumann (1987a) argues strongly that they should. Others have argued that there is at least a case to be made for requiring the assessments to exhibit independence. For example, Bernheim ( 1 986) argues as follows. Aumann has disputed this view [that assessments should exhibit independence] . He argues that there is no a priori basis to exclude any probabilistic beliefs. Cor relation between opponents' strategies may make perfect sense for a variety of reasons. For example, two players who attended the same "school" may have similar dispositions. More generally, while each player knows that his decision does not directly affect the choices of others, the substantive information which leads him to make one choice rather than another also affects his beliefs about other players' choices.
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Yet Aumann's argument is not entirely satisfactory, since it appears to make our theory of rationality depend upon some ill-defined "dispositions" which are, at best, extra-rational. What is the "substantive information" which disposes an in dividual towards a particular choice? In a pure strategic environment, the only available substantive information consists of the features of the game itself. This information is the same, regardless of whether one assumes the role of an outside observer, or the role of a player with a particular "disposition". Other information, such as the "school" which a player attended, is simply extraneous. Such infor mation could only matter if, for example, different schools taught different things. A "school" may indeed teach not only the information embodied in the game it self, but also "something else"; however, differences in schools would then be substantive only if this "something else" was substantive. Likewise, any appar ently concrete source of differences or similarities in dispositions can be traced to an amorphous "something else", which does not arise directly from considera tions of rationality. This addresses Aumann's ideas in the context of Aumann's arguments concerning correlated equilibrium. Without taking a position here on those arguments, it does seem that in the context of a discussion of rationalizability the argument for independence is not valid. In particular, even if one accepts that one's opponents actually choose their strategies independently and that there is nothing substantive that they have in common outside their rationality and the roles they might have in the game, another player's assessment of what they are likely to play could exhibit correlation. Let us go into this in a bit more detail. Consider a player, say Player 3, making some assessment of how Players 1 and 2 will play. Suppose also that Players 1 and 2 act in similar situations, that they have no way to coordinate their choices, and that Player 3 knows nothing that would allow him to distinguish between them. Now we assume that Player 3 forms a probabilistic assessment as to how each of the other players will act. What can we say about such an assessment? Let us go a bit further and think of another player, say Player 4, also forming an assessment of how the two players will play. It is a hallmark of rationalizability that we do not assume that Players 3 and 4 will form the same assessment. (In the definition of rationalizability each strategy can have its own justification and there is no assumption that there is any consistency in the justifications.) Thus we do not assume that they somehow know the true probability that Players 1 and 2 will play a certain way. Further, since we allow them to differ in their assessments it makes sense to also allow them to be uncertain not only about how Players 1 and 2 will play, but indeed about the probability that a rational player will play a certain way. This is, in fact, exactly analogous to the classic problem discussed in probability theory of repeated tosses of a possibly biased coin. We assume that the coin has a certain fixed probability p of coming up heads. Now if an observer is uncertain about p then the results of the coin tosses will not, conditional on his information, be statistically independent. For example, if his assessment was that with probability one half p = 1/4 and with probability one
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B
A
B
A
B
9, 9, 5
B 8, 0, 2 E
w Figure 2.
half p = 3 j4 then after seeing heads for the first three tosses his assessment that heads will come up on the next toss will be higher than if he had seen tails on the first three tosses. Let us be even more concrete and consider the three player game of Figure 2. Here Players 1 and 2 play a symmetric game having two pure strategy equilibria. This game has been much discussed in the literature as the "Stag Hunt" game. [See Aumann (1 990), for example.] For our purposes here all that is relevant is that it is not clear how Players 1 and 2 will play and that how they will play has something to do with how rational players in general would think about playing a game. If players in general tend to "play safe" then the outcome (B, B) seems likely, while if they tend to coordinate on efficient outcomes then (A, A) seems likely. Player 3 has a choice that does not affect the payoffs of Players 1 and 2, but whose value to him does depend on the choices of 1 and 2. If Players 1 and 2 play (B, B) then Player 3 does best by choosing E, while if they play (A, A) then Player 3 does best by choosing W. Against any product dis tribution on the strategies of Players 1 and 2 the better of E or W is better than C for Player 3 . Now suppose that Player 3 knows that the other players were independently randomly chosen to play the game and that they have no further information about each other and that they choose their strategies independently. As we argued above, if he doesn't know the distribution then it seems natural to allow him to have a nondegenerate distribution over the distributions of what rational players commonly knowing the rationality of the others do in such a game. The action taken by Player 1 will, in general, give Player 3 some information on which he will update his distribution over the distributions of what rational players do. And this will lead to correlation in his assessment of what Players 1 and 2 will do in the game. Indeed, in this setting, requiring independence essentially amounts to requiring that players be certain about things about which we are explicitly allowing them to be wrong. We indicated earlier that the set of product distributions over two or more players' strategy sets was not convex. This is correct, but somewhat incomplete. It is possible to put a linear structure on this set that would make it convex. In fact, that is exactly what we do in the proof of the existence of equilibrium below. What we mean is that if we think of the product distributions as a subset of the set of all probability distributions on the profiles of pure strategies and use the linear structure that is natural for that latter set then the set of product distributions is not convex. If instead we use the product of the linear structures on the spaces of distributions on the individual strategy spaces, then the
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set of product distributions will indeed be convex. The nonconvexity however reappears in the fact that with this linear structure the expected payoff function is no longer linear - or even quasi-concave. 2.2.
Strengthening rationalizability
There have been a number of suggestions as to how to strengthen the notion of ra tionalizability. Many of these involve some form of the iterated deletion of (weakly) dominated strategies, that is, strategies such that some other (mixed) strategy does at least as well whatever the other players do, and strictly better for some choices of the others. The difficulty with such a procedure is that the order in which weakly dominated strategies are eliminated can affect the outcome at which one arrives. It is certainly possible to give a definition that unambiguously determines the order, but such a definition implicitly rests on the assumption that the other players will view a strategy eliminated in a later round as infinitely more likely than one eliminated ear lier. Having discussed in the previous section the reasons for rejecting the requirement that a player's beliefs over the choices of two of the other players be independent we shall not again discuss the issue but shall allow correlated beliefs in all of the definitions we discuss. This has two implications. The first is that our statement below of Pearce's notion of cautiously rationalizable strategies will not be his original definition but rather the suitably modified one. The second is that we shall be able to simplify the descrip tion by referring simply to rounds of deletions of dominated strategies rather than the somewhat more complicated notions of rationalizability. Even before the definition of rationalizability, Moulin (1979) suggested using as a solution an arbitrarily large number of rounds of the elimination of all weakly domi nated strategies for all players. Moulin actually proposed this as a solution only when it led to a set of strategies for each player such that whichever of the allowable strate gies the others were playing the player would be indifferent among his own allowable strategies. A somewhat more sophisticated notion is that of cautiously rationalizable strate gies defined by Pearce (1984). The set of such strategies is the set obtained by the following procedure. One first eliminates all strictly dominated strategies, and does this for an arbitrarily large number of rounds until one reaches a game in which there are no strictly dominated strategies. One then has a single round in which all weakly dominated strategies are eliminated. One then starts again with another (arbi trarily long) sequence of rounds of elimination of strictly dominated strategies, and again follows this with a single round in which all weakly dominated strategies are removed, and so on. For a finite game such a process ends after a finite number of rounds. Each of these definitions has a certain apparent plausibility. Nevertheless they are not well motivated. Each depends on an implicit assumption that a strategy eliminated at a later round is much more likely than a strategy eliminated earlier. And this in turn
Ch. 42: Foundations of Strategic Equilibrium
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Y
Z
A
1, 1
1, 0
1,0
B
0, 1
1, 0
2, 0
c
1,0
1, 1
0, 0
D
0, 0
0, 0
1, 1
Figure 3.
depends on an implicit assumption that in some sense the strategies deleted at one round are equally likely. For suppose we could split one of the rounds of the elimination of weakly dominated strategies and eliminate only part of the set. This could completely change the entire process that follows. Consider, for example, the game of Figure 3. The only cautiously rationalizable strategies are A for Player 1 and X for Player 2. In the first round strategies C and D are (weakly) dominated for Player 1 . After these are eliminated strategies Y and Z are strictly dominated for Player 2. And after these are eliminated strategy B is strictly dominated for Player 1 . However, if (A , X) is indeed the likely outcome then perhaps strategy D is, in fact, much less likely than strategy C, since, given that 2 plays X, strategy C is one of Player 1 's best responses, while D is not. Suppose that we start by eliminating just D. Now, in the second round only Z is strictly dominated. Once Z is eliminated, we eliminate B for Player 1, but nothing else. We are left with A and C for Player 1 and X and Y for Player 2. There seems to us one slight strengthening of rationalizability that is well motivated. It is one round of elimination of weakly dominated strategies followed by an arbitrarily large number of rounds of elimination of strictly dominated strategies. This solution is obtained by Dekel and Fudenberg ( 1 990), under the assumption that there is some small uncertainty about the payoffs, by Borgers (1 994), under the assumption that rationality was "almost" common knowledge, and by Ben-Porath ( 1 997) for the class of generic extensive form games with perfect information. The papers of Dekel and Fudenberg and of Borgers use some approximation to the common knowledge of the game and the rationality of the players in order to derive, simultaneously, admissibility and some form of the iterated elimination of strategies. Ben-Porath obtains the result in extensive form games because in that setting a natural definition of rationality implies more than simple ex ante expected utility maximization. An alternative justification is possible. Instead of deriving admissibility, we include it in what we mean by rationality. A choice s is admissibly rational against some conjecture c about the strategies of the others if there is some sequence of conjectures putting pos itive weight on all possibilities and converging to c such that s is maximizing against each conjecture in the sequence. Now common knowledge of the game and of the ad missible rationality of the players gives precisely the set we described. The argument is essentially the same as the argument that common knowledge of rationality implies correlated rationalizability.
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3. The idea of equilibrium
In the previous section we examined the extent to which it is possible to make predic tions about players' behavior in situations of strategic interaction based solely on the common knowledge of the game, including in the description of the game the players' rationality. The results are rather weak. In some games these assumptions do indeed restrict our predictions, but in many they imply few, if any, restrictions. To say something more, a somewhat different point of view is productive. Rather than starting from only knowledge of the game and the rationality of the players and asking what implications can be drawn, one starts from the supposition that there is some established way in which the game will be played and asks what properties this manner of playing the game must satisfy in order not to be self-defeating, that is, so that a rational player, knowing that the game will be played in this manner, does not have an incentive to behave in a different manner. This is the essential idea of a strategic equilibrium, first defined by John Nash (1950, 1951). There are a number of more detailed stories to go along with this. The first was suggested by von Neumann and Morgenstern (1944), even before the first definition of equilibrium by Nash. It is that players in a game are advised by game theorists on how to play. In each instance the game theorist, knowing the player's situation, tells the player what the theory recommends. The theorist does offer a (single) recommendation in each situation and all theorists offer the same advice. One might well allow these recommendations to depend on various "real-life" features of the situation that are not normally included in our models. One would ask what properties the theory should have in order for players to be prepared to go along with its recommendations. This idea is discussed in a little more detail in the introduction of the chapter on "Strategic equilibrium" in this Handbook [van Damme (2002)]. Alternatively, one could think of a situation in which the players have no information beyond the rules of the game. We'll call such a game a Tabula-Rasa game. A player's optimal choice may well depend on the actions chosen by the others. Since the player doesn't know those actions we might argue that he will form some probabilistic as sessment of them. We might go on to argue that since the players have precisely the same information they will form the same assessments about how choices will be made. Again, one could ask what properties this common assessment should have in order not to be self-defeating. The first to make the argument that players having the same infor mation should form the same assessment was Harsanyi (1967-1968, Part III). Aumann (1974, p. 92) labeled this view the Harsanyi doctrine. Yet another approach is to think of the game being preceded by some stage of "pre play negotiation" during which the players may reach a non-binding agreement as to how each should play the game. One might ask what properties this agreement should have in order for all players to believe that everyone will act according to the agreement. One needs to be a little careful about exactly what kind of communication is available to the players if one wants to avoid introducing correlation. Ban'iny (1992) and Forges (1990) show that with at least four players and a communication structure that allows
Ch. 42: Foundations of Strategic Equilibrium
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for private messages any correlated equilibrium "is" a Nash equilibrium of the game augmented with the communication stage. There are also other difficulties with the idea of justifying equilibria by pre-play negotiation. See Aumann (1990). Rather than thinking of the game as being played by a fixed set of players one might think of each player as being drawn from a population of rational individuals who find themselves in similar roles. The specific interactions take place between randomly se lected members of these populations, who are aware of the (distribution of) choices that had been made in previous interactions. Here one might ask what distributions are self-enforcing, in the sense that if players took the past distributions as a guide to what the others' choices were likely to be, the resulting optimal choices would (could) lead to a similar distribution in the current round. One already finds this approach in Nash
(1950).
A somewhat different approach sees each player as representing a whole population of individuals, each of whom is "programmed" (for example, through his genes) to play a certain strategy. The players themselves are not viewed as rational, but they are assumed to be subject to "natural selection", that is, to the weeding out of all but the payoff-maximizing programs. Evolutionary approaches to game theory were introduced by Maynard Smith and Price (1973). For the rest of this chapter we shall consider only interpretations that involve rational players. 3.1.
Self-enforcing plans
One interpretation of equilibrium sees the focus of the analysis as being the actual strate gies chosen by the players, that is, their plans in the game. An equilibrium is defined to be a self-enforcing profile of plans. At least a necessary condition for a profile of plans to be self-enforcing is that each player, given the plans of the others, should not have an alternate plan that he strictly prefers. This is the essence of the definition of an equilibrium. As we shall soon see, in order to guarantee that such a self-enforcing profile of plans exists we must consider not only deterministic plans, but also random plans. That is, as well as being permitted to plan what to do in any eventuality in which he might find himself, a player is explicitly thought of as planning to use some lottery to choose between such deterministic plans. Such randomizations have been found by many to be somewhat troubling. Arguments are found in the literature that such "mixed strategies" are less stable than pure strate gies. (There is, admittedly, a precise sense in which this is true.) And in the early game theory literature there is discussion as to what precisely it means for a player to choose a mixed strategy and why players may choose to use such strategies. See, for example, the discussion in Luce and Raiffa (1957, pp. 74-76). Harsanyi (1973) provides an interpretation that avoids this apparent instability and, in the process, provides a link to the interpretation of Section 3.2. Harsanyi considers a model in which there is some small uncertainty about the players' payoffs. This un certainty is independent across the players, but each player knows his own payoff. The uncertainty is assumed to be represented by a probability distribution with a contin uously differentiable density. If for each player and each vector of pure actions (that
J. Hillas and E. Kohlberg
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is, strategies in the game without uncertainty) the probability that the payoff is close to some particular value is high, then we might consider the game close to the game in which the payoff is exactly that value. Conversely, we might consider the game in which the payoffs are known exactly to be well approximated by the game with small uncertainty about the payoffs. Harsanyi shows that in a game with such uncertainty about payoffs all equilibria are essentially pure, that is, each player plays a pure strategy with probability 1 . Moreover, with probability 1 , each player is playing his unique best response to the strategies of the other players; and the expected mixed actions of the players will be close to an equilibrium of the game without uncertainty. Harsanyi also shows, modulo a small technical error later corrected by van Darnme ( 1 99 1), that any regular equilibrium can be approximated in this way by pure equilibria of a game with small uncertainty about the payoffs. We shall not define a regular equilibrium here. Nor shall we give any of the technical details of the construction, or any of the proofs. The reader should instead consult Harsanyi ( 1973), van Damme (199 1 ), or van Damme (2002). 3.2.
Self-enforcing assessments
Let us consider again Harsanyi's construction described in the previous section. In an equilibrium of the game with uncertainty no player consciously randomizes. Given what the others are doing the player has a strict preference for one of his available choices. However, the player does not know what the others are doing. He knows that they are not randomizing - like him they have a strict preference for one of their available choices - but he does not know precisely their payoffs. Thus, if the optimal actions of the oth ers differ as their payoffs differ, the player will have some probabilistic assessment of the actions that the others will take. And, since we assumed that the randomness in the payoffs was independent across players, this probabilistic assessment will also be independent, that is, it will be a vector of mixed strategies of the others. The mixed strategy of a player does not represent a conscious randomization on the part of that player, but rather the uncertainty in the minds of the others as to how that player will act. We see that even without the construction involving uncertainty about the payoffs, we could adopt this interpretation of a mixed strategy. This interpretation has been suggested and promoted by Robert Aumann for some time [for example, Au mann ( 1 987a), Aumann and Brandenburger ( 1 995)] and is, perhaps, becoming the pre ferred interpretation among game theorists. There is nothing in this interpretation that compels us to assume that the assessments over what the other players will do should exhibit independence. The independence of the assessments involves an additional as sumption. Thus the focus of the analysis becomes, not the choices of the players, but the assess ments of the players about the choices of the others. The basic consistency condition that we impose on the players' assessments is this: a player reasoning through the con clusions that others would draw from their assessments should not be led to revise his own assessment.
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More formally, Aumann and Brandenburger (1995, p. 1 177) show that if each player's assessment of the choices of the others is independent across the other players, if any two players have the same assessment as to the actions of a third, and if these assess ments, the game, and the rationality of the players are all mutually known, then the assessments constitute a strategic equilibrium. (A fact is mutually known if each player knows the fact and knows that the others know it.) We discuss this and related results in more detail in Section 15. 4 . The mixed extension of a game
Before discussing exactly how rational players assess their opponents' choices, we must reflect on the manner in which the payoffs represent the outcomes. If the players are pre sumed to quantify their uncertainties about their opponents' choices, then in choosing among their own strategies they must, in effect, compare different lotteries over the outcomes. Thus for the description of the game it no longer suffices to ascribe a pay off to each outcome, but it is also necessary to ascribe a payoff to each lottery over outcomes. Such a description would be unwieldy unless it could be condensed to a compact form. One of the major achievements of von Neumann and Morgenstern (1944) was the development of such a compact representation ("cardinal utility"). They showed that if a player's ranking of lotteries over outcomes satisfied some basic conditions of consis tency, then it was possible to represent that ranking by assigning numerical "payoffs" just to the outcomes themselves, and by ranking lotteries according to their expected payoffs. See the chapter by Fishburn (1994) in this Handbook for details. Assuming such a scaling of the payoffs, one can expand the set of strategies avail able to each player to include not only definite ("pure") choices but also probabilistic ("mixed") choices, and extend the definition of the payoff functions by taking the ap propriate expectations. The strategic form obtained in this manner is called the mixed extension of the game. Recall from Section 3.2 that we consider a situation in which each player's assess ment of the strategies of the others can be represented by a product of probability dis tributions on the others' pure strategy sets, that is, by a mixed strategy for each of the others. And that any two players have the same assessment about the choices of a third. Denoting the (identical) assessments of the others as a probability distribution (mixed strategy) over a player's (pure) choices, we may describe the consistency condition as follows : each player's mixed strategy must place positive probability only on those pure strategies that maximize the player's payoff given the others' mixed strategies. Thus a profile of consistent assessments may be viewed as a strategic equilibrium in the mixed extension of the game. Let us consider again the example of Figure 1 that we looked at in the introduction. Player 1 chooses the row and Player 2 (simultaneously) chooses the column. The result ing (cardinal) payoffs are indicated in the appropriate box of the matrix, with Player 1 's payoff appearing first.
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What probabilities could characterize a self-enforcing assessment? A (mixed) strat egy for Player 1 (that is, an assessment by 2 of how 1 might play) is a vector (x , 1 - x), where x lies between 0 and 1 and denotes the probability o f playing T. Similarly, a strat egy for 2 is a vector (y, 1 - y). Now, given x, the payoff-maximizing value of y is indicated in Figure 4(a), and given y the payoff-maximizing value of x is indicated in Figure 4(b). When the figures are combined as in Figure 4(c), it is evident that the game possesses a single equilibrium, namely x = 1 /2, y = 1/3. Thus in a self-enforcing as sessment Player 1 must assign a probability of 1/3 to 2's playing L, and Player 2 must assign a probability of 1/2 to Player 1 's playing T. Note that these assessments do not imply a recommendation for action. For example, they give Player 1 no clue as to whether he should play T or B (because the expected payoff to either strategy is the same). But this is as it should be: it is impossible to expect rational deductions to lead to definite choices in a game like Figure 1, because whatever those choices would be they would be inconsistent with their own implications. (Fig ure 1 admits no pure-strategy equilibrium.) Still, the assessments do provide the players with an a priori evaluation of what the play of the game is worth to them, in this case 2/3 to Player 1 and 1/2 to Player 2. The game of Figure 1 is an instance in which our consistency condition completely pins down the assessments that are self-enforcing. In general, we cannot expect such a sharp conclusion. Consider, for example, the game of Figure 5. There are three equilibrium outcomes: (8, 5), (7, 6) and (6, 3) (for the latter, the probability of T must lie between 0.5 and 0.6). Thus, all we can say is that there are three different consistent ways in which the players could view this game. Player 1 's assessment would be: either that Player 2 was (defiy 1 1------.
y
y 1
1 1-------,
.------'
� lr---+----'
!1 1
2
1
X
1 Figure 4.
L
C
Figure 5.
1
2
(c)
(b)
(a)
X
R
1
X
Ch. 42: Foundations of Strategic Equilibrium
161 1
nitely) going to play L, or that Player 2 was going to play C, or that Player 2 was going to play R. Which one of these assessments he would, in fact, hold is not revealed to us by means of equilibrium analysis.
5. The existence of equilibrium
We have seen that, in the game of Figure 1, for example, there may be no pure self enforcing plans. But what of mixed plans, or self-enforcing assessments? Could they too be refuted by an example? The main result of non-cooperative game theory states that no such example can be found.
1 [Nash (1950, 195 1)]. The mixed extension of everyfinite game has at least one strategic equilibrium. T HEOREM
(A game is finite if the player set as well as the set of strategies available to each player is finite.) The proof may be sketched as follows. (It is a multi-dimensional version of Figure 4(c).) Consider the set-valued mapping (or correspondence) that maps each strategy profile, x, to all strategy profiles in which each player's component strat egy is a best response to x (that is, maximizes the player's payoff given that the others are adopting their components of x ). If a strategy profile is contained in the set to which it is mapped (is a fixed point) then it is an equilibrium. This is so because a strategic equilibrium is, in effect, defined as a profile that is a best response to itself. Thus the proof of existence of equilibrium amounts to a demonstration that the "best response correspondence" has a fixed point. The fixed-point theorem of Kakutani ( 1941) asserts the existence of a fixed point for every correspondence from a convex and com pact subset of Euclidean space into itself, provided two conditions hold. One, the image of every point must be convex. And two, the graph of the correspondence (the set of pairs (x , y) where y is in the image of x) must be closed. Now, in the mixed extension of a finite game, the strategy set of each player consists of all vectors (with as many components as there are pure strategies) of non-negative numbers that sum to 1 ; that is, it is a simplex. Thus the set of all strategy profiles is a product of simplices. In particular, it is a convex and compact subset of Euclidean space. Given a particular choice of strategies by the other players, a player's best responses consist of all (mixed) strategies that put positive weight only on those pure strategies that yield the highest expected payoff among all the pure strategies. Thus the set of best responses is a subsimplex. In particular, it is convex. Finally, note that the conditions that must be met for a given strategy to be a best response to a given profile are all weak polynomial inequalities, so the graph of the best response correspondence is closed. S KETCH OF PROOF.
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Hi/las and E. Kohlberg
Thus all the conditions of Kakutani's theorem hold, and this completes the proof of Nash's theorem. D Nash's theorem has been generalized in many directions. Here we mention two. THEOREM 2 [Fan (1952), Glicksberg (1952)]. Consider a strategic form game with finitely many players, whose strategy sets are compact subsets of a metric space, and whose payofffunctions are continuous. Then the mixed extension has at least one strate gic equilibrium. (Here "mixed strategies" are understood as Borel probability measures over the given subsets of pure strategies.) THEOREM 3 [Debreu (1952)]. Consider a strategic form game with finitely many play ers, whose strategy sets are convex compact subsets of a Euclidean space, and whose payofffunctions are continuous. If, moreover, each payofffunction is quasi-concave in the player's own strategy, then the game has at least one strategic equilibrium. (A real-valued function on a Euclidean space is quasi-concave if, for each number a, the set o f points at which the value of the function i s at least a i s convex.) Theorem 2 may be thought of as identifying conditions on the strategy sets and payoff functions so that the game is like a finite game, that is, can be well approximated by finite games. Theorem 3 may be thought of as identifying conditions under which the strategy spaces are like the mixed strategy spaces for the finite games and the payoff functions are like expected utility.
6. Correlated equilibrium
We have argued that a self-enforcing assessment of the players' choices must consti tute an equilibrium of the mixed extension of the game. But our argument has been incomplete: we have not explained why it is sufficient to assess each player's choice separately. Of course, the implicit reasoning was that, since the players' choices are made in ignorance of one another, the assessments of those choices ought to be independent. In fact, this idea is subsumed in the definition of the mixed extension, where the expected payoff to a player is defined for a product distribution over the others' choices. Let us now make the reasoning explicit. We shall argue here in the context of a Tabula-Rasa game, as we outlined in Section 3. Let us call the common assessment of the players implied by the Harsanyi doctrine the rational assessment. Consider the assessment over the pure-strategy profiles by a rational observer who knows as much about the players as they know about each other. We claim that observation of some player's choice, say Player 1 's choice, should not affect the observer's assessment of
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the other players' choices. This is so because, as regards the other players, the observer and Player 1 have identical information and - by the Harsanyi doctrine - also identical analyses of that information, so there is nothing that the observer can learn from the player. Thus, for any player, the conditional probability, given that player's choice, over the others' choices is the same as the unconditional probability. It follows [Aumann and Brandenburger (1995, Lemma 4.6, p. 1 1 69) or Kohlberg and Reny (1997, Lemma 2.5, p. 285)] that the observer's assessment of the choices of all the players must be the product of his assessments of their individual choices. In making this argument we have taken for granted that the strategic form encompasses all the information available to the players in a game. This assumption, of "completeness", ensured that a player had no more information than was available to an outside observer. Aumann ( 1 974, 1987a) has argued against the completeness assumption. His position may be described as follows: it is impractical to insist that every piece of information available to some player be incorporated into the strategic form. This is so because the players are bound to be in possession of all sorts of information about random variables that are strategically irrelevant (that is, that cannot affect the outcome of the game). Thus he proposes to view the strategic form as an incomplete description of the game, indicat ing the available "actions" and their consequences; and to take account of the possibility that the actual choice of actions may be preceded by some unspecified observations by the players. Having discarded the completeness assumption (and hence the symmetry in infor mation between player and observer), we can no longer expect the rational assessment over the pure-strategy profiles to be a product distribution. But what can we say about it? That is, what are the implications of the rational assessment hypothesis itself? Aumann (1987a) has provided the answer. He showed that a distribution on the pure-strategy profiles is consistent with the rational assessment hypothesis if and only if it constitutes a correlated equilibrium. Before going into the details of Aumann's argument, let us comment on the signif icance of this result. At first blush, it might have appeared hopeless to expect a direct method for determining whether a given distribution on the pure-strategy profiles was consistent with the hypothesis: after all, there are endless possibilities for the players' additional observations, and it would seem that each one of them would have to be tried out. And yet, the definition of correlated equilibrium requires nothing but the verifica tion of a finite number of linear inequalities. Specifically, a distribution over the pure-strategy profiles constitutes a correlated equilibrium if it imputes positive marginal probability only to such pure strategies, s , a s are best responses against the distribution on the others' pure strategies obtained by conditioning on s . Multiplying throughout by the marginal probability of s, one obtains linear inequalities (if s has zero marginal probability, the inequalities are vacu ous).
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Consider, for example, the game of Figure 1 . Denoting the probability over the ij th entry of the matrix by Pii , the conditions for correlated equilibrium are as follows: 2p 1 1 �
Pl 2,
P22 � 2p2 1 ,
P21
and
� P1 1 ,
P 12 � P22·
There is a unique solution: P I ! = P2 1 = 1 /6, P l 2 = P22 = 1/3. So in this case, the correlated equilibria and the Nash equilibria coincide. For an example of a correlated equilibrium that is not a Nash equilibrium, consider the distribution l /2(T, L) + l /2(B, R) in Figure 6. The distribution over II's choices obtained by conditioning on T is L with probability 1 , and so T is a best response. Similarly for the other pure strategies and the other player. For a more interesting example, consider the distribution that assigns weight 1/6 to each non-zero entry of the matrix in Figure 7. [This example is due to Moulin and Vial (1978).] The distribution over Player 2's choices obtained by conditioning on T is C with probability 1/2 and R with probability 1 /2, and so T is a best response (it yields 1.5 while M yields 0.5 and B yields 1). Similarly for the other pure strategies of Player 1 , and for Player 2. It is easy to see that the correlated equilibria of a game contain the convex hull of its Nash equilibria. What is less obvious is that the containment may be strict [Aumann (1974)], even in payoff space. The game of Figure 7 illustrates this: the unique Nash equilibrium assigns equal weight, 1/3, to every pure strategy, and hence gives rise to the expected payoffs ( 1 , 1); whereas the correlated equilibrium described above gives rise to the payoffs ( 1 .5 , 1 .5) . Let us now sketch the proof of Aumann's result. By the rational assessment hypoth esis, a rational observer can assess in advance the probability of each possible list of observations by the players. Furthermore, he knows that the players also have the same assessment, and that each player would form a conditional probability by restricting attention to those lists that contain his actual observations. Finally, we might as well assume that the player's strategic choice is a function of his observations (that is, that if L
R
Figure 6. L
C
R
T
0, 0
1, 2
2, 1
M
2, 1
0, 0
1,2
B
1,2
2, 1
0, 0
Figure
7.
Ch. 42: Foundations of Strategic Equilibrium
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the player must still resort to a random device in order to decide between several payoff maximizing alternatives, then the observer has already included the various outcomes of that random device in the lists of the possible observations). Now, given a candidate for the rational assessment of the game, that is, a distribution over the matrix, what conditions must it satisfy if it is to be consistent with some such assessment over lists of observations? Our basic condition remains as in the case of Nash equilibria: by reasoning through the conclusions that the players would reach from their assessments, the observer should not be led to revise his own assessment. That is, conditional on any possible observation of a player, the pure strategy chosen must maximize the player's expected payoff. As stated, the condition is useless for us, because we are not privy to the rational observer's assessment of the probabilities over all the possible observations. However, by lumping together all the observations inducing the same choice, s (and by noting that, if s maximized the expected payoff over a number of disjoint events then it would also maximize the expected payoff over their union), we obtain a condition that we can check: that the choice of s maximizes the player's payoff against the conditional distribution given s . But this is precisely the condition for correlated equilibrium. To complete the proof, note that the basic consistency condition has no implications beyond correlated equilibria: any correlated equilibrium satisfies this condition rela tive to the following assessment of the players' additional observations: each player's additional observation is the name of one of his pure strategies, and the probability dis tribution over the possible lists of observations (that is, over the entries of the matrix) is precisely the distribution of the given correlated equilibrium. For the remainder of this chapter we shall restrict our attention to uncorrelated strate gies. The issues and concepts we discuss concerning the refinement of equilibrium are not as well developed for correlated equilibrium as they are in the setting where sto chastic independence of the solution is assumed.
7.
The extensive form
The strategic form is a convenient device for defining strategic equilibria: it enables us to think of the players as making single, simultaneous choices. However, actually to describe "the rules of the game", it is more convenient to present the game in the form of a tree. The extensive form [see Hart (1992)] is a formal representation of the rules of the game. It consists of a rooted tree whose nodes represent decision points (an appropri ate label identifies the relevant player), whose branches represent moves and whose endpoints represent outcomes. Each player's decision nodes are partitioned into infor mation sets indicating the player's state of knowledge at the time he must make his move: the player can distinguish between points lying in different information sets but cannot distinguish between points lying in the same information set. Of course, the ac tions available at each node of an information set must be the same, or else the player
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could distinguish between the nodes according to the actions that were available. This means that the number of moves must be the same and that the labels associated with moves must be the same. Random events are represented as nodes (usually denoted by open circles) at which the choices are made by Nature, with the probabilities of the alternative branches in cluded in the description of the tree. The information partition is said to have peifect recall [Kuhn ( 1 953)] if the players remember whatever they knew previously, including their past choices of moves. In other words, all paths leading from the root of the tree to points in a single information set, say Player i 's, must intersect the same information sets of Player i and must display the same choices by Player i . The extensive form is "finite" if there are finitely many players, each with finitely many choices at finitely many decision nodes. Obviously, the corresponding strategic form is also finite (there are only finitely many alternative "books of instructions"). Therefore, by Nash's theorem, there exists a mixed-strategy equilibrium. But a mixed strategy might seem a cumbersome way to represent an assessment of a player's behavior in an extensive game. It specifies a probability distribution over complete plans of action, each specifying a definite choice at each of the player's infor mation sets. It may seem more natural to specify an independent probability distribution over the player's moves at each of his information sets. Such a specification is called a
behavioral strategy.
Is nothing lost in the restriction to behavioral strategies? Perhaps, for whatever rea son, rational players do assess their opponents' behavior by assigning probabilities to complete plans of action, and perhaps some of those assessments cannot be reproduced by assigning independent probabilities to the moves? Kuhn's theorem ( 1953) guarantees that, in a game with perfect recall, nothing, in fact, is lost. It says that in such a game every mixed strategy of a player in a tree is equivalent to some behavioral strategy, in the sense that both give the same distribution on the endpoints, whatever the strategies of the opponents. For example, in the (skeletal) extensive form of Figure 8, while it is impossible to re produce by means of a behavioral strategy the correlations embodied in the mixed strat egy 0. 1 TLW + 0. 1 TRY + 0.5BLZ + O . l BLW + 0.2BRX, nevertheless it is possible to con struct an equivalent behavioral strategy, namely ( (0.2, 0. 8 ) (0.5, 0, 0.5), (0. 125 , 0.25, 0, 0.625 ) ) . To see the general validity o f the theorem, note that the distribution over the end points is unaffected by correlation of choices that anyway cannot occur at the same "play" (that is, on the same path from the root to an endpoint). Yet this is precisely the type of correlation that is possible in a mixed strategy but not in a behavioral strategy. (Correlation among choices lying on the same path is possible also in a behavioral strat egy. Indeed, this possibility is already built into the structure of the tree: if two plays differ in a certain move, then (because of perfect recall) they also differ in the informa tion set at which any later move is made and so the assessment of the later move can be made dependent on the earlier move.) ,
1617
Ch. 42: Foundations of Strategic Equilibrium L C
R
B
T
Figure
8. 0
L
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R
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Figure 9.
Kuhn's theorem allows us to identify each equivalence class of mixed strategies with a behavioral strategy - or sometimes, on the boundary of the space of behavioral strate gies, with an equivalence class of behavioral strategies. Thus, in games with perfect recall, for the strategic choices of payoff-maximizing players there is no difference between the mixed extension of the game and the "behavioral extension" (where the players are restricted to their behavioral strategies). In particular, the equilibria of the mixed and of the behavioral extension are equivalent, so either may be taken as the set of candidates for the rational assessment of the game. Of course, the equivalence of the mixed and the behavioral extensions implies the existence of equilibrium in the behavioral extension of any finite game with perfect re call. It is interesting to note that this result does not follow directly from either Nash's theorem or from Debreu's theorem. The difficulty is that the convex structure on the behavioral strategies does not reflect the convex structure on the mixed strategies, and therefore the best-reply correspondence need not be convex. For example, in Figure 9, the set of optimal strategies contains (T, R) and (B, L) but not their behavioral mixture ( � T + � B , � L + � R) . This corresponds to the difference between the two linear struc tures on the space of product distributions on strategy vectors that we discussed at the end of Section 2.1 .
8.
Refinement of equilibrium
Let us review where we stand: assuming that in any game there is one particular assess ment of the players' strategic choices that is common to all rational decision makers ("the rational assessment hypothesis"), we can deduce that that assessment must con-
J. Hillas and E. Kohlberg
1618 0, 0
4, 1
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Figure 10.
stitute a strategic equilibrium (which can be expressed as a profile of either mixed or behavioral strategies). The natural question is: can we go any further? That is, when there are multiple equi libria, can any of them be ruled out as candidates for the self-enforcing assessment of the game? At first blush, the answer seems to be negative. Indeed, if an assessment stands the test of individual payoff maximization, then what else can rule out its candidacy? And yet, it turns out that it is possible to rule out some equilibria. The key insight was provided by Selten ( 1965). It is that irrational assessments by two different players might each make the other look rational (that is, payoff-maximizing). A typical example is that of Figure 10. The assessment (T, R) certainly does not appear to be self-enforcing. Indeed, it seems clear that Player 1 would play B rather than T (because he can bank on the fact that Player 2 - who is interested only in his own payoff - will consequently play L). And yet (T, R) constitutes a strategic equilibrium: Player 1 's belief that Player 2 would play R makes his choice of T payoff-maximizing, while Player 2's belief that Player 1 would play T makes his choice of R (irrelevant hence) payoff-maximizing. Thus Figure 10 provides an example of an equilibrium that can be ruled out as a self-enforcing assessment of the game. (In this particular example there remains only a single candidate, namely (B, L).) By showing that it is sometimes possible to nar row down the self-enforcing assessments beyond the set of strategic equilibria, Selten opened up a whole field of research: the refinement of equilibrium. 8. 1.
The problem ofperfection
We have described our project as identifying the self-enforcing assessments. Thus we interpret Selten's insight as being that not all strategic equilibria are, in fact, self enforcing. We should note that this is not precisely how Selten described the problem. Selten explicitly sees the problem as the prescription of disequilibrium behavior in "un reached parts of the game" [Selten ( 1 975, p. 25)]. Harsanyi and Selten ( 1988, p. 17) describe the problem of imperfect equilibria in much the same way. Kreps and Wilson ( 1982) are a little less explicit about the nature of the problem but their description of their solution suggests that they agree. "A sequential equilibrium provides at each juncture an equilibrium in the subgame (of incomplete information) induced by restarting the game at that point" [Kreps and Wilson ( 1982, p. 864)] .
Ch. 42: Foundations of Strategic Equilibrium
1619 L
R
Figure 1 1 .
Also Myerson seems to view his definition of proper equilibrium as addressing the same problem as addressed by Selten. However he discusses examples and defines a solution explicitly in the context of normal form games where there are no umeached parts of the game. He describes the program as eliminating those equilibria that "may be inconsistent with our intuitive notions about what should be the outcome of a game" [Myerson (1978, p. 73)]. The description of the problem of the refinement of strategic equilibrium as looking for a set of necessary and sufficient conditions for self-enforcing behavior does nothing without specific interpretations of what this means. Nevertheless it seems to us to tend to point us in the right direction. Moreover it does seem to delineate the problem to some extent. Thus in the game of Figure 1 1 we would be quite prepared to concede that the equilibrium (B, R) was unintuitive while at the same time claiming that it was quite self-enforcing. 8.2.
Equilibrium refinement versus equilibrium selection
There is a question separate from but related to that of equilibrium refinement. That is the question of equilibrium selection. Equilibrium refinement is concerned with es tablishing necessary conditions for reasonable play, or perhaps necessary and sufficient conditions for "self-enforcing". Equilibrium selection is concerned with narrowing the prediction, indeed to a single equilibrium point. One sees a problem with some of the equilibrium points, the other with the multiplicity of equilibrium points. The central work on equilibrium selection is the book of Harsanyi and Selten (1988). They take a number of positions in that work with which we have explicitly disagreed (or will disagree in what follows): the necessity of incorporating mistakes; the necessity of working with the extensive form; the rejection of forward-induction-type reasoning; the insistence on subgame consistency. We are, however, somewhat sympathetic to the basic enterprise. Whatever the answer to the question we address in this chapter there will remain in many games a multiplicity of equilibria, and thus some scope for selecting among them. And the work of Harsanyi and Selten will be a starting point for those who undertake this enterprise. 9.
Admissibility and iterated dominance
It is one thing to point to a specific equilibrium, like (T, R) in Figure 10, and claim that "clearly" it cannot be a self-enforcing assessment of the game; it is quite another matter to enunciate a principle that would capture the underlying intuition.
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One principle that immediately comes to mind is admissibility, namely that rational players never choose dominated strategies. (As we discussed in the context of rational izability in Section 2.2 a strategy is dominated if there exists another strategy yielding at least as high a payoff against any choice of the opponents and yielding a higher pay off against some such choice.) Indeed, admissibility rules out the equilibrium (T, R) of Figure 10 (because R is dominated by L). Furthermore, the admissibility principle immediately suggests an extension, iterated dominance: if dominated strategies are never chosen, and if all players know this, all know this, and so on, then a self-enforcing assessment of the game should be unaffected by the (iterative) elimination of dominated strategies. Thus, for example, the equilibrium (T, L) of Figure 1 2(a) can be ruled out even though both T and L are admissible. (See Figure 1 2(b).) At this point we might think we have nailed down the underlying principle separating self-enforcing equilibria from ones that are not self-enforcing. (Namely, that rational equilibria are unaffected by deletions of dominated strategies.) However, nothing could be further from the truth: first, the principle cannot possibly be a general property of self enforcing assessments, for the simple reason that it is self-contradictory; and second, the principle fails to weed out all the equilibria that appear not to be self-enforcing. On reflection, one realizes that admissibility and iterated dominance have somewhat inconsistent motivations. Admissibility says that whatever the assessment of how the game will be played, the strategies that receive zero weight in this assessment neverthe less remain relevant, at least when it comes to breaking ties. Iterated dominance, on the other hand, says that some such strategies, those that receive zero weight because they are inadmissible, are irrelevant and may be deleted. To see that this inconsistency in motivation actually leads to an inconsistency in the concepts, consider the game of Figure 1 3 . If a self-enforcing assessment were unaffected by the elimination of dominated strategies then Player 1 ' s assessment of Player 2's choice would have to be L (delete B and then R) but it would also have to be R -1, -1
4, 1 u
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L 2, 5
L
2 T
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0, 0
-1, -1
(b)
(a) Figure 12.
R
Ch. 42: Foundations of Strategic Equilibrium
1621 L
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1,0
!VI
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Figure 13. 0, 0
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,y D
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ED
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(b)
(a) Figure 14.
(delete M and then L). Thus the assessment of the outcome would have to be both (2, 0) and ( 1 , 0) . To see the second point, consider the game of Figure 14(a). As is evident from the normal form of Figure 14(b), there are no dominance relationships among the strate gies, so all the equilibria satisfy our principle, and in particular those in which Player 1 plays T (for example, (T, 0.5L + 0.5C)). And yet those equilibria appear not to be self-enforcing: indeed, if Player 1 played B then he would be faced with the game of Figure 5, which he assesses as being worth more than 5 (recall that any self-enforcing assessment of the game of Figure 5 has an outcome of either (8, 5) or (7 , 6) or (6, 3) ); thus Player 1 must be expected to play B rather than T. I n the next section, w e shall concentrate on the second problem, namely how to cap ture the intuition ruling out the outcome (5 , 4) in the game of Figure 14(a).
10. Backward induction 1 0. 1.
The idea of backward induction
Selten ( 1 965, 1 975) proposed several ideas that may be summarized as the following
principle of backward induction:
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A self-enforcing assessment of the players' choices in a game tree must be consis tent with a self-enforcing assessment of the choices from any node (or, more generally, information set) in the tree onwards. This is a multi-person analog of "the principle of dynamic programming" [Bellman ( 1 957)], namely that an optimal strategy in a one-person decision tree must induce an optimal strategy from any point onward. The force of the backward induction condition is that it requires the players' assess ments to be self-enforcing even in those parts of the tree that are ruled out by their own assessment of earlier moves. (As we have seen, the equilibrium condition by itself does not do this: one can take the "wrong" move at a node whose assessed probability is zero and still maximize one's expected payoff.) The principle of backward induction indeed eliminates the equilibria of the games of Figure 10 and Figure 1 2(a) that do not appear to be self-enforcing. For example, in the game of Figure 1 2(a) a self-enforcing assessment of the play starting at Player 1 's second decision node must be that Player 1 would play U, therefore the assessment of the play starting at Player 2's decision node must be BU, and hence the assessment of the play of the full game must be BRU, that is, it is the equilibrium (BU, R). Backward induction also eliminates the outcome (5, 4) in the game of Figure 14(a). Indeed, any self-enforcing assessment of the play starting at Player 1 's second decision node must impute to Player 1 a payoff greater than 5, so the assessment of Player 1 's first move must be B. And it eliminates the equilibrium (T, R , D) in the game of Figure 15 (which is taken from Selten (1975)). Indeed, whatever the self-enforcing assessment of the play starting at Player 2's decision node, it certainly is not (R, D) (because, if Player 2 expected Player 3 to choose D, then he would maximize his own payoff by choosing L rather than R). There have been a number of attacks [Basu (1 988, 1990), Ben-Porath ( 1 997), Reny ( 1992a, 1993)] on the idea of backward induction along the following lines. The require ment that the assessment be self-enforcing implicitly rests on the assumption that the 0, 0, 0
u
3 , 2, 2
0, 0, 1
D
U
4, 4, 0
D 1, 1 , 1
· · · · · · · · · · · · · · · · · · ·
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/
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Figure 15.
B
Ch. 42: Foundations of Strategic Equilibrium
1623 2, 2
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Figure 16.
players are rational and that the other players know that they are rational, and indeed, on higher levels of knowledge. Also, the requirement that a self-enforcing assessment be consistent with a self-enforcing assessment of the choices from any information set in the tree onwards seems to require that the assumption be maintained at that information set and onwards. And yet, that information set might only be reached if some player has taken an irrational action. In such a case the assumption that the players are rational and that their rationality is known to the others should not be assumed to hold in that part of the tree. For example, in the game of Figure 16 there seems to be no compelling reason why Player 2, if called on to move, should be assumed to know of Player 1 's rational ity. Indeed, since he has observed something that contradicts Player 1 being rational, perhaps Player 2 must believe that Player 1 is not rational. The example however does suggest the following observation: the part of the tree following an irrational move is anyway irrelevant (because a rational player is sure not to take such a move), so whether or not rational players can assess what would happen there has no bearing on their assessment of how the game would actually be played (for example, in the game of Figure 16 the rational assessment of the outcome is (3 , 3 ) , regardless of what Player 2 ' s second choice might be). While this line o f reasoning is quite convincing in a situation like that of Figure 16, where the irrationality of the move B is self-evident, it is less convincing in, say, Figure 1 2(a). There, the rationality or irrationality of the move B becomes evident only after consideration of what would happen if B were taken, that is, only after consideration of what in retrospect appears "counterfactual" [Binmore (1990)]. One approach to this is to consider what results when we no longer assume that the players are known to be rational following a deviation by one (or more) from a can didate self-enforcing assessment. Reny ( 1 992a) and, though it is not the interpretation they give, Fudenberg, Kreps and Levine ( 1988) show that such a program leads to few restrictions beyond the requirements of strategic equilibrium. We shall discuss this a little more at the end of Section 10.4. And yet, perhaps one can make some argument for the idea of backward induction. The argument of Reny ( 1992a), for example, allows a deviation from the candidate
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equilibrium to be an indication to the other players that the assumption that all of the players were rational is not valid. In other words the players are no more sure about the nature of the game than they are about the equilibrium being played. We may recover some of the force of the idea of backward induction by requiring the equilibrium to be robust against a little strategic uncertainty. Thus we argue that the requirement of backward induction results from a series of tests. If indeed a rational player, in a situation in which the rationality of all players is common knowledge, would not take the action that leads to a certain information set being reached then it matters little what the assessment prescribes at that information set. To check whether the hypothesis that a rational player, in a situation in which the rationality of all players is common knowledge, would not take that action we suppose that he would and see what could arise. If all self-enforcing assessments of the situation following a deviation by a particular player would lead to him deviating then we reject the hypothesis that such a deviation contradicts the rationality of the players. And so, of course, we reject the candidate assessment as self-enforcing. If however our analysis of the situation confirms that there is a self-enforcing as sessment in which the player, if rational, would not have taken the action, then our assessment of him not taking the action is confirmed. In such a case we have no rea son to insist on the results of our analysis following the deviation. Moreover, since we assume that the players are rational and our analysis leads us to conclude that rational players will not play in this part of the game we are forced to be a little imprecise about what our assessment says in that part of the game. This relates to our discussion of sets of equilibria as solutions of the game in Section 1 3 .2. This modification of the notion of backward induction concedes that there may be conceivable circumstances in which the common knowledge of rationality of the players would, of necessity, be violated. It argues, however, that if the players are sure enough of the nature of the game, including the rationality of the other players, that they aban don this belief only in the face of truly compelling evidence, then the behavior in such circumstances is essentially irrelevant. The principle of backward induction is completely dependent on the extensive form of the game. For example, while it excludes the equilibrium (T, L) in the game of Figure 1 2(a), it does not exclude the same equilibrium in the game of Figure 1 2(b) (that is, in an extensive form where the players simultaneously choose their strategies). Thus one might see an inconsistency between the principle of backward induction and von Neumann and Morgenstern's reduction of the extensive form to the strategic form. We would put a somewhat different interpretation on the situation. The claim that the strategic form contains sufficient information for strategic analysis is not a denial that some games have an extensive structure. Nor is it a denial that valid arguments, such as backward induction arguments, can be made in terms of that structure. Rather the point is that, were a player, instead of choosing through the game, required to decide in advance what he will do, he could consider in advance any of the issues that would lead him to choose one way or the other during the game. And further, these issues will
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affect his incentives in precisely the same way when he considers them before playing as they would had he considered them during the play of the game. In fact, we shall see in Section 1 3 .6 that the sufficiency of the normal form substan tially strengthens the implications of backward induction arguments. We put off that discussion for now. We do note however that others have taken a different position. Selten's position, as well as the position of a number of others, is that the reduction to the strategic form is unwarranted, because it involves loss of information. Thus Fig ures 14(a) and 14(b) represent fundamentally different games, and (5 , 4) is indeed not self-enforcing in the former but possibly is self-enforcing in the latter. (Recall that this equilibrium cannot be excluded by the strategic-form arguments we have given to date, such as deletions of dominated strategies, but can be excluded by backward induction in the tree.) 10.2.
Subgame peifection
We now return to give a first pass at giving a formal expression of the idea of back ward induction. The simplest case to consider is of a node such that the part of the tree from the node onwards can be viewed as a separate game (a "subgame"), that is, it con tains every information set which it intersects. (In particular, the node itself must be an information set.) Because the rational assessment of any game must constitute an equilibrium, we have the following implication of backward induction [subgame peifection, Selten ( 1 965)] :
The equilibrium of the full game must induce an equilibrium on every subgame. The subgame-perfect equilibria of a game can be determined by working from the ends of the tree to its root, each time replacing a subgame by (the expected payoff of ) one of its equilibria. We must show that indeed a profile of strategies obtained by means of step-by-step replacement of subgames with equilibria constitutes a subgame-perfect equilibrium. If not, then there is a smallest subgame in which some player's strategy fails to maximize his payoff (given the strategies of the others). But this is impossible, because the player has maximized his payoff given his own choices in the subgames of the subgame, and those he is presumed to have chosen optimally. For example, in the game of Figure 1 2(a), the subgame whose root is at Player 1 's sec ond decision node can be replaced by (4, 1 ) , so the subgame whose root is at Player 2's decision node can also be replaced by this outcome, and similarly for the whole tree. Or in the game of Figure 14(a), the subgame (of Figure 5) can be replaced by one of its equilibria, namely (8, 5), (7, 6) or (6, 3) . Since any of them give Player 1 more than 5, Player 1 's first move must be B . Thus all three outcomes are subgame perfect, but the additional equilibrium outcome, (5, 4) , is not. Because the process of step-by-step replacement of subgames by their equilibria will always yield at least one profile of strategies, we have the following result. THEOREM 4 [Selten ( 1 965)] . Every game
librium.
tree has at least one subgame-peifect equi
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Subgame perfection captures only one aspect of the principle of backward induction. We shall consider other aspects of the principle in Sections 10.3 and 10.4. 1 0.3.
Sequential equilibrium
To see that subgame perfection does not capture all that is implied by the idea of back ward induction it suffices to consider quite simple games. While subgame perfection clearly isolates the self-enforcing outcome in the game of Figure 1 0 it does not do so in the game of Figure 17, in which the issues seem largely the same. And we could even modify the game a little further so that it becomes difficult to give a presentation of the game in which subgame perfection has any bite. (Say, by having Nature first decide whether Player 1 obtains a payoff of 5 after M or after B and informing Player 1 , but not Player 2.) One way of capturing more of the idea of backward induction is by explicitly re quiring players to respond optimally at all information sets. The problem is, of course, that, while in the game of Figure 17 it is clear what it means for Player 2 to respond optimally, this is not generally the case. In general, the optimal choice for a player will depend on his assessment of which node of his information set has been reached. And, at an out of equilibrium information set this may not be determined by the strategies being played. The concept of sequential equilibrium recognizes this by defining an equilibrium to be a pair consisting of a behavioral strategy and a system of beliefs. A system of beliefs gives, for each information set, a probability distribution over the nodes of that infor mation set. The behavioral strategy is said to be sequentially rational with respect to the system of beliefs if, at every information set at which a player moves, it maximizes the conditional payoff of the player, given his beliefs at that information set and the strate gies of the other players. A system of beliefs is said to be consistent with a behavioral strategy if it is the limit of a sequence of beliefs each being the actual conditional dis tribution on nodes of the various information sets induced by a sequence of completely mixed behavioral strategies converging to the given behavioral strategy. A sequential equilibrium is a pair such that the strategy is sequentially rational with respect to the beliefs and the beliefs are consistent with the strategy. 4, 1
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.
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. . . . . . . . .2 . .
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Figure
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The idea of a strategy being sequentially rational appears quite straightforward and intuitive. However the concept of consistency is somewhat less natural. Kreps and Wil son ( 1 982) attempted to provide a more primitive justification for the concept, but, as was shown by Kreps and Ramey (1987) this justification was fatally flawed. Kreps and Ramey suggest that this throws doubt on the notion of consistency. (They also suggest that the same analysis casts doubt on the requirement of sequential rationality. At an unreached information set there is some question of whether a player should believe that the future play will correspond to the equilibrium strategy. We shall not discuss this further.) Recent work has suggested that the notion of consistency is a good deal more natural than Kreps and Ramey suggest. In particular Kohlberg and Reny (1997) show that it follows quite naturally from the idea that the players' assessments of the way the game will be played reflects certainty or stationarity in the sense that it would not be affected by the actual realizations observed in an identical situation. Related ideas are explored by Battigalli (1996a) and Swinkels (1994). We shall not go into any detail here. The concept of sequential equilibrium is a strengthening of the concept of sub game perfection. Any sequential equilibrium is necessarily subgame perfect, while the converse is not the case. For example, it is easy to verify that in the game of Fig ure 17 the unique sequential equilibrium involves I choosing M. And a similar re2, 0
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. . ". . . \/ 2
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}\;J T
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Figure
18.
4, 1
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2
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M 3, 5 T
IN
Figure
19.
2, 0 R
J. Hi/las and E. Kohlberg
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sult holds for the modification of that game involving a move of Nature discussed above. Notice also that the concept of sequential equilibrium, like that of subgame perfec tion, is quite sensitive to the details of the extensive form. For example in the extensive form game of Figure 1 8 there is a sequential equilibrium in which Player 1 plays T and Player 2 plays L. However in a very similar situation (we shall later argue strategically identical) - that of Figure 1 9 there is no sequential equilibrium in which Player 1 plays T. The concepts of extensive form perfect equilibrium and quasi-perfect equilib rium that we discuss in the following section also feature this sensitivity to the details of the extensive form. In these games they coincide with the sequential equilibria. -
1 0.4.
Perfect equilibrium
The sequential equilibrium concept is closely related to a similar concept defined ear lier by Selten ( 1 97 5), the perfect equilibrium. Another closely related concept, which we shall argue is in some ways preferable, was defined by van Damme (1984) and called the quasi-perfect equilibrium. Both of these concepts, like sequential equilibrium, are de fined explicitly on the extensive form, and depend essentially on details of the extensive form. (They can, of course, be defined for a simultaneous move extensive form game, and to this extent can be thought of as defined for normal form games.) When defining these concepts we shall assume that the extensive form games satisfy perfect recall. Myerson ( 1978) defined a normal form concept that he called proper equilibrium. This is a refinement of the concepts of Selten and van Damme when those concepts are applied to normal form games. Moreover there is a remarkable relation between the normal form concept of proper equilibrium and the extensive form concept quasi-perfect equilibrium. Let us start by describing the definition of perfect equilibrium. The original idea of Selten was that however close to rational players were they would never be perfectly rational. There would always be some chance that a player would make a mistake. This idea may be implemented by approximating a candidate equilibrium strategy profile by a nearby completely mixed strategy profile and requiring that any of the deliberately chosen actions, that is, those given positive probability in the candidate strategy profile, be optimal, not only against the candidate strategy profile, but also against the nearby mixed strategy profile. If we are defining extensive form perfect equilibrium, a strategy is interpreted to mean a behavioral strategy and an action to mean an action at some information set. More formally, a profile of behavioral strategies b is a perfect equilib rium if there is a sequence of completely mixed behavioral strategy profiles {b1 } such that at each information set and for each b 1 , the behavior of b at the information set is optimal against b 1 , that is, is optimal when behavior at all other information sets is given by b1 . If the definition is applied instead to the normal form of the game the resulting equilibrium is called a normal form peifect equilibrium. Like sequential equilibrium, (extensive form) perfect equilibrium is an attempt to express the idea of backward induction. Any perfect equilibrium is a sequential equilib-
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rium (and so is a subgame perfect equilibrium). Moreover the following result tells us that, except for exceptional games, the converse is also true. THEOREM 5 [Kreps and Wilson (1982), Blume and Zame (1994)]. For any extensive form, except for a closed set of payoffs of lower dimension than the set of all possi ble payoffs, the sets of sequential equilibrium strategy profiles and perfect equilibrium strategy profiles coincide. The concept of normal form perfect equilibrium, on the other hand, can be thought of as a strong form of admissibility. In fact for two-player games the sets of normal form perfect and admissible equilibria coincide. In games with more players the sets may differ. However there is a sense in which even in these games normal form per fection seems to be a reasonable expression of admissibility. Mertens (1987) gives a definition of the admissible best reply correspondence that would lead to fixed points of this correspondence being normal form perfect equilibria, and argues that this definition corresponds "to the intuitive idea that would be expected from a concept of 'admissible best reply' in a framework of independent priors" [Mertens (1987, p. 15)]. Mertens (1995) offers the following example in which the set of extensive form per fect equilibria and the set of admissible equilibria have an empty intersection. The game may be thought of in the following way. Two players agree about how a certain social decision should be made. They have to decide who should make the decision and they do this by voting. If they agree on who should make the decision that player decides. If they each vote for the other then the good decision is taken automatically. If each votes for himself then a fair coin is tossed to decide who makes the decision. A player who makes the social decision is not told if this is so because the other player voted for him, or because the coin toss chose him. The extensive form of this game is given in Figure 20. The payoffs are such that each player prefers the good outcome to the bad outcome. (In Mertens (1995) there is an added complication to the game. Each player does slightly worse if he chooses the bad outcome than if the other chooses it. However this additional complication is, as Mertens pointed out to us, totally unnecessary for the results.) In this game the only admissible equilibrium has both players voting for themselves and taking the right choice if they make the social decision. However, any perfect equi librium must involve at least one of the players voting for the other with certainty. At least one of the players must be at least as likely as the other to make a mistake in the second stage. And such a player, against such mistakes, does better to vote for the other.
1 0.5.
Perfect equilibrium and proper equilibrium
The definition of perfect equilibrium may be thought of as corresponding to the idea that players really do make mistakes, and that in fact it is not possible to think coherently about games in which there is no possibility of the players making mistakes. On the
J. Hillas and E. Kohlberg
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other hand one might think of the perturbations as instead encompassing the idea that the players should have a little strategic uncertainty, that is, they should not be completely confident as to what the other players are going to do. In such a case a player should not be thought of as being uncertain about his own actions or planned actions. This is (one interpretation of ) the idea behind van Damme's definition of quasi-perfect equilibrium. Recall why we used perturbations in the definition of perfect equilibrium. We wanted to require that players act optimally at all information sets. Since the perturbed strategies are completely mixed all information sets are reached and so the conditional distribution on the nodes of the information set is well defined. In games with perfect recall however we do not need that all the strategies be completely mixed. Indeed the player himself may affect whether one of his information sets is reached or not, but cannot affect what will be the distribution over the nodes of that information set if it is reached - that depends only on the strategies of the others. The definition of quasi-perfect equilibrium is largely the same as the definition of perfect equilibrium. The definitions differ only in that, instead of the limit strategy b being optimal at each information set against behavior given by b 1 at all other informa tion sets, it is required that b be optimal at all information sets against behavior at other information sets given by b for information sets that are owned by the same player who owns the information set in question, and by b 1 for other information sets. That is, the player does not take account of his own "mistakes", except to the extent that they may make one of his information sets reached that otherwise would not be. As we explained above, the assumption of perfect recall guarantees that the conditional distribution on each information set is uniquely defined for each t and so the requirement of optimality is well defined. This change in the definition leads to some attractive properties. Like perfect equi libria, quasi-perfect equilibria are sequential equilibrium strategies. But, unlike perfect
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equilibria, quasi-perfect equilibria are always normal form perfect, and thus admissible. Mertens (1995) argues that quasi-perfect equilibrium is precisely the right mixture of admissibility and backward induction. Also, as we remarked earlier, there is a relation between quasi-perfect equilibria and proper equilibria. A proper equilibrium [Myerson ( 1 978)] is defined to be a limit of £ proper equilibria. An £-proper equilibrium is a completely mixed strategy vector such that for each player if, given the strategies of the others, one strategy is strictly worse than another the first strategy is played with probability at most £ times the probabil ity with which the second is played. In other words, more costly mistakes are made with lower frequency. Van Damme ( 1 984) proved the following result. (Kohlberg and Mertens ( 1 982, 1986) independently proved a slightly weaker result, replacing quasi perfect with sequential.)
A proper equilibrium of a normal form game is quasi-peifect in any extensive form game having that normalform.
THEOREM 6.
In van Damme's paper the theorem is actually stated a little differently referring sim ply to a pair of games, one an extensive form game and the other the corresponding normal form. (It is also more explicit about the sense in which a quasi-perfect equilib rium, a behavioral strategy profile, is a proper equilibrium, a mixed strategy profile.) Thus van Damme correctly states that the converse of his theorem is not true. There are such pairs of games and quasi-perfect equilibria of the extensive form that are in. no sense equivalent to a proper equilibrium of the normal form. Kohlberg and Mertens ( 1986) state their theorem in the same form as we do, but refer to sequential equilibria rather than quasi-perfect equilibria. They too correctly state that the converse is not true. For any normal form game one could introduce dummy players, one for each profile of strategies having payoff one at that profile of strategies and zero otherwise. In any ex tensive form having that normal form the set of sequential equilibrium strategy profiles would be the same as the set of equilibrium strategy profiles originally. However it is not immediately clear that the converse of the theorem as we have stated it is not true. Certainly we know of no example in the previous literature that shows it to be false. For example, van Damme (199 1 ) adduces the game given in extensive form in Figure 2 1 (a) and in normal form in Figure 2 1 (b) to show that a quasi-perfect equilibrium may not be proper. The strategy (BD, R) is quasi-perfect, but not proper. Nevertheless there is a game - that of Figure 22 - having the same normal form, up to duplication of strategies, in which that strategy is not quasi-perfect. Thus one might be tempted to conjecture, as we did in an earlier version of this chapter, that given a normal form game, any strategy vector that is quasi-perfect in any extensive form game having that normal form would be a proper equilibrium of the normal form game. A fairly weak version of this conjecture is true. If we fix not only the equilibrium under consideration but also the sequence of completely mixed strategies converging to it then if in every equivalent extensive form game the sequence supports the equilibrium
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as a quasi-perfect equilibrium then the equilibrium is proper. We state this a little more formally in the following theorem.
An equilibrium a of a normal form game G is supported as a proper equilibrium by a sequence of completely mixed strategies {a k } with limit a ifand only if {a k } induces a quasi-peifect equilibrium in any extensive form game having the normal form G. THEOREM 7 .
S KETCH OF PROOF. This is proved in Hillas ( 1 998c) and in Mailath, Samuelson and Swinkels ( 1 997). The if direction is implied by van Damme's proof. (Though not quite by the result as he states it, since that leaves open the possibility that different extensive forms may require different supporting sequences.) The other direction is quite straightforward. One first takes a subsequence such that the conditional probability on any subset of a player's strategy space converges. (This conditional probability is well defined since the strategy vectors in the sequence are
Ch. 42: Foundations of Strategic Equilibrium
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k assumed to be completely mixed.) Thus {O' } defines, for any subset of a player's strat k egy space, a conditional probability on that subset. And so the sequence {O' } partitions the strategy space S11 of each player into the sets S�, s,;, . . , Sl where s/, is the set of those strategies that receive positive probability conditional on one of the strategies in Sn \(U i <J S�) being played. Now consider the following extensive form. The players move in order of their names, but without observing anything about the choices of those who chose earlier. (That is, they move essentially simultaneously.) Each Player n first decides whether to play one of the strategies in S� or not. If he decides to do so then he chooses one of those strategies. If he decides not to then he decides whether to play one of the strategies in s,; , and so on. The only behavioral strategy in such a game consistent with (the limiting behavior k of ) { O' } has each player choosing at each opportunity to play one of the strategies in s/, rather than continuing with the process, and then choosing among those strategies according to the (strictly positive) limiting conditional (on s/, ) probability distribution. Now, if such a vector of strategies is a quasi-perfect equilibrium, then for sufficiently .
large k, for each player, every strategy in s/, is at least as good as any strategy in s/,' for any j' > j and is assigned probability arbitrarily greater than any such strategy. Thus k D { O' } supports 0' as a proper equilibrium.
However, the conjecture as we originally stated it - that is, without any reference to the supporting sequences of completely mixed strategies - is not true. Consider the game of Figure 23. The equilibrium (A , V) is not proper. To see this we argue as follows. Given that Player 1 plays A, Player 2 strictly prefers W to Y and X to Y. Thus in any .s-proper equilibrium Y is played with at most .s times the probability of W, and also at most E times the probability of X. The fact that Y is less likely than W implies that Player 1 strictly prefers B to C , while the fact that Y is less likely than X implies that Player 1 strictly prefers B to D. Thus in an £-proper equilibrium C and D are both played with at most £ times the probability of B . This in tum implies that Player 2 strictly prefers Z to V, and so there can be no .s-proper equilibrium in which V is played with probability close to 1 . Thus (A , V) is not proper. Nevertheless, in any extensive form game having this normal form there are perfect and quasi-perfect - equilibria equivalent to (A, V). The idea is straightforward and not V
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at all delicate. In order to argue that (A, V) was not proper we needed to deduce, from the fact that Player 2 strictly prefers W to Y and X to Y, that Y is played with much smaller probability than W, and much smaller probability than X. However there is no extensive representation of this game in which quasi-perfection has this implication. For example, consider an extensive form in which Player 2 first decides whether to play W and, if he decides not to, then decides between X and Y . Now if all we may assume is that Player 2 strictly prefers W to Y and X to Y then we cannot rule out that Player 2 strictly prefers X to W. And in that case it is consistent with the requirements of either quasi-perfectness or perfectness that the action W is taken with probability 8 2 and the action Y with probability 8. This results in the strategy Y being played with substantially greater probability than the strategy W. Something similar results for any other way of structuring Player 2's choice. See Hillas ( 1 998c) for greater detail. 10.6.
Uncertainties about the game
Having now defined normal form perfect equilibrium we are in a position to be a lit tle more explicit about the work of Fudenberg, Kreps and Levine ( 1 988) and of Reny ( 1 992a), which we mentioned in Section 10. 1 . These papers, though quite different in style, both show that if an out of equilibrium action can be taken to indicate that the player taking that action is not rational, or equivalently, that his payoffs are not as speci fied in the game, then any normal form perfect equilibrium is self-enforcing. Fudenberg, Kreps and Levine also show that if an out of equilibrium action can be taken to indicate that the game was not as originally described - so that others' payoffs may differ as well - then any strategic equilibrium is self-enforcing.
11.
Forward induction
In the previous section we examined the idea of backward induction, and also combina tions of backward induction and admissibility. If we suppose that we have captured the implications of this idea, are there any further considerations that would further narrow the set of self-enforcing assessments? Consider the game of Figure 24(a) [Kohlberg and Mertens (1982, 1 986)] . Here, every node can be viewed as a root of a subgame, so there is no further implication of "back ward induction" beyond "subgame perfection". Since the outcome (2, 5) (that is, the equilibrium where Player 1 plays T and the play of the subgame is (D, R)) is subgame perfect, it follows that backward induction cannot exclude it. On the other hand, this outcome cannot possibly represent a self-enforcing assess ment of the game. Indeed, in considering the contingency of Player 1 's having played B , Player 2 must take it for granted that Player 1 's subsequent move was U and not D (be cause a rational Player 1 will not plan to play B and then D, which can yield a maximum of 1 , when he can play T for a sure 2). Thus the assessment of Player 1 's second move must be U , and hence the assessment of Player 2's move must be L , which implies that the assessment of the whole game must be (4, 1 ) .
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0, 0
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Notice that this argument does not depend on the order in which Players 1 and 2 move in the subgame. If the order was reversed so that Player 2 moved first in the subgame the argument could be written almost exactly as it has been requiring only that "was U and not D" be changed to "will be U and not D". Thus a self-enforcing assessment of the game must not only be consistent with de ductions based on the opponents' rational behavior in the future (backward induction) but it must also be consistent with deductions based on the opponents' rational behavior in the past (forward induction). A formal definition of forward induction has proved a little elusive. One aspect of the idea of forward induction is captured by the idea that a solution should not be sensitive to the deletion of a dominated strategy, as we discussed in Section 9. Other aspects depend explicitly on the idea of a solution as a set of equilibria. Kohlberg and Mertens (1 986) give such a definition; and Cho and Kreps (1987) give a number of definitions in the context of signaling games, as do Banks and Sobel (1987). We shall leave this question for the moment and return to it in Sections 1 3 .3, 1 3 .5, and 1 3 .6.
12.
Ordinality and other invariances
In the following section we shall look again at identifying the self-enforcing solutions. In this section we examine in detail part of what will be involved in providing an an swer, namely, the appropriate domain for the solution correspondence. That is, if we are asking simply what behavior (or assessment of the others' behavior) is self-enforcing, which aspects of the game are relevant and which are not? We divide the discussion into two parts, the first takes the set of players (and indeed their names) as given, while the second is concerned with changes to the player set. The material in this section and the next is a restatement of some of the discussion in and of some of the results from Kohlberg and Mertens (1 982, 1 986) and Mertens (1 987, 1 989, 1 99 lb, 1 99 1 a, 1 992, 1 995).
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Recall that we are asking only a rather narrow question: what solutions are self enforcing? Or, equivalently, what outcomes are consistent with the individual incentives of the players (and the interaction of those incentives)? Thus if some change in the game does not change the individual incentives then it should not change the set of solutions that are self-enforcing. The following two sections give some formal content to this informal intuition. Before that, it is well to emphasize that we make no claim that a solution in any other sense, or the strategies that will actually be played, should satisfy the same invariances. This is a point that has been made in a number of places in the papers mentioned above, 1 but is worth repeating. It is also worth reminding the reader at this point that we are taking the "traditional" point of view, that all uncertainty about the actual game being played has been explic itly included in the description of the game. Other analyses take different approaches. Harsanyi and Selten (1 988) assume that with any game there are associated error prob abilities for every move, and focus on the case in which all errors have the same small probability. Implicit, at least, in their analysis is a claim that one cannot think coher ently, or at least cannot analyze, a game in which there is no possibility of mistakes. Others are more explicit. Aumann ( 1 987b) argues that to arrive at a strong form of rationality (one must assume that) irrationality cannot be ruled out, that the players ascribe irrationality to each other with small proba bility. True rationality requires 'noise' ; it cannot grow on sterile ground, it cannot feed on itself only. We differ. Moreover, if we are correct and it is possible to analyze sensibly games in which there are no possibilities of mistakes, then such an analysis is, in theory, more general. One can apply the analysis to a game in which all probabilities of error (or irrationality) are explicitly given. 12.1.
Ordinality
Let us now consider a candidate solution of a game. This candidate solution would not be self-enforcing if some player, when he came to make his choices, would choose not to play as the solution recommends. That is, if given his beliefs about the choices of the others, the choice recommended for him was not rational. For fixed beliefs about the choices of the others this is essentially a single-agent decision problem, and the concept of what is rational is relatively straightforward. There remains the question of whether all payoff-maximizing choices, or just admissible ones, are rational, but for the moment we don't need to take a position on that issue. Now consider two games in which, for some player, for any beliefs on the choices of the others the set of that player's rational choices is the same. Clearly if he had the
1
Perhaps most starkly in Mertens (1987, p. 6) where he states "There is here thus no argument whatsoever
according to which the choice of equilibrium should depend only on this or that aspect of the game".
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same beliefs in both games then those recommendations that would be self-enforcing (rational) for him in one game would also be self-enforcing in the other. Now suppose that this is true for all the players. If the beliefs of the players are consistent with the individual incentives in one of the games they will also be in the other, since given those beliefs the individual decision problems have, by assumption, the same solutions in the two games. Thus a combination of choices and beliefs about the others' choices will be self-enforcing in one game if and only if it is self-enforcing in the other. Moreover, the same argument means that if we were to relabel the strategies the conclusion would continue to hold. A solution would be self-enforcing in the one game if and only if its image under the relabeling was self-enforcing in the other. Let us flesh this out and make it a little more formal. Previously we have been mainly thinking of the mixed strategy of a player as the beliefs of the other players about the pure action that the player will take. On the other hand there it is difficult to think how we might prevent a player from randomizing if he wanted to. (It is not even completely clear to us what this might mean.) Thus the choices available to a player include, not only the pure strategies that we explicitly give him, but also all randomizations that he might decide to use, that is, his mixed strategy space. A player's beliefs about the choices of the others will now be represented by a probability distribution over the product of the (mixed) strategy spaces of the other players. Since we are not using any of the structure of the mixed strategy spaces, we consider only probability distributions with finite support. Let us consider an example to make things a little more concrete. Consider the games of Figures 25(a) and 25(b ). The game of Figure 25(a) is the Stag Hunt game of Figure 2 that we discussed in Section 2. 1 . The game of Figure 25(b) is obtained from the game of Figure 25(a) by subtracting 7 from Player 1 's payoff when Player 2 plays L and subtracting 7 from Player 2's payoff when Player 1 plays T. This clearly does not, for any assessment of what the other player will do, change the rational responses of either player. Thus the games are ordinally equivalent, and the self-enforcing behaviors should be the same. There remains the question of whether we should restrict the players' beliefs to be independent across the other players. We argued in Section 2 that in defining rationaliz ability the assumption of independence was not justified. That argument is perhaps not as compelling in the analysis of equilibrium. And, whatever the arguments, the assump tion of independence seems essential to the spirit of Nash equilibrium. In any case the analysis can proceed with or without the assumption. L
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For any such probability distribution the set of rational actions of the player is de fined. Here there are essentially two choices: the set of that player's choices (that is, mixed strategies) that maximize his expected payoff, given that distribution; or the set of admissible choices that maximize his payoff. And if one assumes independence there is some issue of how exactly to define admissibility. Again, the analysis can go ahead, with some differences, for any of the assumptions. For the rest of this section we focus on the case of independence and admissibility. Further, we assume a relatively strong form of admissibility. To simplify the statement of the requirement, the term belief will always mean an independent belief with finite support. We say a player's choice is an admissible best reply to a certain belief p about the choices of the others if, for any other belief q of that player about the others, there are beliefs q arbitrarily close to p and having support containing the support of q , such that the player's choice is expected payoff-maximizing against q . The statement of this requirement i s a little complicated. The idea i s that the choice should be optimal against some completely mixed strategies of the other players close to the actual strategy of the others. However, because of the ordinality requirement we do not wish to use the structure of the strategy spaces, so we make no distinction between completely mixed strategies and those that are not completely mixed. Nor do we retain any notion of one mixed strategy being close to another, except information that may be obtained simply from the preferences of the players. Thus the notion of "close" in the definition is that the sum over all the mixed strategies in the support of either p or q of the difference between p and q is small. One now requires that if two games are such that a relabeling of the mixed strategy space of one gives the mixed strategy space of the other and that this relabeling trans forms the best reply correspondence of the first game into the best reply correspondence of the second then the games are equivalent in terms of the individual incentives and the (self-enforcing) solutions of the one are the same as the (appropriately relabeled) solu tions of the other. Mertens ( 1987) shows that this rather abstract requirement has a very concrete imple mentation. Two games exhibit the same self-enforcing behavior if they have the same pure strategy sets and the same best replies for every vector of completely mixed strate gies; this remains true for any relabeling of the pure strategies, and also for any addition or deletion of strategies that are equivalent to existing mixed strategies. That is, the question of what behavior is self-enforcing is captured completely by the reduced nor mal form, and by the best reply structure on that form. And it is only the best reply structure on the interior of the strategy space that is relevant. If one dropped the assumption that beliefs should be independent one would require a little more, that the best replies to any correlated strategy of the others should be the same. And if one thought of the best replies as capturing rationality rather than admissible best replies, one would need equivalence of the best replies on the boundary of the strategy space as well.
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Changes in the player set
We turn now to questions of changes in the player set. We start with the most obvious. While, as Mertens ( 1 99 1a) points out, the names of the players may well be relevant for what equilibrium is actually played, it is clearly not relevant for the prior question of what behavior is self-enforcing. That is, the self-enforcing outcomes are anonymous. Also it is sometimes the case that splitting a player into two agents does not matter. In particular the set of self-enforcing behaviors does not change if a player is split into two subsets of his agents, provided that in any play, whatever the strategies of the players at most one of the subsets of agents moves. This requirement too follows simply from the idea of self-enforcing behavior. Given the player's beliefs about the choices of the others, if his choice is not rational then it must be not rational for at least one of the subsets, and conversely. Moreover the admissibility considerations are the same in the two cases. It is the actions of Nature and the other players that determines which of the agents actually plays. Now, at least in the independent case, the beliefs of the others over the choices of the player that are consistent with the notion of self-enforcing behavior and the beliefs over the choices of the subsets of the agents that are consistent with the notion of self enforcing behavior may differ. But only in the following way. In the case of the agents it is assumed that the beliefs of the others are the independent product of beliefs over each subset of the agents, while when the subsets of the agents are considered a single player no such assumption is made. And yet, this difference has no impact on what is rational behavior for the others. Which of the agents plays in the game is determined by the actions of the others. And so, what is rational for one of the other players depends on his beliefs about the choices of the rest of the other players and his beliefs about the choices of each agent of the player in question conditional on that agent actually playing in the game. And the restrictions on these conditional beliefs will be the same in the two cases. Notice that this argument breaks down if both subsets of the agents might move in some play. In that case dependence in the beliefs may indeed have an impact. Further, the argument that the decision of the player is decomposable into the individual deci sions of his agents in not correct. We now come to the final two changes, which concern situations in which subsets of the players can be treated separately. The first, called the decomposition property, says that if the players of a game can be divided into two subsets and that the players in each subset play distinct games with no interaction with the players in the other subset - that is, the payoffs of the one subset do not depend on the actions of the players in the other subset - then the self-enforcing behaviors in the game are simply the products of the self-enforcing behaviors in smaller games. Mertens (1 989, p. 577) puts this more succinctly as "if disjoint player sets play different games in different rooms, one can as well consider the compound game as the separate games". Finally, we have the small worlds axiom, introduced in Mertens ( 1 99 lb) and proved for stable sets in Mertens ( 1 992). This says that if the payoffs of some subset of the
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players do not depend on the choices of the players outside this subset, then the self enforcing behavior of the players in the subset is the same whether one considers the whole game or only the game between the "insiders". These various invariance requirements can be thought of as either restrictions on the permissible solutions or as a definition of what constitutes a solution. That is, they de fine the domain of the solution correspondence. Also, the requirements become stronger in the presence of the other requirements. For example, the player splitting requirement means that the restriction to nonoverlapping player sets in the decomposition require ment can be dispensed with. [For details see Mertens (1989, p. 578).] And, ordinality means that in the decomposition and small worlds requirements we can replace the con dition that the payoffs of the subset of players not change with the condition that their best replies not change, or even that their admissible best replies not change.
13.
Strategic stability
We come now to the concept of strategic stability introduced by Kohlberg and Mertens (1 986) and reformulated by Mertens ( 1 987, 1 989, 199 1 a, 199 1b, 1 992) and Govin dan and Mertens ( 1993). This work is essentially aimed at answering the same ques tion as the concepts of Nash equilibrium and perfect equilibrium, namely, what is self enforcing behavior in a game? We argued in the previous section that the answer to such a question should depend only on a limited part of the information about the game. Namely, that the answer should be ordinal, and satisfy other invariances. To this extent, as long as one does not require admissibility, Nash equilibrium is a good answer. Nash equilibria are, in some sense, self-enforcing. However, as was pointed out by Selten ( 1 965 , 1975), this answer en tails only a very weak form of rationality. Certainly, players choose payoff-maximizing strategies. But these strategies may prescribe choices of strategies that are (weakly) dominated and actions at unreached information sets that seem clearly irrational. Normal form perfect equilibria are also invariant in all the senses we require, and moreover are also clearly admissible, even in the rather strong sense we require. How ever, they do not always satisfy the backward induction properties we discussed in Sec tion 9. Other solutions that satisfy the backward induction property do not satisfy the invariance properties. Sequential equilibrium is not invariant. Even changing a choice between three alternatives into two binary choices can change the set of sequential equilibria. Moreover, sequential equilibria may be inadmissible. Extensive form perfect equilibria too are not invariant. Moreover they too may be inadmissible. Recall that in Section 10.4 (Figure 20) we examined a game of Mertens ( 1 995) in which all of the ex tensive form perfect equilibria are inadmissible. Quasi-perfect equilibria, defined by van Damme (1984), and proper equilibria, defined by Myerson (1 978), are both admissible and satisfy backward induction. However, neither is ordinal - quasi-perfect equilibrium depends on the particular extensive form and proper equilibrium may change with the addition of mixtures as new strategies - and proper equilibrium does not satisfy the player splitting requirement.
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We shall see in the next section that it is impossible to satisfy the backward induction property for any ordinal single-valued solution concept. This leads us to consider set valued solutions and, in particular, to strategically stable sets of equilibria. 13. 1.
The requirements for strategic stability
We are seeking an answer to the question: What are the self-enforcing behaviors in a game? As we indicated, the answer to this question should satisfy the various invari ances we discussed in Section 12. We also require that the solution satisfy stronger forms of rationality than Nash equilibrium and normal form perfect equilibrium, the two "non-correlated" equilibrium concepts that do satisfy those invariances. In particu lar, we want our solution concept to satisfy the admissibility condition we discussed in Section 9, and some form of the iterated dominance condition we also discussed in Sec tion 9, the backward induction condition we discussed in Section 10, and the forward induction condition we discussed in Section 1 1 . We also want our solution to give some answer for all games. As we indicated above, it is impossible for a single-valued solution concept to satisfy these conditions. In fact, two separate subsets of the conditions are inconsistent for such solutions. Admissibility and iterated dominance are inconsistent, as are backward induction and invariance. To see the inconsistency of admissibility and iterated dominance consider the game in Figure 26 [taken from Kohlberg and Mertens ( 1 986, p. 101 5)] . Strategy B is dominated (in fact, strictly dominated) so by the iterated dominance condition the solution should not change if B is deleted. But in the resulting game admissibility implies that (T, L) is the unique solution. Similarly, M is dominated so we can delete M and then (T, R) is the unique solution. To see the inconsistency of ordinality and backward induction consider the game of Figure 27(a) [taken from Kohlberg and Mertens ( 1 986, p. 1 0 1 8)] . Nature moves at the node denoted by the circle going left with probability a and up with probability 1 a. Whatever the value of a the game has the reduced normal form of Figure 27(b). (Notice that strategy Y is just a mixture of T and M.) Since the reduced normal form does not depend on a , ordinality implies that the solution of this game should not depend on a. And yet, in the extensive game the unique sequential equilibrium has Player 2 playing L with probability (4 - 3a) / (8 - 4a) . -
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Thus it may be that elements of a solution satisfying the requirements we have dis cussed would be sets. However we would not want these sets to be too large. We are still thinking of each element of the solution as, in some sense, a single pattern of behavior. In generic extensive form games we might think of a single pattern of behavior as being associated with a single equilibrium outcome, while not specifying exactly the out of equilibrium behavior. One way to accomplish this is to consider only connected sets of equilibria. In the definition of Mertens discussed below the connectedness requirement is strengthened in a way that corresponds, informally, to the idea that the particular equilibrium should depend continuously on the "beliefs" of the players. Without a bet ter understanding of exactly what it means for a set of equilibria to be the solution we cannot say much more. However some form of connectedness seems to be required. In Section 13.4 we shall discuss a number of definitions of strategic stability that have appeared in the literature. All of these definitions are motivated by the attempt to find a solution satisfying the requirements we have discussed. As we have just seen such a solution must of necessity be set-valued, and we have claimed that such sets should be connected. For the moment, without discussing the details of the definition we shall use the term strategically stable sets of equilibria (or simply stable sets) to mean one of the members of such a solution. 13.2.
Comments on sets of equilibria as solutions to non-cooperative games
We saw in the previous section that single-valued solutions could not satisfy the re quirements that were implied by a relatively strong form of the notion of self-enforcing. There are two potential responses to this observation. One is to abandon the strong con ditions and view the concept of Nash equilibrium or normal form perfect equilibrium as
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the appropriate concept. The other is to abandon the notion that solutions to noncoop erative games should be single-valued. It may seem that the first is more natural, that perhaps the results of Section 1 3 . 1 tell u s that there i s some inconsistency between the invariances w e have argued for and the backward induction requirements as we have defined them. We take the oppo site view. Our view is that these results tell us that there is something unwarranted in insisting on single-valued solutions. Our own understanding of the issue is very incom plete and sketchy and this is reflected in the argument. Nevertheless the issue should be addressed and we feel it is worthwhile to state here what we do know, however incom plete. Consider again the game of Figure 16. In this game the unique stable set contains all the strategy profiles in which Player 1 chooses T. The unique sub game perfect equilib rium has Player 2 choosing R . Recall our interpretation of mixed strategies as the as sessment of others of how a player will play. In order to know his best choices Player 1 does not need to know anything about what Player 2 will choose, so there is no need to make Player 1 's assessment precise. Moreover from Player 2's perspective one could argue for either L or R equally well on the basis of the best reply structure of the game. They do equally well against T while L is better against BU and R is better against BD. Neither BU nor BD are ever best replies for Player 1, so we cannot distinguish between them on the basis of the best reply correspondence. What then of the backward induction argument? We have said earlier that backward induction relies on an assumption that a player is rational whatever he has done at pre vious information sets. This is perhaps a bit inaccurate. The conclusion here is that Player 1 , being rational, will not choose to play B . The "assumption" that Player 2 at his decision node will treat Player 1 as rational is simply a test of the assumption that Player 1 will not play B . Once this conclusion is confirmed one cannot continue to maintain the assumption that following a choice of B by Player 1 , Player 2 will continue to assume that Player 1 is rational. Indeed he cannot. In this example Player 1 's choices in the equilibrium tell us nothing about his as sessment of Player 2's choices. If there is not something else - such as admissibility considerations, for example - to tie down his assessment then there is no basis to do so. Consider the slight modification of this game of Figure 28(a). In this game it is not so immediately obvious that Player 1 will not choose B . However if we assume that Player 1 does choose B , believes that Player 2 is rational, and that Player 2 still be lieves that Player 1 is rational we quickly reach a contradiction. Thus it must not be that Player 1 , being rational and believing in the rationality of the other, chooses B . Again we can no longer maintain the assumption that Player 1 will be regarded as rational in the event that Player 2's information set is reached. Again both L and R are admissible for Player 2. In this case there is a difference between BU and BD for Player 1 - BD is ad missible while B U is not. However, having argued that a rational Player 1 will choose T there is no compelling reason to make any particular assumption concerning Player 2's assessment of the relative likelihoods of B U and BD. The remaining information we have to tie down Player 1 's assessment of Player 2's choices is Player 1 's equilibrium
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choice. Since Player 1 chooses T, it must be that his assessment of Player 2's choices made this a best response. And this is true for any assessment that puts weight no greater than a half on L. Again this set of strategies is precisely the unique strategically stable set. These examples point to the fact that there may be something unwarranted in speci fying single strategy vectors as "equilibria". We have emphasized the interpretation of mixed strategy vectors as representing the uncertainty in one player's mind about what another player will do. In many games there is no need for a player to form a very pre cise assessment of what another player will do. Put another way, given the restrictions our ordinality arguments put on the "questions we can ask a player" we cannot elicit his subjective probability on what the others will do. Rather we must be satisfied with less precise information. As we said, this is at best suggestive. In fact, if there are more than two players there are real problems. In either of the examples discussed we could add a player with a choice between two strategies, his optimal choice varying in the stable set and not affecting the payoffs of the others. Here too we see a sense in which the set-valued solution does represent the strategic uncertainty in the interaction among the players. However we do not see any way in which to make this formal, particularly since the notion of a "strategic dummy" is so elusive. For example, we could again amend the game by adding another game in which the third player interacts in an essential way with the other two and, say, having nature choose which of the two games will be played. Finally, it is worth pointing out that in an important class of games, the sets of equi libria we consider are small. More specifically, we have the following result. THEOREM 8 [Kohlberg and Mertens (1986)]. For generic extensive games, every con nected set of equilibria is associated with a single outcome (or distribution over endpoints) in the tree.
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This follows from a result of Kreps and Wilson (1 982) and Kohlberg and Mertens ( 1986) that the set of equilibrium probability distributions on endpoints is finite. If there are a finite number of equilibrium distributions then, since the map from strategies to distributions on endpoints is continuous, a connected set of equilibria must all lead to the same distribution on endpoints. 13.3.
Forward induction
We discussed in general terms in Section 1 1 the idea of forward induction; and discussed one aspect of this idea in Section 9. We are now in a position to be a little more explicit about another aspect of the idea. The idea that the solution should not be sensitive to the (iterated) deletion of dom inated strategies was that since it was known that players are never willing to choose such strategies it should not make a difference if such strategies were removed from the game. This suggests another aspect of forward induction. If, in the solution in question, there is some strategy of a player that the player would never be willing to choose then it should not make a difference if such a strategy were removed from the game. In fact, like the deletion of a dominated strategy, it is clear that the deletion of a strategy that a player is never willing to play in the solution must change the situation. (It can, for example, change undominated strategies into dominated ones, and vice versa.) So our intuition is that it should not matter much. This is reflected in the fact that various versions of this requirement have been stated by requiring, for example, that a stable set contain a stable set of the game obtained by deleting such a strategy, or by looking at equilibrium outcomes in generic extensive form games, without being explicit about what out of equilibrium behavior is associated with the solution. There is one aspect of this requirement that has aroused some comment in the liter ature. It is that in deleting dominated strategies we delete weakly dominated strategies. In deleting strategies that are inferior in the solution we delete only strategies that are strictly dominated at all points of the solution. In the way we have motivated the defi nition this difference appears quite naturally. The weakly dominated strategy is deleted because the player would not in any case be willing to choose it. In order to be sure that the player would not anywhere in the solution be willing to choose the strategy it is nec essary that the strategy be strictly dominated at all points of the solution. Another way of saying almost the same thing is that the idea that players should not play dominated strategies might be thought of as motivated by a requirement that the solution should not depend on the player being absolutely certain that the others play according to the solution. Again, the same requirement implies that only strategies that are dominated in some neighborhood of the solution be deleted. 13.4.
The definition ofstrategic stability
The basic ideas of backward induction and forward induction seem at first sight to re quire in an essential way the extensive form; and implicit in much of the literature fol lowing Selten ( 1 965) was that essential insights about what was or was not reasonable
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in a game could not be seen in the normal form. However we have argued in Section 1 2 that only the normal form of the game should matter. This argument had been made originally by von Neumann and Morgenstern ( 1944) and was forcefully reintroduced by Kohlberg and Mertens (1986). In fact, we saw in discussing, for example, the sequential equilibrium concept that even looking explicitly at the extensive form the notion of sequential equilibrium seems sensitive to details of the extensive form that appear irrelevant. Dalkey ( 1953) and Thompson (1952), and more recently Elmes and Reny ( 1 994), show that there is a list of inessential transformations of one extensive form into another such that any two ex tensive form games having the same normal form can be transformed into each other by the application of these transformations. Thus even if we wish to work in the extensive form we are more or less forced to look for solutions that depend only on the normal form. Kohlberg and Mertens reconciled the apparent dependence of the concepts of back ward and forward induction by defining solutions that depended only on the reduced normal form of the game and that satisfied simultaneously rather strong forms of the backward and forward induction requirements, and showing that all games had such solutions. The solutions they defined consisted of sets of equilibria rather than single equilibria, as they showed must be the case. Nevertheless, in generic extensive form games all equilibria in such a set correspond to a single outcome. Kohlberg and Mertens ( 1 986) define three different solution concepts which they call hyper-stable, fully stable, and stable. Hyper-stable sets are defined in terms of perturba tions to the payoffs in the normal form, fully stable sets in terms of convex polyhedral restrictions of the mixed strategy space, and stable sets in terms of Selten-type pertur bations to the normal form of a game. A closed set of equilibria is called strategically stable in one of these senses if it is minimal with respect to the property that all small perturbations have equilibria close to the set. Kohlberg and Mertens explicitly viewed these definitions as a first pass and were well aware of their shortcomings. In particular hyper-stable and fully stable sets may contain inadmissible equilibria, while stable sets might fail to satisfy quite weak versions of backward induction. They give an example, due to Faruk Gul, of a game with such a stable set that does not yield the same outcome as the unique subgame perfect equilibrium. Moreover such sets are often not connected, though there always exist stable sets contained within a connected component of equi libria. In spite of the problems with these definitions this work already shows that the pro gram of defining solutions in the normal form that satisfy apparently extensive form requirements is practical; and the solutions defined there go most of the way to solving the problems. A minor modification of this definition somewhat expanding the notion of a perturbation remedies part of the deficiencies of the original definition of strategic stability. Such a definition essentially expands the definition of a perturbation enough so that backward induction is satisfied while not so much as to violate admissibility, as the definition of fully stable sets by Kohlberg and Mertens does. This modification was known to Kohlberg and Mertens and was suggested in unpublished work of Philip
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Reny. It was independently discovered and called, following a suggestion of Mertens, fully stable sets of equilibria in Hillas ( 1 990). While remedying some of the deficiencies of the original definition of Kohlberg and Mertens this definition does not satisfy at least one of the requirements given above. The definition defines different sets depending on whether two agents who never both move in a single play of a game are considered as one player or two. Hillas ( 1 990) also gave a different modification to the definition of strategic stability in terms of perturbations to the best reply correspondence. That definition satisfied the requirements originally proposed by Kohlberg and Mertens, though there is some prob lem with invariance and an error in the proof of the forward induction properties. This is discussed in Vermeulen, Potters and Jansen (1997) and Hillas et al. (200 1a). Related definitions are given in Vermeulen, Potters and Jansen (1997) and Hillas et al. (200 1b). This definition also fails to satisfy the small worlds axiom. In a series of papers Mertens ( 1 987, 1 989, 1 991b, 1992) gave and developed another definition - or, more accurately, a family of definitions - of strategic stability that sat isfies all of the requirements we discussed above. The definitions we referred to in the previous paragraph are, in our opinion, best considered as rough approximations to the definitions of Mertens. Stable sets are defined by Mertens in the following way. One again takes as the space of perturbations the Selten-type perturbations to the normal form. The stable sets are the limits at zero of some connected part of the graph of the equilibrium correspon dence above the interior of a small neighborhood of zero in the space of perturbations. Apart from the minimality in the definition of Kohlberg and Mertens and the assump tion that the part of the graph was connected in the definition of Mertens, the definition of Kohlberg and Mertens could be stated in exactly this way. One would simply require that the projection map from the part of the equilibrium correspondence to the space of perturbations be onto. Mertens strengthens this requirement in a very natural way, requiring that the pro jection map not only be onto, but also be nontrivial in a stronger sense. The simplest form of this requirement is that the projection map not be homotopic to a map that is not onto, under a homotopy that left the projection map above the boundary unchanged. However such a definition would not satisfy the ordinality we discussed in Section 12. 1 . Thus Mertens gives a number of variants of the definition all involving coarser notions of maps being equivalent. That is, more maps are trivial (equivalent to a map that is not onto) and so fewer sets are stable. These definitions require that the projection map be nontrivial either in homology or in cohomology, with varying coefficient modules. Mertens shows that, with some restrictions on the coefficient modules, such definitions are ordinal. One also sees in this formulation the similarities of the two aspects of forward induc tion we discussed in the previous section. In Mertens's work they appear as special cases of one property of stable sets. A stable set contains a stable set of the game obtained by deleting a strategy that is nowhere in the relevant part of the graph of the equilibrium correspondence played with more than the minimum required probability.
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Govindan and Mertens (1993) give a definition of stability directly in terms of the best reply correspondence. One can think of the set of equilibria as the intersection of the graph of the best reply correspondence with the diagonal. To define stable sets one looks at the graph of the best reply correspondence in a neighborhood of the diagonal. It is required that a map that takes points consisting of a strategy and a best reply to that strategy to points in the strategy space in the following way be essential. The points on the diagonal are projected straight onto the strategy space. Other points are taken to the best reply and then shifted by some constant times the difference between the original strategy vector and the best reply. As long as this constant is large enough the boundary of the neighborhood will be taken outside the strategy space (at which point the function simply continues to take the boundary value). The form of the essentiality requirement in this paper is that the map be essential in Cech cohomology. Loosely, the requirement says that the intersection of the best reply correspondence with the diagonal should be essential. Govindan and Mertens show that this definition gives the same stable sets as the original definition of Mertens in terms of the graph of the equilibrium correspondence with the same form of essentiality. The following theorem applies to the definitions of Mertens and to the definition of Govindan and Mertens. THEOREM 9. Stable sets are ordinal, satisfy the player splitting, the decomposition property, and the small worlds axiom. Every game has at least one stable set. Any stable set is a connected set of normal form peifect equilibria and contains a proper (and hence quasi-peifect and hence sequential) equilibrium. Any stable set contains a stable set of a game obtained by deleting a dominated strategy or by deleting a strategy that is strictly inferior everywhere in the stable set. 13.5.
Strengthening forward induction
In this section we discuss an apparently plausible strengthening of the forward induction requirement that we discussed in Section 1 1 that is not satisfied by the definitions of stability we have given. Van Damme ( 1 989) suggests that the previous definitions of forward induction do not capture all that the intuition suggests. Van Damme does not, in the published version of the paper, give a formal definition of forward induction but rather gives a (weak) property which in my opinion should be satisfied by any concept that is consistent with forward induction . . . The proposed requirement is that in generic 2-person games in which player i chooses between an outside option or to play a game r of which a unique (viable) equilibrium e* yields this player more than the outside option, only the outcome in which i chooses r and e* is played in r is plausible. He gives the example in Figure 29(a) to show that strategic stability does not satisfy this requirement. In this game there are three components of normal form perfect equi-
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libria, (BU, L), { (T, (q, 1 q, 0)) 1 1/3 :S; q :S; 2/3}, and { (T, (0, q , 1 q)) 1 1 /3 :S; q :S; 2/3} . (Player 2's mixed strategy gives the probabilities of L , C, and R, in that order.) All of these components are stable (or contain stable sets) in any of the senses we have discussed. Since (U, L) is the unique equilibrium of the subgame that gives Player 1 more than his outside option the components in which Player 1 plays T clearly do not satisfy van Damme's requirement. It is not clear if van Damme intends to address the question we ask, namely, what outcomes are self-enforcing? (Recall that he doesn't claim that outcomes not satisfying his condition are not self-enforcing, but rather that they are not plausible.) Nevertheless let us address that question. There does not seem, to us, to be a good argument that the outcome in which Player 1 plays T is not self-enforcing. There are two separate "patterns of behavior" by Player 2 that lead Player 1 to play T. To make matters concrete consider the patterns in which Player 2 plays L and C. Let us examine the hypothesis that Player 1 plays T because he is uncertain whether Player 2 will play L or C . If he is uncertain (in the right way) this is indeed the best he can do. Further at the boundary of the behavior of Player 2 that makes T the best for Player 1 there is an assessment of Player 2 's choices that make B U best for Player 1 and another assessment of Player 2' s choices that make BD best for Player 1 . If Player 2 can be uncertain as to whether Player 1 has deviated to B U or to BD then indeed both strategies L and C can be rationalized for Player 2, as can mixtures between them. To be very concrete, let us focus on the equilibrium (T, (�, �. 0)) . In this equilibrium Player 1 is indifferent between T and BD and prefers both to BU. It seems quite con sistent with the notion of self-enforcing that Player 2, seeing a deviation by Player 1 , should not b e convinced that Player 1 had played B U. In a sense we have done nothing more than state that the stable sets in which Player 1 plays T satisfy the original statement of forward induction given by Kohlberg and Mertens ( 1 986). The argument does however suggest to us that there is something miss ing in a claim that the stronger requirement of van Damme is a necessary implication of -
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the idea of self-enforcing. To be much more explicit would require a better developed notion of what it means to have set-valued solutions, as we discussed in Section 1 3 .2. Before leaving this example, we shall address one additional point. Ritzberger ( 1 994) defines a vector field, which he calls the Nash field, on the strategy space. This defini tion loosely amounts to moving in (one of) the direction(s) of increasing payoff. Using this structure he defines an index for isolated zeros of the field (that is, equilibria). He extends this to components of equilibria by taking the sum of indices of regular pertur bations of the Nash field in a small neighborhood of the component. He points out that in this game the equilibrium (BU, L) has index + 1 , and so since the sum of the indices is + 1 the index of the set associated with Player 1 playing T must have index zero. However, as we have seen there are two separate stable sets in which Player 1 plays T. Now these two sets are connected by a further set o f (nonperfect) equilibria. The meth ods used by Ritzberger seem to require us to define and index for this component in its entirety. As we said above both of the connected subsets of normal form perfect equi libria are stable in the sense of Mertens's reformulation. And it would seem that there might be some way to define an index so that these sets, considered separately, would have nonzero index, + 1 and - 1 . Ritzberger comments that "the reformulation of Stable Sets [Mertens (1989)] eliminates the component with zero index" and speculates that "as a referee suggested, it seems likely that sets which are stable in the sense of Mertens will all be contained in components whose indices are non-zero". However his claim that neither of the sets are stable is in error and his conjecture is incorrect. It seems to us that the methods used by Ritzberger force us to treat two patterns of behavior, that are separately quite well behaved, together and that this leads, in some imprecise sense, to their indices canceling out. There seems something unwarranted in rejecting an outcome on this basis. 13.6.
Forward induction and backward induction
The ideas behind forward induction and backward induction are closely related. As we have seen backward induction involves an assumption that players assume, even if they see something unexpected, that the other players will choose rationally in the future. Forward induction involves an assumption that players assume, even if they see some thing unexpected, that the other players chose rationally in the past. However, a distinc tion between a choice made in the past whose outcome the player has not observed, and one to be made in the future goes very much against the spirit of the invariances that we argued for in Section 12. In fact, in a much more formal sense there appears to be a relationship between back ward and forward induction. In many examples - in fact, in all of the examples we have examined - a combination of the invariances we have discussed and backward induc tion gives the results of forward induction arguments and, in fact, the full strength of stability. Consider again the game of Figure 24(a) that we used to motivate the idea of forward induction. The game given in extensive form in Figure 30(a) and in normal form in Figure 30(b) has the same reduced normal form as the game of Figure 24(a). In
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that game the unique equilibrium of the subgame is (BU, L). Thus the unique subgame perfect equilibrium of the game is (BU, L). Recall that this is the same result that we argued for in terms of forward induction in Section 1 1 . This is perhaps not too surprising. After all this was a game originally presented to illustrate in the most compelling way possible the arguments in favor of forward induction. Thus the issues in this example are quite straightforward. We obtain the same result from two rounds of the deletion of weakly dominated strategies. What is a little surprising is that the same result occurs in cases in which the arguments for forward induction are less clear cut. A number of examples are examined in Hillas ( 1 998b) with the same result, including the examples presented by Cho and Kreps ( 1 987) to argue against the strength of some definitions of forward induction and strategic stability. We shall illustrate the point with one further example. We choose the particular ex ample both because it is interesting in its own right and because in the example the identification of the stable sets seems to be driven mostly by the definition of stability and not by an obvious forward induction logic. The example is from Mertens ( 1 995). It is a game with perfect information and a unique subgame perfect equilibrium - though the game is not generic. Moreover there are completely normal form arguments that support the unique subgame perfect equilibrium. In spite of this the unique strategically stable set of equilibria consists of all the admissible equilibria, a set containing much besides the subgame perfect equilibrium. The extensive form of the game is given here in Figure 3 1 (a) and the normal form in Figure 3 1 (b). The unique subgame perfect equilibrium of this game is (T, R) . Thus this is also the unique proper equilibrium - though the uniqueness of the proper equilibrium does involve an identification of duplicate strategies that is not required for the uniqueness of the subgame perfect equilibrium - and so (T, R) is a sequential equilibrium of any extensive form game having this normal form. Nevertheless there are games having
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the same reduced normal form as this game for which (T, R) is not sequential, and moreover the outcome of any sequential equilibrium of the game is not the same as the outcome from (T, R) . Suppose that the mixtures X = 0.9T + O . l M and Y = 0. 86£ + O . l y1 + 0.04y4 are added as new pure strategies. This results in the normal form game of Figure 32(b). The extensive form game of Figure 32(a) has this normal form. It is straightforward to see
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that the unique equilibrium of the simultaneous move subgame is (M, Y) in which the payoff to Player 1 is strictly positive (0.02) and so in any subgame perfect equilibrium Player 1 chooses to play "in" - that is, M, or B , or X - at his first move. Moreover in the equilibrium of the subgame the payoff to Player 2 is strictly negative ( -0. 14) and so in any subgame perfect equilibrium Player 2 chooses to play L at his first move. Thus the unique subgame perfect equilibrium of this game is (M, L ) . This equilibrium is far from the unique backward induction equilibrium o f the orig inal game. Moreover its outcome is also different. Thus, if one accepts the arguments that the solution(s) of a game should depend only on the reduced normal form of the game and that any solution to a game should contain a "backward induction" equi librium, then one is forced to the conclusion that in this game the solution must con tain much besides the subgame perfect equilibrium of the original presentation of the game.
14. An assessment of the solutions
We offer here a few remarks on how the various solutions we have examined stand up. As we indicated in Section 1 2 we do not believe that the kinds of invariance we discussed there should be controversial. (We are aware that for some reason in some circles they are, but have every confidence that as the issues become better understood this will change.) On the other hand we have less attachment to or confidence in our other maintained assumption, that of stochastic independence. We briefly discussed in the introduction our reasons for adopting that assumption in this chapter. Many of the issues we have discussed here in the context of strategic equilibrium have their analogs for correlated equilibrium. However the literature is less well developed - and our own understanding is even more incomplete. The most straightforward, and in some ways weakest, requirement beyond that of equilibrium is admissibility. If one is willing to do without admissibility the unrefined strategic equilibrium seems a perfectly good self-enforcing outcome. Another way of thinking about this is given in the work of Fudenberg, Kreps and Levine ( 1 988). If the players are permitted to have an uncertainty about the game that is of the same order as their strategic uncertainty - that is, their uncertainty about the choices of the others given the actual game - then any strategic equilibrium appears self-enforcing in an otherwise strong manner. There are a number of different ways that we could define admissibility. The standard way leads to the standard definition of admissible equilibrium, that is, a strategic equi librium in undominated strategies. A stronger notion takes account of both the strategy being played and the assumption of stochastic independence. In Section 1 2 . 1 we gave a definition of an admissible best reply correspondence such that a fixed point of such a correspondence is a normal form perfect equilibrium. Again Fudenberg, Kreps and Levine give us another way to think about this. If the players have no uncertainty about
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their own payoffs but have an uncertainty about the payoffs (or rationality) of the oth ers that is of the same order as their strategic uncertainty then any normal form perfect equilibrium is self-enforcing. Reny's ( 1 992a) work makes much the same point. To go further and get some version of the backward induction properties that we discussed in Section 10 one needs to make stronger informational assumptions. We claimed in Section 10. 1 that the idea of an equilibrium being robust to a little strate gic uncertainty in the sense that the strategic uncertainty should be when possible of a higher order than the uncertainty about the game led us towards backward induction. At best this was suggestive. However as we saw in Section 1 3 .6 it takes us much further. In combination with the invariances of Section 12 it seems to imply something close to the full strength of strategic stability. As yet we have seen this only in examples, but the examples include those that were constructed precisely to show the unreasonable ness of the strength of strategic stability. We regard these examples to be, at worst, very suggestive. Mertens ( 1989, pp. 582, 583) described his attitude to the project that led to his work on strategic stability as follows: I should say from the outset that I started this whole series of investigations with substantial scepticism. I had (and still have) some instinctive liking for the bruto Nash equilibrium, or a modest refinement like admissible equilibria, or normal form perfect equilibria. It is not entirely coincidental that we arrive at a similar set of preferred solutions. Our approaches have much in common and our work has been much influenced by Mertens. 15. Epistemic conditions for equilibrium
There have been a number of papers relating the concept of strategic equilibrium to the beliefs or knowledge of the players. One of the most complete and successful is by Aumann and Brandenburger ( 1 995). In the context of a formal theory of interactive belief systems they find sets of sufficient conditions on the beliefs of the players for strategic equilibrium. In keeping with the style of the rest of this paper we shall not develop the formal theory but shall rather state the results and sketch some of the proofs in an informal style. Of course, we can do this with any confidence only because we know the formal analysis of Aumann and Brandenburger. There are two classes of results. One concerns the actions or the plans of the players and the other the conjectures of the players concerning the actions or plans of the other players. These correspond to the two interpretations of strategic equilibrium that we discussed in Section 3 . The first class consists of a single result. It is quite simple; its intuition is clear; and its proof is, as Aumann and Brandenburger say, immediate.
Suppose that each player is rational and knows his own payofffunction and the vector of strategies chosen by the others. Then these strategies form a strategic equilibrium. THEOREM 1 0 .
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The results involving conjectures are somewhat more involved. The reader is warned that, while the results we are presenting are precisely Aumann and Brandenburger's, we are presenting them with a quite different emphasis. In fact, the central result we present next appears in their paper as a comment well after the central arguments. They give central place to the two results that we give as corollaries (Theorems A and B in their paper). THEOREM 1 1 . Suppose that each player knows the game being played, knows that all the players are rational, and knows the conjectures of the other players. Suppose also that (a) the conjectures of any two players about the choices of a third are the same, and (b) the conjectures of any player about the choices of two other players are indepen dent. Then the vector ofconjectures about each player's choiceforms a strategic equilibrium. The proof is, again, fairly immediate. Suppositions (a) and (b) give immediately that the conjectures "are" a vector of mixed strategies, a- . Now consider the conjecture of Player n' about Player n. Player n' knows Player n's conjecture, and by (a) and (b) it is a- . Also Player n' knows that Player n is rational. Thus his conjecture about Player n 's choices, that is, o-n , should put weight only on those choices of Player n that maximize n 's payoff given n 's conjecture, that is, given a- . Thus a- is a strategic equilibrium. COROLLARY 1 2 . Suppose that there are only two players, that each player knows the game being played, knows that both players are rational, and knows the conjecture of the other player. Then the pair of conjectures about each player's choice forms a strategic equilibrium. The proof is simply to observe that for games with only two players the conditions (a) and (b) are vacuous. COROLLARY 1 3 . Suppose that the players have a common prior that puts positive weight on the event that a particular game g is being played, that each player knows the game and the rationality of the others, and that it is common knowledge among the players that the vector ofconjectures takes on a particular value
E.
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where (N \ i, v) is the game v restricted to the set of players N \ i . To be associated with a measure of power in the game, these marginal contributions need to add up to the total resources available to the grand coalitions, i.e., v (N). So we will say that P is a potential function if
L D i P (N, v) = v(N) .
(2)
iEN Moreover, we normalize P to satisfy P (0, v) = 0. Given the mild requirement on p, there seems to be enough flexibility for many potential functions to exist. The remark able result of Hart and Mas-Colell (1989) is that there is in fact only one potential function, and it is closely related to the Shapley value. Specifically, THEOREM [Hart and Mas-Colell (1989)]. There exists a unique potential func tion P. Moreover, the corresponding vector ofmarginal contributions (D 1 P(N, v), . . . , vn P(N, v)) coincides with the Shapley value of the game (N, v).
Let us note that by rewriting Equation (2) we obtain the following recursive equation:
P(N, v)
=
[
]
(1/INI) v(N) + L P(N \ i, v) .
tEN
(3)
Starting with P(0, v) = 0, Equation (3) determines p recursively. It is interesting to note that the potential is related to the notion of "dividends" used by Harsanyi (1963) to extend the Shapley value to games without transferable utility. Specifically, let L T c N arur be the (unique) representation of the game (N, v) as a linear combina tion of unanimity games. In any unanimity game u T on the carrier T, the value of each player in T is the "dividend" dr ay J I T I , and the Shapley value of player i in the game (N, v) is the sum of dividends that a player earns from all coalitions of which he is a member, i.e., L {T ; i ET} dr. Given the definition and uniqueness of the potential function, it follows that P(N, v) '£7 dr, i.e., the total Harsanyi dividends across all coalitions in the game. One can view Hart and Mas-Colell's result as a concise axiomatic characterization of the Shapley value - indeed, one that builds on a single axiom. In the same paper, Hart and Mas-Colell propose another axiomatization of the value by a different but related approach based on the notion of consistency. Unlike in non-cooperative game theory, where the feasibility of a solution concept is judged according to strategic or optimal individual behavior, in cooperative game theory neither strategies nor individual payoffs are specified. A cooperative solution concept is considered attractive if it makes sense as an arbitration scheme for allocating costs or benefits. It comes as no surprise, then, that some of the popular properties used to support solution concepts in this field are normative in nature. The symmetry axiom is a case in point. It asserts that individuals indistinguishable in terms of their contributions =
=
Ch. 53: The Shapley Value
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to coalitions are to be treated equally. One of the most fundamental requirements of any legal system is that it be internally consistent. Consider some value (a single-point solution concept) ¢, which we would like to use as a scheme for allocating payoffs in games. Suppose that we implement ¢ by first making payment to a group of players Z. Examining the environment subsequent to payment, we may realize that we are facing a different (reduced) game defined on the remaining players N \ Z who are still waiting to be paid. The solution concept ¢ is said to be consistent if it yields the players in the re duced game exactly the same payoffs they are getting in the original game. Consistency properties play an important role in the literature of cooperative game theory. They were used in the context of the Shapley value by Hart and Mas-Colell (1989). The difference between Hart and Mas-Colell's notion of consistency and that of the related literature on other solution concepts lies in their definition of reduced game. For a given value ¢, a game (N, v), and a coalition T, the reduced game (T, vr) on the set of players T is defined as follows:
vT (S) v(S U Tc ) - L ¢; (v l suF ) for every S C T, =
i ETc
where v i R denotes the game v restricted to the coalition R. The worth of coalition S in the reduced game vr is derived as follows. First, the players in S consider their options outside T, i.e., by playing the game only with the complementary coalition Tc. This restricts the game v to coalition S U Tc. In this game the total payoff of the members of rc according to the solution ¢ is L ; E T' ¢; ( v I su re ) . Thus the resources left available for the members of S to allocate among themselves are precisely vr (S) . A value ¢ is now said to be consistent if for every game (N, v), every coalition T, and every i E T, one has ¢; (T, vr) ¢; (N, v). Hart and Mas-Colell (1989) show that with two additional standard axioms the con sistency property characterizes the Shapley value. Specifically, =
THEOREM [Hart and Mas-Colell (1989)]. There exists a unique value satisfying sym metry, efficiency and consistency, namely the Shapley value. 3
It is interesting to note that by replacing Hart and Mas-Colell's property with a differ ent consistency property one obtains a characterization of a different solution concept, namely the pre-nucleolus. Sobolev's (1975) consistency property is based on the fol lowing definition of the reduced game:
[
max v(Q U S) - L ¢; (v) v� (S) = QcN\ T 3
iEQ
J
if S # T, 0,
H art and Mas-Colell ( 1989) in fact showed that instead of the efficiency and symmetry axioms it is enough to require that the solution be "standard" for two-person games. i.e., that for such games
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v} (S) = L ¢; (v) if S = T, and v} (S) = 0 if S = 0. i ET
Note that in Sobolev's definition the members of S evaluate their power in the reduced game by considering their most attractive options outside T . Furthermore, the collabo rators of S outside T are paid according to their share in the original game v (and not according to the restricted game as in Hart and Mas-Colell's property). It is surprising that while the definitions of the Shapley value and the pre-nucleolus differ completely, their axiomatic characterizations differ only in terms of the reduced game on which the consistency property is based. This nicely demonstrates the strength of the axiomatic approach in cooperative game theory, which not only sheds light on individual solution concepts, but at the same time exposes their underlying relationships. Hart and Mas-Colell's consistency property is also related to the "balanced contribu tions" property due to Myerson (1977). Roughly speaking, this property requires that player i contribute to player j's payoff what player j contributes to player i's payoff. Formally, the property can be written as follows: BALANCED CONTRIBUTION.
tPj (V ) - tPj ( VI N\j ).
For every two players i and j, ¢; (v) - ¢; ( V IN \j )
=
Myerson (1977) introduced a value that associates a payoff vector with each game v and graph g on the set N (representing communication channels between players). His result implies that the balanced contributions property, the efficiency axiom, and the symmetry axiom characterize the Shapley value. We will close this section by discussing another axiomatization of the value, pro posed by Chun (1989). It employs an interesting property which generalizes the strate gic equivalence property traceable to von Neumann and Morgenstern (1944). Chun's coalitional strategic equivalence property requires that if we change the coalitional form game by adding a constant to every coalition that contains some (fixed) coalition T C N, then the payoffs to players outside S will not change. This means that the extra benefit (or loss if the added constant is negative) accrues only to the members of T . Formally: COALITIONAL STRATEGIC EQUIVALENCE . For all T C N and a E JR, let wi be the game defined by wi (S) = a if T C S and 0 otherwise. For all T C N and a E lit, if v = w + w I, then ¢; ( v) = ¢; ( w) for all i E N \ T.
Von Neumann and Morgenstern's strategic equivalence imposes the same require ment, but only for I T I = 1 . Another similar property proposed by Chun is fair ranking. It requires that if the un derlying game changes in such a way that all coalitions but T maintain the same worth, then although the payoffs of members of T will vary, the ranking of the payoffs within T will be preserved. This directly reflects the idea that the ranking of players' payoffs within a coalition depends solely on the outside options of their members. Specifically:
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Ch. 53: The Shapley Value FAIR RANKING. Suppose that v(S) = w(S) T, (Pi (v) > c/>j (v) if and only if c{>; (w) > c/>j (w).
for every S i= T; then for all i, j E
To characterize the Shapley value an additional harmless axiom is needed. TRIVIALITY. If vo is the constant zero game, i.e., vo(S) = 0 for all S C N, then c/>; (vo) = Ofor all i E N.
The following characterization of the Shapley value can now be obtained: THEOREM [Chun (1989)]. The Shapley value is the unique value satisfying efficiency, triviality, coalitional strategic equivalence, andfair ranking.
6. Sub-classes of games
Much of the Shapley value's attractiveness derives from its elegant axiomatic character ization. But while Shapley's axioms characterize the value uniquely on the class of all games, it is not clear whether they can be used to characterize the value on subspaces. It sounds puzzling, for what could go wrong? The fact that a value satisfies a certain set of axioms trivially implies that it satisfies those axioms on every subclass of games. However, the smaller the subclass, the less likely these axioms are to determine a unique value on it. To illustrate an extreme case of the problem, suppose that the subclass con sists of all integer multiples of a single game v with no dummy players, and no two players are symmetric. On this subclass Shapley's symmetry and dummy axioms are vacuous: they impose no restriction on the payoff function. It is therefore easy to verify that any allocation of v(N) can be supported as the payoff vector for v with respect to a value on this subclass that satisfies all Shapley's axioms. Another problem that can arise when considering subclasses of games is that they may not be closed under oper ations that are required by some of the axioms. A classic example of this is the class of all simple games. It is clear that the additivity axiom cannot be used on such a class, because the sum of two simple games is typically not a simple game anymore. One can revise the additivity axiom by requiring that it apply only when the sum of the two games falls within the subclass considered. Indeed, Dubey (1975) shows that with this amendment to the additivity4 axiom, Shapley's original proof of uniqueness still applies to some subclasses of games, including the class of all simple games. However, even in conjunction with the other standard axioms, this axiom does not yield uniqueness in the class of all monotonic5 simple games. To redress this problem, Dubey (1975) intro duced an axiom that can replace additivity: for two simple games v and v', we define 4
v 1 , v2
5
=
Specifically, one has to require the axiom only for games subclass. Recall that a simple game v is said to be monotonic if v(S)
whose sum belongs to the underlying
I and
S c T implies v (T) = 1 .
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by min{ v, v'} the simple game in which S is winning if and only if it is winning in both v and v'. Similarly, we define by max{v, v'} the game in which S is winning if and only if it is winning in at least one of the two games v and v'. Dubey imposed the property of MODULARITY.
¢ (min{v, v'}) + ¢ (max{v, v'}) = ¢ (v) + ¢ (v').
One can interpret this axiom within the framework of Roth's model in Section 3. Suppose that ¢; (v) stands for the utility of playing i 's position in the game v, and player i is facing two lotteries. In the first lottery he will participate in either the game v or the game v' with equal probabilities. The other lottery involves two more "extreme" games: max{v, v'}, which makes winning easier than with v and v', and min{v, v'}, which makes winning harder. As before, each of these games will be realized with probability 1/2. Modularity imposes that each player be "risk neutral" in the sense that he be indifferent between these two lotteries. Note that min{v, v'} and max{v, v'} are monotonic simple games whenever v and v' are, so we do not need any conditioning in the formulation of the axiom. Dubey char acterized the Shapley value on the class of monotonic simple games by means of the modularity axiom, showing that: THEOREM [Dubey (1975)]. There exists a unique value on the class ofmonotonic6 sim ple games satisfying efficiency, symmetry, dummy, and modularity, and it is the Shapley Shubik value.
When trying to apply a solution concept to a specific allocation problem (or game), one may find it hard to justify the Shapley value on the basis of its axiomatic charac terization. After all, an axiom like additivity deals with how the value varies as a result of changes in the game, taking into account games which may be completely irrelevant to the underlying problem. The story is different insofar as Shapley's other axioms are concerned, because the three of them impose requirements only on the specific game under consideration. Unfortunately, one cannot fully characterize the Shapley value by axioms of the second type only7 (save perhaps by imposing the value formula as an axiom). Indeed, if we were able to do so, it would mean that we could axiomatically characterize the value on subclasses as small as a single game. While this is impossi ble, Neyman (1989) showed that Shapley's original axioms characterize the value on the additive class (group) spanned by a single game. Specifically, for a game v let G (v) denote the class of all games obtained by a linear combination of restricted games of v, i.e., G(v) = {v' s.t. v' = Li k v l s; , where k; are integers and vis; is the game v restricted to the coalition si } . ;
6
The same result was shown by Dubey (1975) to hold for the class of superadditive simple games. A similar distinction can be made within the axiomatization of the Nash solution where the symmetry and efficiency axioms are "within games" while IIA and Invariance are "between games".
7
Ch. 53: The Shapley Value
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Note that by definition the class G(v) is closed under summation of games, which makes the additivity axiom well defined on this class. Neyman shows that: THEOREM [Neyman (1989)]. For any v there exists a unique value on the subclass G(v) satisfying efficiency, symmetry, dummy, and additivity, namely the Shapley value. It is worth noting that Hart and Mas-Colell 's ( 1989) notion of Potential also character izes the Shapley value on the subclass G ( v), since the Potential is defined on restricted games only. 7.
Cooperation structures
One of Shapley's axioms which characterize the value is the symmetry axiom. It re quires that a player's value in a game depend on nothing but his payoff opportunities as described by the coalitional form game, and in particular not on the his "name". In deed, as we argued earlier, the possibility of constructing a unique measure of power axiomatically from very limited information about the interactive environment is doubt less one of the value's most appealing aspects. For some specific applications, however, we might possess more information about the environment than just the coalitional form game. Should we ignore this additional information when attempting to measure the rel ative power of players in a game? One source of asymmetry in the environment can fol low simply from the fact that players differ in their bargaining skills or in their property rights. This interpretation led to the concept of the weighted Shapley value by Shap ley (1953) and Kalai and Samet (1985) (see Chapter 54 in this Handbook). But asym metry can derive from a completely different source. It can be due to the fact that the interaction between players is not symmetric, as happens when some players are orga nized into groups or when the communication structure between players is incomplete, thus making it difficult if not impossible for some players to negotiate with others. This interpretation has led to an interesting field of research on the Shapley value, which is mainly concerned with generalizations. The earliest result in this field is probably due to Aumann and Dreze (1975), who consider situations in which there exists an exogenous coalition structure in addition to the coalitional form game. A coalition structure B = (S1 , . . . , Sm ) is simply a partition of the set of players N, i.e., U SJ = N and S; n SJ = 0 for i =!= j . In this context a value is an operator that assigns a vector of payoffs ¢ (B, v) to each pair (B, v), i.e., a coalition structure and a coalitional form game on N. Aumann and Dreze (1975) imposed the following axioms on such operators. First, the efficiency axiom is based on the idea that by forming a group, players can allocate to themselves only the resources available to their group. Specifically, RELATIVE EFFICIENCY. m, we have ¢J (B ,
Lj E Sk
For every coalition structure B = (S1 , . . . , Sm ) and 1 v) = v(Sk).
:(
k :(
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The remaining three axioms are straightforward extensions of their counterparts in the benchmark model. SYMMETRY. For every permutation n on N and every coalition structure B, we have ¢ (n B, nv) = ncf>(B, v), where n B is the coalition structure with (n B); = nS; . DUMMY. Ifi is a dummy player in v, then c/>; (B, v ) = Ofor all B. ADDITIVITY. c/> (B, v + w) = cf>(B, v) + c/> (B, w) for all B and any pair of games v and w. Aumann and Dreze showed that the above axioms characterize a value uniquely. This value assigns the Shapley value to each player in the game restricted to his group. Specifically, for each coalition S and game v, we define by v i S the game on S given by (vls)(T) = v(T) for all T C S. We now have: THEOREM [Aumann and Dreze (1975)]. For a given set ofplayers N and a coalition structure B, there exists a unique B-value satisfying the four above axioms, namely ¢; (B, v) = ¢; (vI B (i )), where B ( i) denotes the component of B that includes player i, and 1> stands for the Shapley value. Aumann and Dreze then defined the coalition structure of all the major solution con cepts in cooperative game theory, in addition to the Shapley value. In so doing they exposed a startling property of consistency satisfied by all but one of them. The Shapley value is the exception. Hart and Mas-Colell (1989) would later show that the Shapley value satisfies a version of this property (discussed earlier in Section 5). In Aumann and Dreze' s framework, the coalition structure can be thought of as repre senting contractual relationships that affect players' property rights over resources, i.e., players in S E B "own" a total resource v(S), from which transfers to players outside S are forbidden. This is apparent from both the definition and the efficiency axiom that depends on the coalition structure. Roughly speaking, players from one component of the partition cannot share benefits with any player from another component. But in real life coalition formation often takes place merely for strategic reasons without affecting the fundamentals of the economy. A group may form in order to pressure another group, or to enhance the bargaining position of its members vis-a-vis other members without affecting the constraint that all players in N share the total pie v(N). It is thus reason able to ask whether it is possible to extend the Shapley value to this context as well. Owen (1977), Hart and Kurz (1983), and Winter (1989, 1991, 1992) take up this ques tion. Owen proposed a value, like the Shapley value, in which each player receives his expected marginal contribution to coalitions with respect to a random order of players. But while the Shapley value assumes that all orders are of equal probability, the Owen value restricts orders according to the coalition structure. Specifically, for a given coali tion structure B, let us consider a subset of the set of all orders, which includes only
Ch. 53: The Shapley Value
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the orders in which players of the same component of B appear successively. Denote this set of orders fl(B). The set of orders fl(B) can be obtained by first ordering the components of B and then ordering players within each component of the partition. Ac cording to the Owen value, each player is assigned an expected marginal contribution to the coalition of preceding players with respect to a uniform distribution over the set of orders in fl(B). More formally, let B = (S1 , . . . , Sm ) be a coalition structure. Set fl(B) = {n E fl ; if i, j E Sk and n (i) < n(r) < n (j), then r E Sk } , which is the set of all permutations consistent with the coalition structure B. The Owen value of player i in the game v with the coalition structure B is now given by
(Pi(B, v) = (l /lfl(B) I) L [v(p� U i) - v(p�) ] . n Efl(B)
Note that the Owen value satisfies efficiency with respect to the grand coalition, i.e., the total payoff across all players is v(N). This is due to the fact that the total marginal contribution of all players with respect to a fixed order sums up to precisely v(N) . Like the Shapley value, the Owen value can be characterized axiomatically in more than one way, including the way proposed by Owen himself and that of Hart and Kurz (1983). We will introduce here one version that was used in Winter (1989) for the characterization of level structure values, which generalize the Owen value. First note that for a given coalition structure B = (SJ , . . . , Sm ) and game v, we can define a "game between coalitions" in which each coalition S; acts as a player. A coali tion acting as a player will be denoted by [S; ] . Specifically, the worth of the coalition { [ Si l ] , . . . , [S;k]} is v (Si ! U S;2 , . . . , US;k). We will denote this game by v*. We now impose two symmetry axioms: SYMMETRY ACROSS COALITIONS. If players [Sk] and [Sz] are symmetric in the game v*, then the total values for these coalitions are equal, i.e., L sz
We recall that the Owen value satisfies:
With the above axioms we now have: THEOREM [Owen (1977)]. For a given set of players N and a coalition structure B, the Owen value is the unique B-value satisfying symmetry across coalitions, symmetry within coalitions, dummy, additivity, and efficiency.
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Hart and Kurz ( 1983) formulated a different axiom which can be used to characterize the Owen value requiring that: INESSENTIAL GAMES AMONG COALITIONS .
have Lj E Sk c/Jj (B, v)
=
v(Sk).
If v(N) = Lk v(Sk), then for all k we
Hart and Kurz (1983) show that this axiom together with symmetry, additivity, and a carrier axiom combining dummy and efficiency yield a characterization of the Owen value. Hart and Kurz also used the Owen value to model processes of coalition forma tion. Other papers take the Owen value as a starting-point either to suggest alternative ax iomatizations or to develop further generalizations. Winter (1992) used the consistency and the potential approach to characterize the Owen value. Calvo, Lasaga, and Win ter (1996) used Myerson's approach of balanced contributions for another axiomatiza tion of the Owen value. In Owen and Winter (1992), the multilinear extension approach [see Owen (1972)] was used to propose an alternative interpretation of the Owen value based on a model of random coalitions. Finally, Winter (1991) proposed a general solu tion concept, special cases of which are the Harsanyi (1963) solution on standard NTU games and the Owen value for TU games with coalition structures (as well as other non-symmetric generalizations of the Shapley value). As argued earlier, there is an intrinsic difference between Aumann and Dreze's inter pretation of coalition structures and that of Owen. Thinking of coalition structures as unions or clubs, which define an asymmetric "closeness" relation between the various individuals, suggests several alternatives for introducing cooperation structures into the Shapley value framework. One such approach was proposed by Myerson (1977), who used graphs to represent cooperation structures between players. This important paper is discussed thoroughly in Monderer and Samet (see Chapter 54 in this Handbook). Another approach that is closer to Owen's (and indeed generalizes it) was analyzed by Winter (1989). Here cooperation is described by the notion of (hierarchical) level struc tures. Formally, a level structure is a finite sequence of partitions B (BJ , B2, . . . , Bm) such that B; is a refinement of B; +i · That is, if S E B; , then S C T for some T E B;+l · The idea behind level structure is that individuals inherit connections to other in dividuals by association to groups with various levels of commitment. In the context of international trade, one can think of Bm (the coarsest partition) as representing free trade agreements within, say, Nafta, the European Union, Mercusor, etc. Each bloc in this coalition structure is partitioned into countries, each country into federal states or regions, and so on down to the elementary units of households. In order to define the extension of the Owen value within this framework, we adopt the interpretation that the partition Bj represents a stronger connection between players than that of the coarser Bk where k > j . Let us think of a permutation as the order by which players appear to collect their payoffs. To make payoffs dependent on the coop eration structure, we restrict ourselves to orders in which no player i follows player j if =
Ch. 53: The Shapley Value
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there is another player, say k, who is "closer" to player i and who has still not appeared. Formally, we can construct this set of orders inductively as follows: For a given level structure B (B, , . . . , Bm), define =
ITm = {n E IT; for each l, j E S E Bm and i E N, n(l) < n(i) < n(j) implies i E S} and Ilr
=
{n E ITr + l ; for each l, j E S E Br and i E N, n(l) < n(i) < n(j) implies i E S}.
The proposed value gives an expected marginal contribution to each player with re spect to the uniform distribution over all the orders that are consistent with the level structure B, i.e., the orders in IT, . Specifically,
¢; (B, v) = ( 1 /I ITI I) L [ v (p� U i ) - v (p� ) ] .
(4)
n Ell1
Note that when the level structure consists of only one partition (i.e., m = 1), we are back in Owen's framework. Moreover, in contrast to the special case of Owen, in which there is only one game between coalitions, this framework gives rise to m games be tween coalitions, one for each hierarchy level (partition). We denote these games by v 1 , v 2 , . . . , vm . The following axiom is an extension of the axiom of symmetry across coalitions that we defined earlier in relation to the Owen value.
Let B = (B I , . . . , Bm) be a level structure. For each level 1 � i � m, if S, T E B; are symmetric as players ([S] and [T]) in the game vi and if S and T are subsets of the same component in Bj for all j > i, then L rES cf>r (B, v) LrET cf>r (B, v). COALITIONAL S YMMETRY.
=
In order to axiomatize the level structure value, we need another symmetry axiom that requires equal treatment for symmetric players within the same coalition: S YMMETRY WITHIN COALITIONS . If k and j are two symmetric players with respect to the game v, where for every level l � i � m, and any non-singleton coalition S E B1 , then k E S iff j E S, and ¢; (B, v) = c{>j (B, v).
Using straightforward generalizations of the rest of the axioms of the Owen value, it can be shown that: [Winter (1989)]. There exists a unique level structure value satisfying coali tional symmetry, symmetry within coalitions, dummy, additivity, and efficiency. This value is given by (4). THEOREM
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Several other approaches to cooperation structures have been proposed. We have al ready mentioned Myerson (1977), who uses a graph to represent bilateral connections (or communications) between individuals. An interesting application of Myerson's so lution was proposed by Aumann and Myerson (1988). They considered an extensive form game in which players propose links to other players, sequentially. Using the My erson value to represent the players' returns from each graph that forms, they analyze the endogenous graphs that form (given by the subgame perfect equilibria of the link formation game). Myerson (1980) discusses conference structures that are given for mally by an arbitrary collection of coalitions representing a (possibly non-partitional) set of associations to which individuals belong. Derks and Peters (1993) consider a ver sion of the Shapley value with restricted coalitions, representing a set of cooperation constraints. These are given by a mapping p : 2 N 2 N , such that (1) p ( S) c S, (2) S C T implies p(S) C p(T), and (3) p(p(S)) p(S) . This mapping can be interpreted as institutional constraints on the communications between players, i.e., p (S) represents the most comprehensive agreement that can be attained within the set of players S. Van den Brink and Gilles propose Permission Structures, based on the idea that some inter actions take place in hierarchical organizations in which cooperation between two indi viduals requires the consent of their supervisors. Permission structures are thus given by a mapping p : N --+ 2 N , where j E p(i) stands for "j supervises i". The function p im poses exogenous restrictions on cooperation and allows for extension of the Shapley value. I will conclude this section by briefly discussing another interesting (asymmetric) generalization of the value. Unlike the others, this one was proposed by Shapley himself (see also Chapter 32 in this Handbook). Shapley (1977) examines the power of indices in political games, where players' political positions affect the prospects of coalition formation. Shapley's aim was to embed players' positions in an m-dimensional Euclid ean space. The point Xi E lRm of player i summarizes i 's position (on a scale of support and opposition) on each of the relevant m "pure" issues. General issues faced by legis lators typically involve a combination of several pure ones. For example, if the two pure issues are government spending (high or low) and national defense policy (aggressive or moderate), then the issue of whether or not to launch a new defense missile project is a combination of the two pure issues. Shapley's suggestion was to describe general issues as vectors of weights w = (W J , . . . , Wm ) . Note that every vector w induces a nat ural order over the set of players. Specifically, j appears after i if w · Xi > w · xj (where x y stands for the inner product of the vectors x and y). The main point to notice here is that different vectors induce different orders on the set of players. To measure the legislative power of each player in the game, one has to aggregate over all possible (general) issues. Let us therefore assume that issues occur randomly with respect to a uniform distribution over all issues (i.e., vectors w in lRm ). For each order of players n , let e (n) be the probability that the random issue generates the order n . Thus the players' profile of political positions (XJ , xz, . . . , x11) is mapped into a probability dis tribution over the set of all permutations. Shapley's political value yields an expected marginal contribution to each player, where the random orders are given by the prob--+
=
an
·
Ch. 53: The Shapley Value
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ability distribution e . Note that the political value is in the class of Weber's random order values [see Weber (1988)]. A random order value is characterized by a probabil ity distribution over the set of all permutations. According to this value, each player receives his expected marginal contribution to the players preceding him with respect to the underlying probability distribution on orders. The relation between the political value and the Owen value is also quite interesting. Suppose that the vector of positions is represented by m clusters of points in JK2 , where the cluster k consists of the players in Sk , whose positions are very close to each other but further away than those of other players (in the extreme case we could think of the members of Sk as having identical positions). It is pretty clear that the payoff vector that will emerge from Shapley's po litical value in this case will be very close to the Owen value for the coalition structure B (S1 , . . . , Sm). =
8. Sustaining the Shapley value via non-cooperative games
If the Shapley value is interpreted merely as a measure for evaluating players' power in a cooperative game, then its axiomatic foundation is strong enough to fully justify it. But the Shapley value is often interpreted (and indeed sometimes applied) as a scheme or a rule for allocating collective benefits or costs. The interpretation of the value in these situations implicitly assumes the existence of an outside authority - call it a planner which determines individual payoffs based on the axioms that characterize the value. However, situations of benefit (or cost) allocation are, by their very nature, situations in which individuals have conflicting interests. Players who feel underpaid are therefore likely to dispute the fairness of the scheme by challenging one or more of its axioms. It would therefore be nice if the Shapley value could be supported as an outcome of some decentralized mechanism in which individuals behave strategically in the absence of a planner whose objectives, though benevolent, may be disputable. This objective has been pursued by several authors as part of a broader agenda that deals with the interface between cooperative and non-cooperative game theory. The concern of this literature is the construction of non-cooperative bargaining games that sustain various cooperative solution concepts as their equilibrium outcomes. This approach, often referred to in the literature as "the Nash Program", is attributed to Nash's (1950) groundbreaking work on the bargaining problem, which, in addition to laying the axiomatic foundation of the solution, constructs a non-cooperative game to sustain it. Of all the solution concepts in cooperative game theory, the Shapley value is ar guably the most "cooperative", undoubtedly more so than such concepts as the core and the bargaining set whose definitions include strategic interpretations. Yet, perhaps more than any other solution concept in cooperative game theory, the Shapley value emerges as the outcome of a variety of non-cooperative games quite different in structure and interpretation. Harsanyi (1985) is probably the first to address the relationship between the Shapley value and non-cooperative games. Harsanyi' s "dividend game" makes use of the relation
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between the Shapley value and the decomposition of games into unanimity games. To the more recent literature which uses sequential bargaining games to sustain cooperative solution concepts, Gul (1989) makes a pioneering contribution. In Gul's model, play ers meet randomly to conduct bilateral trades. When two players meet, one of them is chosen randomly (with equal probabilities 1 /2- 1 /2) to make a proposaL In a proposal by player i to player j at period t, player i offers to pay r1 to player j for purchasing j 's resources in the game. If player j accepts the offer, he leaves the game with the proposed payoff, and the coalition { i, j} becomes a single player in the new coalitional form game, implying that i now owns the property rights of player j . If j rejects the proposal by i , both players return to the pool of potential traders who meet through random matching in a subsequent period. Each pair's probability of being selected for trading is 2j(n 1 (n 1 - 1)), where n1 is the number of players remaining at period t. The game ends when only a single player is left For any given play path of the game, the payoff of player j is given by the current value of his stream of resources minus the payments he made to the other players. Thus, for a given strategy combination a and a discount factor 8, we have 00
1 u; (a, 8) = :Lu - 8) [ V (Mf) - rf ] 8 , 0
where Mf is the set of players whose resources are controlled by player i at time t and 8 is a discount factor. Gul confines himself to the stationary subgame perfect equilibria (SSPE) of the game, i.e., equilibria in which players' actions at period t depend only upon the allocation of resources at time t. He argues that SSPE outcomes may not be efficient in the sense of maximizing the aggregate equilibrium payoffs of all the players in the economy, but he goes on to show that in any no-delay equilibrium (i.e., an equi librium in which all pairwise meetings end with agreements) players' payoffs converge to the Shapley value of the underlying game when the discount factor approaches 1 . Specifically, THEOREM [Gul ( 1989)]. Let a (8k) be a sequence ofSSPEs with respect to the discount factors { 8k} which converge to 1 as k goes to infinity. If a (8k) is a no-delay equilibrium for all k, then Ui (a (8k), 8k) converges to i 's Shapley value of V as k goes to infinity.
It should be argued that in general the SSPE outcomes of Gul's game do not con verge to the Shapley value as the discount factor approaches 1 . Indeed, if delay occurs along the equilibrium path, the outcome may not be close to the Shapley value even for 8 close to 1 . Gul's original formulation of the above theorem required that a (8k) be efficient equilibria (in terms of expected payoffs). Gul argues that the condition of effi ciency is a sufficient guarantee that along the equilibrium path every bilateral matching terminates in an agreement However, Hart and Levy ( 1999) show in an example that ef ficiency does not imply immediate agreement Nevertheless, in a rejoinder to Hart and
Ch. 53: The Shapley Value
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Levy (1999), Gul (1999) points out that if the underlying coalitional game is strictly convex, 8 then in his model efficiency indeed implies no delay. A different bargaining model to sustain the Shapley value through its consistency property was proposed by Hart and Mas-Colell (1996). 9 Unlike in Gul's model, which is based on bilateral agreements, in Hart and Mas-Colell's approach players submit proposals for payoff allocations to all the active players. Each round in the game is characterized by a set S C N of "active players" and a player i E S who is designated to make a proposal after being randomly selected from the set S. A proposal is a fea sible payoff vector x for the members in S, i.e., L jES xi = v ( S). Once the proposal is made, the players in S respond sequentially by either accepting or rejecting it. If all the members of S accept the proposal, the game ends and the players in S share payoffs according to the proposal. Inactive players receive a payoff of zero. If at least one player rejects the proposal, then the proposer i runs the risk of being dropped from the game. Specifically, the proposer leaves the game and joins the set of inactive players with a probability of 1 - p, in which case the game continues into the next period with the set of active players being S \ i. Or the proposer remains active with probability p, and the game continues into the next period with the same set of active players. The game ends either when agreement is reached or when only one active player is left in the game. Hart and Mas-Colell analyzed the above (perfect information) game by means of its stationary subgame perfect equilibria, and concluded: THEOREM [Hart and Mas-Colell (1996)]. For every monotonic and non-negative 1 0 game v and for every 0 � p < 1, the bargaining game described above has a unique stationary subgame perfect equilibrium (SSPE). Furthermore, if as is the SSPE pay off vector of a subgame starting with a period in which S is the active set ofplayers, then as = ¢ ( v [s) (where ¢ stands for the Shapley value and v[s for the restricted game on S). In particular, the SSPE outcome of the whole game is the Shapley value ofv. A rough summary of the argument for this result runs as follows. Let as, i denote the equilibrium proposal when the set of active players is S and the proposer is i E S. In equilibrium, player j E S with j 'f. i should be paid precisely what he expects to receive if agreement fails to be reached at the current payoff. As the protocol specifies, with probability p we remain with the same set of players and the next period's expected proposal will be as = 1 / [ S[ L i E S as , i . With the remaining probability 1 - p, players i will be ejected so that the next period's proposal is expected to be as \i . We thus obtain the following two equations for as .i : (1) Lj aL = v ( S) (feasibility condition), and 8
S
We recall that a game v is said to be strictly convex if v(S U i) - V (S) > v(T U i) - v(T) whenever T c and S i T. 9 In the original Hart and Mas-Colell (1996) paper, the bargaining game was based on an underlying non transferable utility game. 10 v ( S) ;:;,: 0 for all S C N, and v(T) :( v(S) for T C S.
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E. "Winter
(2) a L = pa� + (1 - p)a�\i (equilibrium condition). Rewriting the second condition we notice that the two of them correspond precisely to the two properties of Myerson (1977), which we discussed in Section 5 and which together with efficiency characterize the value uniquely. In a recent paper, Perez-Castrillo and Wettstein (1999) suggested a game that modi fies that of Hart and Mas-Colell ( 1996) so as to allow the (random) order of proposals to be endogenized. The game runs as follows. Prior to making a proposal there is a bidding phase in which each player i commits to pay a payoff vf to player j . These bids are made simultaneously. The identity of the proposal is determined by the bids. Specifi cally, the proposer is chosen to be the player i for which the difference between the bids made by i and the bids made to i is maximized, i.e., i = argmaxk E N [ L j v£ L j vj] (players' bids to themselves is always zero) . If there is more than one player for which this maximum is attained, then the proposer is chosen from among these play ers with an equal probability for each candidate. Following player i 's recognition to propose, the game proceeds according to Hart and Mas-Colell's protocol, with p = 1 . Namely, upon rejection, player i leaves the game with probability 1 . Perez-Castrillo and Wettstein (1999) show that this game implements the Shapley value in a unique subgame perfect equilibrium (since the game is of a finite horizon, no stationarity re quirement is needed). Almost all the bargaining games that have been proposed in the literature on the im plementation of cooperative solutions via non-cooperative equilibria are based on the exchange of proposals and responses. A different approach to multilateral bargaining was adopted in Winter (1994). Rather than a model in which players make full proposals concerning payoff allocations and respond to such proposals, a more descriptive feature of bargaining situations is sought by assuming that players submit only demands, i.e., players announce the share they request in return for cooperation. A coalition emerges when the underlying resources are sufficient to satisfy the demands of all members. I will describe here a simple version of the Winter ( 1994) model and some of the results that follow. Consider the order in which players move according to their name, i.e., player 1 followed by 2, etc. Each player i in his turn publicly announces a demand d; (which should be interpreted as a statement by player i of agreeing to be a member of any coalition provided that he is paid at least d; ). Before player i makes his demand, we check whether there is a compatible coalition among the i 1 players who already made their demands. A coalition S is said to be compatible (to the underlying game v) if S can satisfy the demands of all its members, i.e., L j E S dj :( v (S) . If compatible coalitions exist, then the largest one (in terms of membership) leaves the game and each of its members receives his demand. The game then proceeds with the set of remaining players. If no such coalition exists, then player i moves ahead and makes his demand. The game ends when all players have made their demands. Those players who are not part of a compatible coalition receive their individually rational payoff. Consider now a game that starts with a chance move that randomly selects an or der with a uniform probability distribution over all orders and then proceeds in ac-
-
Ch. 53: The Shapley Value
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cordance with the above protocol. We call this game the demand commitment game. Winter ( 1994) shows that the demand commitment game implements the Shapley value if the underlying game is strictly convex. THEOREM [Winter (1994)]. For strictly convex (cooperative) games, the demand com mitment game has a unique subgame peifect equilibrium, and each player's equilibrium payoff equals his Shapley value.
Winter (1994) also considers a protocol that requires a second round of bidding in the event that the first round ends without a grand coalition that is compatible. It can be shown that with small delay costs the Shapley value emerges not only as the expected equilibrium outcome for each player, but that the actual demands made by the players in the first round coincide with the Shapley value of the underlying game. Several other papers have followed the same approach in different contexts. Dasgupta and Chiu (1998) discuss a modified version of the Winter (1994) game, which allows for the implementation of the Shapley value in general games. Roughly, the idea is to allow outside transfers (or gifts) that will convexify a non-convex game. A balanced budget is guaranteed by a schedule of taxes dependent on the order of moves. Bag and Winter ( 1999) used a demand commitment-type mechanism to implement stable and ef ficient allocations in excludable public goods. Morelli (1999) modified Winter's ( 1994) model to describe legislative bargaining under various voting rules. Finally, Mutuswami and Winter (2000) used demand mechanisms of the same kind to study the formation of networks. They also noted that if the mechanism in Winter ( 1994) is amended to allow a compatible coalition to leave the game only when it is connected (i.e., only when it includes the last k players for some 1 "( k "( n ) , then the resulting game implements the Shapley value not only in the case of convex games but in all games. 9. Practice
While game theory is thought of as "descriptive" in its attempt to explain social phenom ena by means of formal modeling, cooperative game theory is primarily "prescriptive". It is not surprising that much of the literature on cooperative solution concepts finds its way not into economics journals but into journals of management science and opera tions research. Cooperative game theory does not set out to describe the way individu als behave. Rather, it recommends reasonable rules of allocation, or proposes indices to measure power. The prospect of using such a theory for practical applications is there fore quite attractive, the more so for its single-point solution and axiomatic foundation. In this section, I discuss two areas in which the Shapley value can be (and indeed has been) used as a practical tool: the measurement of voter power and cost allocation. 9. 1.
Measuring states' power in the U.S. presidential election
The procedure for electing a president in the United States consists of two stages. First, each state elects a group of representatives, or "Great Electors", who comprise the Elec-
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toral College. Second, the Electoral College elects the president by simple majority rule. It is assumed that each Great Elector votes for the candidate preferred by the majority of his/her state. Since the Electoral College of each state grows in proportion to its census count, a narrow majority in a densely populated state, like California, can affect an elec tion's outcome more than wide majorities in several scarcely populated states. Mann and Shapley ( 1 962) and Owen ( 1 975) measured the voting power of voters from different states, using the Shapley value together with the interesting notion of compound simple games. Let Mt , M2 , . . . , Mn be a sequence of n disjoint sets of players. Let Wt , w2 , . . . , Wn be a sequence of n simple games defined on the sets M1 , . . . , Mn respectively. And let v be another simple game defined on the set N = { 1 , 2, . . . , n}. We will refer to the players in N as districts. The compound game u = v[w1 , . . . , wn ] is a simple game defined on the set M = Mt U M2 U U Mn by u(S) = v ({ j l wJ (S n MJ) = 1 } ) . In words: We say that S wins in district j if S's members in that district form a winning coalition, i.e., if wi (S n MJ) = 1 . S is said to be winning in the game u if and only if the set of districts in which S is winning is itself a winning coalition in v. In the context of the presidential race, Mj is the set of voters in state j , wi is the simple majority game in state j, and v is the electoral college game. Specifically, the electoral college game can be written as the following weighted majority game [270; P l , . . , ps t ] , where 5 1 stands for the number of states, and Pi is the number of electors nominated by a state i (e.g., 45 for California and 3 for the least populated states and the District of Columbia). In general compound games, Owen has shown that the value of player i is the product of his value in the game within his district and the value of his district in the game v, i.e., ¢; (u) = ¢J (v)¢; (Wj). Since the districts' games are all symmetric simple majority games, the value of each player in the voting game in his state is simply 1 divided by the number of voters. To compute the value of the game v, Owen used the notion of a multilinear extension [see Owen ( 1 972)] . Overall, he found that the power of a voter in a more populated state is substantially greater than that of a voter in a less populated state. For example, California voters enjoy more than three times the power of Washington D.C. voters. Others have used the Shapley value (as well as other indices) to measure political power. Seidmann ( 1987) used it to compare the power of governments in Ireland follow ing elections in the early and mid 80s. He argued that a government's durability greatly depends on the distribution of power across opposition parties, which can be estimated by means of the Shapley-Shubik index. Carreras, Garda-Jurado, and Pacios ( 1993) used the Shapley and the Owen value to evaluate the power of each of the parties in all the Spanish regional parliaments. Fedzhora (2000) uses the Shapley-Shubik index to study the voting power of the 27 regions (oblasts) in the Ukraine in the run-off stage of the presidential elections between 1 994 and 1999. She also compares these indicators to the transfers that Ukrainian governments were making to the different regions. Another interesting application of the Shapley value to political science is due to Rapoport and Golan ( 1 985). Immediately after the election of the tenth Israeli parliament, 21 students of political science, 24 Knesset members, and 7 parliamentary correspondents were · · ·
.
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invited to assess the political power ratios of the 10 parties represented in the Knes set. These assessments were then compared with various power indices, including the Shapley value. The value provided the best fit for 3 1 % of the subjects, but the authors claimed the Banzhaf index performed better. 9.2.
Allocation of costs
The problem of allocating the cost of constructing or maintaining a public facility among its various users is of great practical importance. Young (1 994) (see Chap ter 34 in this Handbook) offers a comprehensive survey of the relation between game theory and cost allocation. An interesting allocation rule for such problems, which is closely related to the Shapley value, emerges from the "Airport Game" of Littlechild and Owen ( 1 973). Specifically, consider n planes of different size for which a runway needs to be built. Suppose that there are m types of planes and that the construction of a runway sufficient to service a type j plane is ci with CJ < c2 < < Cm . Let NJ be the number of planes of type j so that U NJ = N is the set of all planes which need to be serviced. A runway servicing a subgroup of planes S C N will have to be long enough to allow the servicing of the largest plane in S. This gives rise to the following natural cost-sharing game (in coalitional form) defined on the set of planes: c (S) = CJ(S) and c (0) = 0, where j (S) = max{j i S n N1 =f. 0}. Littlechild and Owen's ( 1 973) suggestion was to use the game c to determine the allocation of cost by applying the Shapley value on the game. Note that the game v can be decomposed into m unanimity games. Specifically, define Rk = Rk U Rk+ 1 U U Rm and consider the coalitional form games Vk with Vk (S) = 0 when S n Rk = 0 and vk(S) = q - Ck- 1 otherwise (we set co = 0). It is easy to verify that the sum of the games Vk is exactly the cost-sharing game, i.e., Vt (S) + + Vm (S) = c(S) for every coalition of planes S. The additivity of the Shapley value implies that the value of the game c is ¢ (c) = ¢ ( VJ ) + + ¢ ( Vm ). But each Vk is a unanimity game with ¢; (vk) = 0 for i E N \ Rk and ¢; (vk) = (ck - Ck - t )/ I Rk l for i E Rk. We therefore obtain that the Shapley value of the game c is given by ¢; (c) = (c2 - CJ) j I R t l + (c3 c2)/IR2 I + + (cj - CJ_J)JIRJ I for a plane of type j . The rule suggested by Littlechild and Owen (1973) has the following interesting in terpretation. First, all players share equally the cost of a runway for type 1 planes. Then all players who need a larger facility share the marginal extra cost, i.e., C2 - CJ . Next, all players who need yet a larger runway share equally the cost of upgrading to a run way large enough to service type 3 planes. We now continue in this manner until all the residual costs are allocated, which will ultimately allow for the acquisition of a runway long enough to service all planes. A version of the Airport game was studied by Fragnelli et al. ( 1 999). Their work was part of a research project funded by the European Commission with the aim of de termining cost allocation rules for the railway infrastructure in Europe. Fragnelli et al. realized that the original Airport game is ill-suited to their problem since the mainte nance cost of a railway infrastructure depends on the number of users. They constructed · · ·
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· · ·
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a new game which distinguishes construction costs (which do not depend on the number of users) from maintenance costs, and derived a simple formula for the Shapley value of the game. They also used real data concerning the maintenance of railway infrastruc tures to estimate the allocation of cost among different users. References Aumann, R.J., and J. Dreze ( 1975), "Cooperative games with coalition structures", International Journal of Game Theory 4:217-237. Aumann, R.J., and R.B. Myerson ( 1988), "Endogenous formation of links between players and of coalitions: An application of the Shapley value", in: A.E. Roth, ed., The Shapley Value (Cambridge University Press, Cambridge), 175-19 ! . Aumann, R.J., and L.S. Shapley (1974), Values of Non-Atomic Games (Princeton University Press, Prince ton). Bag, P.K., and E. Winter (1999), "Simple subscription mechanisms for the production of public goods", Journal of Economic Theory 87:72-97. Banzhaf, J.F. III ( 1965), "Weighted voting does not work: A mathematical analysis", Rutgers Law Review 19:3 1 7-343. Calvo, E., J. Lasaga and E. Winter (1996), "On the principle of balanced contributions", Mathematical Social Sciences 3 1 : 171-182. Carreras, F., I. Garda-Jurado and M.A. Pacios (1993), "Estudio coalicional de los parlamentos auton6micos espaiioles de regimen comun", Revista de Estudios Polfticos 82: 152-176. Chun, Y. ( 1989), "A new axiomatization of the Shapley value", Games and Economic Behavior 1 : 1 19-130. Dasgupta, A., and Y.S. Chiu (1998), "On implementation via demand commitment games", International Journal of Game Theory 27: 1 61-190. Derks, J., and H. Peters (1993), "A Shapley value for games with restricted coalitions", International Journal of Game Theory 21 :351-360. Dubey, P. ( 1975), "On the uniqueness of the Shapley value", International Journal of Game Theory 4: 13 1-139. Fedzhora, L. (2000), "Voting power in the Ukrainian presidential elections", M.A. Thesis (Economic Educa tion and Research Consortium, Kiev, Ukraine). Fragnelli, V. , I. Garda-Jurado, H. Norde, F. Patrone and S. Tijs ( 1999), "How to share railway infrastructure costs?", in: I. Garda-Jurado, F. Patrone and S. Tijs, eds., Game Practice: Contributions from Applied Game Theory (Kluwer, Dordrecht). Gul, F. (1989), "Bargaining foundations of Shapley value", Econometrica 57:81-95. Gul, F. (1999), "Efficiency and immediate agreement: A reply to Hart and Levy", Econometrica 67:913-918. Harsanyi, J.C. (1963), "A simplified bargaining model for the n-person cooperative game", International Economic Review 4 : 194-220. Harsanyi, J.C. (1985), "The Shapley value and the risk dominance solutions of two bargaining models for characteristic-function games", in: R.J. Aumann et al., eds., Essays in Game Theory and Mathematical Economics (Bibliographisches Institut, Mannheim), 43-68. Hart, S., and M. Kurz ( 1983), "On the endogenous formation of coalitions", Econometrica 5 1 : 1295-1313. Hart, S., and Z. Levy (1999), "Efficiency does not imply immediate agreement", Econometrica 67:909-91 2. Hart, S., and A. Mas-Colell (1989), "Potential, value and consistency", Econometrica 57:589-614. Hart, S., and A. Mas-Colell (1996), "Bargaining and value", Econometrica 64:357-380. Kalai, E., and D. Samet (1985), "Monotonic solutions to general cooperative games", Econometrica 53:307327. Littlechild, S.C., and G. Owen (1973), "A simple expression for the Shapley value in a special case", Man agement Science 20:99-107. Mann, I., and L.S. Shapley (1962), "The a-priori voting strength of the electoral college", American Political Science Review 72:70-79.
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Monderer, D., and D. Samet (2002), "Variations on the Shapley value", in: R.J. Aumann and S. Hart, eds., Handbook of Game Theory, Vol. 3 (North-Holland, Amsterdam) Chapter 54, 2055-2076. Morelli, M. ( 1999), "Demand competition and policy compromise in legislative bargaining", American Polit ical Science Review 93:809-820. Mutuswami, S., and E. Winter (2000), "Subscription mechanisms for network formation", Mimeo (The Center for Rationality and Interactive Decision-Making, The Hebrew University of Jerusalem). Myerson, R.B. (1977), "Graphs and cooperation in games", Mathematics of Operations Research 2:225-229. Myerson, R.B. (1980), "Conference structures and fair allocation rules", International Journal of Game Theory 9:169-182. Nash, J. (1950), "The bargaining problem", Econometrica 18:155-162. Neyman, A. (1989), "Uniqueness of the Shapley value", Games and Economic Behavior 1 : 1 16-1 18. Owen, G. (1972), "Multilinear extensions of games", Management Science 18:64-79. Owen, G. ( 1975), "Evaluation of a presidential election game", American Political Science Review 69:947953. Owen, G. (1977), "Values of games with a priori unions", in: R. Henn and 0. Moeschlin, eds., Essays in Mathematical Economics and Game Theory (Springer-Verlag, Berlin), 76-88. Owen, G., and E. Winter ( 1992), "The multilinear extension and the coalition value", Games and Economic Behavior 4:582-587. Perez-Castrillo, D., and D. Wettstein ( 1999), "Bidding for the surplus: A non-cooperative approach to the Shapley value", DP 99-7 (Ben-Gurion University, Monaster Center for Economic Research). Rapoport, A., and E. Golan ( 1985), "Assessment of political power in the Israeli Knesset", American Political Science Review 79:673-692. Roth, A.E. (1977), "The Shapley value as a von Neumann-Morgenstern utility", Econometrica 45:657-664. Seidmann, D. (1987), "The distribution of power in Da' il E' ireann", The Economic and Social Review 19:6168. Shapley, L.S. ( 1953), "A value for n-person games", in: H.W. Kuhn and A.W. Tucker, eds., Contributions to the Theory of Games II (Annals of Mathematics Studies 28) (Princeton University Press, Princeton), 307-3 17. Shapley, L.S. (1977), "A comparison of power indices and a nonsymmetric generalization", P-5872 (The Rand Corporation, Santa Monica, CA). Shapley, L.S., and M. Shubik ( 1954), "A method for evaluating the distribution of power in a committee system", American Political Science Review 48:787-792. Sobolev, A.I. (1975), "The characterization of optimality principles in cooperative games by functional equa tions", Mathematical Methods in Social Sciences 6:94-151 (in Russian). Straffin, P.D. (1977), "Homogeneity, independence and power indices", Public Choice 30: 107-1 18. Straffin, P.D. (1994), "Power and stability in politics", in: R.J. Aumann and S. Hart, eds., Handbook of Game Theory, Vol. 2 (North-Holland, Amsterdam) Chapter 32 , 1 127- 1 1 52. Von Neumann, J., and 0. Morgenstern (1944), Theory of Games and Economic Behavior (Princeton Univer sity Press, Princeton). Weber, R. (1988), "Probabilistic values for games", in: A.E. Roth, ed., The Shapley Value (Cambridge Uni versity Press, Cambridge), 101-120. Weber, R. (1994), "Games in coalitional form", in: R.J. Aumann and S. Hart, eds., Handbook of Game Theory, Vol. 2 (North-Holland, Amsterdam) Chapter 36, 1285-1304. Winter, E. (1989), "A value for games with level structures", International Journal of Game Theory 18:227242. Winter, E. (1991), "A solution for non-transferable utility games with coalition structure", International Jour nal of Game Theory 20:53-63. Winter, E. (1992), "The consistency and the potential for values with games with coalition structure", Games and Economic Behavior 4: 1 32-144. Winter, E. ( 1994), "The demand commitment bargaining and snowballing cooperation", Economic Theory 4:255-273.
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Young, H.P. (1985), "Monotonic solutions of cooperative games", International Journal of Game Theory 14:65-72. Young, H.P. (1994), "Cost allocation", in: R.J. Aumann and S. Hart, eds., Handbook of Game Theory, Vol. 2 (North-Holland, Amsterdam) Chapter 34, 1 193-1235.
Chapter 54 VARIATIONS ON THE SHAP LEY VALUE DOV MONDERER
The Technion, Haifa, Israel DOV SAMET
Tel Aviv University, Tel Aviv, Israel Contents 1 . Introduction Preliminaries 3. Probabilistic values 4. Efficient probabilistic values (quasivalues) 5. Weighted values 6. Symmetric probabilistic values (semivalues) 7. Indices of power 8. Values with a social structure References
2,
Handbook of Game Theory, Volume 3, Edited by R.J Aumann and S. Hart © 2002 Elsevier Science B. V. All rights reserved
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D. Monderer and D. Samet
Abstract
This survey captures the main contributions in the area described by the title that were published up to 1997. (Unfortunately, it does not capture all of them.) The variations that are the subject of this chapter are those axiomatically characterized solutions which are obtained by varying either the set of axioms that define the Shapley value, or the domain over which this value is defined, or both. In the first category, we deal mainly with probabilistic values. These are solutions that preserve one of the essential features of the Shapley value, namely, that they are given, for each player, by some averaging of the player's marginal contributions to coalitions, where the probabilistic weights depend on the coalitions only and not on the game. The Shapley value is the unique probabilistic value that is efficient and symmetric. We characterize and discuss two families of solutions: quasivalues, which are efficient prob abilistic values, and sernivalues, which are symmetric probabilistic values. In the second category, we deal with solutions that generalize the Shapley value by changing the domain over which the solution is defined. In such generalizations the so lution is defined on pairs, consisting of a game and some structure on the set of players. The Shapley value is a special case of such a generalization in the sense that it coincides with the solution on the restricted domain in which the second argument is fixed to be the "trivial" one. Under this category we survey mostly solutions in which the structure is a partition of the set of the players, and a solution in which the structure is a graph, the vertices of which are the players.
Keywords
Shapley value, quasivalue, sernivalue, non-symmetric values, probabilistic values, weighted values
JEL classification: C7 1 , D46, D63, D72, D74
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1. Introduction
The variations that are the subject of this discussion are those axiomatically character ized solutions that are obtained by varying the set of axioms which define the Shapley value, or the domain over which this value is defined, or both. We first capture axiomatically, in Section 3, those solutions that preserve one of the essential features of the Shapley value, namely, that they are given, for each player, by some averaging of the player's marginal contributions to coalitions, where the proba bilistic weights depend on the coalitions only and not on the game. We call the family of all such solutions probabilistic values. The Shapley value is the unique probabilistic value that is efficient and symmetric. In the following sections we characterize and dis cuss two families of solutions: quasivalues, which are efficient probabilistic values, and semivalues, which are symmetric probabilistic values. In Section 4 we discuss quasivalues. These solutions are related to the descriptions of the Shapley value as a random-order value: it is the expected marginal contributions of the players when they are randomly ordered. Indeed a solution is a quasivalue if and only if it is a random-order value. The Shapley value is the special random-order value in which the random orders are uniformly drawn. Quasivalues can be equivalently described by choosing at random the "arrival" times of the players, rather than their order, and taking the expected marginal contribution of a player to the players who arrived before him. In particular, the Shapley value is obtained when the arrival times are independent and each is uniformly distributed in the unit in terval. When arrival times are independent, the quasivalue has another representation as a path value: the value can be computed by integrating the derivative of the multilinear extension of the game along a certain path. In Section 5 we characterize and discuss a family of quasivalues - the weighted Shap ley values - which was introduced by Shapley alongside the Shapley value. Semivalues are discussed in Section 6. The symmetry axiom, which plays a central role in the characterization of semivalues, is reflected in the probabilities that define the value: they depend only on the size of coalitions. A semivalue does not, of course, have to be efficient. We show that the specific deviation of a semivalue from efficiency defines it uniquely. That is, if the magnitude of the deviation from efficiency is the same for two semivalues in each game, then they coincide. A probabilistic value defined on the set of all games is necessarily additive. When the set of games is restricted to simple games (that is, games in which the worth of a coalition is either 0 or 1), then additivity cannot be applied. Yet, all the generalizations discussed so far can be axiomatically defined for this class of games when the transfer axiom replaces that of additivity. This is done in Section 7. Section 8 deals with solutions that generalize the Shapley value by changing the do main over which the solution is defined. In such generalizations the solution is defined on pairs, consisting of a game and some structure on the set of players. The Shapley value is a special case of such a generalization in the sense that it coincides with the so lution on the restricted domain in which the second argument is fixed to be the "trivial"
D. Monderer and D. Samet
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one. Under this category we survey mostly solutions in which the structure is a partition of the set of the players, and a solution in which the structure is a graph the vertices of which are the players. 2. Preliminaries
Let N be a finite set with n elements, n � l . Elements of N are called players. Any subset of N is called a coalition. For a coalition S, we denote N \ S by sc . We denote by C = C(N) the set of all coalitions, and by Co = Co(N) the set of all nonempty coalitions. A game on N is a set function v : C -+ R, where R denotes the set of real numbers, with v (0) = 0. We will write v(S U i) for v(S U {i }), and v(S \ i) for v(S \ {i }). A game v is additive if for every pair of disjoint coalitions S, T E C, v(S U T) = v(S) + v (T ) , and i t is superadditive if for each pair of such coalitions, v ( S U T) � v (S) + v ( T ) . I t is monotone if v (S) � v(T) for every S <; T . A game v is simple if v (S) E {0, 1 } for every
S E C.
Player i is a null player in v if v(S U i) - v (S) = 0 for every S <; N \ i . A coalition M is a carrier of v if Me consists of null players. Hence, M is a carrier for v if and only if v(S n M) = v(S) for all S <; N. Player i is a dummy player in v if there exists an a E R such that v(S U i) - v (S) = a for every S <; N \ i . The set of all games is denoted G = G(N) and the set of all additive games is de noted A = A(N). We naturally identify an additive game v E A with the vector x E R N defined by x; = v (i ) for every i E N. For each such x E R N and for every S E C we
denote x (S) = L i ES Xi . A permutation of N is a one-to-one function that maps N onto N. The set of all permutations on N is denoted by 8 = G(N). The set of all one-to-one functions r : N -+ { 1 , . . . , n} is denoted by OR = OR(N). Each r E OR is identified with a com plete order on N. In this order, i comes before j if r (i ) < r(j). For a game v and a permutation e we define e * v E G by e * v (S) = v (e (S)) for all s E C. Let H <; G. A solution on H is a function 1/r : H -+ A. We will use the terms operator and solution interchangeably. The domain of a solution is G whenever it is not explicitly specified otherwise. It is useful to single out a list of possible properties of solutions (axioms) that have been discussed in the literature. (A. l ) 1/r satisfies the additivity axiom or is additive if 1/r (u + v) = 1/r(u) + 1/r (v) for every u, v E G. (A.2) 1/r i s homogeneous if lfr (av) = a lfr (v) for every v E G and a E R. (A.3) 1/r satisfies the linearity axiom, or i s linear, if it i s additive and homogeneous. (A.4) 1/r satisfies the projection axiom, or is a projection, if 1/r(tL) = fL for every
fL E A.
(A.5) 1/r satisfies the null player axiom if lfrv(i ) = 0 for every v E G and every null player i in v . (A.6) 1/r satisfies the dummy player axiom i f 1/rv(i) = a for v E G and for every dummy player i such that v(S U i) - v (S) = a for every S <; N \ i .
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(A.7) 1jr satisfies the positivity axiom, or 1jr is positive, if ljrv(i) � 0 for every i whenever v is a monotone game. (A.8) 1jr satisfies the Milnor axiom, or is a Milnor operator, if for every v and i E N, min
Sc;; N \i
( v(S U i) - v(S) ) � ljrv(i) � Sc;; max ( v(S U i) - v(S) ) . \i N
(A.9) 1jr satisfies the symmetry axiom or is symmetric if 1/r(B*v) = 8*1/rv for every v E G and every permutation e (A. I O) 1jr satisfies the efficiency axiom or is efficient if ljrv(N) = v(N) for every 0
v E G.
(A. l l) 1jr satisfies the carrier axiom if ljrv(M) = v(M) for every v E G and every carrier M of v. We state the above axioms for solutions defined on G, but will sometimes use them for solutions defined on strict subsets of G. In such a case, an axiom is to be understood as holding whenever possible. Thus, for example, if 1jr is a solution on H c G, the projection axiom should be read as 1/rtt = f.t whenever f.t E H n A . 3. Probabilistic values
For a finite set M we denote by L1(M) the set of all probability measures on M. That is, p E L1(M) if p : M ---+ [0, 1] and Lm E M p(m) = 1 . We denote by Ep the expectation operator with respect to p. ; ; The set of all subsets o f N \ i i s denoted b y C = C (N). For each game v, w e denote ; ; by D v the set function defined on C i by D v(S) = v(S U i) - v(S). We call D ; v(S) the marginal contribution of i to S. A solution 1jr is called a probabilistic value, if for each player i there exists a probabil ity p i E L1 (Ci) , such that ljrv(i) is the expected marginal contribution of i with respect to p i . Namely,
ljrv(i) = EP (D ; v ) ;
=
L p; (S) ( v(S U i) - v(S) ) .
SEC;
(3. 1 )
The Shapley value, which inspired this definition, is the probabilistic value defined n 1 by the probabilities p i defined by p i (S) = 1/ (n ( � )) , where s denotes the number of elements in S. We denote the Shapley value by q; = q; N . It can be easily verified that the probability measures p i in (3 . 1) are uniquely determined by 1jr . Probabilistic values can be axiomatically characterized as follows. THEOREM 1 . Let 1jr be a solution. Then the following three assertions are equivalent: (1) 1jr is a probabilistic value. (2) 1jr is a linear positive operator that satisfies the dummy axiom. (3) 1jr is a linear Milnor operator.
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Weber ( 1 988) proved the equivalence of ( 1 ) and (2). Monderer ( 1 988, 1 990) proved the equivalence of (2) and (3). Shapley ( 1 953a) proved that the Shapley value is the unique solution that satisfies the additivity, symmetry and carrier axioms. 1 The following is one of many other charac terizations of the Shapley value. We single out this particular characterization [which is proved in Weber ( 1 988)] because it can serve as the basis for a unified approach to analyzing the variations on the Shapley value.
The Shapley value is the unique solution which is a symmetric and effi cient probabilistic value.
THEOREM 2 .
The main variations on the Shapley value retain the condition of being a probabilistic value (i.e., that a player's value depends linearly and positively on his marginal con tributions), while relaxing one of the other two conditions in Theorem 2. We call an efficient probabilistic value a quasivalue, and a symmetric probabilistic value a semi value. Quasivalues are discussed in Sections 4 and 5, semivalues in Sections 6 and 7. REMARKS .
( 1) As probabilistic values are characterized by axioms that do not mix values of dif ferent players, they can be defined and characterized for each player separately. Such an approach was taken by Weber ( 1 988). (2) Weber ( 1 988) proved that Theorem 1 holds for solutions defined on the class of monotone games. One can deduce from Monderer (1988) and from a remark in Weber that the equivalence of ( 1 ) and (3) in Theorem 1 continues to hold for solutions defined on the class of superadditive games.
4. Efficient probabilistic values (quasivalues) We begin with an important characterization of quasivalues. For each T the operator cp r by
E OR we define
cp r v(i) = v({j E N: r (j ) :( r (i) } ) - v( {j E N: r (j ) < r (i) }) . If we think of T as the order in which players enter the "room" where the game takes place, then cp r (i) is the marginal contribution of i to the coalition of players that pre b ceded him. For every b E i1 (0R) we define a solution cp by
cpb v (i )
=
L cp r v(i)b(r). T EOR
1 He proved this result for solutions defined on the set of superadditive games, with finite support, on a universal set of players; but his proof, as is well known, can also be used to characterize the Shapley value on the space G(N) of all games on N.
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Thus, (l v (i) is the expected marginal contribution of i, where the order of entering the room is selected according to b. A solution 1/J is called a random-order value if there exists b E LJ.(OR) such that 1/1 = cpb . It is easy to verify that every random-order value is a quasivalue. Indeed, given b E LJ.(OR) we can define for each i E N and S E C ; ,
pi (S) = b ({ r E OR: r (j) < r (i) iff j E S}).
(4. 1)
If we insert the pi 's from (4. 1) into (3 . 1 ), the resulting probabilistic value 1/J coincides with cp b . The following theorem of Weber ( 1988) shows that the converse is also true. THEOREM 3 . A solution is a quasivalue if and only if it is a random-order value. Note that the Shapley value is a random-order value with b(r) = l j n ! for every r E OR.2 It can be easily verified that for every b E LJ.(OR) there exist n random variables (z; );E N that are jointly distributed in the cube [0, l ] N with a density function and there fore with absolutely continuous marginal distribution functions (a; );E N , such that for every r E OR
b(r) = prob(Zcl (i ) < Zr-l (j) for every 1 :::;; i < j :::;; n).
(4.2)
This enables us to interpret cp b in terms of random arrival times, as follows. We think of the value of the random variable z; as the time of arrival of player i in the "room". The marginal contribution of i, when he arrives at time t, is his marginal contribution to the set of other players who arrived up to time t. Define
N(t) = {j E N: ZJ :::;; t}.
(4.3)
Then N(t) is a random coalition in C and it can be easily verified that
cpb v(i)
=
11 E (v (N(t) U i ) - v (N(t) \ i ) lz;
=
t) da; (t),
(4.4)
where the integral in (4.4) is the Riemann-Stieltjes integral. Thus, we have shown that a solution 1/J is a random-order value, if and only if there exist n random variables 2 Observe that for sufficiently large n the affine map b ---+
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D. Monderer and D. Samet
(z; ) ; E N , which are jointly distributed in the cube [0, 1] N , and have a density function, such that 1{fv(i) is given by the right-hand side of (4.4), where (a; ); E N are the marginal
distribution functions. Owen (1972) defined the multilinear extension of a game v. Each coalition S can be identified with an extreme point, e s, of the cube [0, 1] N , where es is the characteristic function of S. Therefore, each game v can be considered as a function on the extreme points of the cube and hence it can be uniquely extended to a multilinear function on the cube, which is denoted by iJ . Owen showed that if the random variables that define b via (4.2) are independent, then for every i : (4.5) In particular, the Shapley value is determined by n i.i.d. random variables that are uni formly distributed in [0, 1 ] . Therefore,
q;v(i) =
lo1 - (te) av
0
ax;
dt ,
where e = eN . The presentation of the solution q; b , in (4.5), gives rise to another natural solution. A solution 1(;' is a path value if there exists an absolutely continuous nondecreasing path a : [0, 1 ] --+ [0, 1] N with a(O) = 0 and a (1) = e such that for every game v and every player i , 1{fv(i) is defined by the right-hand side of (4.5). Alternatively, 1(;' is a path value if there exists b E /'.. (OR) such that 1(;' = q; b , and there exist n independent random variables that are distributed in [0, 1 ] with absolutely continuous distribution functions such that (4.2) holds. It can be verified that not every quasivalue is a path value. REMARKS .
( 1 ) Gilboa and Monderer (199 1 ) discussed quasivalues defined on subspaces of games. They gave necessary and sufficient conditions that ensure that such qua sivalues can be extended to quasivalues on G. (2) Monderer (1 992) used the concept of quasivalue in order to provide a game theoretic approach to the stochastic choice problem. He used Theorem 3 to con firm a conjecture of Block and Marschak ( 1960), which had already been proved before by Falmagne ( 1 978). The relationship between the stochastic choice prob lem and quasivalues was further analyzed by Gilboa and Monderer ( 1 992). 5. Weighted values
Weighted values comprise a special class of quasivalues, and more specifically of path values. We describe two forms of such values. Both were introduced by Shapley (1953a,
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Variations on the Shapley Value
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1 953b ), alongside the standard Shapley value, and were axiomatized by Kalai and Samet (1987). Let w E R'j_+ (that is, w; > 0 for every i E N). Recall that a unanimity game us, for S E Co, is defined by us (T) = 1 for T 2 S and us(T) = 0 otherwise, and that the set of unanimity games, for all S E Co, is a basis for the linear space G of games. The positively weighted value cpw (which is called the w-value) is the unique linear operator satisfying for i E S for i fJ_ S for each unanimity game us. If w; = * for every i E N, then cp w is the Shapley value. Owen (1972) has shown that positively weighted values are path values, the path a being defined by a;
(t) = twi
for every t E
[0, 1 ] .
(5. 1)
Note that the mapping w --+ cp w is homogeneous of degree zero, that is, cp J..w = cp w for every 'A > 0. Therefore, we can restrict our attention to vectors w in the relative interior of !J.(N), which we call weight vectors. We turn now to an axiomatic charac terization of positively weighted values. A solution 1fr is strictly positive if for every monotone game v without null players, 1frv(i) > 0 for every i E N. It is easily verified that a probabilistic value 1fr is strictly positive, if and only if p i (S) > 0 for every i and ; every S E C , where {p i : i E N} is the collection of probability measures that define 1fr via (3 . 1 ). A coalition S is said to be a coalition ofpartners or a p-type coalition, in the game v, if, for every T c S and each R s; sc , v(R U T) = v(R) . A solution 1fr satisfies the partnership axiom if, whenever S is a p-type coalition,
1fr v(i)
=
1fr ( 1frv (S) u s ) (i)
for every i
E S.
The proof of the following theorem is given in Kalai and Samet ( 1987): THEOREM 4. A solution is a positively weighted value, if and only if it is a strictly positive probabilistic value that satisfies the partnership axiom. The set of positively weighted values is not closed in the space of solutions. Obvi ously every solution which is the limit of positively weighted values is a probabilistic value that satisfies the partnership axiom. We turn now to characterize such solutions by s introducing the concept of weight systems. A weight system is a vector w = (w ) S ECo in x S ECa !J. (S) that satisfies (5.2)
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whenever i E S s; T and w T (S) > 0. Note that any weight vector co can be identified with the weight system w, where w[ = wC�l for every i E S E Co. The set of all weight systems is denoted by W. For w E W the weighted value q;w (which is called the w value) is defined as the unique linear operator which is defined for every unanimity game us by
{
q;wus(i) = w[ ' for i E S, 0, for i ¢. S.
(5.3)
The w-value of a unanimity game in (5.3) can be interpreted as follows. Every weight system w defines an ordered partition (Nt , . . . , Nk) of N (where k depends on w). The players in N1 are those players i for whom wt > 0. The coalition Nh , for 1 < h :( k, ? iN consists of all the players i in cUi'�} Nz) c ' for which w;u � ,)c 0. For S E Co, let h = h (S) be the first index for which S n Nh =/= 0. Then q;w allocates among the players in S n Nh their share, us(N) 1 , proportional to their weights in w N" while q;w us (i ) = 0 for all other players. To show that every weighted value is a path value, Kalai and Samet (1987) defined independent random variables z; , i E N with pairwise disjoint supports, such that if ih E Nh , for h = 1 , , k, then prob(z;1 > z;2 > > z;k ) = 1 . The distribution of each z; is defined on its support in [0, 1] by a linear transformation of the distribution given
=
· · ·
. . .
W;
I =!
1
T HEOREM 5. A solution is a weighted value ifand only if it is a quasivalue that satisfies the partnership axiom.
Hart and Mas-Colell ( 1989) presented two new approaches to value theory: the po tential approach and the consistency approach. They showed that these approaches are naturally extended to the case of positively weighted values. Here we present their theory using a slightly different notation. For a game v and a coalition S we denote the restriction of v to S by v5. That is, v5 E G(S) and v5 (T) v(T) for every T E C(S) . Let AG = Usc N G(S). A function P : AG --+ R is a co-potentialfunction, for a weight vector co E L1 (N) (that is, co; > 0 for every i E N), if for every v E G =
P (v0) = 0 and l_)P (v5) - P (v5 \i )) = v(S), for every S =/= 0. i ES
(5.4)
Note that if (5.4) holds, then it also holds for every v E G(T) for S s; T C N. Hart and Mas-Colell proved that there exists a unique co-potential function, denoted by P and called the co-potential function of G. It satisfies w,
lo l
Pw ( Vs ) = v(a(t)es) dt t 0
,
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2065
N
where a(t) (i) = twi for every i E N, and where for x , y E R , xy is the vector defined by (xy); = x;y; for every i E N . The e-potential function, Pe , is called the potential function (or simply the potential) of G . Note that Pw v = E(t J01 v (N(t ) ) dt) , where N (t) is given by (4.3) with the random variables (z; );EN defined by (5 . 1). Hart and Mas-Colell also proved: THEOREM 6 .
Let w E Ll (N) be a weight vector. Thenfor every game v and every i E N,
We now present the consistency property of weighted values. A function 1j; : AG ---+ UsEe A (S) is called an extended solution if 1j;v5 E A (S) for every S E C. Thus, an extended solution is a collection of solutions 1j;5 on G (S) such that 1j;5 v5 = 1j;v5 for every coalition S. Let 1/1 be an extended solution and let T s; S. We define the reduced game vf·o/l E G(T) as follows:
v f·o/\F) = v(F U (S \ T)) - L 1/f v FU ( S\ T ) (i ) , for all F s; T . i E S\ T An extended solution 1j; is consistent if for every game v E G, coalition S, and coali
tion
T s; S,
'''vT(S ,
'I'
10r every J. E
+
T.
Hart and Mas-Colell prove that for a fixed weight vector w, the w-weighted Shapley value, regarded in a natural way as an extended solution (that is, (({!w)S = (j!ws , where w5 is the restriction of w to S), is consistent. Moreover, they proved that if an extended solution 1j; is consistent and coincides with the w-weighted extended value on two person games, then it is the w-weighted extended value on AG. They further used the consistency axiom to give an axiomatic characterization for positively weighted values. THEOREM 7. An extended solution is a w-value,for some weight vector w, ifand only if it is consistent, andfor two-person games it is efficient, TU-invariant3, and monotonic.
We end this discussion with a result concerning the weighted values of a game as a set-valued solution. Whatever the meaning of the weight vector w, it reflects certain properties or information about the players, not derived from the game, but rather given exogenously. If we lack this information, we can consider the less informative set-valued solution W(v) = {({!wv: w E W}. Let C(v) denote the core of v and let AC(v) be the anti-core of v, which consists of all x E A such that x (S) :( v(S) for every S and x (N) = v(N). Monderer, Samet, and Shapley (1992) proved:
3
That is ,
1/f (av [i , j) + fL{i,j))
=
mjrv{i,j) + fL{i,j), for every i i= j, a > 0 and fL E A.
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THEOREM 8 .
For every game v,
C(v) U AC(v) <; W ( v) . That there is such a general relationship between the core and values is somewhat surprising in light of the difference in concept behind these solutions. Theorem 8 gen eralizes a theorem of Weber ( 1988), who proved that C(v) <; Q(v), where Q(v) is the Weber set, i.e., the set of all lfr(v) where 1/r ranges over all quasivalues. REMARKS .
( 1 ) Monderer, Samet and Shapley ( 1 992) discussed geometrical and topological properties of the set of weight systems W (which is homeomorphic to the set of weighted values, because the map w -+ cp w is one-to-one). W can easily be seen to be homeomorphic to the set of conditional systems that was discussed in the literature on non-cooperative game theory [e.g., see Myerson ( 1 986) and McLennan ( 1 989a, 1 989b)]. The main theorem of McLennan ( 1 989b) states that this set is homeomorphic to a ball. Monderer, Samet and Shapley (1992) proved that for a strictly convex game v, W is mapped homeomorphically onto the core of v by the mapping w -+ cp w v. This gives a new proof for McLennan's theorem. Moreover, the well-studied geometry of the core of strictly convex games [see Shapley (197 1 )] is reflected in the structure of W. (2) Shapley ( 1 9 8 1 ) proposed a family of weighted cost allocation schemes which, when translated into the language of cooperative game theory, yield the dual weighted values. The dual game, v * , of a game v is defined by: v* (S) = v(N) v(Sc) . For w E W, the dual w-weighted value is defined by: cplj!v = cp w v*. Shapley gave an axiomatization for solutions 1/r : G x W - + A of the form (v, w) -+ cp':;v. Kalai and Samet (1987) axiomatized the family of dual-weighted values, explored the relationship between weighted and dual-weighted values, and used this relationship in order to give an axiomatization of the Shapley value that does not use the symmetry axiom. Dual-weighted values are also discussed in Monderer, Samet and Shapley (1992), where it is shown that a theorem analo gous to Theorem 8 can be stated for these solutions.
6. Symmetric probabilistic values (semivalues)
As semivalues are not efficient (except for the Shapley value), they cannot be considered as mechanisms for benefit or cost sharing. However, they can be interpreted as indices of power. The structure of power is usually described by a simple game, and therefore it is of interest to study semivalues as solutions defined on such games. But before doing so, we first analyze semivalues on all of G. Recall that the probabilities (p i ) i EN, where p i E L1(C i ) , by which a probabilistic value 1/r can be expressed, as in (3 . 1 ) , are uniquely determined by 1/r . If in addition 1/r is
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symmetric, i.e., it is a semivalue, then it can easily be seen that p i (S) depends only on the number of elements of S. That is, there exists a function .B : {0, 1 , . . . , n - 1 } -+ [0, 1] such that for every i,
p1 (S) = f3(s),
(6. 1)
where s denotes the number of elements in S. Obviously,
( n 1) L f3 (s) � = 1 .
n -l
(6.2)
s=O
Thus, we get the following characterization of semivalues given in Dubey, Neyman and Weber (198 1): THEOREM
9. A solution 1/ f is a semivalue if and only if there exists a function
f3 : {0, 1 , . . . , n - 1 } -+ [0, 1], that satisfies (6.2) such that 1/f = cp/3, where for every game v and player i, cpf3 v(i) = L f3(s) ( v( S U i) - v(S) ) , SEC;
i E N.
(6.3)
Moreover, the correspondence f3 -+ cp/3 is one-to-one. 1 Note that the Shapley value is cp/3 for f3(s) = n 1 (" � ) Banzhaf value4 is the semivalue defined by f3(s) = 2 �1 "
for every
0 ::::;; s ::::;; n - 1 . The
•
By removing the efficiency axiom from the list of axioms defining the Shapley value, we obtain the family of semivalues. Thus, for a semivalue 1/f , 1/ f v(N) is not necessarily v(N). It is interesting, then, to note that it is precisely what 1/f gives to the grand coali tion N, in the various games, which determines 1/f . This is what we state and prove next. THEOREM
1 0. Let 1/ft and 1/f2 be two semivalues5 that satisfy for each game v,
1/J't v(N) = 1/f2v(N).
(6.4)
Then 1/ft = 1/f2 See Section 7. This theorem can be proved for more general solutions, namely, for non-positive semivalues which are linear, symmetric solutions that satisfy the dummy axiom. Roth (1977) proved a special case of this stronger version of Theorem 10. He showed that the Banzhaf value is the only non-positive semivalue 1j.r that satisfies lj.rv(N ) cpbz v(N ) for every game v, where cpbz is the Banzhaf value. 4
5
=
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PROOF. Let S be the linear space of operators spanned by the set of semivalues. By Theorem 9, S is n-dimensional. For every game v we define a linear functional fv : S � R , by fv (lfr) = 1/fv(N) . Fix an order r on N, and for k = 1 , . . . , n, let Vk be , fvn are the unanimity game on { r- 1 (1), . . . , r-1 (k) } . It is easily verified that fv1 , linearly independent. Therefore, if 1/f E S and fvk ( 1/f) = 0 for k = 1 , . . . , n, then 1/f = 0. Thus, if 1fr 1 and 1/f2 satisfy (6.4), then 1fr 1 - 1f;'2 = 0. D • • •
Dubey, Neyman and Weber (198 1) characterized semivalues defined on games with finite support on an infinite universe of players. Let U be an infinite set of players. Denote by G 1 = G 1 ( U) the space of all games on U with a finite carrier, and by A f = A f ( U) the space of all additive games in G 1 . A solution, in this context, is a function 1/f : G 1 � A 1 . The symmetry of a solution is defined as before, with slight and obvious changes. Probabilistic solutions are defined as follows. For every finite subset of players M and for every v E G (M), denote by vM the natural extension of v to C(U), that is, VM (T) = v(T n M) for T s; U . Every solution 1/f on Gt defines a solution 1/f M on G(M) by 1/f M v(i) = 1/fvM (i) for every v E G(M) and every i E M. We say that 1/f is a probabilistic value on G( U) if for every finite subset of players, M, 1/f M is a probabilistic value. Defining a semivalue as a probabilistic and symmetric solution, note that 1/f is a semivalue if and only if 1/f M is a semivalue for every finite coalition M.
1 1 . Let U be an infinite set ofplayers and let 1/f be a solution on G f. Then 1/f is a semivalue if and only if there exists a Borel probability measure � on [0, 1] such that for every finite coalition N with n elements, and for every v E G 1 with carrier N, 1/fv = 1/f�; v, where, THEOREM
1/f�; v(i) = L ,B" (s) ( v(S U i) - v(S) ) , S � N \i
i E N,
with (6.5)
Moreover, the correspondence � � 1/ft; is one-to-one. Note that (6.5) can be rewritten in terms of random independent arrival times like (4.4),
1/fv(i) =
fo1 E (v (N(t) U i ) - v (N(t) \ i ) lzi = t)
d�(t),
where N(t) is the random set defined in (4.5) with respect to n i.i.d. random variables z1, . , Zn , uniformly distributed on [0, 1]. Observe, however, that unlike (4.4) the inte gration here is with respect to � and not Zi . . .
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REMARKS . ( 1 ) Dubey, Neyman and Weber ( 1 9 8 1 ) discussed continuity properties of semivalues on G ! · For every v E G f the variation norm of v is: ll v /1 = inf{u (N) + w (N) } , where the supremum is over all monotone games u , w E G f such that u - w = v. With this norm both G t and A f are normed spaces. The continuity of semivalues is defined with respect to this norm. It is proved that a semivalue 1h is continuous if and only if � is absolutely continuous with respect to the Lebesgue measure m on [0, 1], and its Radon-Nikodym derivative with respect to m is bounded, when considered as a member of L 00 (0, 1 ) , that is, I I i/!, lloo < oo. Moreover, the map ping g --+ 1/f� (g l , where � (g) (A) = fA g dm , is an isometry in the natural sense. (2) Monderer ( 1 988) discussed semivalues defined on symmetric subspaces of G. He proved that every such semivalue can be extended (not necessarily in a unique way) to a semivalue on G . Note that if 1/f is a semivalue and Gl/r is the subspace of games v for which 1/f v (N) = v(N) , then the restriction of 1jf to any symmetric subspace of Gl/r is a Shapley value on this subspace. Monderer's theorem proves that every Shapley value on a symmetric subspace of games is obtained in this manner.
7.
Indices of power
We denote the class of simple games by SG. A coalition S is a winning coalition in a simple game v if v (S) 1 , and it is a losing coalition if v(S) = 0. Of special interest are monotone simple games. In such games, a superset of a winning coalition is a winning coalition, and a subset of a losing coalition is a losing one. 6 This property reflects the natural idea, especially for voting games, that by adding members to a coalition it be comes stronger. The set of monotone simple games is denoted by MSG and MSG \ {0} is denoted by MSG+ . The set of superadditive simple games is denoted by SSG, and sse+ is SSG \ {O} . A solution defined on a subset of simple games is called an index ofpower. We refer the reader to Dubey and Shapley (1979), and Owen (1982) for a partial reference to articles about the use of indices of power in various areas, such as political science, law and electrical engineering. Other than the linearity axiom, all axioms in Section 2 are applicable (and make sense with respect) to indices of power. We now present the transfer axiom, introduced by Dubey (1975), 7 which in some sense substitutes for the linearity axiom for indices of power. For any two games v and w we define the games ( v v w) and ( v 1\ w) by: ( v v w) (S) = max(v (S) , w(S)) and (v 1\ w) (S) = min(v(S) , w (S)) . Note that both games are simple whenever v and w are simple games. =
6 Many authors refer to a game v E SG as a 0-- 1 game, and reserve the notion "simple game" for what we called a monotone simple game, and some of those authors even exclude the game v = 0 from the set of simple games. 7 The name was suggested by Weber (1988).
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(A. l 2) A solution 1j; defined on a set M of simple games satisfies the transfer axiom if for every v, w E M for which v v w and v 1\ w belong to M, lj; (v)
+ lj; (w) = lj; (v v w) + lj; (v 1\ w).
The next theorem, which is proved in Weber (1988), characterizes probabilistic val ues, and hence quasivalues and semivalues on simple games. THEOREM 12. An index ofpower defined on the set of simple games, or on the set of monotone simple games, is a probabilistic value if and only if it satisfies the transfer, dummy, and positivity axioms. Consequently, an index ofpower is a quasivalue on these sets of games if and only if it satisfies the transfer, dummy, positivity and efficiency axioms, and it is a semivalue if and only if it satisfies the transfer, dummy, positivity and symmetry axioms. Note that every probabilistic value 1j; that is defined on a set of simple games, which contains the unanimity games, can be uniquely extended to a probabilistic value on G . Moreover, i f 1j; i s a quasivalue o n such a set o f simple games, then its extension i s a quasivalue on G and if 1j; is a semivalue, then its extension is a semivalue. This follows from the fact that the symmetry and efficiency axioms are "linear" axioms, in the sense that if they hold for a symmetric linear base of games then they hold for every game. The extension property of power indices implies, in particular, that every quasivalue on simple games is a random-order value (see Theorem 3). Representations of indices of power as random-order values, or probabilistic values, have a special flavor. Let v =f. 0 be a simple monotone game, and let r E OR. The pivot for r in v is the unique player j = j, for whom the set Rj , of players preceding j in r , i s a losing coalition, while Rj U j i s winning. Thus, for a probability measure b o n the set of orders d(OR), q; b v (i ) is the b-probability that i is the pivot. In particular, the Shapley value [which is also called the Shapley-Shubik index of power - see Shapley and Shubik (1954)] of a player i in v is his probability of being the pivot when all orders have equal probability. We say that i is a swing for a coalition S in v, if v(S) = 0 and v(S U i) = 1 . Let 1j; be a probabilistic value defined by the probabilities p 1 , . . . , pn on c 1 ' . . . ' en ' respectively. Then lj; v (i ) is the p i -probability that i is a swing for a random coalition of the other voters. 8 The Banzhaf index of power [Banzhaf ( 1965)] has been extensively discussed in the literature. Dubey and Shapley (1979) characterized the Banzhaf index on MSG+ in the same way that Roth (1977) characterized the Banzhaf value on G (see Footnote 5). Lehrer ( 1988) gave another characterization, based on the behavior of an extended index of power when applied to games in which coalitions of the original game are amalga mated into one player, as follows. For every 0 C T C N choose a player iy in T. Let v
8
See Straffin (1988) for other probabilistic interpretations of the Shapley-Shubik and tbe Banzhaf indices of power.
Ch. 54: Variations on the Shapley Value
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be a simple game and let T be a nonempty coalition. Define two games V[ TJ and v [h on the set of players T c U {i T } as follows: V[ T J(S) = v[ h CS) = v(S) for every S :;:; T c , and for every such S,
V[ T J (S U i T ) = v(S U T)
and
v (T1 (S U i T ) =
max
0c B c;.T
v(S U B) .
THEOREM 13. Let 1j! be an extended semivalue on the sets of simple games on subsets of N, 9 and let Ci T ) 0 c T c N be a collection ofplayers such that i T E T. Then 1/1 is the Banzhafindex ofpower if and only iffor every two-player coalition T = { i, j},
Moreover,
1/1
is the Banzhaf index ifand only iffor each such T,
1j! v(i) + 1j! v(j) � 1/l v ['·h Ci T ). REMARKS .
(1)
Einy (1987) characterized semivalues on MSG ( U ) , when U is an infinite universe set of players. Actually he proved the "semivalues" part of Theorem 12 in such a setup. In his papers (1987, 1988), he also provided a formula for the Banzhaf index of power of a monotone simple game v in terms of the minimal winning coalitions of this game. (2) For every positive weight vector w = (W ! , . . . , w n ) , and 0 < c < 1 we define the weighted majority game, Vw , c with a quota c as follows:
Vw , c (S) =
{ �:
if w(S) � i f w(S) <
c; c.
Dubey and Shapley (1979) have proved the following intriguing equality. For any probabilistic index of power 1/1, Jd 1/lvw , c (i) de = w; . 8. Values with a social structure
In all the variations on the Shapley value discussed in this section, there is some struc ture on the set of the players which is involved in determining the value of a game. All are extensions of the Shapley value, because when the structure on N is the "trivial" one, where triviality depends on the type of structure used in the solution, then the value is the Shapley value. That is, 1/f : Us e N SG(S) � a semivalue for every S c;_ N.
9
eN A(S), 1/fv E A(S) for v E SG(S), and the restriction of 1/f to SG(S) is Us-
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Aumann and Dreze (1975) were the first to introduce such a structure into the study of value, by changing the efficiency axiom. Instead of assuming that the grand coali tion, N, forms and allocates its worth, v(N), among its players, they assumed that there is a fixed partition rr = { B 1 , . . . , Bm } of N, which they called a coalitional struc ture, such that each coalition, Bb forms and allocates its value, v(Bk) , among its members. Using the following axioms they characterized a unique solution called the
n-value.
A solution 1/f is n-efficient if for every game v, ljf v(Bk) = v(Bk) for all 1 ::::; k ::::; m. A permutation e is n-invariant if e (Bk) = Bk for every k, and 1/f is n-symmetric if ljf (e*v) = e*ljfv for every rr -invariant permutation e . Aumann and Dreze (1975) proved:
14. There exists a unique probabilistic value cp rr which is rr -symmetric and rr -efficient, called the rr -value, and it is given by: THEOREM
(8.1) Note that by (8.1), for i E Bk, cp rr v(i) depends only on the game v Bk . Hence, the power of a player i E Bk to negotiate with players outside Bk is not taken into account when computing his share of v(Bk). Myerson (1977) studied a value, called thefair allocation rule, where the social struc ture is given by a graph with vertex set N. An arc (i, j) in graph g can be thought of as signifying some social relation, say a neighborhood or a channel of communication, between players i and j. For each coalition S, we denote by g I s the restriction of the graph g to the vertices in S. By rrg l s we denote the partition of S into the connected components of g I s . The axiomatization of the fair allocation rule is accomplished by fixing the game v, and varying the social structure. A function ¢ : GR -+ A, where GR i s the set o f all graphs on N, is afair allocation rule if it satisfies two properties: it is relatively efficient, by which we mean that ¢g(S) = v(S) for every connected component of g; and it is fair in the sense that for any two graphs g and h such that h = g U {(i, j)},
¢h(i) - ¢g(i) = ¢h(j) - ¢g(j). Thus, ¢ is fair if, by adding an arc to a graph, the players forming this arc equally gain or lose. Myerson (1977) proved:
1 5 . For any game v E G, there exists a unique function ¢m : GR -+ A which is a fair allocation, and it is given by,
THEOREM
Ch. 54: Variations on the Shapley Value
2073
where cp is the Shapley value, and v / g is the game which satisfies for each coalition S, (vJg)(S)
=
L v(T).
TE7rg[s
Note that each of the coalitions in the partition of N to its connected components, i.e., the partition JTg I N , forms and allocates its worth among its members. Moreover, like the case of the n-value, the value of a player, in the fair allocation rule, is deter mined solely by the restriction of the game to the coalition in JTg i N , of which he is a member. The following solution, introduced by Owen (1977), is based on a coalition structure, i.e., a partition JT = { B t , . . . , Bm } of N. But unlike the previous two solutions, it is a quasivalue, and as such it is efficient in the usual sense, that is, the players allocate the worth of the grand coalition N. By Theorem 3, a quasivalue is a random-order value, b and the value proposed by Owen (1977) is the random-order value cp J[ , defined by the probability distribution bn E L1 (0R) , which we describe next. We say that an order r is consistent with JT if for each k = 1 , . . , m, all players in Bk anive "together" in the following sense. If i anives before j, and both are in Bk, then all the players aniving between i and j are also in Bk . Let bn E L1 (0R) be the probability measure that is uniformly distributed over all orders that are consistent with JT . That is, it assigns the probability !hJ · ·hm ! to each such order, where bk denotes the number of m players in Bk . This value, called the JT -coalitional structure (cs) value was characterized in various setups by Owen (1977) and by Hart and Kurz (1983). We denote it by cp�s and need some additional notations before we can present a modified version of their axiomatization. Let JT be a partition of N. For each v E G we define a partitional game Vn , the players of which are the coalitions in JT , as follows. For each F 5; JT , .
We say that B 1 and Bz are symmetric players in Vn i f for every F 5; JT that does not con tain Bt and Bz, Vn (F U {Bt }) = Vn (F U {Bz}). A solution 1jr satisfies the JT -coalitional symmetry axiom if ljrv(Bt ) = ljrv(Bz) whenever Bt and Bz are symmetric players in Vn . The proof of the following theorem can be derived from Owen ( 1977) and Hart and Kurz
(1983):
1 6 . Let 1jr be a solution and let JT be a partition. Then 1jr is the JT coalitional structure value if and only if 1jr is a random-order value that satisfies the JT -symmetry and the JT -coalitional symmetry axioms.
THEOREM
Owen (1977) and Hart and Kurz (1983) explored some interesting properties of the Owen value. One such property is that the total value of a coalition B in JT equals the
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Shapley value of the player B in the game Vn . That is,
cp�s v (B )
=
f{! Vn (B) = L f{!Vn (i). iEB
The next question is how do the players in B distribute the total amount available to them, f{! Vn (B). To answer this, we define for B E JT a game V(n, B) in G(B). Let S s; B and denote by JTB I S the partition refining JT by splitting B into S and B \ S. The game V(n, B) is defined by:
THEOREM
1 7 . For each i E B,
Owen showed that Theorem 17 holds for i E S if JTB IS is defined as the partition of obtained from JT by replacing B with S (i.e., the players in B \ S simply disappear). Hart and Kurz showed that Theorem 17 also holds when we define the game V(n, B) by defining JTB I S as the partition obtained from JT by replacing B with S and the singletons of B \ S .
N \ (B \ S)
REMARKS .
(1)
Myerson (1980) extended the notion of the cooperation graph to a cooperation hypergraph, where a link can be formed among the members of a coalition with more than two players. (2) Amer and Carreras (1995) extended the notion of the cooperation graph to the weighted cooperation graph, called a cooperation index. More formally, a coop eration index is a function w : C � [0, 1] , where w (S) may be interpreted as the probability of a communication link among the members of S. Naturally, it is required that w ({i}) = 1 for every player i . Amer and Carreras (1997) further discussed cooperation indices in the context of weighted Myerson values. (3) Owen (1977) and Hart and Kurz (1983) gave an axiomatic characterization of the coalitional structure value as an operator which is defined on pairs (v, rr ) , using axioms that consider changes in the values of such an operator when both the game and the partition are changed. (4) McLean (1991) defined and analyzed coalitional structure random-order values, as follows. Let JT be a partition of N and let b E L1 (0R) . b is consistent with JT if b ( r) = 0 for every order r which is not consistent with JT . A random order value is a JT random order coalitional structure value if it is defined by some b E L1 (0R) which is consistent with JT . Levy and McLean (1989) defined and characterized weighted coalitional structure values.
Ch. 54: Variations on the Shapley Value
(5) (6)
(7) (8)
2075
McLean and Ye (1996) developed the theory of coalitional structure semivalues for finite games. Winter (1992) characterized the Owen value by a consistency property which generalizes the one given in Hart and Mas-Colell (1989). He also generalized the potential approach and showed that the Owen value, Myerson value and A-D value can be developed through this approach. Winter (1989) developed the theory of values defined on pairs (v, p), where v E G and p = (n 1 , . . . , nm ) is a decreasing sequence of partitions. 1 0 Lucas (1963) and Myerson (1976) dealt with values of generalized games, where the worth of a coalition also depends on the partition into coalitions generated by the players.
References Amer, R., and F. Carreras
(1995), "Games and cooperation indices", International Journal of Game Theory
3:239-258. Amer, R., and F. Carreras (1997), "Cooperation indices and weighted Shapley values", Mathematics of Oper ations Research 22:955-968. Aumann, R.J., and J.H. Dreze (1975), "Cooperative games with coalition structures", International Journal of Game Theory 4:217-237. Banzhaf, J.F., Ill (1965), "Weighted voting does not work: A mathematical analysis", Rutgers Law Review
19:3 17-343.
Block, H.D., and J. Marschak ( 1960), "Random ordering and stochastic theories of responses", Cowles Foun dation Paper, No. 147. Dubey, P. (1975), "On the uniqueness ofthe Shapley value", International Journal of Game Theory 4: 13 1-139. Dubey, P., and L.S. Shapley (1979), "Mathematical properties of the Banzhaf power index", Mathematics of Operations Research 4:99-!3!. Dubey, P., A. Neyman and R.J. Weber (198 1), "Value theory without efficiency", Mathematics of Operations Research 6: 122-128. Einy, E. ( 1987), "Semivalues of simple games", Mathematics of Operations Research 12:185-192. Einy, E. (1988), "The Shapley value on some lattices of monotonic games", Mathematical Social Sciences
15:1-10. Falmagne, J.C. (1978), "A representation theorem for finite random scale systems", Journal of Mathematical Psychology 1 8:52-72. Gilboa, I., and D. Monderer ( 1991), "Quasivalues on subspaces of games", International Journal of Game Theory 19:353-363. Gilboa, I., and D. Monderer (1992), "A game theoretic approach to the binary stochastic choice problem", Journal of Mathematical Psychology 36:555-572. S., and M. Kurz ( 1983), "Endogenous formation of coalitions", Econometrica 5 1:589--614. Hart, S., and A. Mas-Colell ( 1989), "Potential, value and consistency", Econometrica 57:589-614. Kalai, E., and D. Samet (1987), "On weighted Shapley values", International Journal of Game Theory 16:205-
Hart,
222. Lehrer, E. (1988), "An axiomatization of the Banzhaf value", International Journal of Game Theory 17:89-99. Levy, A., and R.P. McLean ( 1989), "Weighted coalition structure values", Games and Economic Behavior
1 :234-249. 10
The idea for dealing with such values appears in Owen (1977).
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Lucas, W.F. (1963), "n-Person games in partition function form", Naval Research Logistics Quarterly 10:281298. McLean, R.P. (1991), "Random order coalition structure values", International Journal of Game Theory 20: 109-1 27. McLean, R.P., and T.X. Ye (1996), "Coalition structure semivalues of finite games", Rutgers University. McLennan, A. (1989a), "Consistent systems in noncooperative game theory", International Journal of Game Theory 18: 141-174. McLennan, A. (1989b), "The space of conditional systems is a ball", International Journal of Game Theory 18:127-140. Monderer, D. (1988), "Values and semivalues on subspaces of finite games", International Journal of Game Theory 17:321-338. Monderer, D. (1990), "A Milnor condition for nonatomic Lipschitz games and its applications", Mathematics of Operations Research 15:714-723. Monderer, D. (1992), "The stochastic choice problem: A game theoretic approach", Journal of Mathematical Psychology 36:547-554. Monderer, D., D. Samet and L.S. Shapley (1992), "Weighted Shapley values and the core", International Journal of Game Theory 21:27-39. Myerson, R.B. (1976), "Values of games in partition function form", International Journal of Game Theory 6:23-31 . Myerson, R.B. (1977), "Graphs and cooperation i n games", Mathematics of Operations Research 2:225-229. Myerson, R.B. (1980), "Conference structures and fair allocation rules", International Journal of Game Theory 9:169-182. Myerson, R.B. (1986), "Multistage games with communication", Econometrica 54:323-358. Owen, G. (1972), "Multilinear extension of games", Management Sciences 5:64-79. Owen, G. (1977), "Values of games with a priori unions", in: R. Henn and 0. Moeschlin, eds., Mathematical Economics and Game Theory (Springer, Berlin). Owen, G. (1982), Game Theory, 2nd edn. (Academic Press, Orlando). Roth, A.E. (1977), "Bargaining ability, the utility of playing a game, and models of coalition formation", Journal of Mathematical Psychology 16:153-160. Roth, A.E., ed. (1988), "The Shapley Value (Cambridge University Press, Cambridge). Shapley, L.S. (1953a), "A value for n-person games", Annals of Mathematics Studies 28:307-3 17. Shapley, L.S. (1953b), "Additive and non-additive set functions", PhD Thesis (Dept. of Mathematics, Prince ton University). Shapley, L.S. (197 1), "Cores of convex games", International Journal of Game Theory 1 : 1 1-26. Shapley, L.S. (1981), "Discussant's comments: Equity considerations in traditional full cost allocation prac tices", in: Proceedings of the University of Oklahoma Conference on Cost Allocation, April l981, 13 1-136. Shapley, L.S., and M. Shubik (1954), "A method for evaluating the distribution of power in a committee system", American Political Science Review 48:787-792. Straffin, P.D., Jr. ( 1988), "The Shapley-Shubik and Banzhaf power indices as probabilities", in: A.E. Roth, ed., The Shapley Value (Cambridge University Press, Cambridge) 71-8 1 . Weber, R.J. (1988), "Probabilistic values for games", in: A.E. Roth, ed., The Shapley Value (Cambridge University Press, Cambridge) 101-119. Winter, E. (1989), "A value for games with level structures", International Journal of Game Theory 18:227240. Winter, E. (1992), "The consistency and potential for values of games with coalition structure", Games and Economic Behavior 4:132-144.
Chapter 55 VALUES OF N ON-TRANSFERABLE UTILITY GAMES RICHARD P. McLEAN
Department ofEconomics, NJ Hall, Rutgers University, New Brunswick, NJ, USA Contents
1.
Notation and definitions 1 . 1 . General notation 1 .2. Transferable utility games 1 . 3 . Non-transferable utility games
2.
Values and bargaining solutions 2. 1 . Random order values 2.2. Weighted Shapley values 2.3. The Shapley value
3.
2.4. Bargaining solutions on r/J
The Shapley NTU value 3 . 1 . The definition of the Shapley NTU value 3.2. An axiomatic approach to the Shapley NTU value 3.3. Non-symmetric generalizations of the Shapley NTU value
4. The Harsanyi solution 4. 1 . The Harsanyi NTU value 4.2. The Harsanyi solution 4.3. A comparison of the Shapley and Harsanyi NTU value
5.
Multilinear extensions for NTU games 5 . 1 . Multilinear extensions and Owen's approach to the NTU value 5.2. A non-symmetric generalization 5.3. A comparison of the Harsanyi, Owen and Shapley NTU approaches
6.
The Egalitarian solution 6.1. Egalitarian values and Egalitarian solutions 6.2. An axiomatic approach to the Egalitarian value 6.3. An axiomatic approach to the Egalitarian solution
7.
Potentials and values of NTU games 7 . 1 . Potentials and the Egalitarian value of finite games 7.2. Potentials and the Egalitarian solution of large games
8.
Consistency and values of NTU games 8 . 1 . Consistency and TU games: The Shapley value
Handbook of Game Theory, Volume 3, Edited by R.J. Aumann and S. Hart © 2002 Elsevier Science B. V. All rights reserved
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8.2. Consistency and NTU games: The Egalitarian value 8.3. The consistent value for hyperplane games 8.4. The consistent value for general NTU games 8.5. Another comparison of NTU values 9. Final remarks 9 . 1 . Values and cores of convex NTU games 9 .2. Applications of values for NTU games with a continuum of players 9.3. NTU values for games with incomplete information References
2110 21 10 21 13 21 15 2116 21 16 2117 2 1 17 2118
Abstract
This chapter surveys a class of solution concepts for n-person games without transfer able utility - NTU games for short - that are based on varying notions of "fair division". An NTU game is a specification of payoffs attainable by members of each coalition through some joint course of action. The players confront the problem of choosing a payoff or solution that is feasible for the group as a whole. This is a bargaining problem and its solution may be reasonably required to satisfy various criteria, and different sets of rules or axioms will characterize different solutions or classes of solutions. For trans ferable utility games, the main axiomatic solution is the Shapley Value and this chapter deals with values of NTU games, i.e., solutions for NTU games that coincide with the Shapley value in the transferable utility case. We survey axiomatic characterizations of the Shapley NTU value, the Harsanyi solution, the Egalitarian solution and the non symmetric generalizations of each. In addition, we discuss approaches to some of these solutions using the notions of potential and consistency for NTU games with finitely many as well as infinitely many players.
Keywords
non-transferable utility, Shapley value, Harsanyi solution, Egalitarian solution, consistency, potential, consistent value
JEL classification: C710, C780
Ch. 55: Values ofNon-Transferable Utility Games
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1. Notation and definitions 1.1.
General notation
Throughout this chapter, we work with a set of players N = { 1 , 2, . . . , n}. If S ::::; N, then l S I denotes the cardinality of S and ms is the l S I -dimensional Euclidean space whose axes are labelled with the members of S. The non-negative (positive) orthant of ms will be denoted by ffi� (ffi�+). Similarly, the non-positive orthant -ffi� will be written ffi�. N If X E m ' then xs E ms is the "projection" of X onto ms . The vector (1' . . . ' 1) in ms N will be denoted I . The "indicator vector" X T E m is defined by X� = 1 if i E T and s i i x� = 0 if i ¢ T. We write x � y if x � y for all i and x > y if x i > yi for all i . If X and U belong to ffi S and t E ffi, then X U denotes the inner product, tx denotes N scalar multiplication, and X * u E ms is defined by (x * u ) i = x i u i . If A ' B ::::; m ' N y E m and t E ffi then A + B is the Minkowski sum of A and B , A + y = A + {y}, y * A = {y * x I x E A } and tA = {tx I x E A}. The Pareto efficient boundary of A is denoted a A and the complement of A is written ""' A . If A is convex, then A is smooth if there exists a unique supporting hyperplane at each point of a A . A set A is comprehensive if x E A and x � y imply that y E A. To simplify the notation we will not distinguish between i and { i }. The symbol "\" denotes set theoretic subtraction. •
1.2.
Transferable utility games
N A TU game is a set function v : 2 --* ffi satisfying v(0) = 0. (If X is a set, then 2x denotes the power set of X.) The 2n - 1 -dimensional vector space of TU games is de i N noted by G N . If 1/1 : G N --* m is a function and S ::::; N, define (lj!v) (S) = L i E S lj! (v) . A game v is monotonic if v (S) � v(T) whenever S ::::; T . For each nonempty T ::::; N, the unanimity game ur is defined by ur (S) = 1 if T ::::; S and ur (S) = 0 otherwise. (It is well known that the collection {ur }0# T <; N is a basis for G N .) Player i E N is a null player in v if v (S U i) - v (S) = 0 whenever S ::::; N\i, while player i E N is a dummy player in v if v(S U i) - v (S) = v (i ) whenever S ::::; N\i . Players i and j are substitutes in v if v(S U i) = v (S U j) whenever S ::::; N\{i, j}. A coalition S ::::; N is flat in v if v (S) = L i E S v (i ) . The collection of n! orderings of the elements of N will be denoted S?N . If R = (ii , . . . , in ) is an element of S?N and i = ik, let X i (R) = {i], . . . , ik-d be the set of players that precede i in R . (Xi (R) = 0 if i = i 1 .) The marginal contribution of i in R i i in the game v is defined as i1 i (R, v) = v (X (R) U i) - v (X (R) ) . Finally, let M(S?N) denote the set of probability measures on [2N . 1.3.
Non-transferable utility games
An NTU game V is a correspondence that assigns to each nonempty S ::::; N a nonempty subset V (S) s:; ms. If V is an NTU game, let V* (S) = V (S) x {oN \ S } for each S ::::; N. (In many treatments, an NTU game i s defined a s V* rather than V . ) If V and W are NTU
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N
games, t E ffi and A E ffi , then (V + W)(S) = V (S) + W(S), (A * V) (S) = As * V(S), and (tV)(S) = tV(S) . Define rN to be the set of NTU games V for which each set V (S) is a proper, convex, closed, comprehensive subset of ffi 5 . In the discussion that follows, we will need to consider certain subclasses of rN whose members satisfy some of the following additional restrictions: (1) V(N) is smooth; (2) if x , y E a V(N) and x ;? y , then x = y ; N (3) for each S � N, there exists an x E m such that V* (S) � V (N) + x ; (4) if {xm}�= I is a sequence in V (S) with Xm+l ;? Xm for each m, then there exists y E ffi S SUCh that Xm :S;; y for all m; (5) for each S � N, the set of extreme points of V (S) i s bounded. The main subclasses of rN that we study are listed now: T� consists of those games in TN satisfying conditions ( 1 ), (2) and (3) above,
f� consists of those games in rN satisfying conditions (1), (2), (3) and (5) above, r� consists of those games in rN satisfying conditions (1) and (2) above, T� consists of those games in TN satisfying condition (4). Condition (1) above is a regularity condition that guarantees that utility is locally transferable according to uniquely determined rates of transfer. Condition (2) is a "non levelness" assumption and guarantees that a normal vector to a boundary point of V (N) has positive components. Condition (3) is a weak version of monotonicity while (4) is a weak boundedness assumption. Condition (5) is important for the existence of some of the concepts we discuss below. If v E G N, then the NTU game corresponding to v is defined by
V (S)
=
{x E
ffi5
; I I> :S;; v(S) 1 ES
}·
If V corresponds to v, we will sometimes say that v generates V. For each T � N, the NTU game corresponding to the unanimity game u T will be denoted UT . The collection of all NTU games corresponding to TU games is denoted rJU and, obviously, rJu ::; �1 r
N.
N
For any NTU game v E rN, let dv E m be the vector with d� = max{xi I xi E V(i)} for each i. The collection of v E rN with dv E V(N) is denoted r�R . A game v E r�R is a Pure Bargaining Problem if
V (S) = {x E ffi 5 I xi
:s;;
d� for each i E s }
for each S ::f= N, dv tj. a V (N) and the set {x E bargaining problems in T�R is denoted T/J .
V (N) I x ;? dv} is compact. The class of
Ch. 55: Values ofNon-Transferable Utility Games
2.
208 1
Values and bargaining solutions
We will record several solutions for TU games and bargaining problems whose NTU extensions are the focus of this chapter. We will also provide statements of character ization theorems since they will be referenced in the sketches of proofs of their NTU generalizations. 2. 1.
Random order values
For each p E as follows : cp
M (QN )' the p-Shapley value is the linear function cpp G N :
-+
mN defined
� ( v ) = L p(R)!l, (R , v ) . RE Q
A p-Shapley value is also called a random order value or a quasivalue and four axioms characterize the class of such values. In the statements of the axioms, 1/r : G N -+ fi N is a function. Sl: (Linearity) For every V J , v2 E G N and real numbers oq and a2 ,
S2: (Efficiency) For each v E G N , (1/rv) (N) = v(N) . S3: (Null Players) If i is a null player in v, then 1/r ' (v) = 0. S4: (Monotonicity) If v is monotonic, then 1/r ' (v) ;;:: 0 for all i E N. PROPOSITION 2. 1 . 1 [Weber (1988)]. An operator 1/r : G N
only if there exists p E M(Q) such that 1/r = CfJp · 2.2.
-+ fi N satisfies Sl-S4 ifand
Weighted Shapley values
We now turn to a special class of p-Shapley values, the weighted Shapley values. If w E ffi�+· let p w E M(Q) be the probability measure defined as follows: for each
R = Cit , i2 , . . . , in ) E S?N , p w (R) =
fi[ w ik ;t W i1 J k= l 1= 1
Hence, orderings are constructed according to the following procedure. Player i E N is chosen to be at the end of the line with probability w ' I L j EN wj . Given that the players in S are already in line, player i E N\S is added to the front of the line with probability w i I L j EN\ S w j . Define the weighted Shapley value with weight vector w (or simply the
w-Shapley value) as the operator CfJp w : G N -+ m N . For simplicity, we will denote C{Jp w
as CfJw ·
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The w-Shapley value was defined in Shapley (1953a) and axiomatized in Kalai and Samet (1987) where it is called the "weighted Shapley value for simple weight systems". The w-Shapley value satisfies S 1-S4 but two other axioms are required to characterize the class of w-Shapley values. First, S4 must be strengthened to guarantee that each R E QN occurs with positive probability. S4' : (Positivity) If v E G N is monotonic and contains no null players, then l{r i (v) > 0 for all i E N. The second axiom guarantees that the probability measure on orders has the special form p w for some w E lR�+ . A coalition T s; N is a partnership in v if v (S U R) = v (A) whenever S s; T, S -=!= T and A s; N\ T. For example, T is a partnership in u T . SS: (Partnership) If T is a partnership in v , then l{r i ( v) = l{r i [(l{rv)(T)ur] for each i E T. Informally, Axiom S 5 states that the behavior of a partnership T in a game v is the same as the behavior of T in u T in the sense that the fraction of (1{r v) (T) received by i E T is the same as i 's fraction of the one unit being allocated to T in u T . 2.2. 1 [Kalai and Samet (1987)]. An operator 1{r : G N ll'iN satisfies Axioms Sl, S2, S3, S4' an� S5 if and only if there exists w E lR�+ such that 1{r = cpw . PROPOSITION
_,.
REMARK 2.2.2. Axioms S 1 , S2, S3, S4 and S5 characterize the weighted Shapley values for "general weighted systems" [see Kalai and Samet (1987)].
2.3.
The Shapley value
If w E lR�+ and w i = w i for all i, j E N, then p w (R) = � for each R E Q . In this case, we write cpw simply as cp and refer to cp as the Shapley value. An additional axiom is required to guarantee equal weights. S6: (Substitution) If i and j are substitutes in v, then l{r i ( v) = l(r i ( v) . 2.3 . 1 [Shapley (1953b)]. An operator l{r : G N ioms Sl, S2, S3 and S6 if and only if 1{r = cp.
PROPOSITION
2.4.
_,.
ffiN satisfies Ax
Bargaining solutions on r/J
Let w = ( w 1 , w 2 , . . . , w n ) E ffi�+ be a vector of weights. For a game V E r/J , define
aw (V) = arg max n (x i - d� t; . x EV(N) i E N x ).dv The set aw (V) is a singleton because V(N) is convex and the function
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is the weighted Nash Bargaining Solution with the weight vector w or simply the w NBS. One can characterize the w-NBS on rjj axiomatically and a proof may be found in Roth (1979) or Kalai (1977a). If the components of the weight vector are identical, then we write aw simply as a and refer to a as the Nash Bargaining Solution or NBS for short. The NBS was defined and axiomatically characterized in Nash (1950) and a detailed discussion may be found in Chapter 24. We will be interested in another solution for bargaining problems. Again, let w E ffi�+' V E rJ and define Tw (
V) := dv + t * (V, w)w
where t* (V, w) = max[t I dv + t w E V(N)] . The function Tw : r!J ---+ ffiN is the w proportional solution with weight vector w. If the components of w are all equal, then we write Tw simply as T and refer to T as the (symmetric) proportional solution for V. The w-proportional solution was defined and axiomatized in Kalai (1977b). 3.
The Shapley NTU value
3. 1.
The definition of the Shapley NTU value
Let V E rN and for A E ffi�+' define
V)JS) = sup [A 5
·
z Iz
E V ( S) ] .
The TU game V). is definedfor V if V). (S) is finite for each S. 3 . 1 . 1 . A vector X E !R N is a Shapley NTU value payoff for v E rN if there exists A E ffi�+ such that: (1) x E V(N), (2) V). is defined for V, (3) A * cp(V).). DEFINITION
X =
Condition (1) states that x is feasible for the grand coalition. Conditions (2) and (3) are best understood by first looking at a two-person game in r!J. In this case, conditions (1), (2) and (3) are equivalent to:
and
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These are precisely the two conditions that characterize (x 1 , x2) as the NBS of V. In tuitively, an arbitrator might desire to choose a payoff (x 1 , x2) E V ( l , 2) that is "util itarian" in the sense that x 1 + x 2 = max[z 1 + z2 I z E V ( 1 , 2)] and "equitable" in the sense that x 1 - d& = x2 - d� (i.e., net utility gains are equal). In general, these two criteria cannot be satisfied simultaneously. However, it may be possible to rescale the utility of the two players and achieve a balance of utilitarianism and equity in terms of the rescaled utilities. This is precisely what is accomplished by the NBS. Hence, conditions (1) and (2) of Definition 3.1.1 provide an extension of utilitarianism to the n-player problem while condition (3) is an extension of equity. (The indefinite article is used here because other extensions of equity and efficiency are possible as shown in Sections 4 and 6 below.) An alternative definition of Shapley NTU payoff is possible using the idea of dividend in Harsanyi (1963).
3 . 1 .2. A vector x E ffi N is a Shapley NTU value payoff for V E rN if there exists A E ffi�+ and a collection of numbers {ay }0# T <;N such that: (1) x E V(N), (2) VJc is defined for V, (3) Ai x i = L T<;N: iET a y and L i ES L T<;S : i ET a T = VJc (S) whenever S <; N. DEFINITION
The equivalence of Definitions 3. 1 . 1 and 3.1.2 follows from the fact that
L L ay = L I T ia y . T <; S
iES T<;S: i ET
[See Aumann and Shapley (1974), Appendix A, Note 5.] The numbers a y are in terpreted as dividends that will be paid to the members of T. Suppose that the proper subcoalitions of S have all declared their dividends. Then each i E S collects L TcS: S#T , i ET a y and condition (3) of Definition 3.1.2 requires that each member of S receive a dividend as equal to
1
[
V (S) S I I Jc
-L
L
i ES T <; S: S# T, i ET
J
ay .
Since each player in S receives the same dividend, condition (3) is an alternative view of the equity requirement of a Shapley value payoff. From Definition 3 . 1 . 1 , we see that, if V corresponds to v E G N , then A must be a positive multiple of 1 N so that cp ( v) is the unique Shapley NTU value payoff for V . The Shapley NTU value is also an extension of the NBS to general NTU games in the sense that these solutions coincide for V E r!J whenever cP (V) -=!= 0 . PROPOSITION
3 . 1 .3 . lfV E r/J , then cP (V) = {a(V)}.
Ch. 55: Values ofNon-Transferable Utility Garnes
2085
PROOF. Using an argument from convex programming, it can be shown that x = rr(V) if and only if there exists 17 E ffi�+ such that (i) 17 1 (x i - d � ) = 17 i (x i - d? ) for all i, j E N and (ii) 17 x = max [17 z I z E V(N)]. Let V E rJ and suppose y E
·
A 1 (x 1 - 4) = q) (v;,.) - A i d� =
� [v;,. (N) - LEN v;,. (j) ] ]
for each i, (i) and (ii) are satisfied and it follows that x = rr(V). Conversely, suppose that x = rr(V). Then there exists A E ffi�+ such that (i) and (ii) are satisfied. Letting v;,. (N) = max[A z I z E V(N)] and v;,.(S) = L i ES A i d� for each S 1- N, we conclude that v;,. is defined for V. Since A; x i = cp i ( v;,.), it follows that x is a Shapley NTU value D payoff for V. ·
3.2. An axiomatic approach to the Shapley NTU value In the previous section, it was shown that the set of Shapley NTU value payoffs co incides with the Shapley value for the TU games and the Nash Bargaining Solution for pure bargaining games. Thus, it is natural to synthesize the axiomatic characteriza tion of these two solutions in an effort to construct a characterization for the Shapley NTU value. This approach is precisely the method of attack in Aumann (1985a) and Kern (1985). Let cP : rJ --+ rnN be the correspondence that associates with each v in rJ the set of NTU value payoffs. The solution correspondence cP will be referred to as the Shapley correspondence. Let fJ be the set of NTU games V E rJ such that
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The next three lemmas [all found in Aumann (1985a)] provide the essential steps in the characterization of the Shapley correspondence. LEMMA 3 .2 . 1 . If lJ! : f� -+ ffi satisfies Axioms AO-A4 then lJ! ( V ) = {cp(v)} whenever V corresponds to the TU game v. PROOF. For each real number a, let va correspond to the TU game av. Using AO, A l , A2 and A3, we deduce that lJ! ( V 0) = {0}. Using Axioms A l and A2 again, it can be shown that lJ! (V - 1 ) + lJ! (V) = {0} so AO implies that lJ! (V - 1 ) = - lJ! (V) and each is single-valued. Combining A4 with the results above, we deduce that lJ! (a V) = alJ! (V) for all scalars a. Using A3 it follows that lJ!(U¥) = alJ! (Ur) = {a X T / IT I } = {cp(aur) }. Since { u rl0# T <;N i s a basis for G N w e can express each v E G N as v L.0#T <; N ar u T so that V = L T u? for the NTU game V corresponding to v . Applying AI and A2, we deduce that
=
(
)
{ cp(v) } _L) cp(arur) } = L w (u?) s; lJ! L u? = lJ! (V). T T T Since lJ!( V ) is a singleton set, lJ! (V) = {cp(v)}. =
D
LEMMA 3 .2.2. If lJ! satisfies axioms AO-A4, then lJ! (V) s;
A*Y
[lJ! (V) + {O}] n a (V(N) + V 0 (N)) = [lJ!(V) + w (v0 )] n a (v (N) + V0 (N)) s; lJ! (v + V0) = { cp(VJc) } . E
Thus A * y = cp(vJc) and we conclude that y E
D
LEMMA 3 .2.3. /f iJI satisfies Axioms AO-AS, then
PROOF.
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Since A * y E cJ> (W), it follows that W E f� and Axiom AO implies that l[r(W) =f- 0. Again, let V0 correspond to the TU game that is identically zero so that V;.. = W + V 0 and a(W(N) + V 0 (N)) = aW(N). Using Lemma 3.2. 1 and A2 and A1, we get
{¢(v;.. ) } = W(V;.. ) = w ( w + v0) 2 [W(W) + {0}] n a (W(N) + V 0 (N)) = W(W) n aW(N) = W(W). Having already shown that W(W) =f- 0, we conclude that W(W) = {cp(v;.. )} = {A * y}. Since A * V(N) s; W(N) and A s * V(S) = W(S) for S =f- N, we can use A5 and A4 to deduce that {A * y} = W(W) n [A * V(N)] s; W(A * V) = A * W(V) and, therefore, D y E W (V). As the reader may verify, the Shapley correspondence cJ> satisfies Axioms AO-A5 and the main characterization result for the Shapley NTU value can now be stated. 3 .2.4 [Aumann (1985a)]. A correspondence l[r : f� --+ ffiN satisfies Ax ioms AO-A5 if and only if l[r = 4> .
THEOREM
If A5 is dropped from the axiomatization, then one obtains the following result as a consequence of Lemma 3.2.2. 3 .2.5 [Aumann (1985a)]. The Shapley correspondence cJ> satisfies Ax ioms AO-A4 and if l[r is any other correspondence satisfying Axioms AO-A4, then W(V) s; cJ> (V).
THEOREM
In Kern (1985), an alternative version of the Independence of Irrelevant Alternatives (IIA) is offered. AS': (Transfer Independence of Irrelevant Alternatives) Let V and W be games and let x E l[r (V). If there exists a A E ffi�+ and a collection {�s} s� N with �N = x such that for each S s; N, �s E a V(S) n aW(S), A s is normal to V (S) at �s and A s is normal to W(S) at �s, then x E l[r(W). As a general rule, IIA-type axioms stipulate that, once a solution has been determined, we can alter the game "far away" from the solution and obtain a new game with the same solution. Indeed, this aspect of Nash's Bargaining Solution has been criticized on a variety of grounds and has resulted in alternatives to the NBS in which IIA has been replaced by other axioms. Axiom A5' has a local flavor as well. If �s E a V (S) n a W (S) and A s is normal to V(S) and W(S) at �s, then av(S) and aW(S) are "similar near �s". This is certainly true for S = N because of the smoothness assumption. When V and W are related in this way, the axiom requires that x E l[r (W) if x E l[r (V). 3 .2.6. Ij l[r : f� --+ ffiN satisfies Axioms AO-A4 andA5', then cJ> (V) s; W(V), for all V E Tt. LEMMA
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PROOF. Let y E
=
It is easy to verify that the Shapley correspondence
3.2.8. Theorem 3.2.7 is essentially Theorem S.S of Kern (198S). However, Kern uses only Axioms A4 and AS' and a third axiom requiring that lft(V) {cp(v ) } whenever V corresponds to the TU game v. Lemma 3.2.1 makes clear the relationship between these two axiom systems. REMARK
=
REMARK 3 .2.9. In the statements of Theorems 3.2.4 and 3.2.7, the domain of the cor respondence lJt is f� and some may find this an aesthetic drawback. However, there are smaller domains that one could work with. For example, the characterization the orems will still be true if one works with the smaller collection r�. For v E r� , a fixed point argument can be used to verify that AO holds. For a more detailed discussion see Aumann (198Sa). The existence question is also treated at length in Kern (198S) where conditions different from smoothness and nonlevelness of V (N) are identified that guarantee the existence of Shapley NTU value payoffs.
3.3. Non-symmetric generalizations of the Shapley NTU value In the axiomatic characterization of the Shapley value, the Nash Bargaining Solution and the Shapley NTU value, some type of symmetry postulate is required. Basically, these symmetry axioms all guarantee that two players who affect the game in the same way will receive the same payoff. However, one can envision circumstances in which two players who are identical in terms of the data of the game might nonetheless be treated asymmetrically in the arbitrated solution. For example, two workers with the same productivity may be treated differently because one has attained seniority. Thus, an arbitrator might assign weights to the individuals that reflect the relative "importance" of each person in the resolution of the bargaining problem. Our next problem is to define and axiomatize a weighted Shapley NTU value that generalizes the symmetric
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NTU value defined above and that coincides with the w-NBS on r!J and the w-Shapley value on r;cu . 3 . 3 . 1 . Let w E ffi�+. A vector x E ffi N is a w-Shapley NTU value payoff for V E rN if there exists A. E ffi�+ such that: (1) X E V(N), (2) VJc is defined for V, (3) A * X =
The correspondence
LEMMA 3 . 3.2. Suppose that a correspondence tJ! : rJ -+ ffi N satisfies Axioms AO, AI, A2, A3' and A4. Then there exists an operator 1jJ : G N -+ rn N satisfying Sl, S2 and S3 such that tJ!(V) {ljl(v)} whenever V corresponds to the TU game v. =
Suppose tJt satisfies the axioms and let {qr }0# T <;N be the collection specified by Axiom A3'. For each 0 # T s; N, define ljl(ur) = qr and linearly extend 1jJ to all of G N. The resulting 1jJ is obviously linear and AI implies that 1jJ is efficient. It can be verified using an induction argument that the null player axiom is satisfied. Using the same argument as that used in the proof of Lemma 3.2. 1, we conclude that tJ!(V) is a singleton whenever V corresponds to a TU game. Since PROOF.
{ ljl(v) } = 2J ar 1/l (u T ) } =
T
it follows that tJt ( V) = { 1jJ ( v)} .
LT tJt (U�T ) s; tJt (LT U? )
=
tJ!(V), 0
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3 . 3. 3 . If lf/ : r� � ffiN satisfies Axioms AO, Al, A2, A3', A4, A6 and A7, then there exists a w E ffi�+ such that lf/ (V) = {
Suppose V corresponds to v E G N. Using Lemma 3.3.2, we conclude that lf/(V) = {1/f(v)} where 1/J : G N � ffi N satisfies S 1, S2 and S3. Since lf/ satisfies A6
PROOF.
and A7, it follows that 1/1 satisfies S4' and S5 so applying Proposition 2.2. 1 yields the D rewk Using Lemma 3.3.3 above and the techniques of Lemmas 3.2.2 and 3.2.3, we can prove the following generalization of Theorem 3.2.4. 3 . 3 .4 [Levy and McLean (1991)). A correspondence lf/ : f'� � ffiN satis fies Axioms AO, Al, A2, A3', A4, A5, A6 and A7 if and only if there exists a w E ffi�+ such that lf/(V) = c:t>w . THEOREM
REMARK 3 . 3 . 5 . Theorem 3.3.4 can be extended to the case of "general weight sys tems" of Kalai and Samet (1987). This extension is accomplished by weakening the positivity axiom A6 (to the monotonicity axiom A6' below) and using Theorem 2 in Kalai and Samet ( 1987) instead of Proposition 2.2. 1.
The w-Shapley correspondence obviously coincides with the w-Shapley value if V corresponds to a game v E G N . Furthermore, the w-Shapley correspondence is an ex tension of the w-NBS as well and we record a generalization of Proposition 3 . 1 .3 that utilizes an essentially identical proof. PROPOSITION
3 . 3 .6. lf w E ffi�+ and V E r/J, then c:t> w (V) = {aw (V)}.
In Section 2. 1, we presented the notion of random order value for TU games as a generalization of the w-Shapley value. In a similar way, Levy and McLean ( 1991) define and axiomatize a random order NTU value that generalizes the weighted Shapley NTU value discussed above and thus construct a taxonomy of NTU values with and without symmetry. Recall that M ( QN ) is the set of probability measures on QN and
The set of all p-Shapley NTU value payoffs of a game V is denoted c:t> p (V) and the correspondence that assigns to each V the set of p-Shapley NTU value payoffs is
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called the p-Shapley correspondence. Our characterization of cJ> P weakens the positivity Axiom A6 to A6'. A6' : (Monotonicity) If v is a monotonic TU game and if V is the NTU game corre sponding to v, then x E ffi� whenever x E tJi(V). Suppose a correspondence tJi satisfies Al, A2, A3', A4 and A6'. By combining Lemma 3.3.2 and Proposition 2.1 . 1, we deduce that there exists p E M(Q) such that tJi (V) = {
4 . The Harsanyi solution
4.1. The Harsanyi NTU value A payoff configuration is a collection {xs}07'orc; N with xs E ms for each S. For sim plicity, we will denote a payoff configuration by {xs}. If v E GN and S � N, then v i s represents the restriction of v to 2 s and
4 . 1 . 2. Definition 3 . 1 . 1 for the Shapley NTU value can be restated so as to facilitate a comparison with the Harsanyi NTU value. In particular, a vector x E ffiN is a Shapley NTU value payoff for V E rN if there exists a payoff configuration {xs} with x N = x and A E ffi�+ such that: (1 * ) xs E a V (S) for each S, (2*) A s · xs = maxzEV (S) A s · z for each S, (3*) A * x =
Comparing the definitions of the Shapley and Harsanyi NTU values, we see that both solutions are defined in terms of a comparison vector A E ffi�+' a payoff configuration {xs} with xs E a V (S) for each S, and a TU game VA. where iiA.(S) = A s · xs. The Shapley NTU value requires that VA. (S) = maxzEV(S) A 5 · z for every S � N (so that VA. = VA.) while the Harsanyi value requires this only for S = N. On the other hand, the Harsanyi
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NTU value requires that A s xs = rp ( VJc Is) for every S <; N while the Shapley NTU value requires this only for S = N. If N = { 1 , 2}, then the two definitions are identical and both coincide with the NBS. In light of the discussion in Section 3 . 1 , conditions (2) and (3) in Definition 4. 1 . 1 are, respectively, the extensions of the utilitarian and equity properties of the NBS. Comparing these conditions with 2* and 3* in the definition of the Shapley NTU value, we see that the Shapley NTU value emphasizes coalitional utilitarianism while the Harsanyi NTU value emphasizes coalitional equity. The definition of Harsanyi NTU value can also be restated in terms of dividends in the spirit of Definition 3 . 1 .2. For a proof of the equivalence of the two definitions, see Section 2.2 in Imai (1983). *
4. 1 .3 . A vector x E rn:N is a Harsanyi NTU value payoff for V E rN if there exists a payoff configuration {xs} with XN = x, a vector A E ffi:�+ and a collection of numbers {aT}G#TC:: N such that: (1) xs E a V (S) for each S, (2) A · x = maxzE V(N) A · z, (3) A; x� = L T c; S : i ET aT for each S <; N and each i E S. DEFINITION
As in Definition 3 . 1 .2, the number aT is interpreted as a dividend that each player in T receives as a member of T. If coalition S were to form, then each i E S would receive the payoff x�. Condition (1) requires that xs be efficient and condition (3) requires that i 's rescaled payoff A ; x� be equal to his accumulated dividends. From Definition 4. 1 . 1 , it is easy to verify that, if V corresponds to v E G N , then rp ( v) is the unique Harsanyi NTU value payoff for V . To see this, note that if x is a Harsanyi value payoff, then the associated vector A must be a positive scalar multiple of l N . If {xs} is the associated payoff configuration, then ls · xs = v(S) so VJc is a positive multiple of v and, therefore, X N = x = rp(v) . Conversely, define xs = rp(vls). Then {x s} satisfies the three conditions of the definition and xN = rp (v) is the Harsanyi NTU value payoff. Like the Shapley NTU value, the Harsanyi NTU value is an extension of the NBS to general NTU games in the sense that the Harsanyi value coincides with the NBS for
V E rjJ .
PROPOSITION 4 . 1 .4. Suppose that V E r{ lfy is a Harsanyi NTU value payoffvector for V, then y = a (V). PROOF. Suppose y is a Harsanyi NTU value payoff for V . Then Definition 4.1.3 implies that there exist numbers {aT}o#T C:: N and a payoff configuration {xs} with xs E a V (S) for all S such that XN = y and
Aix� = L aT T C:: S : i ET
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for each i E S � N. Hence, Ah/n = auJ so auJ = A.id� . Let S = {i, j }. Then )._i x1 = a{i J + as = 'Aid� + as so x1 = 4 + as/'A; . Since xs and (d� , d�) are both in CJ V(S) it follows that as = 0. Using this argument inductively, we conclude that as = 0 if 1 < lS I < n. Hence, x1 = d� whenever 1 < lSI < n. Therefore,
j ' i (XNi - di ) V - aN - A' (X Nj - d Vj )
A
for all i, j E N . Combining this fact with condition (2) in Definition 4. 1.3, we can use the argument in the proof of Proposition 3 .1.3 and conclude that XN = y is the NBS for V. D 4.2.
The Harsanyi solution
Harsanyi's general approach for NTU games can be given an axiomatic foundation if we take the point of view that the entire payoff configuration should be treated as the outcome. DEFINITION 4.2. 1 . A payoff configuration {xs} is a Harsanyi solution for V E rN if there exists 'A E Dr�+ such that: (1) xs E CJV(S) for all S � N , (2) 'A · XN = maxzEV(N) 'A · z , (3) 'A s * xs = cp(ih l s) for all S � N where fh(S) = 'A s · xs.
Note that we are distinguishing between a Harsanyi solution payoff configuration and a Harsanyi NTU value payoff vector. (In the literature, the term "Harsanyi solution" is ordinarily used for both of these concepts.) Obviously, it follows from the defini tion that if {xs} is a Harsanyi solution, then XN is a Harsanyi NTU value payoff. The correspondence that associates with each V E rJ the set of Harsanyi solution payoff configurations for V is called the Harsanyi solution correspondence and is denoted H. Let fJ be the set of V E rJ such that H(V) # 0. In the axioms below, U, V and W are games and t/f : fJ � D s c N R s is a payoff configuration valued correspondence. HO: (Non-Emptiness) t/f(V) # 0. Hl : (Efficiency) If {xs} E t/f (V) then xs E a V(S) for each S � N. H2: (Conditional Additivity) If U = V + W, {xs} E t/f(V), {ys} E t/f(W) and xs + Ys E CJU(S) for each S, then {xs + ys} E t/f (V). H3: (Unanimity Games) If T � N is nonempty, then t/f(Uy) {zs} where zs = Xf/ I T I if T � S and zs = 0 otherwise. s H4: (Scale Covariance) t/f(a * V) = {{a * xs} I {xs} E t/f(V)} for every a E Dr�+ · HS: (Independence of Irrelevant Alternatives) If {xs} E t/f (W) and if xs E V(S) � W(S) for each S, then {xs} E t/f(V). The axioms for the Harsanyi solution correspondence on fJ can be readily compared with those for the Shapley correspondence on fk. Axioms HO-H5 are payoff config=
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uration analogues of AO-A5 and they are used in Hart (1985a) to prove the following characterization theorem. THEOREM 4.2.2 [Hart (1985a)]. A correspondence t/1 : f� --+ fl s c N R 5 satisfies Axioms HO-H5 if and only if t/1 H. =
For v E G N , let h(v) = {xs} where xs cp (v ls). If V corresponds to v, then {h(v)} so rJ/ :; f� . (See the discussion preceding Proposition 4. 1 .4.) Now the proof proceeds in several steps: Step 1 : Since t/1 satisfies HO-H4, it follows that
H ( V)
=
=
The Harsanyi correspondence can be axiomatically characterized on r� if H (V) is not required to be nonempty. To accomplish this, H3 must be strengthened and an additional axiom must be imposed. H3 + : (Unanimity Games) For each real number c and nonempty T :; N, let Ufj be the game corresponding to the TU game cuT. Then t/I(Ufj) {zs} where zs (c/ I T I )x f if T :; S and zs 0 otherwise. H6: (Zero Inessential Games) If 0 E B V (S) for each S, then {0} E t/I(S). =
=
=
4.2.3 [Hart (1985a)]. A correspondence t/1 : r� --+ fl s -cN R 5 satisfies Axioms H1, H2, H3+ , H4, H5 and H6 if and only if t/1 H.
THEOREM
=
We conclude this section with a brief discussion of a weighted generalization of the Harsanyi solution analogous to the w-Shapley NTU value. DEFINITION 4.2.4. Let w E ffi�+ · A payoff configuration {xs} is a w-Harsanyi solu tion for V E rN if there exists A E ffi�+ and a collection of numbers {aT }0#T<; N such that: (1) xs E B V (S), (2) A · XN = maxzE V(N) A · z , (3) ;J * x� = w ; LTc; S : i E T CY T ·
If the w 1 are all equal then the w-Harsanyi solution reduces to the Harsanyi solution. Axiom H3 (or H3+) imposes symmetry on the NTU value. If H3 is generalized in the same way that A3 is generalized to A3', then the class of weighted Harsanyi solutions can be axiomatized. Other nonsymmetric generalizations of the Harsanyi solution [in cluding NTU extensions of the TU coalition structure values studied in Owen (1977),
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Hart and Kurz (1983) and Levy and McLean (1989)] are discussed in Winter (1991) and Winter (1992).
4.3. A comparison of the Shapley and Harsanyi NTU value EXAMPLE 4.3 . 1 . Let N = { 1 , 2, 3 } and for each c E [0, 1 /2], consider the game V8 defined as: Vc(i) = {x E R[ i ] I x :::;; 0} ; V�: (1, 2) = {x E R[ l ,2} I (x 1 , x2) :::;; c1. 1)} ; V8 (1 , 3) = {x E R[1 ,3} I (x 1 , x 3 ) :::;; (c, 1 - c )} ; V�: (2, 3) = {x E R l 2 ·3l I (x2 , x3) :::;; (c , 1 - c )}; V8 (N) = {x E R N I x :::;; y for some y E C}; where C = convex hull {(1/2, 1 /2 , 0) , (c, 0, 1 - c ) , (0, c, 1 - c)}.
This game appears in Roth (1980) and has generated a great deal of discussion in the literature [see Aumann (1985b), Roth (1986), Aumann (1986) and Hart (1985b)] which we summarize here. The Shapley NTU value is ( � , � , �) with comparison vector A = ( 1, 1 , 1) and the Harsanyi NTU value is C1 - �. 1 - �. ¥) with the same A. Es sentially, Roth argues that ( 1 , 1 , 0) is the only reasonable outcome in this game since ( 1 , �) is preferred by { 1 , 2} to any other attainable payoff and { 1 , 2} can achieve (!, 1) without the aid of player 3. In this example, ( � , �, �) is the NTU value because the TU game V)c corresponding to V8 is symmetric although the NTU game V8 is certainly not symmetric. Aumann counters that there are circumstances in which ( �, � , �) is an expected outcome of bargaining. If coalitions form randomly, then 1 might be willing to strike a bargain with 3 in order to guarantee a positive payoff since, if he were to reject an offer by 3, then 2 might strike a bargain with 3 leaving 1 with nothing. A similar argument holds for 2. This argument is less compelling when c is very small. Hence, one might argue that 3 should get something but that the payoff to 3 should decline as c declines. In fact, it is precisely the Harsanyi NTU value that is close to ( � , ! , 0) for small c and, therefore, may better reflect the underlying structure. EXAMPLE 4.3 .2. Let N = { 1 , 2, 3 } and for each c E [0, .2] consider a three-person, two-commodity pure exchange economy with initial endowments a1 = (1 - c, 0), a2 = (0, 1 - c) and a3 = (c, c ) and utility functions U J (X , y) = min{x, y } = u 2 (x , y) and u 3 (x, y) = ( 1 /2)(x + y). The NTU game V8 associated with this economy is: VE (i) = {x E Rli ) I x i :::;; 0} if i = 1 , 2; V£ (3) = {x E R l3l I x3 :::;; c}; VE ( 1 , 2) = {X E R { I ' 2} I X I + X 2 :::;; 1 - c , X 1 :::;; 1 - c, X 2 :::;; 1 - c}; <..:::: l +£ x i < c x3 -... <..:::: 1 +£ } if i = 1 ' 2· V8 (i 3) = {x E R l i· 3l I (x i ' x3) -... 2 i N V8 (N) = {x E R ix 1 + x2 + x3 :::;; 1 , x :::;; 1 for each 2i } . '
'
�
'
'
This example i s a simplified version of one that appears in Shafer (1980) and i s due to Hart and Kurz (1983). The payoffs at the unique Harsanyi NTU value (evaluated at the
R.P. McLean
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=
corresponding allocation) with comparison vector (1, 1 , 1) are: u 1 ( 1 - ¥- , 1 - ¥- ) = 1 - ¥-, u 2 ( � - ¥-, 1 - ¥-) 1 - ¥- , and u 3 (¥ , ¥ ) = ¥ · The payoffs at the unique Shapley NTU value (evaluated at the corresponding allocation) with comparison vector 5 - 12 5t: • 12 5 - 12 5t: ) = 12 5 - 12 5e • u 2 ( 12 5 - 12 5e • 12 5 - 12 5e ) = 12 5 - 12 5e • and ( 1 , 1 , 1) are.. U J ( 12 1 5s + t: 5s 5 ) = 61 + 6 U 3 ( 61 + 6 · •6 6
Shafer argues that the Shapley NTU value is not a reasonable solution in this "economic" example. If £ = 0, then player 3 has no endowment yet he receives a nonzero allocation because of the "utility producing" properties of his utility function. The com putation of the Shapley NTU value depends upon the assumption that utility is trans ferable and this is clearly at variance with the usual notions of ordinality in general equilibrium theory. However, one can use the same reasoning as found in the previous example and argue that if coalitions form randomly, then 3 might reasonably receive a nonzero payoff though such a payoff must be small if £ is small. This is precisely the property exhibited by the Harsanyi NTU value in this example. What conclusion is to be drawn from these examples? In Hart (1985a), a case has been made that the Harsanyi NTU value is a good solution concept for NTU games with a small number of players while the Shapley NTU value is appropriate for "large" games. The latter statement seems justified in light of the equivalence principle discussed in Chapter 36 and Remark 7.2.7 below. 5. Multilinear extensions for NTU games
5.1. Multilinear extensions and Owen 's approach to the NTU value Another approach to the value of an NTU game is provided in Owen (1972b) and is based on the concept of multilinear extension. Let v E G N and define fv : [0, 1]" -+ m as
[
]
fv(XJ , . . . , Xn ) = L n X; n (1 - x;) v(S). Sr;N iES i�S The function fv is called the multilinear extension of v (MLE for short) and is defined in Owen (1972a). For each i E N, it can be shown that
a� = a
(1 - Xj ) [v(S u i) - v(S)] � s L n Xj n r; N\i jES u su; I
so that
ax·
afv (t, . . . , t) I
=
" L
Sc;._N\i
t i S I (1 - t) n-I S I- 1 [ v(S U i) - v (S) J.
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Upon integrating with respect to Lebesgue measure, one obtains
11 ofv (t, . . . , t) dt 0 O Xi
=
i (v).
In Owen (1972b), the MLE is defined in an analogous way for NTU games. For each V E rN , recall that V * (S) = V (S) {ON I S } and (abusing notation slightly) define x
[Tl
J
xi flo - xi ) v * cs) . tv ex ] , . . . , xn ) = L S�N iES i#S For each x E [0, 1]", fv (x) is the sum of comprehensive, convex sets and is there fore comprehensive and convex. Suppose that there exists a differentiable function Fv : ffi N X rn N -+ ffi with the property that ofv (x) = {u E rn N I Fv (u , x) = 0} when ever x > 0. That is, Fv implicitly defines the Pareto boundary of fv (x) . To construct a value for V, Owen makes the following informal argument. Bargaining takes place over time. At time t E [0, 1], player i has accumulated a payoff of u; (t) as a result of a "com mitment" of Xi (t) up to time t. If the vector u(t) is Pareto optimal given x(t) > 0, then Fv (u(t), x(t)) = 0. If the payoff and commitments of j E N\i are fixed at time t , then a change in i 's commitment from Xi (t) to xi (t + 1'-,.t) should result in a corresponding change in i 's payoff so as to maintain Pareto optimality. This means that
If ri is the rate of change of commitment of player i, then (assuming x, u and Fv are differentiable) we are led to the following system of differential equations for i E N :
-[ :;� }i (t), 3Fv
it i (t)
=
Xi (t) U i (O)
= =
auj
ri (t), Xi (O) = 0.
The initial conditions are interpreted to mean that at t = 0, each player is committed and has accumulated a payoff of zero. If (£, u) is a solution to this system with x(t) > 0 for 0 < t ::( 1 , then for each such t, Fv (u(t), x(t)) 0 so that u (t) is Pareto optimal for fv (x(t)). In particular, u(1) is Pareto optimal for fv (x(1)) = fv (1, . . . , 1) = V (N) . Owen refers to u(l) as a "gener alized value" and has shown that both the Shapley value and the NBS are generalized values for the appropriate systems of differential equations. =
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PROPOSITION 5 . 1 . 1 [Owen (1972b)]. Let V correspond to v E G N . Then for each x > 0, Fv (u, x) = Li EN Ui - fv(x). Furthermore, ifri (t) = 1 for each i, then there is a solution (x, u) to the system of differential equations with it (I) = cp(v ) . PROOF. If fv is the MLE of v, then fv (x) = {u E lRN I L iE N Ui � fv(x) whenever x > 0 so that Fv (u, x) = L iE N ui - fv(x). Hence, ��� = - ��� and ��� = 1 so
dui dt
ofv (xi (t) ) OXi
ofv(t, . . . , t) OXi
Upon integrating and using the fact that u(O) = 0, we obtain 0
PROPOSITION 5 . 1 .2 [Owen (1972b)]. Let V E r/j with dv = 0. Then for each x > 0, Fv (u, x) = g(u f fltE N x;). Furthermore, ifn (t) = 1 for each i, then there is a solution (x , u) to the system of differential equations with u(1) = rr (v).
PROOF. Since dv = 0, it follows that fv (x!, . . . , xn) = [H E N xi]V(N). If the Pareto optimal surface of V(N) is defined by g(u) = 0, then Fv (u, x) = g(ujfl iE N x;) when ever x > 0. Thus,
iJ Fv OUi
8i (u f fliE N xi ) nj E N Xj
where g; is the partial derivative of g with respect to the ith argument. In addition,
Now let u* be the NBS for V. Since V(N) is convex, it follows that g(u*) = 0 and u7g; (u*) = ujgj (u*) for each i, j E N. It is now easy to verify that x; (t) = t and u; (t) = t" u7 solve the system of differential equations. Since u(l) = u*, we have the 0 desired result. REMARK 5 . 1 . 3 . In the case of Proposition 5 . 1 . 1, it is not difficult to show that the initial value problem associated with Fv has a unique solution (i.e., the one given in the proof of the proposition) and, therefore, the Shapley value is the unique "gener alized value" of an NTU game derived from a TU game. The uniqueness question is more complicated in the case of Proposition 5.1.2 due to the behavior of the function Fv (u , x) = g(uj fliE N x;) near (u, x) (0, 0) . However, Hart and Mas-Colell (1991) =
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have shown that there is a unique solution to the initial value problem associated with this Fv as well. Hence, there is a unique generalized value for games in rJ which is precisely Nash's Bargaining Solution.
5.2. A non-symmetric generalization In the previous section, r; (t) was interpreted as the rate of change of commitment of i. In Owen (1972a), it also i s interpreted as the density function of a random variable that describes the time at which i joins a coalition. In Propositions 5 . 1 . 1 and 5 . 1.2, this random variable is assumed to be uniformly distributed on [0, 1] but there are other possibilities. Let w = ( w 1 , . . . , w n ) E ffi�+ and define r; (t) = wit w ; I . For this density, larger values of the weight parameter correspond to higher likelihoods of "late arrival". -
PROPOSITION 5 .2. 1 . Let V correspond to v E G N· Then for each x
> 0, Fv (u, x) = L; EN u; - fv (x). Furthermore, if r; (t) = wit wi -I for each i, then there is a solution (x . u) to the system of differential equations with u(1) = 'Pw (v).
PROOF. Following the proof of Proposition 5 . 1 . 1 , we deduce that
u;"" (1) =
lnl 0
.
V fv (t w 1 , . . . , t wn )w i t W1 - l dt.
Since this integral is q;� (v) [Owen (1972a) or Kalai and Samet (1987)], the result fol lows. D PROPOSITION 5 .2.2. Let
0, Fv (u, x) = w g(ujfl i EN x;). Furthermore, ifr; (t) = wi t i-l for each i, then there is a solution (x , u) to the system of differential equations with u(1) = CTw (V ) . V
E rj$ with dv = 0. Then for each x
>
PROOF. As in Proposition 5 . 1 .2, let dv = 0 and let the Pareto optimal surface of V (N) be given by g(u) = 0. Then {u*} = crw ( V ) if and only if g(u*) 0 and =
u* u* g; (u*) ---..L = g1 (u*) ---..!..,w' wJ for each i, j E N. Letting W = L EN wi , it is now easy to verify that x; (t) = t w ; and i u; (t) = tw ur solve the system of differential equations with r; (t) = w it wi - i for each D i E N. Since u; (1) = u7, we have the desired result.
5.3. A comparison of the Harsanyi, Owen and Shapley NTU approaches Although the solution derived from the MLE coincides with the Shapley value on TU games and the NBS on a class of pure bargaining games, it does not coincide with the
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NTU value or the Harsanyi value for general NTU games. To see this, consider the following example from Owen (1972b).
5. 3 . 1 . Let N = { 1 , 2, 3} and define V as follows: V(l , 2, 3) = {u E ffiN i ul + uz + u3 :( 1, u; :( l , u; + uJ :( l for all i, j}; V(l, 2) = {u E ffi l l . 2 l I U J + 4u 2 :( 1 , U J :( 1 , u 2 :( 1 14} ; v (S) = { u E m 5 I u; :::;;; 0 for all i E S} otherwise.
EXAMPLE
The value for V computed using the MLE is u(l) = (0.5185, 0.4757, 0.0058). How ever, VJc is defined for A. = ( 1 , 1 , 1) and VJc (N) = 1 , v;,(l, 2) = 1 and VJc (S) = 0 oth erwise, so the Shapley NTU value payoff is {(1 12, 112, 0)}. Furthermore, the Harsanyi NTU payoff is (2I 5, 2I 5, 1 I 5) with A. = ( 1 , 1 , 1) and the associated payoff configuration is {xs} where XN = (215, 215, 115), X [ J , 2 } = (115, 1 15) and xs = 0 otherwise. 6. The Egalitarian solution
6.1. Egalitarian values and Egalitarian solutions Up to now, this chapter has been concerned with solutions for NTU games that yield the NBS and Shapley value on appropriate subclasses. However, one could take any pair of "related solutions" for pure bargaining games and TU games and look for a common extension to a larger class of NTU games. This program has been carried out in Kalai and Samet (1985) where they define the w-Egalitarian solution for NTU games which coincides with the w-Shapley value on r};u and the w-proportional solution r w on r!J . There are actually two possible approaches to the Egalitarian solution, depending on whether one defines a "solution" to be a payoff vector in mN or a payoff configuration in n s� N ffi 5 . DEFINITION 6. 1 . 1 . Let w E ffi�+· A payoff configuration {xs} is a w-Egalitarian so lution for V E rN if there exists a collection of numbers {ar }0# T �N such that: (1) xs E a V (S) for each S s; N, (2) x� = w ; L T � S : i E T ar for all i E S and all S s; N.
6. 1 .2. A payoff vector x E ffi N is a w-Egalitarian value payoff for V E rN if there exists a w-Egalitarian solution payoff configuration {xs} for V with X N = x.
DEFINITION
REMARK 6. 1 .3. Comparing Definitions 4 . 2.4 and 6. 1 . 1 , we see that (at least struc turally) a w-Egalitarian solution "looks like" a w-Harsanyi solution. As the reader may verify, the following more precise comparison can be made. If {xs} is a w-Egalitarian solution for V and 1 N XN = max[lN z I z E V (N)], then {xs} is a w-Harsanyi so lution with comparison vector IN . On the other hand, every (symmetric) Harsanyi ·
·
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solution happens to be a w-Egalitarian solution. In particular, a payoff configuration {xs} is a (symmetric) Harsanyi solution for V if and only if there exists A E m�+ such that A · XN = max[A z I z E V (N)] and {xs} is a w-Egalitarian solution with W = ( 1/A 1 , . . . , 1 /A ) . ·
11
The motivation for the definitions of Egalitarian solution and value is best understood by considering a two-person game V E r!J . In this case, a payoff vector x = (x 1 , x 2 ) is a (symmetric) Egalitarian value payoff if:
Therefore, x is precisely the (symmetric) proportional solution r (V) for the two person bargaining problem as described in Section 2.4 [see Kalai (1977b)]. Geometri cally, x is the intersection of the 45° line emanating from the point (dt , d�) and the boundary of V ( 1, 2). In our discussion of the NBS, we stated that an arbitrator in ideal circumstances would like to choose a solution (x 1 , x 2 ) that satisfies utilitarianism (i.e., x 1 + x 2 = max[z 1 + z2 I z E V(1, 2)]) and equity (i.e., x 1 - 4 = x 2 - 4). Since these are generally incompatible, the NBS seeks a point (x 1 , x 2 ) for which there exists some A E m�;l such that A * x will simultaneously satisfy these two criteria in the rescaled problem A * V. By comparison, the proportional solution maintains the equity condition that x 1 - dt = x 2 - d� but dispenses with utilitarianism by requiring that x merely be efficient for N. Hence, conditions (1) and (2) of Definition 6.1 . 1 extend respectively the notions of efficiency and equity as they are interpreted in the proportional solution. When V corresponds to a TU game v, it follows immediately from the definitions that the w-Egalitarian, w-Shapley and w-Harsanyi NTU value payoffs for V are identical and coincide with the w-Shapley value for V. Therefore, the w-Egalitarian value is a solution concept that extends the w-Shapley value and the w-proportional solution to the class of general NTU games. 6. 1 .4. Definitions 6. 1 . 1 and 6.1.2 translate into an algorithm for comput ing w-Egalitarian values and solutions. For each i, define CY[i J = d� j w • For each S with lSI � 2, inductively define as as max[t I zs(t ) E V(S)] where z � (t ) = w i [t + L T <; S , T #S: i ET ar ] . If this process is well defined for each S s; N, then {zs(as)} is a w-Egalitarian solution payoff configuration and Z N (aN) is a w-Egalitarian value payoff. REMARK
1
6.2. An axiomatic approach to the Egalitarian value The correspondence that associates with each V the set of w-Egalitarian value pay offs will be denoted E w . In this section, we will provide an axiomatization of the w Egalitarian value correspondence on the class r� (see Section 1 .3). From Remark 6. 1 .4, it follows that Ew(V) can consist of at most one element. Since Ew (V) =1- 0 for each v E r�, we will treat E w : r� m N as a function. -+
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To state the axioms, we need a few more definitions and we follow the terminology in Hart (1985c). If F : r� --+ mN is a function, then the vector (F(V))7 E m7 will be written F7 (V). A coalition T s; N is a carrier for V if V (S) = V (S n T) x m�\ T whenever S n T =f:. 0 . A game is individually rational monotonic (IR monotonic for short) if V+ (S) x V+(i ) s; V+(S U i) where V+(T) = {x E V(T) I x i ;?: d� , for each i E T}. If 0 =/= T s; N and a E mN with a i = 0 for i fj. T, define the NTU game Vr,a as follows: Vr , a (S) = {a s } + Dl� if T s; S and Vr,a(S) Dl� otherwise. In the axioms below, F : r� --+ m N is a function and V and W are NTU games. El: (Individual Rationality) If V is IR monotonic then F i (V) ;?: d� for each i E N. s E2: (Carrier) If S is a carrier for V, then F (V) E oV(S). E3: (Additivity of Endowments) If a E mN with a i = 0 for i fj. T and if F(Vr,a) = a, then F(V + Vr,a) = F(V) + a. E4: (Egalitarian Monotonicity) If T s; N, V(T) s; W(T) and V(S) = W(S) for all S =/= T, then F7 (V) � F7 (W). E5: (Continuity) F is continuous with respect to the Hausdorff topology on r�. E l states that the payoff to player i must be individually rational if whenever i joins S s; N\i, such an arrangement does not eliminate individually rational payoffs that were available to i and S separately. E2 combines an efficiency property and a null player property. Since any superset T of a carrier S is also a carrier of V, it follows that F7 (V) E oV(T) whenever S s; T. In particular, F(V) E oV(N). If S is a carrier of V and i fj. S, then E2 implies that Fi (V) � 0. E2 is an NTU extension of a similar axiom used in the characterization of the TU Shapley value [see Shapley (1953b)] . Axiom E3 has the following interpretation. Let a E m N with a i = 0 for i fj. T and let V be an NTU game and define W = V + Vr,a · Then W(S) V (S) + {a s } if T s; S and W(S) = V(S) otherwise. Hence, W simply translates the payoffs of a superset S of T by a s and leaves the attainable sets of other coalitions unchanged. If a is acceptable in the sense that F(Vr,a) = a, then E3 requires that the solution for W be precisely F(V) + a. Egalitarian monotonicity states that the members of a coalition T should be no worse off if their attainable set expands while those of all other coalitions remain unchanged. Finally, E5 is a regularity condition. The main characterization theorem for the Egalitarian value may now be stated. =
=
6.2. 1 [Kalai and Samet (1985)]. A function F : T� --+ mN satisfies El-E5 if and only if there exists w E Dl�+ such that F Ew.
THEOREM
=
6.3. An axiomatic approach to the Egalitarian solution The Shapley and Harsanyi NTU values do not satisfy E3 [see Hart (1985c), p. 3 1 8] or E4 [see Kalai and Samet (1985)] . In addition, the Shapley and Harsanyi correspondences have closed graphs but are not continuous. There is also another important distinction between these solutions (first indicated in Kalai (1977b) for aw vs Tw ) that lies in the way they respond to the rescaling of utilities. The Shapley NTU value and the Harsanyi
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solution are scale covariant in the sense of Axioms A4 and H4 while the Egalitarian value and the Egalitarian solution satisfy such a condition only for rescalings that put the same weight on each individual's utility. Despite these differences, the Harsanyi solution and the Egalitarian solution are more closely related than one might believe at first glance. Recall that Theorem 4.2.3 states that a payoff configuration valued correspondence satisfies H1, H2, H3+, H4, H5 and H6 on rJ if and only if it is the Harsanyi solution. The next result shows that by drop ping H4 (scale covariance) and enlarging the domain of the other axioms to all of rN , we can actually characterize the Egalitarian solution in terms of the axioms used in the characterization of the Harsanyi solution. Let Ew denote the payoff configuration val ued correspondence that associates with each V E rN the set of w-Egalitarian solution payoff configurations for V. The symmetric Egalitarian solution correspondence will be written simply as E. THEOREM 6.3 . 1 [Samet (1985)] . A correspondence IJ! : TN --+ fl s c-N ffi 5 satisfies HI, H2, H3 +, H5 and H6 if and only if iJ! = E. We can characterize Ew on rN if we modify the unanimity games axiom H3+ in an appropriate way. H3 w : (Unanimity Games) For each real number c and nonempty T s; N, let Uf be the game corresponding to the TU game cur . Then IJ! ( Uf ) = {zs} where z� (c w i I L } E T w i ) if i E T s; S and z� = 0 otherwise. =
THEOREM 6.3.2 [Samet (1985)]. A correspondence IJ! : TN --+ fl s-c N ffi5 satisfies HI,
H2, H3 w , H5 and H6 if and only if IJ! = Ew . 7. Potentials and values of NTU games
7.I. Potentials and the Egalitarian value offinite games In Hart and Mas-Colell (1989), the notion of potential for TU games was introduced. These authors used the potential to provide a new axiomatization of the Shapley value for TU games using the concepts of reduced game and consistency and we briefly review their approach next. In this section and in Section 8 below, let G denote the set of all pairs (N, v ) where N is a finite set and v is a TV game in G N . If ( N, v) is a TU game and S s; N, we will write (S, vis) simply as (S, v). In a similar fashion, let r denote the set of all pairs (N, V) where N is a finite set and V is a NTU game in rN satisfying the nonlevelness condition (3) in Section 1 .3. In addition, let r T U denote the games (N, V) where v E r;;u .
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Let w { w i } be a collection of positive weights. A w-potential on G is a function G -+ ffi satisfying 0 and
Pw :
=
Pw(0, v) = _2]Pw(N, v) - Pw(N\i, v)] = v(N).
i EN
The following result provides a characterization of the weighted TU Shapley value in terms of potentials. THEOREM 7 . 1 . 1 [Hart and Mas-Colell (1989)]. There exists one and only one w potential on G. Furthermore,
for all i
E N.
The definition of potential can be extended to NTU games. A w-potential on r is a function : r -+ ffi satisfying V) = 0 and
Pw Pw (0, (w;[Pw(N, V) - Pw(N\i, V)]) i EN E BV(N) whenever N i= 0.
THEOREM 7 . 1 . 2 [Hart and Mas-Colell (1989)]. There exists one and only one w potential on r. Furthermore,
w; [ Pw(N, V) - Pw(N\i, V) ] = E� (N, V) for all i E N. Recall (see Remark 6. 1 .3) that a payoff configuration {xs} is a (symmetric) Harsanyi solution for V if and only if there exists A ffi�+ such that A · X N = max[A · z I and {xs} is a w-Egalitarian solution with w (1/A1 , . . . , 1/A11). Hence, we have the following characterization of Harsanyi NTU value payoffs in terms of potentials. If (1/A 1 , . . , 1/A11). A ffi�+' let A
V(N)] E
E
-I :=
zE
=
.
THEOREM 7 . 1 .3 [Hart and Mas-Colell (1989)]. A payoff vector x ffi N is a Harsanyi if and only if there exists a collection of NTU value payoff vector for the game positive weights {A; } such that
(N, V)
Ai x i =
PA (N, V) - P'A (N\i, V) -1
-1
for each i
E
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and
A x = max A z. ·
z E V (N)
·
7.2. Potentials and the Egalitarian solution of large games An NTU game form is a correspondence V : ffi� � ffi and a finite type NTU game with a continuum of players is defined as a pair (a, V) where a E ffi�+ and V : ffi� � ffi is a game form. A game form has the following interpretation. Each profile x E ffi+ is interpreted as a "coalition" and xi represents the total "number" (i.e., mass) of players of type i in the coalition. If a E V (x), then a i represents the per capita payoff to the players of type i in the coalition x . Hence, a i xi is the total payoff to the type i players in coalition x . A solution function for a game form V i s a mapping � : ffi�+ � ffin such that � (x) E a v (x) for each X E ffi�+ . In particular, � i (x) is interpreted as the payoff to each type i player and the total payoff to the players of type i is therefore x i � i (x) . The characterization of the Egalitarian solution for finite NTU games in terms of a potential function presented in Theorem 7 . 1 .2 suggests an extension to the case of large games with finitely many types. If V is a game form, then a solution function � : ffi�+ � ffi" for V is Egalitarian if � admits a potential, i.e., if there exists a differentiable function P : ffi�+ � ffi such that
� (x) = V P(x) for each x E ffi�+ . Thus, we are ideally interested in the existence of a differentiable function p such that v P(x) E a V (x) for each X E ffi�+· Unfortunately [as shown in Hart and Mas-Colell ( l995a)], such a function may not exist, even for "well-behaved" game forms. This necessitates a weaker definition of solution and potential. DEFINITION 7 .2. 1 . Let V be a game form. A generalized solutionfunction for V is a mapping � : ffi�+ � ffin such that � (x) E a V (x) for almost all x E ffi�+.
7 .2.2. A function P : ffi� + � ffi is a Lipschitz potential for V if P is Lip schitz continuous and V P (x) E a V (x) for each x E ffi�+ at which P is differentiable.
DEFINITION
Every Lipschitz potential for V defines a generalized solution since (by the classical Rademacher theorem) a Lipschitz continuous function is differentiable almost every where. Even if we content ourselves with Lipschitz potentials (and their induced gener alized solutions), we are still left with the problem of choosing a reasonable candidate from among the different Lipschitz potentials that may exist for a given game form. Tak: ing their cue from certain problems in the theory of partial differential equations [see, e.g., Fleming (1969)], Hart and Mas-Colell (1995a) introduce the variational potential and show that, while not necessarily the unique Lipschitz potential, it does satisfy many
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desirable properties and is the "right" potential if one approaches the problem from an asymptotic viewpoint. To construct the variational potential, let v : !R'j_+ X mn --+ ffi u { 00} denote the support function of the set V (x), i.e.,
v(x, p) = sup p · a. a EV(x)
Next, let AC(x) denote the set of absolutely continuous functions y : [0, 1] --+ !R'j_+ U { 0} satisfying y (0) = 0 and y ( 1) = x . The variational potential for V is defined for each x E !R'j_+ as follows:
P* (x) =
min
t v(y(t), y (t) ) dt.
y(·)EA C (x) lo
In order for P* (x) to be well defined (and have other desirable properties), the game form V must satisfy certain restrictions. Let Q denote the set of game forms V ffi'j_ --+ ffi satisfying (A. l)*-(A.3)* and (A.4)-(A.9) in Hart and Mas-Colell (1995b). These as sumptions include the familiar requirements that, for each x E ffi'j_, V (x ) be a nonempty, closed, convex, comprehensive, proper subset of mn . However, these assumptions also guarantee that the support function of the set V (x) satisfies certain regularity require ments, including differentiability in x . :
THEOREM 7 . 2 . 3 [Hart and Mas-Colell (1995a)]. Let V E Q . For each x E !R'j_+, P * (x ) is well defined and the resulting function P* : ffi'j_+ ffi is Lipschitz continuous hence differentiable almost everywhere. Furthermore, V P* (x) E {) V (x ) whenever P* is dif ferentiable at x. --+
To illustrate the variational potential, suppose that V is a TU game form defined by
where f : ffi'j_+ !R is a (sufficiently well behaved) differentiable function. For each t, the support function v evaluated at (x, p) = (y (t), y (t)) is finite only if for each i , y i (t) = f3(t)y i (t) for some f3 (t) ;:;::: 0. If we choose the normalization f3 (t) = Ijt, then Yx = tx is a solution to this system of differential equations satisfying the boundary conditions Yx (0) = 0 and Yx (1) = x . Therefore, --+
( x (t) , yx (t) ) = i f(tx)
v y
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so that
P * (x) = and we obtain
aP* axi
2107
lo l l v(yx (t), yx (t) ) dt = -f(tx) dt 1 t 1 0
- (a) =
0
af 1 ax 1 i (ta dt 0
)
for each a E ffi�+ · For finite NTU games generated by TU games, the Egalitarian solu tion coincides with the TU Shapley value. In the case of NTU game forms, the Egali tarian solution of a game form generated by a finite-type nonatomic TU game coincides with the Aumann-Shapley value and is obtained by the familiar diagonal formula. In summary, each V E Q admits at least one Lipschitz potential, the variational po tential P*, and the variational potential can be used to define a generalized solution for the game (a, V). The variational potential may not be the unique Lipschitz potential even for well-behaved game forms [see Section V.8 of Hart and Mas-Colell (1995a)] so a given game form may have different generalized solutions corresponding to differ ent Lipschitz potentials. However, the variational potential does exhibit several features that indicate that it is the "right" definition of Egalitarian solution, especially for appli cations. Some of these are summarized in the following result: 7 .2.4 [Hart and Mas-Colell (1995a)]. Suppose that V E Q. (i) If Q is a Lipschitz potentialfor V, then P* (x) � Q(x) for all x E ffi�+. (ii) If V admits a continuously differentiable potential P, then P = P*. (iii) If V is a hyperplane game form, then the variational potential P* is the unique Lipschitz potentialfor V.
THEOREM
A complete specification of an Egalitarian solution requires that we define the solu tion for all profiles x E ffi�+' even those at which P* is not differentiable. If we are willing to allow the solution to be set-valued at points at which P* is not differentiable, then the Lipschitz continuity of P* suggests that Clarke's generalized gradient can be used to complete the description of an Egalitarian solution based on the variational po tential. 7 .2.5. For each a E ffi�+ and each V E Q, the Egalitarian solution E* (a, V) is defined as
DEFINITION
E* (a, V) = (V c P* (a) + ffi';_] n B V(a) where v c P * (x) denotes the Clarke generalized gradient at x [see Clarke (1983)].
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Hart and Mas-Colell (1995a) prove that v c P * (x) = {V P * (x)} whenever P * is dif ferentiable at x . Since * * v c P * (x) = conv { y I y = lim k V P (xk), Xk � x and P is differentiable at each Xk } , it follows that [Vc p * (x) + tll� ] n a v (x) =I= 0 if P * is not differentiable at X . Another strong justification for the variational Egalitarian solution is provided in Hart and Mas-Colell (1995b) where they develop an asymptotic approach to the def inition of Egalitarian solution for a game form V . In the spirit of the asymptotic value of nonatomic TU games, the game form V is approximated by a sequence of finite NTU games { Vr}. If the associated sequence of Egalitarian solutions { E ( Vr)} has a limit, then that limit is defined as the "asymptotic Egalitarian solution" of the game form V. Let a E tll�+ be a profile in which each a; is a positive integer. For each positive integer r, construct a finite NTU game (Nr , Vr) as follows. The player set Nr con sists of ra; players of type i and the total number of players is INr I = r(L; a; ) . For each S c; Nr , let m; (S) denote the number of type i players in S and let f.l/ (S) := (m 1 (S)jr, . . . , m11 (S)/r). Hence, f.l/ (Nr) = a. The set Vr (S) is defined as follows: z E Vr (S) if and only if there exists (a, , . . . , a11 ) E V(f.l.' (S)) such that zJ = a; whenever player j E N,. is of type i . As r increases the finite games (Nr , Vr) become better ap proximations of the continuum game (a, V). Since the potential preserves the symme tries of V,, it is completely determined by the values of f.i.r (S). To define the potential, let
L, = { (Z J , . . . , Z11) I for each i, 0 :(, z; :(, a; and rz; is an integer} . The potential of (Nr , Vr) is a function P,. L, � tR satisfying Pr (0) = 0 and for each z E L, , 11 • P ( r (z) - Pr (z (i) )) "i = i E aV(z) DP,. (z) = ( D; P, (z) ) i = i .= :
where Z (i)
_
-
(Z I , . . . , z i - 1 , Z i - r1 , Z i + l , . . . , z" )
if z ; ;?: 1/ r and z (i) = z otherwise. THEOREM 7.2.6 [Hart and Mas-Colell (1995b)]. Suppose that V E Q. Then for each a E m�+: (a) lim,_. 00 (1/r)Pr (a) = P * (a). (b) If P * is differentiable at a, then limr --+ 00 D P, (a) = V P * (a). (c) lim,._. 00 dist[D P, (a), E * (a, V )] = 0, where the dist[DPr (a), E * (a, V)] := inf{ II DP, (a) - � Il l � E E * (a, V)}.
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Property (a) states that the potentials of the approximating games converge to the variational potential P* of (a, V) for every a E m�+ - If P* is differentiable at a, then E * (a, V) { V P * (a) } . Property (b) says that if P * is differentiable at a, then the se quence of Egalitarian values for the approximating sequence of games converges to precisely the single Egalitarian value of (a, V) as defined by the variational potentiaL If P * is not differentiable at a, then the sequence of Egalitarian values for the approxi mating sequence of games may not converge. However, every convergent subsequence of {DP, (a)}� 1 will have a limit in [V c P* (a) + ffi�_] n oV (a) . =
REMARK 7. 2. 7 . Given the relationship between the Harsanyi NTU value and the Egal itarian value for finite player games (see Remark 6. 1 .3), it is natural to propose an anal ogous notion of Harsanyi value for finite type NTU games with a continuum of players using the variational potentiaL This is accomplished in Hart and Mas-Colell ( 1996b) where the authors show that the Walras equilibria of a large economy need not be Harsanyi value allocations. This stands in contrast to the Value Equivalence Principle: in an economy with many agents and smooth preferences the Walras equilibria and the Shapley NTU value allocations coincide. For the special class of "diagonal Harsanyi value payoffs", Imai ( 1 983) does provide an equivalence result for finite-type market games.
8. Consistency and values of NTU games
8.1. Consistency and TU games: The Shapley value Many solutions of TU games have axiomatic characterizations that employ some notion of "reduced game" and "consistency". [For a survey see Driessen (1991).] Let (N, v) be a TU game, let S be a subset of N and let 1/f be a solution. Loosely speaking, a reduced game v f is a game with player set S derived from v by paying the players outside S their payoffs according to 1/f . A solution 1/f is consistent if the payoff to a player in S when 1/f is applied to vf is the same as that player's payoff when 1/f is applied to the original game v. There are several possible constructions of a reduced game and Hart and Mas-Colell (1989) provide a definition that is appropriate for the study of the Shapley value. In particular, Hart and Mas-Colell show that the Shapley value is the unique solution that is "standard" for two-person games and is consistent with respect to their definition of reduced games. Let G and r be defined as in Section 7.1 above. A solutionfunction 1/f on G is a map ping that associates with each (N, v) E G a payoff vector 1/f(N, v) = (1/J i (N, v)) i E N E ffiN . A solution function 1/f on r is a mapping that associates with each (N, V) E r a payoff vector 1/f (N, V) (1/J i (N, V)) i EN E ffi N _ If (N, v) is a TU game, 1/f is a solution function on G, and T s; N, define the reduced game (T, v�) as follows: =
4 (S)
=
{ 0,v (s u (N\T)) - LiEN\T 1/fi (s u (N\T), v) ,
if 0 # S s; T; if s 0. =
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A solution function 1/J on G is consistent if for each game ( N, v) and each T s; N,
1/J ; (T, v� ) = 1/J i (N, v) for all i E T. THEOREM 8. 1 . 1 [Hart and Mas-Colell (1989)]. A solution function 1/J on G is con sistent and coincides with the Shapley value on two-person games if and only if 1/J(N, v) = q;(N, v) whenever (N, v) E G.
8.2. Consistency and NTU games: The Egalitarian value The definition of reduced game and consistency can be extended to NTU games in a natural fashion. For each solution function 1jJ on r, each game (N, V) E r and each T s; N, define the reduced game (T, VJ) as follows:
VJ (S) = { x E
�s 1 (x, (1/Jj (s u (N\T), v)j EN\T)
E
v (s u (N\T)) } .
A solution function 1/J on r i s consistent if, for each (N, V ) and T s; N, 1/r; (T, VJ) = 1/r ; (N, V) for all i E T. The definitions of reduced game and consistency coincide with those given in Sec tion 8.1 if V corresponds to a TU game. The following result extends Theorem 8. 1 . 1 to the NTU case. THEOREM 8 .2. 1 [Hart and Mas-Colell (1989)]. A solution function 1/J on r is consis tent and coincides with the proportional solution on two-person games if and only if
1/J(N, V) = E(N, V) whenever (N, V) E r.
8.3. The consistent value for hyperplane games Since the NTU definition of consistency reduces to TU consistency, we can apply The orem 8. 1 . 1 and make a simple observation: any solution function on r that satisfies consistency and coincides with the TU Shapley value on two-player TU games must coincide with the TU Shapley value on the class ofTU games. Theorem 8.2. 1 illustrates this idea since any solution that is proportional on two-player games must coincide with the Shapley value on two-player TU games. A natural question now arises: does there exist a solution function on r that satisfies consistency and coincides with the Nash Bargaining Solution for two-player problems? If this question has an affirmative answer, then the resulting solution would be different from the Egalitarian solution function but would still coincide with the TU Shapley value on TU games. In fact, Maschler and Owen (1992) have shown that such a solution function does not exist. In particular, they
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have shown that consistency, symmetry, efficiency and scale covariance are incompat ible. However, there does exist a solution function that extends the TU Shapley Value and the NBS as we will see in Section 8.4 below. In order to construct this solution, we need to extend the concept of marginal contri bution to NTU games. Let (N, V) be an NTU game. Suppose that R (i 1 , . . , in ) is an element of Q and recall that d� max{x i I x; E V(i)}. The marginal contribution of ik in R in the game V is defined inductively as follows: =
=
.
and
for each k � 2. Rearranging the numbers .1 i 1 (R, V), . . . , Ll in (R, V) yields the marginal worth vector
Since o (R, V) E a V (N) for each ordering R, a natural random order value for V is de fined by L RE.Q �o(R, V). However, this point is not guaranteed to be in a V(N) unless a v (N) is convex. If V E r T U , then a V(N) is convex and .1 ; (R, V) .1 ; (R, v) (as de fined in Section 1 .2) where v generates V . In this case, L RE .Q �o(R, V) is precisely the Shapley value payoff vector for the TU game that generates V. We will present a random order solution for general NTU games in the next section. First, we will con sider a class of games introduced in Maschler and Owen (1989) on which they have defined a random order value that satisfies a weakened version of consistency. An NTU game (N, V) is a hyperplane game if for each S s; N, there exists a vector as E m!+ and a real number b(S) such that =
V (S)
= {x E ms I as · x
:S;
b(S) } .
The set of all hyperplane games in r is denoted r H . Hyperplane games are NTU games associated with "prize games" [see Hart (1994)]. The consistent value on r H is the solution function K : r H -+ mn defined as
K (N, V)
=
1 o(R, V). '"'""" �" RE.Q n!
Note that K (N, V) E a V (N) if (N, V) E rH since every marginal worth vector o(R, V) E a V(N) and a V (N) is convex. Furthermore, K (N, V) ((J(N, v) if V E r T U and v is the TU game that generates V . =
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The consistent value on r H does not satisfy the Hart-Mas-Colell definition of con sistency but it does satisfy two weaker notions of consistency. DEFINITION 8. 3 . 1 . A solution function 1ft on r is completely consistent if for each (N, V) E r,
L
S<;N :ISI=k, iES
1/l i (
f C� � 1) ; l 1ft (N , V ) for all i E N s, v ) =
whenever 1 � k � IN I .
DEFINITION 8.3.2. A solution function 1ft on r i s globally consistent if for each (N, V) E r,
L 1/ti ( f ) = 2n -I 1/t i (N , V) for all i E N. s, v
Se N S#0
Clearly, consistency implies complete consistency and complete consistency implies global consistency. Complete consistency is defined in Maschler and Owen (1989) (where it is called consistency) and global consistency is defined in Orshan (1992). Global consistency can be used to axiomatize the consistent value on r H . Players i and j are substitutes in V if for every S s; N\ { i, j} , the following condition holds: for every E ffi 5 and for every � E ffi,
xs
xs � ) E V (S U j).
(xs, �) E V(S U i) if and on1y if (
,
If V corresponds to a TU game v, then i and j are substitutes in V if and only if they are substitutes in v as defined in Section 1 .2. If a E m N , let (N, v::dd) be the game defined by
v::dd(S) =
{x E ffi5 I Li ES Xi � z=iES a; } ·
In the axioms below, 1ft is a solution function on r. Cl: (Efficiency) 1/t(N, V) E B V (N). C2: (Symmetry) 1/ti (N, V ) = 1jf i (N, V) whenever i and j are substitutes in V . C3: (Scale Covariance) For each A E ffi�+ and each a E ffiN ,
( 1/t N, A * V + VJ:dd) = A * 1/t(N, V) + {a} .
THEOREM 8 . 3 . 3 [Maschler and Owen (1989), Orshan (1992)]. A solution function 1ft on r H satisfies Cl, C2, C3 and global consistency if and only ifl/t K. =
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Hart (1994) provides yet another axiomatization of K on r H and to present that result, we need one more definition. For each i and each S containing i, define:
Note that D i (S, V) is uniquely determined by the condition (D i (S, V), 1/f (S\i, V)) E
a ves).
(Marginality) If (N, V) and (N, W) are games and if Di (S, V) = Di (S, W) for each S <; N, then 1/f i (N, V) = ljf i (N, W) . If 1jf is efficient and if V corresponds to a TU game, then D i (S, V) = v(S) v ( S\i ) and C4 coincides with the marginality axiom for TU games found in Young (1988). In fact, the next result is an extension of Theorem 1 in Young (1988) to the hyperplane games. C4:
-
8 . 3 .4 [Hart (1994)]. A solutionfunction 1jf on r H satisfies Cl, C2 and C4 if and only if 1jf = K .
THEOREM
8.4.
The consistent value for general NTU games
We can now use the consistent value for hyperplane games as a tool to extend the con sistent value to a class of general NTU games. Let (N, V) be an NTU game and let A = {As}s� N be a collection with As E m!+ · We will call such a collection a compari son system. If sup[As · z I z E V (S)] is finite for each S, then we can define a hyperplane game VA where, for each S <; N,
VA (S) = {x E m s I AS . X � sup [As . z I z E V(S)] } . In this case, we say that the hyperplane game VA is definedfor V. If VA is defined for V, then the restriction (S, VA) is a hyperplane game with player set S and it follows that K(S, VA) E a VA (S) for each S <; N. As with the other solution concepts we have presented, we will distinguish between values as payoff vectors and solutions as payoff configurations. DEFINITION 8.4. 1 . A payoff configuration {xs} is a consistent solution payoff config uration for (N, V) if there exists a comparison system A = {As}s� N such that (i) xs E V (S) for all S; (ii) VA is defined for V; (iii) xs = K (S , VA) for all S. DEFINITION 8.4.2. A payoff vector x E ffin is a consistent value payoff for (N, V) if there exists a consistent solution payoff configuration {x s} such that x N = x.
Every hyperplane game (hence every game generated by a TU game) has a unique consistent payoff configuration and a unique consistent value. If (N, V) is a hyperplane
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game and if {as} s� N is the associated collection of normals, then VA is defined for V if and only if A = {tas}s� N for some t > 0. Hence, VA = V whenever VA is defined, so the only possible consistent solution is the payoff configuration { K (S, V)}s� N · Since V is a hyperplane game, it follows that K (S, V) E B V (S) for each S and we conclude that { K ( S, V)} s� N is the unique consistent solution payoff configuration for V and that K (N, V) is the unique consistent value payoff. An alternative characterization of consistent solution payoff configurations is possi ble. THEOREM 8.4.3 [Hart and Mas-Colell (1996a)]. Let (N, V) be an NTU game. A payoff configuration {x s} is a consistent solution payoffconfigurationfor V ifand only if there exists a comparison system A = {As}s� N such that: (i) xs E B V (S) for each S c; N; (ii) VA is definedfor V and AS · xs = m [A s z I z E V (S)] for each S c; N; (iii) For each S c; N andfor each i E S, ax
·
8.4.4. Suppose that V E r!J and suppose that {xs} is a consistent solution payoff configuration for V with an associated comparison system {AS} s� N . Condi tions (i) and (ii) imply that xs = (d� ); E S for each S c; N, S i= N and condition (iii) REMARK
implies that
L A� (x:v - d�) = L A� (x� - d0.
} EN \i
j EN \i
Therefore, A�(x� - d�) = * L} EN A�(x� - d�) for each i E N. Applying condition (ii) to S = N, we conclude (as in the proof of Proposition 3.1.3) that XN = o-(V). Hence, the consistent value coincides with the Shapley value on the TU games and the Nash Bargaining Solution on the pure bargaining games. Maschler and Owen (1992) provide an existence theorem for the consistent value using a fixed-point argument and Owen (1994) presents an example in which the consistent value is not a singleton set. The consistent value does not satisfy consistency on r (since the consistent value does not satisfy consistency on rH). In addition, Owen (1994) shows that the consistent value does not satisfy complete consistency on r. A consistency based axiomatization on a suitably large class of NTU games that contains r H and coincides with K on r H remains an open problem.
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8.5. Another comparison of NTU values Recall that, in the game V8 of Example 4.3.1, the Shapley NTU value payoff is (1, 1 . 1 ) with comparison vector A = ( 1 , 1 , 1) and that the Harsanyi NTU value payoff is ( ! � . ! - � . ¥- )with the same A. Roth argues that ( � , ! , 0) is the only reasonable outcome while Aumann counters that there are circumstances in which ( �, 1 , �) is reasonable as an expected outcome, though this argument is less compelling when c is very small. Hence, one might argue that 3 should get something but that the payoff to 3 should decline as declines, a feature captured by the Harsanyi NTU value in this example. The unique consistent value payoff vector is ( � + � � + � � ( 1 - c)) with comparison system A { 1 s}. This solution can be justified using an argument based on a repeated bargaining model. Suppose that, with equal probability, one player is chosen randomly to propose a feasible payoff in V8 (N) and suppose that player i has been chosen. If i 's first-round proposal is unanimously accepted, then the bargaining game ends and the players receive their specified payoffs. If at least one player rejects i 's proposal, then i exits the game with probability p and stays in the game with probability 1 - p . If S is the set of remaining players (so S = N with probability 1 - p and S N\i with probability p), then a member of S (possibly i if S = N) is chosen with equal probability to propose a feasible payoff in V8 ( S) in the second round, and the process repeats itself. When a player exits, he receives a payoff of 0 and never reenters the bargaining process. In order to obtain payoffs of 1 / 2 in this bargaining game, players 1 and 2 need the agreement of player 3 (since ( ! , ! , 0) E Vc: (N)) or they need player 3 to have exited the game (since c!. !) E VE ( l , 2)) . If 2 exits before 1 and 3, however, then 1 ends up with a payoff of c (if he strikes a bargain with 3) or 0 (if he fails to strike a bargain with 3). Hence, 1 and 2 are concerned with striking a deal before 3 exits the bargaining process. In the extreme case of c = 0, player 3 is in a powerful position. Players 1 and 2 may be willing to make a relatively large first-round concession to 3 (which 3 accepts) in order to avoid a payoff of 0 if 3 is not the first player to exit. This is reflected in the consistent value payoff of ( � , � , � ) in the game Vo. In the other extreme case in which c = 1 , players 1 and 2 can guarantee themselves a payoff of 1 / 2 each, with or without the presence of player 3. Again, this situation is reflected in the consistent value payoff of ( ! , ! , 0) in the game V1 and supports Roth's (static) argument in favor of C! , !, 0). Finally, the game V1;2 is completely symmetric and the consistent value of ( � , 1 , �) reflects this symmetry. Note that, unlike the Shapley NTU value, the consistent value specifies symmetric payoffs only when the underlying NTU game itself is symmetric. The justification for the consistent value in this example as the outcome of a dynamic bargaining model extends to general NTU games. In Hart and Mas-Colell (1996a), the authors present a dynamic bargaining model as described above and prove that, for a large class of NTU games (basically the class Tk), every limit point (as p --+ 0) of a sequence of subgame perfect equilibrium payoff configurations is a consistent value payoff configuration. If the NTU game is a pure bargaining game, then every subgame £
,
,
=
=
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perfect equibrium payoff vector of the associated dynamic game converges to the Nash Bargaining Solution. If V corresponds to a TU game v, then the associated dynamic game has a unique subgame perfect equibrium payoff vector that converges to the Shap ley value of v. 9.
Final remarks
9.1. Values and cores of convex NTU games As we have repeatedly emphasized, the values and solutions presented in this chapter are all based on axiomatized notions of fairness. An important alternative concept is that of the core, which is defined in terms of coalitional stability. Let V be an NTU game. A vector x E mN can be improved upon if there exists an S t; N and y E V (S) such that / > x i , for each i E S. 9 . 1 . 1 . A vector x E mN is a core payoff for V if x E V (N) and x cannot be improved upon. The set of core payoffs of V is denoted core(V). DEFINITION
If V corresponds to v E G N , then x E core(V) if and only if l s x s � v(S) for each S t; N and l N x :( v(N). If v E G N , we will write core(v). In the TU case, it is not generally tme that
·
U
pEM (Q )
Weber (1988) has shown that core(v) t; u p E M ( .Q ) (/Jp (v) for any game v E G N . lchi ishi (198 1) proved that if the converse is trne, then v is convex. We are interested in the extent to which these results generalize to convex NTU games and the p-Shapley value correspondence tP P . A game V E rN is cardinal convex if V * (S) + V * (T) 5; V * (S U T) + V * (S n T) [see Sharkey (198 l)]. lt is easy to show that if v is a convex TU game and if V is the NTU game derived from v , then V is cardinal
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convex. It is also straightforward to show that if V is cardinal convex and if v;. is defined V , then v;. is a convex TU game. Define the inner core of an NTU game V to be incore(V) = { x E V (N) I v;. is defined for some A E ffi�+ and A x E core(v;.) } . *
By combining the definitions with the results for TV games mentioned above, we con clude that for any NTU game V , incore(V) s; U p EM(Q) (/Jp (v) and in fact we have: 9. 1 .2 [Levy and McLean (1991)] . lf V is a cardinal convex game, then U p E M ( S?)
PROPOSITION
It is well known that incore(V) s; core(V) for any V so Proposition 9. 1 .2 generalizes results in Kern (1985, Theorem 4.4) and Ichiishi (1983, p. 148). Although incore(V) s; core(V), this containment can be strict if V does not correspond to a TU game. In fact, incore(V) may be a nonempty, proper subset of core(V) even if V is cardinal convex. For an example see Levy and McLean (1991).
9.2. Applications of values for NTU games with a continuum ofplayers Many political and economic situations are best modelled as games with infinitely many players. The value theory of such games comprises an extensive literature beginning with the fundamental study of Aumann and Shapley (1974). Values of NTU games with a continuum of players have been studied in several contexts. Value allocations are discussed extensively in Chapter 36 and, for economies with a continuum of agents (and satisfying a number of other regularity conditions), a "value equivalence" principle holds: the set of Walras allocations is equal to the set of value allocations. Another economic application of the NTU value arises in the theory of "Power and Taxes" as originally presented in Aumann and Kurz (1977a, 1977b). [See also Aumann, Kurz and Neyman ( 1987) and Osborne (1984) and the references cited therein.]
9.3. NTU values for games with incomplete information In many bargaining problems, individuals possess private information that affects their evaluation or utility of different outcomes. In the presence of incomplete information, the resolution of bargaining is complicated by the fact that the arbitrator must request that the agents reveal their types before he can recommend an outcome. Of course, the agents will behave strategically and will "misrepresent" themselves to the arbitra tor in a way favorable to themselves. For treatments that generalize the NBS to the incomplete information case, see Harsanyi and Selten (1972) and Myerson (1984a). In Myerson (1984b), the NTU value is extended to cooperative games with incomplete information in a way that generalizes the results of Myerson (1984a) for the pure bar gaining case.
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Weber, R. (1988), "Probabilistic values for games", in: A. Roth, ed., The Shapley Value: Essays in Honor of Lloyd Shapley (Cambridge University Press, Cambridge) 101-1 19. Winter, E. (1991), "On nontransferable utility games with coalition structure", International Journal of Game Theory 20:53-64. Winter, E. (1992), "On bargaining position descriptions in NTU games: Symmetry versus asymmetry", Inter national Journal of Game Theory 2 1 : 1 91-1 1 1 . Young, H.P. (1988), "Individual contribution and just compensation", in: A. Roth, ed., The Shapley Value: Essays in Honor of Lloyd Shapley (Cambridge University Press, Cambridge) 267-278.
Chapter 56 VALUES OF GAMES WITH I N FINITELY M ANY PLAYERS ABRAHAM NEYMAN
Institute ofMathematics and Centerfor Rationality and Interactive Decision Theory, The Hebrew University ofJerusalem, Jerusalem, Israel
Contents 1 . Introduction 1.1. An outline of the chapter
2. 3. 4. 5.
Prelude: Values of large finite games Definitions The value on pNA and bv' NA Limiting values 5.1. The weak asymptotic value 5.2. The asymptotic value
6. The mixing value 7. Formulas for values 8. Uniqueness of the value of nondifferentiable games 9. Desired properties of values 9.1. Strong positivity 9 .2. The partition value 9.3. The diagonal property
10. Semivalues 1 1 . Partially symmetric values 1 1. 1 . Non-symmetric and partially symmetric values 1 1.2. Measure-based values
12. The value and the core 13. Comments on some classes of games 13.1. Absolutely continuous non-atomic games 1 3.2. Games in pNA 13.3. Games in bv'NA
References
Handbook of Game Theory, Volume 3, Edited by R.I. Aumann and S. Hart © 2002 Elsevier Science B. V All rights reserved
2123 2123 2125 2127 2129 2133 2 134 2135 2 140 2140 2150 2151 2151 2151 2151 2152 2154 2 155 2157 2159 2161 2161 2161 2161 2162
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Abstract
This chapter studies the theory of value of games with infinitely many players. Games with infinitely many players are models of interactions with many players. Often most of the players are individually insignificant, and are effective in the game only via coalitions. At the same time there may exist big players who retain the power to wield single-handed influence. The interactions are modeled as cooperative games with a continuum of players. In general, the continuum consists of a non-atomic part (the "ocean"), along with (at most countably many) atoms. The continuum provides a convenient framework for mathematical analysis, and approximates the results for large finite games well. Also, it enables a unified view of games with finite, countable, or oceanic player-sets, or indeed any mixture of these. The value is defined as a map from a space of cooperative games to payoffs that satisfies the classical value axioms: additivity (linearity), efficiency, symmetry and pos itivity. The chapter introduces many spaces for which there exists a unique value, as well as other spaces on which there is a value. A game with infinitely many players can be considered as a limit of finite games with a large number of players. The chapter studies limiting values which are defined by means of the limits of the Shapley value of finite games that approximate the given game with infinitely many players. Various formulas for the value which express the value as an average of marginal con tribution are studied. These value formulas capture the idea that the value of a player is his expected marginal contribution to a perfect sample of size t of the set of all players where the size t is uniformly distributed on [0, 1]. In the case of smooth games the value formula is a diagonal formula: an integral of marginal contributions which are expressed as partial derivatives and where the integral is over all perfect samples of the set of play ers. The domain of the formula is further extended by changing the order of integration and derivation and the introduction of a well-crafted infinitesimal perturbation of the perfect samples of the set of players provides a value formula that is applicable to many additional games with essential nondifferentiabilities. Keywords
games, cooperative, coalitional form, transferable utility, value, continuum of players, non-atomic
JEL classification: D70, D71, D63, C71
Ch. 56: Values of Games with Infinitely Many Players
2123
1. Introduction
The Shapley value is one of the basic solution concepts of cooperative game theory. It can be viewed as a sort of average or expected outcome, or as an a priori evaluation of the players' expected payoffs. The value has a very wide range of applications, particularly in economics and po litical science (see Chapters 32, 33 and 34 in this Handbook). In many of these appli cations it is necessary to consider games that involve a large number of players. Often most of the players are individually insignificant, and are effective in the game only via coalitions. At the same time there may exist big players who retain the power to wield single-handed influence. A typical example is provided by voting among stockholders of a corporation, with a few major stockholders and an "ocean" of minor stockholders. In economics, one considers an oligopolistic sector of firms embedded in a large pop ulation of "perfectly competitive" consumers. In all these cases, it is fruitful to model the game as one with a continuum of players. In general, the continuum consists of a non-atomic part (the "ocean"), along with (at most countably many) atoms. The contin uum provides a convenient framework for mathematical analysis, and approximates the results for large finite games well. Also, it enables a unified view of games with finite, countable, or oceanic player-sets, or indeed any mixture of these.
1.1. An outline of the chapter In Section 2 we highlight a sample of a few asymptotic results on the Shapley value of finite games. These results motivate the definition of games with infinitely many players and the corresponding value theory; in particular, Proposition 1 introduces a formula, called the diagonal formula of the value, that expresses the value (or an approximation thereof) by means of an integral along a specific path. The Shapley value for finite games is defined as a linear map from the linear space of games to payoffs that satisfies a list of axioms: symmetry, efficiency and the null player axiom. Section 3 introduces the definitions of the space of games with a continuum of players, and defines the symmetry, positivity and efficiency of a map from games to payoffs (represented by finitely additive games) which enables us to define the value on a space of games as a linear, symmetric, positive and efficient map. The value does not exist on the space of all games with a continuum of players. Therefore the theory studies spaces of games on which a value does exist and moreover looks for spaces for which there is a unique value. Section 4 introduces several spaces of games with a continuum of players including the two spaces of games, pNA and bv'NA, which played an important role in the development of value theory for non atomic games. Theorem 2 asserts that there exists a unique value on each of the spaces pNA and bv'NA as well as on pNA . In addition there exists a (unique) value of norm 1 on the space 'NA. Section 4 also includes a detailed outline of the proofs. A game with infinitely many players can be considered as a limit of finite games with a large number of players. Section 5 studies limiting values which are defined by 00
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means of the limits of the Shapley value of finite games that approximate the given game with infinitely many players. Section 5 also introduces the asymptotic value. The asymptotic value of a game v is defined whenever all the sequences of Shapley values of finite games that "approximate" v have the same limit. The space of all games of bounded variation for which the asymptotic value exist is denoted ASYMP. Theorem 4 asserts that ASYMP is a closed linear space of games that contains bv' M and the map associating an asymptotic value with each game in ASYMP is a value on ASYMP. Section 6 introduces the mixing value which is defined on all games of bounded variation for which an analogous formula of the random order one for finite games is well defined. The space of all games having a mixing value is denoted MIX. Theorem 5 asserts that MIX is a closed linear space which contains pNA, and the map that associates a mixing value with each game v in MIX is a value on MIX. Section 7 introduces value formulas. These value formulas capture the idea that the value of a player is his expected marginal contribution to a perfect sample of size t of the set of all players where the size t is uniformly distributed on [0, 1]. A value formula can be expressed as an integral of directional derivative whenever the game is a smooth function of finitely many non-atomic measures. More generally, by extending the game to be defined over all ideal coalitions - measurable [0, 1]-valued functions on the (measurable) space of players - this value formula provides a formula for the value of any game in pNA. Changing the order of derivation and integration results in a formula that is applicable to a wider class of games: all those games for which the formula yields a finitely additive game. In particular, this modified formula defines a value on bv'NA or even on bv' M. When this formula does not yield a finitely additive game, applying the directional derivative to a carefully crafted small perturbation of perfect samples of the set of players yields a value formula on a much wider space of games (Theorem 7). These include games which are nondifferentiable functions (e.g., a piecewise linear function) of finitely many non-atomic probability measures. Section 8 describes two spaces of games that are spanned by nondifferentiable func tions of finitely many mutually singular non-atomic probability measures that have a unique value (Theorem 8). One is the space spanned by all piecewise linear functions of finitely many mutually singular non-atomic probability measures, and the other is the space spanned by all concave Lipschitz functions with increasing returns to scale of finitely many mutually singular non-atomic probability measures. The value is defined as a map from a space of games to payoffs that satisfies a short list of plausible conditions: linearity, symmetry, positivity and efficiency. The Shap ley value of finite games satisfies many additional desirable properties. It is continu ous (of norm 1 with respect to the bounded variation norm), obeys the null player and the dummy axioms, and it is strongly positive. In addition, any value that is obtained from values of finite approximations obeys an additional property called diagonality. Section 9 introduces such additional desirable value properties as strong positivity and diagonality. Theorem 9 asserts that strong positivity can replace positivity and linearity in the characterization of the value on pNAoc, and thus strong positivity and continuity can replace positivity and linearity in the characterization of a value on pNA. A strik-
Ch. 56: Values of Games with Infinitely Many Players
2125
ing property of the value on pNA, the asymptotic value, the mixing value, or any of the values obtained by the formulas described in Section 7, is the diagonal property: if two games coincide on all coalitions that are "almost" a perfect sample (within with respect to finitely many non-atomic probability measures) of the set of players, their values coincide. Theorem 1 1 asserts that any continuous value is diagonal. Section 10 studies semivalues which are generalizations of the value that do not nec essarily obey the efficiency property. Theorem 13 provides a characterization of all semivalues on pNA. Theorems 14 and 15 provide results on asymptotic semivalues. Section 1 1 studies partially symmetric values which are generalizations of the value that do not necessarily obey the symmetry axiom. Theorems 15 and 16 characterize in particular all the partially symmetric values on pNA and pM that are symmetric with respect to those symmetries that preserve a fixed finite partition of the set of players. A corollary of this characterization on pM is the characterization of all values on pM. A partially symmetric value, called a J-L-value, is symmetric with respect to all symme tries that preserve a fixed non-atomic population measure JL and (in addition to linearity, positivity and efficiency), obeys the dummy axiom. Such a partially symmetric value is called a J-L-value. Theorem 18 asserts the uniqueness of a J-L-value on pNA(JL) . Theo rem 19 asserts that an important class of nondifferentiable markets have an asymptotic J-L-value. Games that arise from the study of non-atomic markets, called market games, obey two specific conditions: concavity and homogeneity of degree 1 . The core of such games is nonempty. Section 1 2 provides results on the relation between the value and the core of market games. Theorem 20 asserts the coincidence of the core and the value in the differentiable case. Theorem 21 relates the existence of an asymptotic value of nondif ferentiable market games to the existence of a center of symmetry of its core. Theo rems 22 and 23 describe the asymptotic J-L-value and the value of market games with a finite-dimensional core as an average of the extreme points of the core. £
2. Prelude: Values of large finite games
This section introduces several asymptotic results on the Shapley value ljr of finite games. These results can serve as an introduction to the theory of values of games with infinitely many players, by motivating the definitions of games with a continuum of players and the corresponding value theory. We start with the introduction of a representation of games with finitely many play ers. Let £ be a fixed positive integer and f : JR.f ---+ JR. a function with / (0) = 0. Given W I , . . . , we E JR.t, define the n-person game v = [f; W I , . . . , we] by
A special subclass of these games is weighted majority games where £ = 1, 0 < q < 2:: 7= 1 Wi and f (x) = 1 if x � q and 0 otherwise. The asymptotic results relate to se-
2126
A. Neyman
quences of finite games where the function f is fixed and the vector of weights varies. If the function f is differentiable at x, the directional derivative at x of the function f in the direction y E JRe is fy (x) := limH o+ (f(x + 8y) - f(x))/8. The following result follows from the proofs of results on the asymptotic value of non-atomic games. It can be derived from the proof in Kannai ( 1966) or from the proof in Dubey (1980).
1 . Assume that f is a continuously differentiable function on [0, l] e . For every 8 > 0 there is o > 0 such that if W), . . . , Wn E JR� with (the normalization) L7= 1 Wi = ( 1 , . . . , 1) and max7= I I wi ll < o, then PROPOSITION
where v = [f; w 1 , . . . , Wn ]. Thus,for every coalition S ofplayers,
1 1/r v(S) - Ia I fw(S) (t, . . . , t) dt l < 8, where 1/ f v(S) = LiES 1/r v(i) and w(S) = LiES w;. The limit of a sequence of games of the form [ f ; w7 , . . . , w�k ], where f is a fixed function and w1 , . . . , w�k is a sequence of vectors in JR� with L7! 1 w � = ( 1 , . . . , 1) and max7! 1 llwi II ---* k -+oo 0 can be modeled by a 'game' of the form f o (P., J , . . . , P.,e ) where p., = (p., 1 , . . . , p.,e) is a vector of non-atomic probability measures defined on a measurable space (I, C) . If f is continuously differentiable, the limiting value formula thus corresponds to:
cp(f o p., )(S) =
!ol fJL(S) (t, . . . , t) dt .
The next results address the asymptotics of the value of weighted majority games. The n-person weighted majority game v = [q; WJ , . . . , Wn] with W I , . . . , Wn E JR+ and 0 < q < L;:l w; , is defined by v(S) = 1 if Li ES w; )' q and 0 otherwise.
2 [Neyman (1981a)]. For every s > 0 there is K > 0 such that if v = [q ; W J , . . . , Wn] is a weighted majority game with W I , . . . , Wn E JR+, L7= 1 w; = 1 and K max7= 1 w; < q < 1 - K max?= I w;, PROPOSITION
1 1/r v(S) - Lz ES w; I < 8
for every coalition S C { 1, . . . , n}.
2127
Ch. 56: Values of Games with Infinitely Many Players
In particular, if Vk [qk ; w}, . . . , w�k ] is a sequence of weighted majority games with w� ): 0, L7! w� ----+ k---+oo 1 , and qk ----+ k---+oo q with 0 < q < 1 , then for any sequence 1 of coalitions Sk c { 1 , . . . , nk}, 1/rvk (Sk) - Li ESk w� ----+ k ---+ oo 0. The limit of such a se quence of weighted majority games is modeled as a game of the form v = Jq o JL where /1 is a non-atomic probability measure and Jq (x) 1 if x ): q and 0 otherwise. The non-atomic probability measure JL describes the value of this limiting game: =
=
q;v(S)
=
JL(S).
The next result follows easily from the result of Berbee (1981).
PROPOSITION 3 . Let W J , . . . , Wn , . . . be a sequence ofpositive numbers with 00
_L w; = l i= l
and 0 < q < 1. There exists a sequence of nonnegative numbers 1/r;, i ): 1, with L �J Vri = 1, such thatfor every t: > 0 there is a positive constant 8 > 0 and a positive integer no such that if v = [q'; U J , . . . , u n ] is a weighted majority game with n ): no, / q' - q / < 8 and L7=1 / w; - u; / < 8, then
Equivalently, let Vk = [qk ; w}, . . . , w�k ] be a sequence of weighted majority games with L7! w� ----+ k---+ oo 1 and wJ ----+ k ---+ oo WJ and qk ----+ k ---+ oo q. Then the Shapley value 1 of player i in the game Vk converges as k ----+ oo to a limit 1/r; and L �J 1/r; 1 . The limit of such a sequence of weighted majority games can be modeled as a game of the form v = fq o JL where f1 is a purely atomic measure with atoms i 1 , 2, . . . and JL(i) = w; . The sequence 1/r; describes the value of this limiting game: =
=
q;v(S)
=
L 1/r; . i ES
3. Definitions
The space of players is represented by a measurable space (I, C). The members of set I are called players, those of C - coalitions. A game in coalitional form is a real-valued function v on C such that v(0) = 0. For each coalition S in C, the number v(S) is interpreted as the total payoff that the coalition S, if it forms, can obtain for its mem bers; it is called the worth of S. The set of all games forms a linear space over the field of real numbers, which is denoted G. We assume that (!, C) is isomorphic to
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A. Neyman
([0, 1], B), where !3 stands for the Borel subsets of [0, 1]. A game v is .finitely additive if v(S U T) = v(S) + v (T) whenever S and T are two disjoint coalitions. A distribution of payoffs is represented by a finitely additive game. A value is a mapping from games to distributions of payoffs, i.e., to finitely additive games that satisfy several plausible conditions: linearity, symmetry, positivity and efficiency. There are several additional desirable conditions: e.g., continuity, strong positivity, a null player axiom, a dummy player axiom, and so on. It is remarkable, however, that a short list of plausible condi tions that define the value suffices to determine the value in many cases of interest, and that in these cases the unique value satisfies many additional desirable properties. We start with the notations and definitions needed formally to define the value, and set forth four spaces of games on which there is a unique value. A game v is monotonic if v(S) � v(T) whenever S :J T; it is of bounded variation if it is the difference of two monotonic games. The set of all games of bounded variation (over a fixed measurable space (I, C)) is a vector space. The variation of a game v E G, II v II , is the supremum of the variation of v over all increasing chains S 1 C S2 C C Sn in C . Equivalently, I v II = inf{ u (I) + w (I) : u , w are monotonic games with v = u w} (inf0 = oo). The variation defines a topology on G ; a basis for the open neighborhood of a game v in G consists of the sets { u E G: llu v II < 8} where 8 varies over all positive numbers. Addition and multiplication of two games are continuous and so is the multiplication of a game and a real number. A game v has bounded variation if II v I I < oo . The space of all games of bounded variation, BV, is a Banach algebra with respect to the variation norm II II, i.e., it is complete with respect to the distance defined by the norm (and thus it is a Banach Space) and it is closed under multiplication which is continuous. Let Q denote the group of automorphisms (i.e., one-to-one measurable mappings 8 from I onto I with 8 - I measurable) of the underlying space (I, C). Each 8 in Q induces a linear mapping 8* of BV (and of the space of all games) onto itself, defined by (8* v) (S) = v(8S). A set of games Q is called symmetric if 8* Q = Q for all 8 in Q. The space of all finitely additive and bounded games is denoted FA; the subspace of all measures (i.e., countably additive games) is denoted M and its subspace consisting of all non-atomic measures is denoted NA. The space of all non-atomic el ements of FA is denoted AN. Obviously, NA c M C FA C BV, and each of the spaces NA, AN, M, FA and BV is a symmetric space. Given a set of games Q, we denote by Q+ all monotonic games in Q, and by Q 1 the set of all games v in Q+ with v(l) = 1 . A map cp : Q --? B V i s called positive if cp(Q+) C BV+ ; symmetric if for every 8 E Q and v in Q, 8* v E Q implies that cp( 8* v) = 8* (cpv) ; and efficient if for every v in Q, (cpv) (I) = v(l). · · ·
-
-
1 [Aumann and Shapley (1974)]. Let Q be a symmetric linear subspace of BV. A value on Q is a linear map cp : Q FA that is symmetric, positive and efficient.
DEFINITION
--?
The definition of a value is naturally extended also to include spaces of games that are not necessarily included in BV. Thus let Q be a linear and symmetric space of games.
Ch. 56: Values of Games with Infinitely Many Players
2129
A value on Q is a linear map from Q into the space of finitely additive games that is symmetric, positive (i.e., cp ( Q + ) C BV+ ) , efficient, and cp ( Q n BV) C FA.
DEFINITION 2.
There are several spaces of games that have a unique value. One of them is the space of all games with a finite support, where a subset T of I is called a support of a game v if v(S) = v (S n T) for every S in C . The space of all games with finite support is denoted FG.
THEOREM 1
[Shapley ( 1 953a)] .
There is a unique value on FG.
The unique value 1jJ on FG is given by the following formula. Let T be a finite support of the game v (in FG). Then, 1jfv (S) = 0 for every S in C with S n T = 0, and for t E T, 1
[v (P� U {t}) - v(P?�)], 1jfv({t}) = - " n'· � n where n is the number of elements of the support T, the sum runs over all n! orders of T, and P1n is the set of all elements of T preceding t in the order R . Finite games can also be identified with a function v defined on a finite subfield rr of C: given a real-valued function v defined on a finite subfield rr of C with v (0) = 0, the Shapley value of the game v is defined as the additive function 1jJ : rr -+ JR. defined by its values on the atoms of rr ; for every atom a of rr ,
1
R[ v (P7: U a ) - v (P7:)], 1jfv(a) = - " n! � where n is the number of atoms of JT , the sum runs over all n! orders of the atoms of rr , and PJ: i s the union of all atoms of IT preceding a in the order R.
4.
The value on pNA and bv'NA
Let pNA (pM, pFA) be the closed (in the bounded variation norm) algebra generated by NA (M, FA, respectively). Equivalently, pNA is the closed linear subspace that is generated by powers of measure in NA 1 [Aumann and Shapley ( 1974), Lemma 7 .2] . Similarly, pM (pFA) is the closed algebra that is generated by powers of elements in M (FA). Let II lloo be defined on the space of all games G by
In the definition of ll v ll oo , we use the common convention that inf 0 = oo. The defi nition of II lloo is an analog of the C 1 norm of continuously differentiable functions; e.g., if f is continuously differentiable on [0, 1 ] and {L is a non-atomic probability
2130
A. Neyman
measure, II / o J.tlloo = max{ l /' (t) l : 0 � t � 1 } + max{ l / (t) l : 0 � t � 1 } . Note that I u v II oo � I u II oo II v II oo for two games u and v. The function v r+ II v II oo defines a topol ogy on G; a basis for the open neighborhood of a game v in G consists of the sets { u E G : II u - v II 00 < c:} where c: varies over all positive numbers. The space of games pNA 00 is defined as the I I 00 closed linear subspace that is generated by powers of mea sures in NA 1 . Obviously, pNA 00 C pNA ( II II 00 � II II ) . Both spaces contain many games of interest. If J.t is a measure in NA 1 and f : [0, 1 ] -+ JR. is a function with f (0) = 0, then f o J.t E pNA if and only if f is absolutely continuous [Aumann and Shapley ( 1974), Theorem C] and in that case II f o J.t II = f01 I !' (t) I dt where ! ' is the (Radon-Nikodym) derivative of the function f, and f o J.t E pNA 00 if and only if f is continuously differ entiable on [0, 1 ] . Also, if J.t = (J.t 1 , , J.tn ) is a vector of NA measures with range {J.t(S) : S E C} = : R(J.t), and f is a C 1 function on R(J.t) with / (J.t (0)) = 0, then f o J.t E pNA 00 [Aumann and Shapley ( 1 97 4 ), Proposition 7 . 1 ] . Let bv'NA (bv' M , bv'FA) b e the closed linear space generated b y games of the form f o J.t where f E bv' = : {f : [0, 1 ] -+ JR. I f is of bounded variation and continuous at 0 and at 1 with / (0) = 0}, and J.t E NA 1 (J.t E M 1 , J.t E FA 1 ) Obviously, pNA00 C pNA C bv'NA c bv' M c bv' FA. . • .
.
THEOREM 2 [Aumann and Shapley ( 1 974), Theorems A, B; Monderer (1990), Theo rem 2 . 1] . Each of the spaces pNA, bv'NA, pNA 00 has a unique value. The proof of the uniqueness of the value on these spaces relies on two basic argu ments. Let J.t E NA 1 and let Q (J.t) be the subgroup of Q of all automorphisms 8 that are J.t -measure-preserving, i.e., J.t (8 S) = J.t(S) for every coalition S in C . Then the set of all finitely additive games u that are fixed points of the action of Q (J.t) (i.e., 8 u = u * for every 8 in Q (J.t)) are games of the form aJ.t, a E JR. (here we use the standard ness assumption that (/, C) is isomorphic to ([0, 1 ] , B)). Therefore, if Q is a symmetric space of games and cp : Q -+ FA is symmetric, J.t E NA 1 , and f : [0, 1 ] -+ JR. is such that f o J.t E Q, then 8 (/ o J.t) = f o J.t for every 8 E Q (J.t) . It follows that the finitely * additive game cp(f o J.t) is a fixed point of Q (J.t) , which implies that cp(f o J.t) = aJ.t. If, moreover, cp is also efficient, then cp(f o J.t) = f(1)J.t. The spaces pNA and bv'NA are closed linear spans of games of the form f o J.t. The space pNA00 is in the closed linear span of games of the form f o J.t with f o J.t in pNA00. Therefore, each of the spaces pNA 00 , pNA and bv' NA has a dense subspace on which there is at most one value. To complete the uniqueness proof, it suffices to show that any value on these spaces is indeed continuous. For that we apply the concept of internality. A linear subspace Q of BV is said to be reproducing if Q = Q + - Q+. It is said to be internal if for all v in Q,
II v ii
=
inf{ u (l)
+ w (l): u, w E Q + and v u - w}. =
Clearly, every internal space i s reproducing. Also any linear positive mapping cp from a closed reproducing space Q into BV is continuous, and any linear positive and efficient
2131
Ch. 56: Values of Games with Infinitely Many Players
mapping
n
1 L .fi (1)t-t; (S) = s--+ lim 0+ 8 i=l
In
0
1 -s
[n
n ]
L .fi (t + 8t-t; (S) ) - L f; (t) i=l
i=l
dt .
By Lyapunov's Theorem, applied to the vector measure (t-tl , . . . , tJ11) , for every 0 :'( t :'( 1 - 8 there are coalitions T C R with f-ti (T) = t and f-ti (R) = t + 8f-ti (S). Therefore, if v is monotonic,
n
n
v(R) - v(T) = L f; (t + 8 {-t; (S) ) - L f; (t) � 0,
i=l
i=l
q;v(S) � 0. I n particular, if 0 = L7= 1 f; o f-ti - L' = l gi o VJ where /; , gi E bv' and f-ti , VJ E NA 1 , L7= 1 .f; (1)t-t; = L' =l gi (1)vi , and thus also q;v is
and therefore
well defined, i.e., it is independent of the presentation. Obviously,
q;v(I) = v(l)
2 13 2
A.
Neyman
a positive integer, f; E bv' and � i E NA 1 . We next show that cp is of norm 1 and thus has a (unique) extension to a continuous value (of norm 1) on bv'NA. For each u E FA, llu II = supS EC lu( S) I + lu(Sc ) 1 . Therefore, it is sufficient to prove that for every coalition s E C, l cpv(S) I + lcpv(Sc ) I ::( I v II . For each positive integer m let So c SJ c s2 c . . c Sm and S0 c Sf C · · · C S1� be measurable subsets of S and sc = I \ S respectively with .
�i (SJ ) = m�J �i (S) and �; (Sj ) = m�! � i (Sc ) . For every 0 ::( t ::( m� l let It be a mea surable subset of I \ (Sm U S1�1) with � i Ut ) = t. Define the increasing sequence of coali tions To c T1 C · · c hn by To = I1 , T21- 1 = It U SJ U Sj _ 1 and T21 = I1 U Si U Sj·, j = 1 , . . . , m. Obviously, ·
2m ll v ll � � ] v(TJ ) - v(Tj - J ) I j =l
Set E =
m�J . Note that
and similarly
As
11
11
i=l
i =l
and
n
11
v(T2J) - v(T2j- J ) = L f; (t + 2jE ) - L fi (t + 2jc; - E�; ( Sc )), i=l i =l
2133
Ch. 56: Values of Games with Infinitely Many Players
we deduce that for each fixed
]'�[Ef;(t I � [E f; (t
t
0 � � 8,
Ef;(t l E (t ] I ""
+ 'j HJ' ; (S) ) -
+
+ sj ) -
+ jel
f; + j' - st<; ( S' ))
11 " 11
and therefore [
5. Limiting values
Limiting values are defined by means of the limits of the Shapley value of the finite games that approximate the given game. Given a game v on (/, C) and a finite measurable field n, we denote by vJT the re striction of the game v to the finite field n . The real-valued function vJT (defined on n )
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can be viewed as a finite game. The Shapley value for finite games by
1/Jv:n:
on rr is given
1 (1/Jv:n: ) (a) = I l:] v(P� U a ) - v(P�)], n. R where n is the number of atoms of rr , the sum runs over all orders R of the atoms of rr , and P!}- i s the union of all atoms preceding a in the order R .
5.1. The weak asymptotic value The family of measurable finite fields (that contain a given coalition T) is ordered by inclusion, and given two measurable finite fields rr and rr ' there is a measurable finite field rr " that contains both rr and rr 1 • This enables us to define limits of functions of finite measurable fields. The weak asymptotic value of the game v is defined as limn 1/Jv:n: whenever the limit of 1/Jv:n: (T) exists where rr ranges over the family of measurable finite fields that include T. Equivalently,
3 . Let v be a game. A game q;v is said to be the weak asymptotic value of v if for every S E C and every 8 > 0, there is a finite subfield rr of C with S E rr such that for any finite subfield rr ' with rr ' :J rr , 11/J v:n:' (S) - q;v(S) I < 8 . DEFINITION
REMARKS. (1) Let v b e a game. The game v has a weak asymptotic value if and only if for every S in C and 8 > 0 there exists a finite subfield rr with S E rr such that for any finite subfield rr' with rr' :J rr , 11/J v:n:' (S) - 1/Jv:n: (S) I < 8 . (2) A game v has at most one weak asymptotic value. (3) The weak asymptotic value q;v of a game v is a finitely additive game, and llq;vll � II v II whenever v E BV.
3 [Neyman (1988), p. 559]. The set ofall games having a weak asymptotic value is a linear symmetric space of games, and the operator mapping each game to its weak asymptotic value is a value on that space. If ASYMP* denotes all games with bounded variation having a weak asymptotic value, then ASYMP* is a closed subspace of BV with bv'FA c ASYMP*.
THEOREM
REMARK S . The linearity, symmetry, positivity and efficiency of 1/J (the Shapley value for games with finitely many players) and of the map v r-+ V:n: imply the linearity, symmetry, positivity and efficiency of the weak asymptotic value map and also im ply the linearity and symmetry of the space of games having a weak asymptotic value. The closedness of ASYMP* follows from the linearity of 1/J and from the in equalities 1 1 1/J V:n: I � I V:n: I � I v 11 . The difficult part of the theorem is the inclusion bv' FA c ASYMP*, on which we will comment later. The inclusion bv'FA c ASYMP* implies in particular the existence of a value on bv'FA.
Ch. 56: Values of Games with Infinitely Many Players
2 135
5.2. The asymptotic value The asymptotic value of a game v is defined whenever all the sequences of the Shapley values of finite games that "approximate" v have the same limit Formally: given T in C, a T -admissible sequence is an increasing sequence (n 1 , 1r2 , . . .) of finite subfields of C such that T E 1r 1 and U ; JT; generates C.
DEFINITION 4. A game cpv is said to be the asymptotic value of v, if limk -+ oo 1frv1fk (T) exists and = cpv(T) for every T E C and every T -admissible sequence (n;)� 1 . REMARKS . (1) A game v has an asymptotic value if and only if limk-+ oo 1frv1fk (T) exists for every T in C and every T -admissible sequence P = (nk) � 1 , and the limit is independent of the choice of P. (2) For any given v, the asymptotic value, if it exists, is clearly unique. (3) The asymptotic value cpv of a game v is finitely additive, and llcpvll :( II v ii when ever v E BV. (4) If v has an asymptotic value cpv then v has a weak asymptotic value, which = cpv. (5) The dual of a game v is defined as the game v * given by v * (S) = v(I) - v(I\S). Then 1fr v; = 1fr vJ[ = 1fr ( v -+:/ )J[ , and therefore v has a (weak) asymptotic value if and only if v * has a (weak) asymptotic value if and only if ( vj_v* ) has a (weak) asymptotic value, and the values coincide. The space of all games of bounded variation that have an asymptotic value is denoted ASYMP. THEOREM 4 [Aumann and Shapley (1974), Theorem F; Neyman (1981a); and Ney man (1988), Theorem A]. The set of all games having an asymptotic value is a linear symmetric space ofgames, and the operator mapping each game to its asymptotic value is a value on that space. ASYMP is a closed linear symmetric subspace of BV which contains bv' M. REMARKS. (1) The linearity and symmetry of 1/r (the Shapley value for games with finitely many players), and of the map v --+ vJ[ , imply the linearity and symmetry of the asymptotic value map and its domain. The efficiency and positivity of the asymptotic value follows from the efficiency and positivity of the maps v --+ vJ[ and vJ[ --+ lfrvJ[ . The closedness of ASYMP follows from both the linearity of 1/r and the inequalities 11 1/rvJ[ II :s; II v II J[ :s; I v 1 1 (2) Several authors have contributed to proving the inclusion bv' M c ASYMP. Let FL denote the set of measures with at most finitely many atoms and Ma all purely atomic measures. Each of the spaces, bv'NA , bv'FL and bv'Ma is defined as the closed subspace of B V that is generated by games of the form f o fL where f E b v ' and /L is a probability measure in NA 1 , FL 1 , or M� respectively.
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Kannai ( 1966) and Aumann and Shapley ( 1 974) show that pNA c ASYMP. Art stein (1971) proves essentially that pMa c ASYMP. Fogelman and Quinzii ( 1 980) show that pFL C ASYMP. Neyman (198 l a, 1979) shows that bv'NA c ASYMP and bv'FL C ASYMP respectively. Berbee (198 1 ), together with Shapley ( 1 962) and the proof of Neyman [( 1 98 l a), Lemma 8 and Theorem A] , imply that bv' Ma c ASYMP. It was further announced in Neyman ( 1979) that Berbee's result implies the existence of a partition value on bv' M. Neyman ( 1988) shows that bv' M c ASYMP. (3) The space of games, bv' M, is generated by scalar measure games, i.e., by games that are real-valued functions of scalar measures. Therefore, the essential part of the inclusion bv' M C ASYMP is that whenever f E bv' and JJ, is a probability measure, f o fJ, has an asymptotic value. The tightness of the assumptions on f and JJ, for f o M to have an asymptotic value follows from the next three remarks. (4) pFA eJ_ ASYMP [Neyman ( 1988), p. 578]. For example, JJ,3 eJ_ ASYMP whenever M is a positive non-atomic finitely additive measure which is not countably additive. This illustrates the essentiality of the countable additivity of M for f o M to have an asymptotic value when f is a polynomial. (5) For 0 < q < 1 , we denote by /q the function given by /q (x) = 1 if x ;?: q and /q (x) = 0 otherwise. Let JJ, be a non-atomic measure with total mass 1 . Then /q o M has an asymptotic value if and only if JJ, is positive [Neyman ( 1 988), Theorem 5 . 1 ] . There are games of the form /q o JJ,, where JJ, is a purely atomic signed measure with JJ, (I) = 1 , for which I! o M does not have an asymptotic value [Neyman ( 1 988), Example 5 .2]. 2
These two comments illustrate the essentiality of the positivity of the measure JJ,. (6) There are games of the form f o JJ,, where M E NA 1 and f is continuous at 0 and 1 and vanishes outside a countable set of points, which do not have an asymptotic value [Neyman ( 1 98 l a), p. 210]. Thus, the bounded variation of the function f is essential for f o JJ, to have an asymptotic value. (7) The set of games DIAG is defined [Aumann and Shapley (1974), p. 252] as the set of all games v in BV for which there is a positive integer k, measures M l , . . . , Mk E NA 1 , and E > 0, such that for any coalition S E C with Mi (S) - fJ, j (S) � E for all 1 � i, j � k, v(S) = 0. DIAG is a symmetric linear subspace of ASYMP [Aumann and Shapley (1974)] . (8) If f-J, J , !J,2 , JJ, 3 , are three non-atomic probabilities that are linearly independent, then the game v defined by v(S) = min[JJ, I (S) , M2 (S), M 3 (S)] is not in ASYMP [Au mann and Shapley ( 1 974), Example 1 9.2]. (9) Games of the form /q o JJ, where JJ, E M 1 and 0 < q < 1 are called weighted majority games; a coalition S is winning, i.e., v(S) = 1 , if and only if its total JJ,-weight is at least q . The measure JJ, is called the weight measure and the number q is called the quota. The result bv' M C ASYMP implies in particular that weighted majority games have an asymptotic value. ( 10) A two-house weighted majority game v is the product of two weighted maj ority games, i.e., v = (fq o JJ,)(fc o v), where JJ, , v E M 1 , 0 < c < 1 , and 0 < q < 1 . If JJ, , v E
Ch. 56: Values ofGames with Infinitely Many Players
2137
NA 1 with fL =/:. v and q =/:. 1 /2, v has an asymptotic value if and only if q =/:. c. Indeed, if q < c, (fq o tL)Uc o v ) - (fc o v) E DIAG and therefore (fq fL ) (fc v) E bv'NA + DIAG c ASYMP, 0
0
and if q = c =/:. 1 /2, (fq o fL)(fc o v) � ASYMP [Neyman and Tauman ( 1 979), proof of Proposition 5.42] . If fL has infinitely many atoms and the set of atoms of v is dis joint to the set of atoms of fL, (fq o fL)(fc o v) has an asymptotic value [Neyman and Smorodinski ( 1 993)]. ( 1 1) Let A denote the closed algebra that is generated by games of the form f o fL where fL E NA 1 and f E bv' is continuous. It follows from the proof of Proposition 5.5 in Neyman and Tauman ( 1 979) together with Neyman ( 1 98 l a) that A C ASYMP and that for every u E A and v E bv'NA, uv E ASYMP. The following should shed light on the proofs regarding asymptotic and limiting val ues. Let n be a finite subfield of C and let v be a game. We will recall formulas for the Shapley value of finite games by applying these formulas to the game Vn . This will enable us to illustrate proofs regarding the limiting properties of the Shapley value 1/f Vn of the finite game Vn . Let A (n) , or A for short, denote the set of atoms of n . The Shapley value of the game Vn to an atom a in A is given by
where lA I stands for the number of elements of A, the summation ranges over all orders R of the atoms in A , and paR is the union of all atoms preceding a in the order R . An alternative formula for 1/fvn could b e given by means of a family Xa , a E A (n), of i.i.d. real-valued random variables that are uniformly distributed on (0, 1 ) . The values of Xa , a E A (n), induce with probability 1 an order R on A ; a precedes b if and only if Xa < Xb . As Xa , a E A , are i.i.d., all orders are equally likely. Thus, identifying sets with their indicator functions, and letting I (Xa :::;; X b) denote the indicator of the event Xa ::( Xb, the value formula is
Thus, by conditioning on the value of Xa , and letting S(t), of all atoms b of n for which Xb :::;; t, we find that
1/fvn (a ) =
fo l E (v(S(t) U a) - v(S(t) \ a) )
dt .
0 :::;; t :::;; 1 , denote the union
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A. Neyman
Assume now that the game v is a continuously differentiable function of finitely many n non-atomic probability measures; i.e., v = t o /L, where IL = (/L 1 , . . . , !Ln) E (NA 1 ) and t : R(IL) --+ R is continuously differentiable. Then
n at "v(S(t) U a) - v(S(t) \ a) = � ( ) i (a), a �:: . Y !L i = l 51 where y is a point in the interval [/L(S(t) \ a), IL(S(t) U a)]. Thus, in particular, La En lv(S(t) U a) - v(S(t) \ a) l is bounded. For any scalar measure v, v(S(t)) is the sum L a EA(n ) v(a)/ (Xa � t), i.e., it is a sum of IA(n)l independent random variables with expectation tv(a) and variance t(l t)v 2 (a) . Therefore, E(v(S(t))) = tv(!) and
Var(S(t)) = t(l - t) L v2(a) � t(l - t)v(I) max { v(a): a E A (n) } . a EA(H)
Therefore, if IL = (/L 1 , . . . , !Ln) is a vector of measures, and (7rk)� 1 is a sequence of finite fields with max { I /LJ (a) l : 1 � j � n, a E A (nk)} --+k -+oo 0, /L(S(t)) converges in distribution to tIL (I); so if T is a coalition with T E 1Tk for every k, then
" [ v (Sk (t) u a ) - v( Sk(t) \ a)] �
a cT a EA (Hk)
--+
n at " k -+C!O � - (t !L (/ ) ) IL i (T), ax·
i=l
1
and therefore limk -+ CJO lJ; vnk (T) exists and
For any T-admissible sequence (7rk)� 1 and non-atomic measure v, max{v(a): a E A(nk)} --+k-+oo 0. Therefore any game v = t o /L, where t is a continuously differen tiable function defined on the range of /L, {/L(S): S E C}, has an asymptotic value
The following tool which is of independent interest will be used later on to show how to reduce the proof of bv'NA C ASYMP (or bv' M C ASYMP) to the proof that tq o 1L E ASYMP where tq (x) = l (x � q), 0 < q < 1 , and 1L E NA 1 (or 1L E M 1 ).
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2139
Let v be a monotonic game. For every a > 0, let va denote the indicator of v � a, i.e., va (S) = I (v(S) � a), i.e., va 00 (S) = 1 if v(S) � a and 0 otherwise. Note that for every coalition S, v(S) = J0 va (S) da . Then for every finite field n , 1/fvrr = J000 1/f (va)rr da = J;(J) 1/f(va)rr da . Therefore if (7rk)� 1 is a T-admissible se quence such that for every a > 0, limk--+oo 1/f(va)rrk (T) exists, then from the Lebesgue's bounded convergence theorem it follows that limk--+oo 1/fvrrk (T) exists and
lov(I) ( lim 1/fv�k (T) ) da.
1/fvrrk (T) = 0 klim --+ oo
k --+oo
Thus, if for each a > 0, va = I ( v � a) has an asymptotic value cpva, then v also has an asymptotic value cpv, which is given by J000 cpva (T) da = cpv(T). The spaces ASYMP and ASYMP* are closed and each of the spaces bv'NA, bv'M and bv'FA is the closed linear space that is generated by scalar measure games of the form f o fL with f E bv' monotonic and fL E NA 1 , fL E M I , fL E FA I respectively. Also, each monotonic f in bv' is the sum of two monotonic functions in bv', one right continuous and the other left continuous. If f E bv', with f monotonic and left continuous, then (f o /L)* = g o fL, with g E bv' monotonic and right continuous. Therefore, to show that bv'NA c ASYMP (bv' M c ASYMP or bv'FA c ASYMP*), it suffices to show that f o fL E ASYMP (or E ASYMP*) for any monotonic and right continuous function f in bv' and fL E NA 1 (/L E M I or fL E FA 1 ). Note that
(f o /L) a (S) = I (/(tL (S)) � a) = l (tL (S) � inf{ x : f(x) � a }) = /q (/L(S) ) ,
where q = inf{x: f(x) � a}. Thus, in view of the above remarks, to show that bv'NA c ASYMP (or bv' M c ASYMP), it suffices to prove that /q o fL E ASYMP for any 0 < q < 1 and fL E NA I (or fL E M I ), where /q (x) = 1 if x � q and = 0 otherwise. The proofs of the relations /q o fL E ASYMP for 0 < q < 1 and fL E NA I (or fL E M1 or fL E M1 ) rely on delicate probabilistic results, which are of independent mathematical
interest. These results can be formulated in various equivalent forms. We introduce them as properties of Poisson bridges. Let X 1 , X 2 , . . . be a sequence of i.i.d. random variables that are uniformly distributed on the open unit interval (0, 1). With any finite or infinite sequence w = (wi);'= I or w = (wi)� I of positive numbers, we associate a stochastic process, Sw (-), or S(-) for short, given by S(t) = L i w; l(Xi :::;; t) for 0 :::;; t :::;; 1 (such a process is called a pure jump Poisson bridge). The proof of Jq o fL E ASYMP for fL E NA 1 uses the following result, which is closely related to renewal theory.
4 [Neyman (198la)]. For every > 0 there exists a constant K 0, such that if W I , . . . , Wn are positive numbers with L:;'= I w; = 1, and K max;'=! w; < q < 1 - K max;'= I w; then PROPOSITION
n
L: J w; - Prob (S(Xi) E [q , q + w;) ) j < c . i =I
c
>
2140
A.
The proof of Berbee (1981).
fq o
t-t
E
ASYMP
Neyman
for t-t E Ma uses the following result, due to
PROPOSITION 5 [Berbee (1981)] . For every sequence (w; )� 1 with w; 0, and every 0 < q < 2::� 1 w;, the probability that there exists 0 � t � 1 with S(t) = q is 0, i.e., >
Prob (::i t
,
S(t )
=
q) = 0.
6. The mixing value
An order on the underlying space I is a relation R on I that is transitive, irreftexive, and complete. For each order R and each s in I , define the initial segment I (s; R) by I (s ; R) = {t E I : sRt}. An order R is measurable if the a -field generated by all the initial segments I (s; R) equals C . Let ORD be the linear space of all games v in BV such that for each measurable R, there exists a unique measure rp(v; R) satisfying rp (v ; R) ( I (s; R) ) =
v (I (s ; R) )
for all s
E
I.
Let t-t be a probability measure on the underlying space (I, C). A t-t-mixing sequence is a sequence (8 1 , 82 , . . .) of {L-measure-preserving automorphisms of (I, C), such that for all S, T in C, limk--.cxJ t-t (S n 8k T) = t-t (S) t-t ( T) . For each order R on I and auto morphism 8 in Q we denote by 8 R the order on I defined by 8s8R8t {:} sRt. If v is absolutely continuous with respect to the measure t-t we write v « f-t .
DEFINITION 5. Let v E ORD. A game rpv is said to be the mixing value of v if there is a measure f.-t v in NA 1 such that for all t-t in NA 1 with f.-tv « t-t and all t-t-mixing sequences (8 1 , 82 , . . . ) , for all measurable orders R, and all coalitions S, limk-+ oo rp (v ; 8k R)(S) exists and = (rpv) (S) . The set of all games in ORD that have a mixing value is denoted MIX.
THEOREM 5 [Aumann and Shapley ( 1 974), Theorem E] . MIX is a closed symmetric linear subspace of BV which contains pNA, and the map rp that associates a mixing value rpv to each v is a value on MIX with l l rp ll � 1. 7. Formulas for values
Let v be a vector measure game, i.e., a game that is a function of finitely many measures. Such games have a representation of the form v = f o f.-t, where t-t = (t-t 1 , , f.-tn ) is a vector of (probability) measures, and f is a real-valued function defined on the range R(t-t) of the vector measure t-t with f (0) = 0. . • •
Ch. 56: Values of Games with Infinitely Many Players
2141
When f is continuously differentiable, and fL = (fl, 1 , . . , f.Ln ) is a vector of non atomic measures with f.L i (/) =I= 0, then v = f o fL is in pNA00 ( C pNA c ASYMP) and the value (asymptotic, mixing, or the unique value on pNA) is given by Aumann and Shapley ( 1974, Theorem B), .
(where
ff-L(S) is the derivative of f in the direction of f.L (S) ); i.e.,
"
(S) IP Cf o /L) ( S) = " L... /Li i =l
af fo' (ttL ! (I), 0
ax· l
. . . , l f.Ln (!) ) dt .
The above formula expresses the value as an average of derivatives, that is, of marginal contributions. The derivatives are taken along the interval [0, f.L (/) ] , i.e., the line seg ment connecting the vector 0 and the vector f.L(/) . This interval, which is contained in the range of the non-atomic vector measure f.L, is called the diagonal of the range of f.L, and the above formula is called the diagonal formula. The formula depends on the differentiability of f along the diagonal and on the game being a non-atomic vector measure game. Extensions of the diagonal formula (due to Mertens) enable one to apply (variations of) the diagonal formula to games with discontinuities and with essential nondiffer entiabilities. In particular, the generalized diagonal formula defines a value of norm 1 on a closed subspace of BV that includes all finite games, bv'FA, the closed algebra generated by bv'NA, all games generated by a finite number of algebraic and lattice op erations 1 from a finite number of measures, and all market games that are functions of finitely many measures. The value is defined as a composition of positive linear symmetric mappings of norm 1 . One of these mappings is an extension operator which is an operator from a space of games into the space of ideal games, i.e., functions that are defined on ideal coalitions, which are measurable functions from (/, C) into ([0, 1 ] , B). The second one is a "derivative" operator, which is essentially the diagonal formula obtained by changing the order of integration and differentiation. The third one is an averaged derivative; it replaces the derivatives on this diagonal with an average of deriv atives evaluated in a small invariant perturbation of the diagonal. The invariance of the perturbed distribution is with respect to all automorphisms of (/, C). First we illustrate the action and basic properties of the "derivative" and the "averaged derivative" mappings in the context of non-atomic vector measure games. The derivative operator [Mertens ( 1980)] provides a diagonal formula for the value of vector measure games in bv'NA; let fL = (/LJ , . . , f.L11 ) E (NA+yz and let f : R(tL) -+ lP?. .
1 Max and min.
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Neyman
f(O) 0 and f o fL in bv'NA. Then f is continuous at {L(/) and M(0) 0, and the f o fL is given by l 1 1-£ cp(f o {L)(S) = s-->lim -0+ £ 0 [f (tM(l) + £ M(S)) - f (tM(l))] dt 1 1 1 -s = lim s--+0+ 2£ [f (t M(/) +£M(S)) - f (t M(/) - £M(S))]dt . The limits of integration in these formulas, 1 -£ and £ , could be replaced by 1 and 0 respectively whenever f is extended to lR11 in such a way that f remains continuous at {L(/) and M(0). n Let fL = (JLJ, . . . , Mn) E (NA + ) be a vector of measures. Consider the class F(M) of all functions f : R(M) --+ with f(O) = 0, f continuous at 0 and fJ.,(/), f o fL of with value of
=
=
8
lR
bounded variation, and for which the limit
11-£ [f (tJL (I) +£ M(S)) - f (t fL(l) - £M(S))]dt £-+0 £ exists for all S in C. Note that f E F(M) whenever f is continuously differentiable with f(O) = 0. An example of a Lipschitz function that is not in F(M) where fL is a vector of 2 linearly independent probability measures is f(xi, x2) = (xt - x2) sin log l x 1 - x2 l· Denote this limit by Dv( S) and note that Dv is a function of the measures M I , . . . , fLn; i.e., Dv f o fL for some function f that is defined on the range of fL. The derivative n operator D acts on all games of the form f o {L, fL E (NA+) , and f E F(fJ.,) ; it is a value of norm 1 on the space of all these games for which D(f o M) E FA. Let v = f o {L, where fL E (NA + ) n and f E F(JL ). Then D(f oM) is of the form f o {L, where f is linear on every plane containing the diagonal [0, fJ.,(/)], /(M(/)) f({L( l )) and II f o fL II � II f o fL 11 . There is no loss of generality in assuming that the measures M I , . . . , fLn are independent (i.e., that the range of the vector measure R(M) is full dimensional), and then we identify f with its unique extension to a function on lR11 that is linear on every plane containing the interval [0, {L(/)]. The averaging operator expresses the value of the game f o fL (= D(f o M) for f in F({L)) as an average, E(l:7=1 ;{ (z)Mi), where z is an n-dimensional random variable whose distribution, P is strictly stable of index 1 and has a Fourier transform � to be described below. When Ml, . . . , fLn are mutually singular, z = (Z I , . . . , Zn ) is a vector of 1 2£
lim -
=
=
fL,
independent "centered" Cauchy distributions, and then
Ch. 56: Values ofGames with Infinitely Many Players
2 143
+
Otherwise, let v E NA be such that f-L 1 , . . . , f.-L are absolutely continuous with respect n to v; e.g., v = "'£ f-L; . Define Nrt : JRn � JRn by
Then z is an n-dimensional random variable whose distribution has a Fourier transform �z ( Y ) = exp( - Nrt (y)) . Let Q be the linear space of games generated by games of the form v = f o f-L where + f-L = (/-Ll , . . . , f.-L ) E (NA )n is a vector of linearly independent non-atomic measures n and f E F(f.-L). The space Q is symmetric and the map rp : Q � FA given by
is a value of norm 1 on Q and therefore has an extension to a value of norm 1 on the closure Q of Q. The space Q contains bv'NA, the closed algebra generated by b v'NA, all games gen erated by a finite number of algebraic and lattice operations from a finite number of + non-atomic measures and all games of the fonn f o f-L where f-L E (NA )n and f is a continuous concave function on the range of f-L . To show that Ep'" ( /rt (S) (Z)) i s well defined for any f in F(f.-L), one shows that j is Lipschitz and therefore on the one hand [j(z + ot-L (S)) - j (z)]jo is bounded and continuous, and on the other hand, given S in C, for almost every z (with respect to the Lebesgue measure on lRn) frt(S) (z) exists and is bounded (as a function of z). The distribution Prt is absolutely continuous with respect to Lebesgue measure and therefore Ep'" (jrt(S) (z)) is well defined and equals lim8 _,.o Ep'" ([](z + of.-L (S)) - ](z)] / o) (using the Lebesgue bounded convergence theorem). To show that Ep'" (jrt(S) (z)) is finitely additive in S, assume that S, T are two disjoint coalitions. For any o > 0, let Pt be the translation of the measure PM by -of.-L(S) , i.e.,
Since Pt converges in norm to PM as o � 0 (i.e., IJ Pt - PM /I � 8--->0 0), and [ f(z + of.-L(T)) - ] (z)]jo is bounded, it follows that lim Ep'" ( [ j (z + o f.-L (S) 8--->0 =
+ of.-L (T) )
lim E ps ([l(z + of.-L(T) ) 8---> 0 '"
-
l(z + of.-L (S) ) ] /o)
](z) ] / 0 )
= lim Ep'" ([l(z + of.-L (T) ) - ](z) ] Jo) = Ep11 (JM
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A. Neyman
Therefore
Ep�' ( 1/1- (SU T ) (Z)) =
lim Ep�' ( [ l(z + D JL(S)
8 -+ 0
+ D JL(T) )
- f (z + 8 JL(S)) + f(z + D JL(S) ) - ] (z) ]/8) = Epl' (lfl- (S) (Z)) + Epl' (lfl- ( T ) (Z)). We now turn to a detailed description of the mappings used in the construction of a value due to Mertens on a large space of games. Let B(I, C) be the space of bounded measurable real-valued functions on (I, C), and B i (I, C ) = {f E B (I, C): 0 � f � 1 } the space of "ideal coalitions" (measurable fuzzy sets). A function v on B i (I, C) is constant sum if i:i(f) + i:i ( 1 - f) = ii ( 1 ) for every f E B t (I, C). It is monotonic if for every f, g E B i (I, C) with f ) g, i:i (f) ) i:i (g). It is finitely additive if i:i(f + g) = i:i(f) + ii (g) whenever f , g E B t (I, C) with f + g � 1 . It is of bounded variation if it is the difference of two monotonic functions, and the variation norm II v II IBV (or II v II for short) is the supremum of the variation of v over all increasing sequences 0 � !1 � h � � 1 in B t U, C ) . The symmetry group Q acts on B t (l, C) by (8d)(s) = f(8s). · · ·
An extension operator is a linear and symmetric map 1/f from a linear symmetric set of games to real-valued functions on Bt (I, C), with (1/fv) (O) = 0, so that (1/fv) ( 1 ) = v(l), 11 1/f v iiiBv � II v ii , and 1/f v is finitely additive whenever v is finitely additive, and 1/f v is constant sum whenever v is constant sum.
DEFINITION 6.
It follows that an extension operator is necessarily positive (i.e., if v is monotonic so is 1/fv) and that every extension operator 1/f on a linear subspace Q of BV has a unique extension to an extension operator 1(1 on Q (the closure of Q). One example of an extension operator is Owen's ( 1972) multilinear extension for fi nite games. For another example, consider all games that are functions of finitely many non-atomic probability measures, i.e., Q = {f o (JL 1 . . . , JLn ): n a positive integer, JL; E NA 1 and f : R(JLJ , . . . , JLn ) � JR. with f(O) = 0}, where R(JL 1 , . . . , JLn ) = {(JLt (S), . . . , JLn ( S)): S E C} is the range of the vector measure (JL 1 , . . . , f-Ln ) . Then Q is a linear and symmetric space of games. An extension operator on Q is defined as follows: for JL = (JL 1 , . . . , JLn ) E (NA 1 ) 11 and g E B t (/, C) , let JL(g) = (j1 g df-L J , . . . , J1 g dJLn ) and define 1/f (f o JL)(g) /(JL(g)). Then it is easy to verify that 1/f is well defined, lin ear, and symmetric, that 1/fv ( 1 ) = v(I), 1/f v is finitely additive whenever v is, and 1/f v is constant sum whenever v is. The inequality 11 1/fv iiiBv � ll v ll follows from the re sult of Dvoretsky, Wald, and Wolfowitz [(195 1), p. 66, Theorem 4], which asserts that 1 � fk � 1 in B t (I, C), for every JL E (NA ) 11 and every sequence 0 � ft � h � there exists a sequence 0 c St c S2 c c Sk C I with JL(f;) = JL(S;). This ex tension operator satisfies additional properties: ( 1/f v) (S) = v(S), 1/f ( v u ) = ( 1/f v) ( 1/fu), 1/f(f o v) = f o 1/fv whenever f : JR. � JR. with f (O) = 0, 1/f (v 1\ u) = (1/fv) 1\ (1/fu) and 1/f (v v u) = (1/fv) v (1/fu). The NA-topology on B(l, C) is the minimal topology =
· · ·
· · ·
Ch. 56: Values of Games with Infinitely Many Players
2 145
for which the functions f f.-i(f) are continuous for each f.-i E NA. For every game v E pNA, ljrv is NA-continuous. Moreover, as the characteristic functions Is E Bi (I, C), S E C, are NA-dense in B� (I, C), ljrv is the unique NA-continuous function on Bi (I, C) with 1jr v (S) = v (S) . Similarly, for every game v in pNA' - the closure in the supremum norm of pNA - there is a unique function v Bi (I, C) � JR. which is NA -continuous and v (S) = v(S) . Moreover, the map v v is an extension operator on pNA' and it satis fies: vu = vii, v 1\ ii = v 1\ u and v v ii = v V u for all v, u E pNA' and (f ; v) = f o v whenever f : JR. � JR. is continuous with f(O) = 0. Another example is the following natural extension map defined on bv' M. Assume that f.-i E M 1 with a set of atoms A = A(f.-L). Let v = f o f.-i with f E bv' and f.-i E NA 1 • For g E Bi (I, C) define ljrv(g) = E(f({-i(gAc + LaE A Xaa))) where Ac = I \ A and (Xa) aEA are independent {0, 1 }-valued random variables with E(X0) = g(a). If v = 2::::7=1 Vi , v; = /; o f.-ii with f.-i i E M 1 and /; E bv' we define ljrv(g) = 2::::7= vj (g). It is 1 (i.e., it is possible to show that 11 1/r v IIIBV � I v II and thus in particular v is well defined independent of the representation of v) and can be extended to the closure of all these finite sums, i.e., it defines an extension map on bv' M. The space bv'NA is a subset of both bv' M and Q and the above two extensions that were defined on Q and on bv' M coincide on bv'NA, and thus induce (by restriction) an extension operator 1jr on bv'NA and on pNA. The extension on bv'NA can be used to provide formulas for the values on bv'NA and on pNA. Given v in Q, we denote by v the extension (ideal game) lfrv. We also identify S with the indicator of the set S, xs, and the scalar t also stands for the constant func tion in Bi (I, C) with value t . Then, if v E pNA, for almost all 0 < t < 1 , for every coalition S, the derivative a v (t, S) =: lim0_,.o[v (t) + 8s) - v (t)]/8 exists [Aumann and Shapley (1974), Proposition 24.1] and the value of the game is given by 1--+
:
1--+
((JV(S) =
fo 1 a v(t, S) dt.
For any game v in bv'NA, the value of v is given by
1 1 [v(t + 8 S) - v (t)] dt 1 1-18 1 [v(t + 8S) - v(t - 8S)] dt.
1 lim 8__,. 0+ 8 0 1 = lim 8_,. 0 28 1 8 1
qJv (S) =
-
-
o
More generally, given an extension operator 1jr on a linear symmetric space of games Q, we extend v = ljrv, for v in Q, to be defined on all of B(I, C) by v (f) = v(max[O, min(l , f)]) (and then we can replace the limits 18 1 and 1 - 1 8 1 in the above integrals with 0 and 1 respectively, and define the derivative operator 1jrD by 'I' D V- cX ) = 1.liD fn ,,, o-+O 0
1 v (t + 8x ) - v(t - 8x ) dt 28
2146
A. Neyman
whenever the integral and the limit exists for all x in B(I, C). The integral exists whenever v has bounded variation. The derivative operator is positive, linear and sym metric. Assuming continuity in the NA-topology of v at 1 and 0, 1/JD also satisfies 1/!Dv (a + f3 x ) = av(l) + /3 [1/!Dii](X) for every a, f3 E JR, so that 1/JDv is linear on every plan in B(I, C) containing the constants and 1/JD is efficient, i.e., 1/JDii(1) = v(1). The following [due to Mertens (1988a)] will (essentially) extend the previously men tioned extension operators to a large class of games that includes bv'FA and all games of the form f o J1, where J1, = (Ji,I , . . . , Ji,n ) E NA. Intuitively we would like to define 1/!Ev(f) for f in Bi (I, C) as the "limit" of the expectation v (S) with respect to a dis tribution P of a random set that is very similar to the ideal set f. Let F be the set of all triplets a = (n, v, E) where n is a finite subfield of C, v is a finite set of non-atomic elements of FA and E > 0. F is ordered by a = (n, v, E) � a' = (n', v', 81) if n' ::> n , v' ::> v and 8 1 � E. (F, �) is filtering-increasing with respect to this order, i.e., given a = (n, v, E) and a' = (n', v', 8 1 ) in F there is a'' = (n", v", 8 11 ) with a'' ?: a and a" ?: a' . For any a = (n, v, 8 ) in F and f in Bi (I, C), Pa,f is the set of all probabilities P with finite support on C such that I L SEC SP(S) - ! I == I Ep (S) - fl < E uniformly on I (i.e., for every t E I, I L SEC P(S)I (t E S) - f(t) l < c), and such that for any TI , T2 in n with TI n T2 = 0, TI n S is independent of T2 n S, and for all J1, in VJi,(S n TI) = j71 f dfl,. That Pa,f =/= 0 follows from Lyapunov's convexity theorem on the range of a vector of non-atomic elements of FA. For any game v, and any f E Bi(I, C), let v (f) = lim sup Ep (v(S)) == alim sup L v(S) P (S) cxEF PEPa,J EF PEPa,J SEC and Q(j) = lim inf Ep (v(S)) . cxEF PEPa.j The definition of v is extended to all of B(I, C) by v (f) = v (max(O, min(1 , f))). Note that if v is a game with finite support, then for any f in Bi (I, C), v (f) = Q(f) coincides with Owen's multilinear extension, i.e., letting T be the finite support of v,
[
J
v (f) = L Ti t (s ) n (1 - f(s )) v (S) SET\S SeT sES and whenever v is a function of finitely many non-atomic elements of FA, VI , . . . , Vn , i.e., v = g o (V I , . . . , vn) where g is a real-valued function defined on the range of (vi , . . , Vn ) , then for any f E Bi (I, C) .
Ch. 56: Values of Games with Infinitely Many Players
2147
Let V8 = {X E Bi (I, C): sup x - infx � 8 }. Let D be the space of all games v in BV for which � sup{v(x ) - .!!.(x) : x E Ys} --+ 0 as 8 --+ 0+. Obviously, D :J Ds where Ds is the set of all games v for which v(x ) = .!!. ( X) for any x E V8 • Next we define the derivative operators cpD ; its domain is the set of all games v in D for which the integrals (for sufficiently small 8 > 0) and the limit
. 1 1 v(t + rx ) - v (t - rx) dt
[cpDv](x) = hm
r-+0
o
2r
exist for every x in B ( I , C). The integrals always exist for games in BV. The limit, on the other hand, may not exist even for games that are Lipschitz functions of a non-atomic signed measure. However, the limit exists for many games of interest, including the concave functions of finitely many non-atomic measures, and the algebra generated by bv'FA. Assum ing, in addition, some continuity assumption on v at 1 and 0 (e.g., v (f) --+ iJ(1) as inf(f) 1 and v (f) v (0) as sup(/) --+ 0) we obtain that cpD v obeys the follow ing additional properties: CfJDV(X) + CfJDV(l - x ) = v(l) whenever v is a constant sum; [cpDv](a + bx) = a v(1) + b[cpDv](x) for every a, b E R --+
--+
THEOREM 6 [Mertens (1988a), Section 1]. Let Q = {v E Dom cpD: CfJDV E FA}. Then Q is a closed symmetric space that contains DIFF, DIAG and bv'FA and cpD : Q --+ FA is a value on Q. For every function w : B ( I , C) --+ lR in the range of C(JD, and every two elements x, h in B(l, C), we define Wh (X) to be the (two-sided) directional derivative of w at X in the direction h, i.e.,
8-+0
w, (x) = lim [ w(x + 8 h) - w(x - 8 h) ] /(28 ) whenever the limit exists. Obviously, when w is finitely additive then wh (X) w (h). For a function w in the range of cpD and of the form w = f o f.L where f.L = (f.L 1 , . . , f.Ln ) is a vector of measures in NA, f is Lipschitz, satisfying f (af.L(I) + bx) = af (f.L(I)) + bf (x) and for every y in L (R(tL)) the linear span of the range of the vector measure f.L - for almost every x in L (R(tL)) the directional derivative of f at x in the direction y, jy (x) exists, and then obviously wh (X) = ftJ.,(hJ (/L(X)). The following theorem will provide an existence of a value on a large space of games as well as a formula for the value as an average of the derivatives (cpD o cpE v) h (x) Wh (X) where x is distributed according to a cylinder probability measure on B(l, C) which is invariant under all automorphisms of (I, C ) . We recall now the concept of a cylinder measure on B(I, C). The algebra of cylinder sets in B(l, C) is the algebra generated by the sets of the firm f.L -I (B) where B is any Borel subset of JRn and f.L = (f.L 1 , . . . , f.Ln ) is any vector of measures. A cylinder probability is a finitely additive probability P on the algebra of =
.
-
=
2148
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Neyman
cylinder sets such that for every vector measure JL = (JL 1 , . . . , fLn ) , P o JL - ! is a count ably additive probability measure on the Borel subsets of lRn . Any cylinder probability P on B (I, C) is uniquely characterized by its Fourier transform, a function on the dual that is defined by F (JL) = Ep (exp(iJL(X)). Let P be the cylinder probability measure on B(I, C) whose Fourier transform Fp is given by Fp (JL) = exp ( IIJL I I ) . This cylinder measure is invariant under all automorphisms of (/, C). Recall that (/, C) is isomorphic to [0, 1] with the Borel sets, and for a vector of measures JL = (JL 1 , , JLn) , the Fourier transform of P o fL - I , FP o p. - is given by FPo fL- 1 (y) = exp( -NIL (y)), and P o JL- l is absolutely continuous with respect to the Lebesgue measure, with moreover, continuous Radon-Nikodym derivatives. Let Q M be the closed symmetric space generated by all games v in the domain of CfJD such that either cpD v E FA, or cp D ( v) is a function of finitely many non-atomic measures. -
. • .
I
THEOREM 7 [Mertens (1988a), Section 2] . Let v E Q M . Then for every h in B(l, C), (cpD (v) )h (X) exists for P almost every x and is P integrable in x, and the mapping of each game v in Q M to the game cp v given by
j
cpv(S) = (cp D (v) ) 5 (x) dP(x) is a value ofnorm 1 on Q M . REMARKS. The extreme points of the set of invariant cylinder probabilities on B(l, C) have a Fourier transform Fm,a (JL) = exp (im JL ( 1 ) cr ii JL II ) where m E JR, cr ;;;:, 0. More precisely, there is a one-to-one and onto map between countably additive measures Q on lR lR_+ and invariant cylinder measures on B(l, C) where every measure Q on 1R_ JR+ is associated with the cylinder measure whose Fourier transform is given by -
x
x
The associated cylinder measure is nondegenerate if Q(r = 0) = 0, and in the above value formula and theorem, P can be replaced with any invariant nondegenerate cylin der measure of total mass 1 on B (I, C). Neyman (2001 ) provides an alternative approach to define a value on the closed space spanned by all bounded variation games of the form f o JL where JL is a vector of non atomic probability measures and f is continuous at JL (0) and JL(/), and Q M . This alternative approach stems from the ideas in Mertens (1988a) and expresses the value as a limit of averaged marginal contributions where the average is with respect to a distribution which is strictly stable of index 1 . For any lRn -valued non-atomic vector measure JL we define a map cpt from Q(JL) the space of all games of bounded variation that are functions of the vector measure JL and are continuous at JL(0) and at JL(/) - to BV. The map cp! depends on a small positive constant 8 > 0 and the vector measure fL = (JL 1 , . . , JLn ) . .
Ch. 56: Values of Games with Infinitely Many Players
2149
For o > 0 let h (t) = I (3o � t < 1 - 3o) where I stands for the indicator function. The essential role of the function h is to make the integrands that appear in the integrals used in the definition of the value well defined. Let JL = (JL 1 , . . . , f.Ln ) be a vector of non-atomic probability measures and f : R(JL) -+ lR be continuous at JL(0) and JL(J) and with f o JL of bounded variation. It follows that for every x E 2R(JL) - JL(J), S E C, and t with fo (t) = 1 , tJL(J) + ox and tJL(I) + ox + OJL(S) are in R(JL) and therefore the functions t r-+ fo (t)f(tJL(I) + ox) and t r-+ h (t)f (tJL(I) + ox + oJL(S)) are of bounded variation on [0, 1 ] and thus in particular they are integrable functions. Therefore, given a game f o fL E Q(JL), the function Ff, J-i , defined on all triples ( o, x, S) with o > 0 sufficiently small (e.g., o < 1 /9), x E JRn with ox E 2R(JL) - JL (I), and S E C by
is well defined. Let P� be the restriction of P"' to the set of all points in {x E JRn : ox E 2R(JL) JL (I)}. The function x Ff, ( o, x, S) is continuous and bounded and therefore the function ((Jt defined on Q(JL) x C by r-+
((J� (f o JL, S) = {
jAF(J-i)
11
Ft,"' (o, x, S) dP� (x),
where AF(JL) stands for the affine space spanned by R(JL), is well defined. The linear space of games Q(JL) is not a symmetric space. Moreover, the map ((Jt violates all value axioms. It does not map Q (JL) into FA, it is not efficient, and it is not symmetric. In addition, given two non-atomic vector measures, JL and v, the op erators ((Jt and ((J� differ on the intersection Q(JL) n Q(v). However, it turns out that the violation of the value axioms by ((J� diminishes as o goes to 0, and the difference ((Jt (f o JL) - ((J � (go v) goes to 0 as o 0 whenever f o JL = g o v . Therefore, an appropri ate limiting argument enables us to generate a value on the union of all the spaces Q (JL). Consider the partially ordered linear space C of all bounded functions defined on the open interval (0, 1 /9) with the partial order h >- g if and only if h (o) ;:? g(o) for all sufficiently small values of 8 > 0. Let L : £ -+ lR be a monotonic (i.e., L (h) ;:? L(g) whenever h >- g) linear functional with L(l) = 1 . Let Q denote the union of all spaces Q (JL) where JL ranges over all vectors of fi nitely many non-atomic probability measures. Define the map ((J : Q JRC by ((JV(S) = L (((Jt ( v, S)) whenever v E Q (JL). It turns out that ((J is well defined, ((J v is in FA (when ever v E Q) and ((J is a value of norm 1 on Q. As ((J is a value of norm 1 on Q and the continuous extension of any value of norm 1 defines a value (of norm 1 ) on the closure, we have: -+
-+
PROPOSITION
6. ((J is a value of norm 1 on Q.
A.
2150
Neyman
8. Uniqueness of the value of nondifferentiable games
The symmetry axiom of a value implies [see, e.g., Aumann and Shapley (1974), p. 139] that if = (M , . . . , f.Ln ) is a vector of mutually singular measures in NA 1 , [0, l ] n -+ IR?., and v = f o is a game which is a member of a space Q on which a value
f.L
f:
f.L
(f)f.L; . La; i=l
I:7= I (f) f(1, (f) f f.L f.L
f (f)
The efficiency of rp implies that a; = . . . , 1), and if in addition is symmetric in the n coordinates all the coefficients a; are the same, i.e., a; = * 1 , . . . , 1). The results below will specify spaces of games that are generated by vector measure games v = o where is a vector of mutually singular non-atomic probability measures and f [0, l ] n lR?. and on which a unique value exists. Denote by M� the linear space of function on [0, 1] k spanned by all concave Lip = 0 and schitz functions [0, 1] k -+ lR?. with whenever is in . the interior of [0, 1]k , y E IR?.t and 0 < :(; 1 . M% is the linear space of all continuous piecewise linear functions on [0, 1] k with = 0.
f(
:
f:
-+
f(O) t
f
f(O)
fy(x) :(; fy(tx)
x
DEFINITION 7 [Haimanko (2000d)]. CONs (respectively, LINs ) is the linear span of games of the form o where for some positive integer k, f E M� (respectively, E M�) and = 1 , . . , f.Lk) is a vector of mutually singular measures in NA 1 .
f f.L f.L (f.L
f
.
The spaces CONs and LINs are in the domain of the Mertens value; therefore there is a value on CONs and on LlN5 • If E M� or f E M%, the directional derivative exists whenever is in the interior of [0, 1] k and y E JR?.k , and for each fixed x , the directional derivative of the function y in the direction z E IR?.k,
f
x
1.
lm
E--+0+
r+
fy+cz (X ) - fy (x) '
fy (x)
jy (x)
£
af (x;
exists and is denoted y; z). If f.L = (f.L 1 , . . . measures in NA 1 and rpM is the Mertens value,
, f.Lk) is a vector of mutually singular
fa aj (th; f.L (S)) dt 1
rpM (f o f.l,) (S) =
X;
where lk = (1, . . . , 1) E JR?.k and X = (X 1 , . . . , Xk ) is a vector of independent random variables, each with the standard Cauchy distribution. THEOREM 8 [Haimanko (2000d) and (2001b)]. The Mertens value is the unique value on CONs and on L/N5•
Ch. 56: Values of Games with Infinitely Many Players 9.
2151
Desired properties of values
The short list of conditions that define the value, linearity, positivity, symmetry and efficiency, suffices to determine the value on many spaces of games, like pNA00, pNA and bv'NA. Values were defined in other cases by means of constructive approaches. All these values satisfy many additional desirable properties like continuity, the projection axiom, i.e., cpv = v whenever v E FA is in the domain of cp, the dummy axiom, i.e., cpv(Sc ) 0 whenever S is a carrier of v, and stronger versions of positivity. =
9.1. Strong positivity One of the stronger versions of positivity is called strong positivity: Let Q <;::; BV. A map cp : Q ---+ FA is strongly positive if cpv(S) � cp w ( S) whenever v, w E Q and v(T U S') v(T) � w(T U S') - w(T) for all S' ::;; S and T in C. It turns out that strong positivity can replace positivity and linearity in the character ization of the value on spaces of "smooth" games. THEOREM 9 [Monderer and Neyman ( 1 98 8), Theorems 8 and 5]. (a) Any symmetric, efficient, strongly positive map from pNA00 to FA is the Aumann Shapley value. (b) Any symmetric, efficient, strongly positive and continuous map from pNA to FA is the Aumann-Shapley value. 9.2. The partition value Given a value cp on a space Q, it is of interest to specify whether it could be interpreted as a limiting value. This leads to the definition of a partition value. A value cp on a space Q is a partition value [Neyman and Tauman ( 1979)] if for every coalition S there exists a T -admissible sequence (nk ) � 1 such that cpv(T) = limk--+oo 1/fvlfk (T). It follows that any partition value is continuous (with norm � 1) and coincides with the asymptotic value on all games that have an asymptotic value [Neyman and Tau man ( 1 979)] . There are however spaces of games that include ASYMP and on which a partition value exists. Given two sets of games, Q and R, we denote by Q * R the closed linear and symmetric space generated by all games of the form uv where u E Q and v E R. THEOREM 1 0 [Neyman and Tauman ( 1 979)] . There exists a partition value on the closed linear space spanned by ASYMP, bv'NA bv'NA and .A * bv'NA * bv'NA. *
9.3. The diagonal property A remarkable implication of the diagonal formula is that the value of a vector measure game f o 11- in pNA or in bv'NA is completely determined by the behavior of f near the
2152
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Neyman
diagonal of the range of fL. We proceed to define a diagonal value as a value that depends only on the behavior of the game in a diagonal neighborhood. Formally, given a vector of non-atomic probability measures fL = (/L . . . , f.Ln ) and 8 > 0, the 8 fL diagonal neighborhood, D(f.L, 8 ), is defined as the set of all coalitions S with l fL; (S) - fLi (S) I < 8 for all 1 � i, j � n. A map fP from a set Q of games is diagonal if for every 8 fL diagonal neighborhood D(f.L, 8) and every two games v, w in Q n BV that coincide on V(f.L, 8) (i.e., v (S) = w (S) for any S in V(f.L, 8)), ({I V = fP W. There are examples of non-diagonal values [Neyman and Tauman (1976)] even on reproducing spaces [Tauman ( 1977)]. However,
1,
-
-
THEOREM 1 1 [Neyman (1977a)]. Any continuous value (in the variation norm) is di agonal. Thus, in particular, the values on (pNA and) bv'NA [Aumann and Shapley (1974), Proposition 43. 1], the weak asymptotic value, the asymptotic value [Aumann and Shap ley (1974), Proposition 43. 1 1], the mixing value [Aumann and Shapley (1974), Propo sition 43.2], any partition value [Neyman and Tauman (1979)] and any value on a closed reproducing space, are all diagonal. Let DIAG denote the linear subspace of BV that is generated by the games that vanish on some diagonal neighborhood. Then DIAG C ASYMP and any continuous (and in particular any asymptotic or partition) value vanishes on DIAG. 10. Semivalues
Semivalues are generalizations and/or analogies of the Shapley value that do not (neces sarily) obey the efficiency, or Pareto optimality, property. This has stemmed partly from the search for value function that describes the prospects of playing different roles in a game. DEFINITION 8 [Dubey, Neyman and Weber (1981)]. A semivalue on a linear and sym metric space of games Q is a linear, symmetric and positive map 1/1 : Q -+ FA that satisfies the projection axiom (i.e., 1/fv = v for every v in Q n FA). The following result characterizes all semivalues on the space FG of all games having a finite support. THEOREM 1 2 [Dubey, Neyman and Weber (1981), Theorem 1]. (a) For each probability measure � on [0, 1 ], the map 1/1� that is given by 1/f� v(S) =
fo 1 av(t, S) d� (t)
,
Ch. 56: Values of Games with Infinitely Many Players
2 1 53
where v is Owen 's multilinear extension of the game v, defines a semivalue on FG. Moreover, every semivalue on FG is of this form and the mapping � 1/f$ is 1-1. (b) 1/f$ is continuous (in the variation norm) on G if and only if � is absolutely continuous w.r.t. the Lebesgue measure A on [0, 1 ] and d�/dA E L 00 [0, 1]. In that case, l l1/f$ I IBV = l i d� /dA I I oo· __,.
Let W be the subset of all elements g in L 00 [0, 1 ] such that J01 g(t) dt = 1 . THEOREM 1 3 [Dubey, Neyman and Weber ( 1 98 1 ), Theorem 2]. For every g in W, the map 1/fg : pNA FA defined by __,.
1/fg v(S) =
fo1 av(t, S)g(t) dt
is a semivalue on pNA. Moreover, every semivalue on pNA is of this form and the map g 1/fg maps W onto the family ofsemivalues on pNA and is a linear isometry. __,.
Let We be the subset of all elements g in W which are (represented by) continuous functions on [0, 1 ] . We identify W (We) with the set of all probability measures � on [0, 1] which are absolutely continuous w.r.t. the Lebesgue measure A on [0, 1 ] , and their Radon-Nikodym derivative d� jdA E W ( E We) . DEFINITION 9 . Let v be a game and let � be a probability measure on [0, 1 ] . A game cpv is said to be the weak asymptotic � -semivalue of v if, for every S E C and every E > 0, there is a finite subfield JT of C with S E JT such that for any finite subfield JT 1 with JT 1 ::> JT , REMARKS. ( 1 ) A game v has a weak asymptotic �-semivalue if and only if, for every S in C and every E > 0, there is a finite subfield JT of C with S E JT such that for any subfield JT 1 with 7T 1 ::> JT , 1 1/f� Vrr' (S) - 1/f� Vrr (S) I < E. (2) A game v has at most one weak asymptotic � -semivalue. (3) The weak asymptotic �-semivalue cpv of a game v is a finitely additive game and ll cpv ll :( ll v ll ll d�/dA II oo whenever � E W. Let � be a probability measure on [0, 1 ] . The set of all games of bounded variation and having a weak asymptotic �-semivalue is denoted ASYMP* (�). THEOREM 14. Let � be a probability measure on [0, 1]. (a) The set ofall games having a weak asymptotic � -semivalue is a linear symmetric space of games and the operator that maps each game to its weak asymptotic � -semivalue is a semivalue on that space.
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(b) ASYMP* (�) is closed {:} � E W {:} pFA C ASYMP*(�) {:} pNA C ASYMP* (�). (c) � E We :::::} bv'NA C bv'FA c ASYMP* (�). The asymptotic � -semivalue of a game v is defined whenever all the sequences of the �-semivalues of finite games that "approximate" v have the same limit. DEFINITION 1 0. Let � be a probability measure on [0, 1 ] . A game cp v is said to be the asymptotic � -semivalue of v if, for every T E C and every T -admissible sequence (Jri ) � 1 , the following limit and equality exists: lim 1/f� v;rr:k (T) = cp v(T). oo
k -+
REMARKS. (1) A game v has an asymptotic �-semivalue if and only if for every S in C and every S-admissible sequence 'P = (ni )� 1 the limit, limk-+oo 1/f� v:rr; (S), exists and is independent of the choice of 'P . (2) For any given v, the asymptotic � -semivalue, if it exists, is clearly unique. (3) The asymptotic �-semivalue cp v of a game v is finitely additive and i i cp v i \ Bv � ll v iiBv l\d�/dA. I \ oo (when � E W). (4) If v has an asymptotic � -semivalue cp v, then v has a weak asymptotic � -semivalue (= cpv) . The space of all games of bounded variation that have an asymptotic � -semivalue is denoted ASYMP(�). THEOREM 1 5 . (a) [Dubey (1980)]. pNA00 C ASYMP(�) and if cp v denotes the asymptotic � semivalue ofv E pNA 00 , then llcp v lloo � 2 11 v lloo· (b) pNA C pM c ASYMP(�) whenever � E W. (c) bv'NA C bv' M C ASYMP(�) whenever � E We. 11. Partially symmetric values
Non-symmetric values (quasivalues) and partially symmetric values are generalizations and/or analogies of the Shapley value that do not necessarily obey the symmetry axiom. A (symmetric) value is covariant with respect to all automorphisms of the space of players. A partially symmetric value is covariant with respect to a specified subgroup of automorphisms. Of particular importance are the trivial group, the subgroups that preserve a finite (or countable) partition of the set of players and the subgroups that preserve a fixed population measure. The group of all automorphisms that preserve a measurable partition II, i.e., all au tomorphisms e such that es S for any S E II, is denoted Q(II). Similarly, if ll is a a--algebra of coalitions (II c C) we denote by Q(II) the set of all automorphisms e =
Ch. 56:
2 155
Values of Games with Infinitely Many Players
such that () S = S for any S E n. The group of automorphisms that preserve a fixed measure JL on the space of players is denoted Q (JL) . 11.1. Non-symmetric and partially symmetric values DEFINITION 1 1 . Let Q be a linear space of games. A non-symmetric value (quasi value) on Q is a linear, efficient and positive map 1/f : Q � FA that satisfies the projec tion axiom. Given a subgroup of automorphisms H, an H-symmetric value on Q is a non-symmetric value 1/f on Q such that for every () E H and v E Q, 1/f (()* v) ()* ( 1/f v). Given a a-algebra n, a fl-symmetric value on Q is a Q(fl)-symmetric value. Given a measure JL on the space of players, a tJ,-value is a Q (JL) -symmetric value. =
coalition structure is a measurable, finite or countable, partition n of 1 such that every atom of n is an infinite set. The results below characterize all the Q(fl) symmetric values, where n is a fixed coalition structure, on finite games and on smooth games with a continuum of players. The characterizations employ path values which are defined by means of monotonic increasing paths in Bt ( 1, C). A path is a monotonic function y : [0, 1] � Bt (l, C) such that for each s E 1, y (O)(s) 0, y (l)(s) = 1 , and the function t �--+ y (t) (s) is continuous at 0 and 1 . A path y is called continuous if for every fixed s E 1 the function t �--+ y (t)(s) is continuous; NA-continuous if for every JL E NA the function t � tJ,(y (t)) is continuous; and fl-symmetric (where n is a par tition of 1) if for every 0 � t � 1 , the function s y (t) (s) is fl -measurable. Given an extension operator 1/f on a linear space of games Q (for v E Q we use the notation v = 1/f v) and a path y , the y -path extension of v, iiy , is defined [Haimanko (2000c)] by A
=
�--+
vy ( f) = v (r * cn )
where y* : Bt (l, C) � Bt (I, C) is defined by y*(f)(s) = y (f(s))(s). For a coalition S E C and v E Q, define
cp� (v) (S) =
1 1 1-£ [ v (tl + sS) -
-
E
c
y
iiy (tl)
] dt
and DIFF(y) is defined as the set of all v E Q such that for every S E C the limit ( v) (S) = limc:-+0+ cp� ( v) (S) exists and the game
PROPOSITION 7 [Haimanko (2000c)]. DIFF(y) is a closed linear subspace of Q and the map
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THEOREM 1 6 [Haimanko (2000c)]. (a) y is NA-continuous <==} pNA C DIFF(y). (b) Every non-symmetric value on pNA(A), A E NA 1 , is a mixture ofpath values. (c) Every II -symmetric value (where II is a coalition structure) on pNA (on FG) is a mixture of II -symmetric path values. REMARKS [Haimanko (2000c)] . (1) There are non-symmetric values on pNA which are not mixtures of path values. (2) Every linear and efficient map 1/J pNA -+ FA which is II -symmetric with respect to a coalition structure II obeys the projection axiom. Therefore, the projection axiom of a non-symmetric value is not used in (c) of Theorem 16. The above characterization of all II -symmetric values on pNA(A) as mixtures of path values relies on the non-atomicity of the games in pNA. A more delicate construction, to be described below, leads to the characterization of all II -symmetric values on pM. :
For every game v in b v ' M there is a measure A. E M1 such that v E bv' M(A). The set of atoms of A is denoted by A(A.). Given two paths y , y', define for v E bv' M(A), f E B�(l, C)
where y" y [l A(A)] + y'[A(A)] and v -+ v is the natural extension on bv' M. It can be verified that vy,y' is well defined on bv' M, i.e., the definition is independent of the measure A. For a coalition S E C and v E bv' M, define =
-
We associate with bv' M and the pair of paths y and y' the set DIFF(y, y') of all games v E bv' M such that for every S E C the limit
=
THEOREM 1 7 [Haimanko (2000c)]. (a) DIFF(y, y') is a closed linear subspace of bv' M and the map
1--+
1--+
REMARKS [Haimanko (2000c)]. (1) If y is NA-continuous and the discontinuities of the functions t y'(t) (s), s E /, are mutually disjoint, pM C DIFF(y, y') . In particu lar if y and y' are continuous, pM C DIFF(y, y') and therefore if in addition y (t) and y'(t) are constant functions for each fixed t, y and y' are Q-symmetric and thus
Ch. 56: Values ofGames with Infinitely Many Players
2 1 57
defines a value of norm 1 on pM. These infinitely many values on pM were discovered by Hart (1973) and are called the Hart values. (2) A surprising outcome of this systematic study is the characterization of all values on pM. Any value on pM is a mixture of the Hart values. (3) If n is a coalition structure, any n -symmetric linear and efficient map 1jJ : bv' M -+ FA obeys the projection axiom. Therefore, the projection axiom of a non symmetric value is not used in the characterization (c) of Theorem 17. Weighted values are non-symmetric values of the form rpy where y is a path de scribed by means of a measurable weight function w : I -+ 18?.++ (which is bounded away from 0): for every y (s)(t) = t w (s) . Hart and Monderer (1997) have success fully applied the potential approach of Hart and Mas-Colell (1989) to weighted val ues of smooth (continuously differentiable) games: to every game v E pNA00 the game (Pwv)(S) = J01 v(t�S) dt is in pNA00 and rpy v(S) = (Pw v) w s U). In addition they define an asymptotic w-weighted value and prove that every game in pNA00 has an asymptotic w-weighted value. Another important subgroup of automorphisms is the one that preserves a fixed (pop ulation) non-atomic measure fJ .
11.2. Measure-based values Measure-based values are generalizations and/or analogues of the value that take into account the coalition worth function and a fixed (population) measure fJ. This has stemmed partly from applications of value theory to economic models in which a given population measure fJ is part of the models. Let fJ be a fixed non-atomic measure on (!, C) . A subset of games, Q, is tJ-symmetric if for every 8 E 9(/L) (the group of tJ-measure-preserving automorphisms), 8* Q = Q. A map rp from a tJ-symmetric set of games Q into a space of games is tJ-symmetric if rp8* v = 8*rpv for every 8* in 9(tJ) and every game v in Q. DEFINITION 1 2 . Let Q be a tJ-symmetric space of games. A tJ-value on Q is a map from Q into finitely additive games that is linear, tJ-symmetric, positive, efficient, and obeys the dummy axiom (i.e., rpv(S) = 0 whenever the complement of S, sc , is a carrier of v).
The definition of a tJ-value differs from the definition of a value in two aspects. First, there is a weakening of the symmetry axiom. A value is required to be covariant with the actions of all automorphisms, while a tJ-value is required to be co variant only with the actions of the tJ-measure-preserving automorphism. Second, there is an additional axiom in the definition of a tJ-value, the dummy axiom. In view of the other axioms (that rpv is finitely additive and efficiency), this additional axiom could be viewed as a strengthening of the efficiency axiom; the tJ-value is required to obey rpv(S) = v(S) for any carrier S of the game v, while the efficiency axiom requires it REMARKS .
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A. Neyman
only for the carrier S = I. In many cases of interest, the dummy axiom follows from the other axioms of the value [Aumann and Shapley (1974), Note 4, p. 18] . Obviously, the value on pNA obeys the dummy axiom and the restriction of any value that obeys the dummy axiom to a JL-symmetric subspace is a f.,t-value. In particular, the restriction of the unique value on pNA to pNA(JL), the closed (in the bounded variation norm) alge bra generated by non-atomic probability measures that are absolutely continuous with respect to JL, is a JL-value. It turns out to be the unique JL-value on pNA(f.L).
THEOREM 18 [Monderer (1986), Theorem A]. There exists a unique JL-value on pNA(JL); it is the restriction to pNA(JL) of the unique value on pNA. REMARKS. In the above characterization, the dummy axiom could be replaced by the projection axiom [Monderer (1986), Theorem D]: there exists a unique linear, JL symmetric, positive and efficient map cp : pNA (f.L) -+ FA that obeys the projection axiom;
it is the unique value onpNA(JL). The characterizations of the JL-Value onpNA(JL) is de rived from Monderer [(1986), Main Theorem], which characterizes all JL-symmetric, continuous linear maps from pNA (f.L) into FA.
Similar to the constructive approaches to the value, one can develop corresponding constructive approaches to f.,t-values. We start with the asymptotic approach. Given T in C with JL(T) > 0, a JL-T -admissible sequence is a T -admissible sequence (n 1 , n2 , . . . ) for which limk__,.00(min{JL(A): A is an atom of JTk}/ max{JL(A): A is an atom of nk}) = 1 .
DEFINITION 1 3 . A game cp v is said to be the JL-asymptotic value of v if, for every T in C with JL(T) > 0 and every JL- T -admissible sequence (JTi ) � 1 , the following limit and equality exists 1/fvnk (T) = cpv(T). klim --> 00
REMARKS. (1) A game v has a JL-asymptotic value if and only if for every T in C with JL(T) > 0 and every JL-T -admissible sequence P = (JTi ) � 1 ' the limit, limk-->oo 1/fvnk (T) exists and is independent of the choice of P. (2) For a given game v, the JL-asymptotic value, if it exists, is clearly unique. (3) The JL-asymptotic value cpv of a game v is finitely additive and II cpv I :(; I v II when ever v has bounded variation. The space of all games of bounded variation that have a JL-asymptotic value is de noted by ASYMP[JL].
THEOREM 19 [Hart (1980), Theorem 9.2]. The set of all games having a JL-asymptotic value is a linear JL-symmetric space of games and the operator mapping each game to its JL-asymptotic value is a JL-value on that space. ASYMP[JL] is a closed JL-symmetric
Ch. 56:
Values of Games with Infinitely Many Players
2159
linear subspace of BV which contains allfunctions of the form f o (f.LJ , . . . , f.Ln ) where
f.L l , . . . , f.Ln are absolutely continuous w.r.t. f.L and with Radon-Nikodym derivatives,
df.L i /df.L, in L2 (JL) and f is concave and homogeneous ofdegree 1. 12. The value and the core
The Banach space of all bounded set functions with the supremum norm is denoted BS, and its closed subspace spanned by pNA is denoted pNA'. The set of all games in pNA' that are homogeneous of degree 1 (i.e., v(af) = a v(f) for all f in B { (I, C) and all a in [0, 1 ]), and superadditive (i.e., v(S U T) ;;:: v(S) + v (T) for all S, T in C), is denoted H'. The subset of all games in H' which also satisfy v ;;:: 0 is denoted H� . The core of a game v in H' is non-empty and for every S in C, the minimum of JL(S) when f.L runs over all elements JL in core ( v) is attained. The exact cover of a game v in H' is the game min{JL(S): JL E Core(v)} and its core coincides with the core of v. A game v E H' that equals its exact cover is called exact. The closed (in the bounded variation norm) subspace of BV that is generated by pNA and DIAG is denoted by pNAD. THEOREM 2 0 [Aumann and Shapley (1974), Theorem I]. The core of a game v in pNAD n H' (which obviously contains pNA n H') has a unique element which coincides with the (asymptotic) value of v.
2 1 [Hart (1977a)] . Let v E H� . lfv has an asymptotic value cpv, then cpv is the center of symmetry of the core of v. The game v has an asymptotic value cpv when either the core ofv contains a single element v (and then cp v = v), or v is exact and the core of v has a center of symmetry v (and then cp v = v).
THEOREM
(1) The second theorem provides a necessary and sufficient condition for the asymptotic value of an exact game v in H� to exist: the symmetry of its core. There are (non-exact) games in H� whose cores have a center of symmetry which do not have an asymptotic value [Hart (1977a), Section 8]. (2) A game v in H� has a unique element in the core if and only if v E pNAD. (3) The conditions of superadditivity and homogeneity of degree 1 are satisfied by all games arising from non-atomic (i.e., perfectly competitive) economic markets (see Chapter 35 in this Handbook). However, the games arising from these markets have a finite-dimensional core while games in H� may have an infinite-dimensional core. REMARKS .
The above result shows that the asymptotic value fails to exist for games in H� when ever the core of the game does not have a center of symmetry. Nevertheless, value theory can be applied to such games, and when it is, the value turns out to be a "center" of the core. It will be described as an average of the extreme points of the core of the game v E H� .
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A. Neyman
Let v E H� have a finite-dimensional core. Then the core of v is a convex closed subset of a vector space generated by finitely many non-atomic probability mea sures, v 1 , . . . , Vn . Therefore the core of v is identified with a convex subset K of JR" , by a = (a 1 , . . . , a11) E K if and only if L, a;f.L; E Core(v). For almost every z in lR" there exists a unique a(z) = (a i (Z) . . . , a11(z)) in K with L.7=1 a; (z)z; = min{ L,7= 1 a; z; : (a 1 , . . . , an ) E K} . ,
THEOREM 2 2 [Hart (1980)]. If vl , . . . , v11 are absolutely continuous with respect to f.L and dv;/df.L E L2 (f.L), then v E ASYMP[f.L] and its f.L-asymptotic value cp0 v is given by
where N is a normal distribution on ]R" with 0 expectation and a covariance matrix that is given by EN (ZiZj ) =
! ( dv· ) ( dv ·) df.L . d; d: I
Equivalently, N is the distribution on JR" whose Fourier transform cpN is given by
THEOREM 23 [Mertens (1988b)]. The Mertens value of the game v E H� with a finite dimensional core, cpv, is given by
where G is the distribution on JR" whose Fourier transform C(JG is given by (/JG (y) exp( - Nv (Y)), v = (vi , . . . , V11 ) and Nv is defined as in the section regarding formulas for the value. =
It is not known whether there is a unique continuous value on the space spanned by all games in H� which have a finite-dimensional core. We will state however a result that proves uniqueness for a natural subspace. Let MF be the closed space of games spanned by vector measure games f o f.L E H� where f.L = (f.L 1 , . . . , f.Lk) is a vector of mutually singular measures in NA 1 .
THEOREM 24 [Haimanko (200l a)] . The Mertens value is the only continuous value on Mp.
Ch. 56: Values of Games with Infinitely Many Players
2 16 1
13. Comments on some classes of games
13.1. Absolutely continuous non-atomic games A game v is absolutely continuous with respect to a non-atomic measure 11o if for every E > 0 there is 8 > 0 such that for every increasing sequence of coalitions S1 C Sz C C Szk with 2::(= 1 tJo(Sz;) - tJo (Sz; - 1 ) < 8, 2:7= 1 l v (Sz; ) - v (Szi-J ) I < E . A game v is absolutely continuous if there is a measure 11o E NA+ such that v is ab solutely continuous with respect to f.-L. A game of the form f o 11o where 11o E NA 1 is absolutely continuous (with respect to tJo) if and only if f : [0, f.-L(l)] ---+ lR is absolutely continuous. An absolutely continuous game has bounded variation and the set of all absolutely continuous games, AC, is a closed subspace of BV [Aumann and Shapley ( 1974)]. If v and u are absolutely continuous with respect to 11o E NA + and v E NA + re spectively, then vu is absolutely continuous with respect to 11o + v. Thus AC is a closed subalgebra of BV. It is not known whether there is a value on AC. · · ·
13.2. Games in pNA The space pNA played a central role in the development of values of non-atomic games. Games arising in applications are often either vector measure games or approximated by vector measure games. Researchers have thus looked for conditions on vector (or scalar) measure games to be in pNA. Tauman (1982) provides a characterization of vector measure games in pNA. A vector measure game v = f o 11o with 11o E (NA 1 ) is in pNA if and only if the function that maps a vector measure v with the same range as fJo to the game f o v, is continuous at /Jo, i.e., for every E > 0 there is 8 > 0 such that if v E (NA ) has the same range as the vector measure 11o and 2:;'= 1 IIIlo ; - v; II < 8 then I I / o fJo - f o v ii < E . Proposition 10.17 of [Aumann and Shapley (1974)] provides a sufficient condition, expressed as a condition on a continuous non-decreasing real valued function f IR11 IR, for a game of the form f o 11o where 11o is a vector of non-atomic measures to be in pNA . Theorem C of Aumann and Shapley (1974) asserts that the scalar measure game f o 11o (tJo E NA 1 ) is in pNA if and only if the real-valued function f : [0, 1] ---+ lR is absolutely continuous. Given a signed scalar measure 11o with range [-a, b] (a, b > 0), Kohlberg (1973) provides a sufficient and necessary condition on the real-valued function f : [-a, b] ---+ lR for the scalar measure game f 11o to be in pNA . 11
1
''
:
---+
o
13.3. Games in bv'NA Any function f E bv' has a unique representation as a sum of an absolutely continuous function lac that vanishes at 0 and a singular function r in bv', i.e., a function whose variation is on a set of measure zero. The subspace of all singular functions in bv' is denoted s'. The closed linear space generated by all games of the form f o 1-L where
2162
A.
Neyman
f E s' and f.-L E NA 1 is denoted s'NA. Aumann and Shapley (1974) show that bv'NA is the algebraic direct sum of the two spaces pNA and s'NA, and that for any two games, u E pNA and v E s'NA, llu + v ii = llu ll + llvll- The norm of a function f E bv', 11 / 11 , is defined as the variation of f o n [0, 1]. If f.-L E NA 1 and f E bv' then II f o f.-L II = II f 11 . Therefore, if Cfi )� 1 is a sequence of functions in bv' with I:� 1 II fi II < oo and (J.-Li )� 1 is a sequence of non-atomic probability measures, the series I:� 1 fi o f.-Li converges to a game v E bv'NA. If the functions fi are absolutely continuous the series converges to a game in pNA. However, not every game in pNA has a representation as a sum of scalar measure games. Ifthe functions fi are singular, the series I:� 1 fi o f.-Li converges to a game in s'NA, and every game v E s'NA is a countable (possibly finite) sum, v = I:� 1 fi o J.-Li , where fi E s' with II v ii = L� I ll fi II and f.-Li E NA 1 • A simple game is a game v where v (S) E {0, 1 } . A weighted majority game is a (simple) game of the form v (S) =
{�
if J.-L(S) ;? q if J.-L(S) < q
v (S) =
{�
if J.-L(S) > q if J.-L(S) � q
or
where f.-L E FA+ and 0 < q < f.-L ( /). The finitely additive measure J.-L is called the weight measure and q is called the quota. Every simple monotonic game in bv' NA is a weighted majority game, and the weighted measure is non-atomic. References Artstein, Z.
( 197 1 ), "Values of games with denumerably many players", International Journal of Game Theory 1:27-37. Aubin, J. (1981), "Cooperative fuzzy games", Mathematics of Operations Research 6: 1-13. Aumann, R.J. (1975), "Values of markets with a continuum of traders", Econometrica 43:61 1-646. Aumann, R.J. (1978), "Recent developments in the theory of the Shapley value", in: Proceedings of the International Congress of Mathematicians, Helsinki, 994-1003. Aumann, R.J., R.J. Gardner and R. Rosenthal (1977), "Core and value for a public goods economy: An example", Journal of Economic Theory 15:363-365. Aumann, R.J., and M. Kurz (1977a), "Power and taxes", Econometrica 45: 1 1 37-1 161. Aumann, R.J., and M . Kurz ( 1977b), "Power and taxes in a multi-commodity economy", Israel Journal of Mathematics 27: 185-234. Aumann, R.J., and M. Kurz (1978), "Power and taxes in a multi-commodity economy (updated)", Journal of Public Economics 9:139-161 . Aumann, R.J., M . Kurz and A. Neyman (1987), "Power and public goods", Journal of Economic Theory 42: 108-127. Aumann, R.J., M. Kurz and A. Neyman (1983), "Voting for public goods", Review of Economic Studies 50:677-693.
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Aumann, R.J., and M. Perles (1965), "A variational problem arising in economics", Journal of Mathematical Analysis and Applications 1 1 :488-503. Aumann, R.J., and U. Rothblum (1 977), "Orderable set functions and continuity III : Orderability and absolute continuity", SIAM Journal on Control and Optimization 1 5 : 1 56-- 1 62. Aumann, R.J., and L.S. Shapley ( 1974), Values of Non-atomic Games (Princeton University Press, Princeton, NJ). Berbee, H. ( 1 98 1), "On covering single points by randomly ordered intervals", Annals of Probability 9:520528. Billera, L.J. (1976), "Values of non-atomic games", Mathematical Reviews 5 1 :2093-2094. Billera, L.J., and D.C. Heath ( 1982), "Allocation of shared costs: A set of axioms yielding a unique proce dure", Mathematics of Operations Research 7:32-39. Billera, L.J., D.C. Heath and J. Raanan ( 1 978), "Internal telephone billing rates: A novel application of non atomic game theory", Operations Research 26:956-965. Billera, L.J., D.C. Heath and J. Raanan (198 1), "A unique procedure for allocating common costs from a production process", Journal of Accounting Research 19: 1 85-196. Billera, L.J., and J. Raanan (1981), "Cores of non-atomic linear production games", Mathematics of Opera tions Research 6:420-423. Brown, D., and P.A. Loeb (1977), "The values of non-standard exchange economies", Israel Journal of Math ematics 25:7 1-86. Butnariu, D. (1983), "Decompositions and ranges for additive fuzzy measures", Fuzzy Sets and Systems 3 : 1 35-155. Butnariu, D. ( 1987), "Values and cores of fuzzy games with infinitely many players", International Journal of Game Theory 1 6:43-68. Champsaur, P. ( 1 975), "Cooperation versus competition", Journal of Economic Theory l l :394-417. Cheng, H. (1981), "On dual regularity and value convergence theorems", Journal of Mathematical Economics 6:37-57. Dubey, P. ( 1 975), "Some results on values of finite and infinite games", Ph.D. Thesis (Cornell University). Dubey, P. ( 1980), "Asymptotic semivalues and a short proof of Kannai's theorem", Mathematics of Operations Research 5:267-270. Dubey, P., and A. Neyman ( 1984), "Payoffs in nonatomic economies: An axiomatic approach", Econometrica 52: 1 1 29-1 1 50. Dubey, P., and A. Neyman (1 997), "An equivalence principle for perfectly competitive economies", Journal of Economic Theory 75: 3 14--344. Dubey, P., A. Neyman and R.J. Weber (1981), "Value theory without efficiency", Mathematics of Operations Research 6 : 1 22-128. Dvoretsk:y A., A. Wald and J. Wolfowitz (195 1), "Relations among certain ranges of vector measures", Pacific Journal of Mathematics 1 :59-74. Fogelman, F., and M. Quinzii (1980), "Asymptotic values of mixed games", Mathematics of Operations Re search 5:86-93. Gardner, R.J. (1 975), "Studies in the theory of social institutions", Ph.D. Thesis (Cornell University). Gardner, R.J. (1977), "Shapley value and disadvantageous monopolies", Journal of Economic Theory 1 6:5 135 17. Gardner, R.J. (1981), "Wealth and power in a collegial polity", Journal of Economic Theory 25: 353-366. Gilboa, I. ( 1989), "Additivization of nonadditive measures", Mathematics of Operations Research 14: l-17. Guesnerie, R. ( 1 977), "Monopoly, syndicate, and Shapley value: About some conjectures", Journal of Economic Theory 1 5 : 235-25 1 . Haimanko, 0 . (2000a), "Value theory without symmetry", International Journal of Game Theory 29:45 1-468. Haimanko, 0. (2000b), "Partially symmetric values", Mathematics of Operations Research 25:573-590. Haimanko, 0. (2000c), "Nonsymmetric values of nonatomic and mixed games", Mathematics of Operations Research 25:591-605.
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Haimanko, 0. (2000d), "Values of games with a continuum of players", Ph.D. Thesis (The Hebrew University of Jerusalem). Haimanko, 0. (200la), "Payoffs in non-differentiable perfectly competitive TU economies", Journal of Eco nomic Theory (forthcoming). Haimanko, 0. (200lb), "Cost sharing: The non-differentiable case", Journal of Mathematical Economics 35:445-462. Hart, S. (1973), "Values of mixed games", International Journal of Game Theory 2:69-85. Hart, S. (1977a), "Asymptotic values of games with a continuum of players", Journal of Mathematical Eco nomics 4:57-80. Hart, S. (1977b), "Values of non-differentiable markets with a continuum of traders", Journal of Mathematical Economics 4: 103-1 1 6. Hart, S. (1979), "Values of large market games", in: S. Brams, A. Schotter and G. Schwodiauer, eds., Applied Game Theory (Physica-Verlag, Heidelberg), 1 87-197. Hart, S. (1980), "Measure-based values of market games", Mathematics of Operations Research 5: 197-228. Hart, S. (1982), "The number of commodities required to represent a market game", Journal of Economic Theory 27: 163-169. Hart, S., and A. Mas-Cole!! (1989), "Potential, value and consistency", Econometrica 57:589-614. Hart, S ., and A. Mas-Cole!! ( 1995a), "Egalitarian solutions of large games. I. A continuum of players", Math ematics of Operations Research 20:959-1002. Hart, S., and A. Mas-Cole]] (1995b), "Egalitarian solutions of large games. II. The asymptotic approach", Mathematics of Operations Research 20: I 003-1022. Hart, S., and A. Mas-Cole]] (1996), "Harsanyi values of large economies: Nonequivalence to competitive equilibria", Games and Economic Behavior 13:74-99. Hart, S., and D. Monderer (1997), "Potential and weighted values of nonatomic games", Mathematics of Operations Research 22:619-630. Hart, S., and A. Neyman (1988), "Values of non-atomic vector measure games: Are they linear combinations of the measures?", Journal of Mathematical Economics 17:31-40. Imai, H. ( 1983a), "Voting, bargaining, and factor income distribution", Journal of Mathematical Economics 1 1 :2 1 1-233. Imai, H. (1983b), "On Harsanyi's solution", International Journal of Game Theory 12:161-179. Isbell, J. (1975), "Values of non-atomic games", Bulletin of the American Mathematical Society 83:539-546. Kannai, Y. ( 1966), "Values of games with a continuum of players", Israel Journal of Mathematics 4:54-58. Kemperman, J.H.B. ( 1985), "Asymptotic expansions for the Shapley index of power in a weighted majority game", mimeo. Kohlberg, E. (1973), "On non-atomic games: Conditions for f a f1- c pNA", International Journal of Game Theory 2:87-98. Kuhn, H.W., and A.W. Tucker ( 1950), "Editors' preface to Contributions to the Theory of Games (Vol. I), in: Annals of Mathematics Study 24 (Princeton University Press, Princeton, NJ). Lefvre, F. (1994), "The Shapley value of a perfectly competitive market may not exist", in: J.-F. Mertens and S. Sorin, eds., Game-Theoretic Methods in General Equilibrium Analysis (Kluwer Academic, Dordrecht), 95-104. Lyapunov, A. (1940), "Sur les fonction-vecteurs completement additions", Bull. Acad. Soc. URRS Ser. Math. 4:465-478. Mas-Cole!!, A. (1977), "Competitive and value allocations of large exchange economies", Journal of Eco nomic Theory 14:419--438. Mas-Cole!!, A. ( 1980), "Remarks on the game-theoretic analysis of a simple distribution of surplus problem", International Journal of Game Theory 9: 125-140. McLean, R. (1999), "Coalition structure values of mixed games", in: M.H. Wooders, ed., Topics in Math ematical Economics and Game Theory: Essays in Honor of Robert J. Aumann (American Mathematical Society, Providence, RI), 105-120.
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McLean, R.P., and W.W. Sharkey (1998), "Weighted Aumann-Shapley pricing", International Journal of Game Theory 27:5 1 1-523. Mertens, J.-F. ( 1980), "Values and derivatives", Mathematics of Operations Research 5 :523-552. Mertens, J.-F. ( 1988a), "The Shapley value in the non-differentiable case", International Journal of Game Theory 17: 1-65. Mertens, J.-F. (l988b), "Nondifferentiable TU markets: The value", in: A.E. Roth, ed., The Shapley Value, Essays in Honor of Loyd S. Shapley (Cambridge University Press, Cambridge), 235-264. Mertens, J.-F. ( 1990), "Extensions of games, purification of strategies, and Lyapunov's theorem", in: J.J. Gab szewicz, J.-F. Richard and L. Wolsey, eds., Economic Decision-Making: Games, Econometrics and Opti misation. Contributions in Honour of J.H. Dreze (Elsevier Science Publishers, Amsterdam), 233-279. Mertens, J.-F., and A. Neyman (2001), "A value on AN", Center for Rationality DP-276 (The Hebrew Uni versity of Jerusalem). Milchtaich, I. ( 1 997), "Vector measure games based on measures with values in an infinite dimensional vector space", Games and Economic Behavior 24:25-46. Milchtaich, I. (1995), "The value ofnonatomic games arising from certain noncooperative congestion games", Center for Rationality DP-87 (The Hebrew University of Jerusalem).
1
Milnor, J.W., and L.S. Shapley ( 1 978), "Values of large games. II: Oceanic games", Mathematics of Opera tions Research 3:290-307. Mirman, L.J., and A. Neyman ( 1 983), "Prices for homogeneous cost functions", Journal of Mathematical Economics 12:257-273. Mirman, L.J., and A. Neyman ( 1984), "Diagonality of cost allocation prices", Mathematics of Operations Research 9:66-74. Mirman, L.J., J. Raanan and Y. Tauman (1982), "A sufficient condition on f for f o JL to be in pNAD", Journal of Mathematical Economics 9:25 1-257. Mirman, L.J., D. Samet and Y Tauman (1 983), axiomatic approach to the allocation of fixed cost through prices", Bell Journal of Economics 14: 139- 1 5 1 . Mirman, L.J., and Y. Tauman (1981), "Valeur de Shapley et repartition equitable des couts de production", Cahiers du Seminaire d'Econometrie 23 : 121-1 5 1 . Mirman, L.J., and Y Tauman ( 1 982a), "The continuity of the Aumann-Shapley price mechanism", Journal of Mathematical Economics 9:235-249. Mirman, L.J., and Y. Tauman ( 1 982b), "Demand compatible equitable cost sharing prices", Mathematics of Operations Research 7:40-56. Monderer, D. ( 1986), "Measure-based values of non-atomic games", Mathematics of Operations Research 1 1 :321-335. Monderer, D. ( 1988), "A probabilistic problem arising in economics", Journal of Mathematical Economics 17:321-338. Monderer, D. ( 1989a), "Asymptotic measure-based values of nonatomic games", Mathematics of Operations Research 14:737-744. Monderer, D. ( 1989b), "Weighted majority games have many JL-values", International Journal of Game The ory 1 8:321-325. Monderer, D. (1 990), "A Milnor condition for non-atomic Lipschitz games and its applications", Mathematics of Operations Research 15:7 14-723. Monderer, D., and A. Neyman ( 1 988), "Values of smooth nonatomic games: The method of multilinear ap proximation", in: A.E. Roth, ed., The Shapley Value: Essays in Honor of Lloyd S. Shapley (Cambridge University Press, Cambridge), 217-234 (Chapter 15). Monderer, D., and W.H. Ruckle (1 990), "On the symmetry axiom for values of non-atomic games", Interna tional Journal of Mathematics and Mathematical Sciences 1 3 : 165-170. Neyman, A. (1977a), "Continuous values are diagonal", Mathematics of Operations Research 2:338-342. Neyman, A. ( 1 977b), "Values for non-transferable utility games with a continuum of players", TR 35 1 , S.O.R.I.E. (Cornell University).
"An
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Neyman, A. (1979), "Asymptotic values of mixed games", in: 0. Moeschlin and D. Pallaschke, eds., Game Theory and Related Topics (North-Holland, Amsterdam), 7 1-8 1 . Neyman, A . (198la), "Singular games have asymptotic values", Mathematics of Operations Research 6:205212. Neyman, A. (198 lb), "Decomposition of ranges of vector measures", Israel Journal of Mathematics 40:54--6 5. Neyman, A. (1982), "Renewal theory for sampling without replacement", Annals of Probability 10:464--481. Neyman, A. (1985), "Semi-values of political economic games", Mathematics of Operations Research 10: 390--402. Neyman, A. (1988), "Weighted majority games have asymptotic value", Mathematics of Operations Research 13:556-580. Neyman, A. (1989), "Uniqueness of the Shapley value", Games and Economic Behavior 1 : 1 16-1 18. Neyman, A. (1 994), "Values of games with a continuum of players", in: J.-F. Mertens and S. Sorin, eds., Game-Theoretic Methods in General Equilibrium Analysis, Chapter IV (Kluwer Academic Publishers, Dordrecht), 67-79. Neyman, A. (2001), "Values of non-atomic vector measure games", Israel Journal of Mathematics 124:1-27. Neyman, A., and R. Smorodinski (1993), "The asymptotic value of the two-house weighted majority games", mimeo. Neyman, A., and Y. Tauman (1976), "The existence of nondiagonal axiomatic values", Mathematics of Oper ations Research 1 :246-250. Neyman, A., and Y. Tauman (1979), "The partition value", Mathematics of Operations Research 4:236-264. Osborne, M.J. (1984a), "A reformulation of the marginal productivity theory of distribution", Econometrica 52:599--630. Osborne, M.J. (1984b), "Why do some goods bear higher taxes than others?", Journal of Economic Theory 32:301-316. Owen, G. (1972), "Multilinear extension of games", Management Science 18:64--79. Owen, G. (1982), Game Theory, 2nd edn. (Academic Press, New York), Chapter 12. Pallaschke, D., and J. Rosenmliller (1997), "The Shapley value for countably many players", Optimization 40:351-384. Pazgal, A. (1991 ), "Potential and consistency in TU games with a continuum of players and in cost allocation problems", M.Sc. Thesis (Tel Aviv University). Raut, L.K. (1997), "Constmction of a Haar measure on the projective limit group and random order values of nonatomic games", Journal of Mathematical Economics 27:229-250. Raut, L.K. (1999), "Aumann-Shapley random order values of non-atomic games," in: M.H. Wooders, ed., Topics in Mathematical Economics and Game Theory: Essays in Honor of Robert J. Aumann (American Mathematical Society, Providence, RI), 121-136. Reichert, J., and Y. Tauman (1985), "The space of polynomials in measures is internal", Mathematics of Operations Research 10: 1--6. Rosenmliller, J. (1971), "On core and value", in: R. Henn, ed., Operations Research-Verfahren (Anton Hein, Meisenheim), 84-- 104. Rosenthal, R.W. (1976), "Lindahl solutions and values for a public goods example", Journal of Mathematical Economics 3:37-41 . Rothblum, U . (1973), "On orderable set functions and continuity I", Israel Journal o f Mathematics 16:375397. Rothblum, U. ( 1977), "Orderable set functions and continuity II: Set functions with infinitely many null points", SIAM Journal on Control and Optimization 15: 144-155. Ruckle, W.H. (1982a), "Projections in certain spaces of set functions", Mathematics of Operations Research 7:314--3 18. Ruckle, W.H. (1982b), "An extension of the Aumann-Shapley value concept to functions on arbitrary Banach spaces", International Journal of Game Theory 1 1 : 105-116. Ruckle, W.H. (1988), "The linearizing projection: Global theories", International Journal of Game Theory 17:67-87.
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Samet, D. ( 1981 ), Syndication in Market Games with a Continuum of Traders, DP 497 (KGSM, Northwestern University). Samet, D. ( 1984), "Vector measures are open maps", Mathematics of Operations Research 9:471-474. Samet, D. ( 1987), "Continuous selections for vector measures", Mathematics of Operations Research 12:536543. Samet, D., and Y. Tauman ( 1982), "The determination of marginal cost prices under a set of axioms", Econo metrica 50:895-909. Samet, D., Y. Tauman and I. Zang ( 1 984), "An application of the Aumann-Shapley prices to the transportation problems", Mathematics of Operations Research 9:25-42. Santos, J.C., and J.M. Zarzuelo ( 1998), "Mixing weighted values for non-atomic games", International Journal of Game Theory 27:331-342. Santos, J.C., and J.M. Zarzuelo (1 999), "Weighted values for non-atomic games: An axiomatic approach", Mathematical Methods of Operations Research 50:53-63. Shapiro, N.Z., and L.S. Shapley ( 1 978), "Values of large games I: A limit theorem", Mathematics of Opera tions Research 3: 1-9. Shapley, L.S. ( 1953a), "A value for n-person games", in: H.W. Kuhn and A.W. Tucker, eds., Contributions to the Theory of Games, Vol. II (Princeton University Press, Princeton, NJ), 307-3 17. Shapley, L.S. ( 1953b), "Additive and non-additive set functions", Ph.D. Thesis (Princeton University). Shapley, L.S. ( 1 962), "Values of games with infinitely many players", in: M. Maschler, ed., Recent Advances in Game Theory (Ivy Curtis Press, Philadelphia, PA), 1 1 3- 1 1 8. Shapley, L.S. ( 1 964), "Values of large games VII: A general exchange economy with money", RM 4248 (Rand Corporation, Santa Monica). Shapley, L.S., and M. Shubik ( 1969), "Pure competition, coalitional power and fair division", International Economic Review 10:337-362. Sroka, J.J. ( 1993), "The value of certain pNA games through infinite-dimensional Banach spaces", Interna tional Journal of Game Theory 22: 1 23-139. Tauman, Y. ( 1 977), "A nondiagonal value on a reproducing space", Mathematics of Operations Research 2:33 1-337. Tauman, Y. ( 1 979), "Value on the space of all scalar integrable games", in: 0. Moeschlin and D. Pallaschke, eds., Game Theory and Related Topics (North-Holland, Amsterdam), 107-1 1 5. Tauman, Y. (198la), "Value on a class of non-differentiable market games", International Journal of Game Theory 10: 155-162. Tauman, Y. (198lb), "Values of market games with a majority rule", in: 0. Moeschlin and D. Pallaschke, eds., Game Theory and Mathematical Economics (North-Holland, Amsterdam), 103-122. Tauman, Y. (1 982), "A characterization of vector measure games in pNA", Israel Journal of Mathematics 43:75-96. Weber, R.J. (1979), "Subjectivity in the valuation of games", in: 0. Moeschlin and D. Pallaschke, eds., Game Theory and Related Topics (North-Holland, Amsterdam), 129-136. Weber, R.J. (1988), "Probabilistic values for games", in: A.E. Roth, ed., The Shapley Value, Essays in Honor of Lloyd S. Shapley (Cambridge University Press, Cambridge), 1 0 1- 1 1 9. Wooders, M.H., and W.R. Zame ( 1 988), "Values of large finite games", in: A.E. Roth, ed., The Shapley Value, Essays in Honor of Lloyd S. Shapley (Cambridge University Press, Cambridge), 195-206. Young, H.P. ( 1 985), "Monotonic solutions of cooperative games", International Journal of Game Theory 14:65-72. Young, H.P. ( 1 985), "Producer incentives in cost allocation", Econometrica 53:757-765.
Chapter 57 VALUES OF PER FECT LY COM PETITIVE ECONOM IES*
SERGIU HART Center for Rationality and Interactive Decision Theory, Department of Economics, and Department ofMathematics, The Hebrew University ofJerusalem, Jerusalem, Israel Contents
Introduction The model 3. The Value Principle 4. Proof of Value Equivalence 1. 2.
4. 1 . 4.2.
5.
Value allocations are competitive allocations Competitive allocations are value allocations
Generalizations and extensions 5. 1 .
The non-differentiable case The geometry of non-atomic market games 5.3. Other non-atomic TU-values 5.4. Other NTU-values 5.5. Limits of sequences 5.6. Other approaches 5. 7. Ordinal vs. cardinal 5.8. Imperfect competition 5.2.
References
*The author thanks Robert J. Aumann and Andreu Mas-Colell for their comments. Handbook of Game Theory, Volume 3, Edited by R.J Aumann and S. Hart © 2002 Elsevier Science B. V. All rights reserved
2171 2 1 72 2 175 2 176 2 1 76 2 1 77 2 1 78 2 178 2 1 78 2 1 80 2 1 80 2181 2181 2 1 82 2 1 82 2182
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Abstract
Perfectly competitive economies are economic models with many agents, each of whom is relatively insignificant. This chapter studies the relations between the basic economic concept of competitive (or Walrasian) equilibrium, and the game-theoretic solution con cept of value. It includes the classical Value Equivalence Theorem together with its many extensions and generalizations, as well as recent results. Keywords
perfect competition, value, Shapley value, competitive equilibrium, value equivalence principle
JEL classification: C7, D5
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1. Introduction
This chapter is devoted to the study of economic models with many agents, each of whom is relatively insignificant. These are referred to as peifectly competitive models. The basic economic concept for such models is the competitive (or Walrasian) equi librium, which prescribes prices that make the total demand equal to the total supply, i.e., under which the markets clear. The fact that each agent is negligible implies that he cannot singly affect the prices, and so he takes them as given when finding his optimal consumption, or "demand". Game-theoretic concepts, particularly "cooperative game-theoretic" solutions (like core and value), are supported by very different considerations from the above. First, such solutions are based on the "coalitional game" which, in assigning to each coalition the set of possible allocations to its members, abstracts away from specific mechanisms (like prices). And second, they prescribe outcomes that satisfy certain desirable criteria or postulates (like efficiency, stability, or fairness). Nevertheless, it turns out that game-theoretic solutions may yield the competitive outcomes. The prime example is the Core Equivalence Theorem (see the chapter of An derson (1992) in this Handbook) which states that, in perfectly competitive economies, the core coincides with the set of competitive allocations. Here we will study another central solution concept, the (Shapley) value (see the chapter of Winter (2002) in this Handbook). The value, originally introduced by Shap ley (1953) on the basis of a number of simple axioms, 1 turns out to measure the "av erage marginal contributions" of the players. It is perhaps best viewed as an "expected outcome" or an "a priori utility" of playing the game. The first work on values of perfectly competitive economies is due to Shapley (1964) (see also Shapley and Shubik (1969)), who considered differentiable exchange economies (or markets) with transferable utilities (TU), and showed that, as the num ber of players increases by replication, the Shapley values converge to the competitive equilibrium. To extend this result, two developments were needed. One was to develop the theory of value for games with a continuum of players (the "limit game") and, in particular, to obtain the Shapley (1964) result in this framework; see Aumann and Shapley (1974) (and the chapter of Neyman (2002) in this Handbook). The other was to define the con cept of value for general non-transferable utility (NTU) games; see Harsanyi (1963) and Shapley (1969) (and the chapter of McLean (2002) in this Handbook). This culminated in the general Value Equivalence Theorem of Aumann (1975): In a differentiable econ omy with a continuum of traders, the set of Shapley (1969) value allocations coincides with the set of competitive allocations. Next, the non-differentiable case was studied by Champsaur ( 197 5) (limit of replicas) and by Hart (1977b) (a continuum of traders), who showed that one direction always 1 Efficiency, symmetry, dummy player and additivity.
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Table 1 The Value Principle Limit of replicas
Differentiable
TU
(Equivalence)
NTU
General
TU NTU
(Inclusion)
Shapley (1964) Mas-Colell (1977) Champsaur (1975)
Continuum of agents
Aumann and Shapley (1974) Aumann (1975) Hart (1977b)
holds - all value allocations are competitive - but not necessarily the converse. Together with the work of Mas-Colell (1977) (limit of replicas in the NTU case), the picture was completed, resulting in the Value Principle: In perfectly competitive economies, every Shapley value allocation is competitive, and the converse holds in the differentiable case; see Table 1 . In the TU-case, the concept of value is clear and undisputed (of course, its defin ition in the case of a continuum of players entails many difficulties). This is not so in the NTU-case. There are different ways of extending the Shapley TU-value, and which one is more appropriate is a matter of some controversy (see the references in Subsection 5.4). The Value Principle stated above was exclusively established for the Shapley ( 1969) NTU-value (also known as the "A.-transfer value"). Recently, Hart and Mas-Colell ( 1996a) studied the Harsanyi (1963) NTU-value in perfectly compet itive economies, and showed that these value allocations may be different from the competitive ones, even in the differentiable case (and that, moreover, this is a robust phenomenon). Since the Harsanyi NTU-value is no less appropriate than the Shapley NTU-value (and possibly even more so), this casts doubt on the general validity of the Value Principle, in particular when the economy exhibits substantial NTU features; see Subsection 5.4. The chapter is organized as follows: Section 2 presents the basic model of an ex change economy with a continuum of agents, together with the definitions of the appro priate concepts. The Value Principle results are stated in Section 3. An informal (and hopefully instructive) proof of the Value Equivalence Theorem is provided in Section 4. Section 5 is devoted to additional material, generalizations, extensions and alternative approaches.2 2. The model
As we saw in the Introduction, there are various ways to model "perfectly competitive" economies. One is to consider sequences of economies with finitely many agents, where the number of agents increases to infinity; see Subsection 5.5. Another is to study di rectly the "limit" economy with a continuum of agents. Indeed, to model precisely the 2 Precise definitions and statements are given for one main result (the Value Equivalence Theorem with a continuum of traders); the rest of the material is presented in a less formal manner.
Ch.
57:
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intuitive notion that individual's influence on the whole economy is negligible, one needs infinitely many participants. As Aumann (1964, p. 39) says, " . . . the most natural model for this purpose contains a continuum of participants, similar to the continuum of points on a line or the continuum of particles in a fluid". Our basic model is that of a pure exchange economy (or market) with a continuum of traders, and consists of the following: an
El The commodities (or goods): .£ = 1 , 2, . . . , L ; the consumption set of each agent is thus3 IR� . E2 The space of traders: A measure space (T, l.t, f.l), where: T is the set of traders (or agents); l.t, a cr-field of subsets of T, is the set of coalitions; and f.l, a probabil ity measure4 on l.t, is the population measure (i.e., for every coalition S E l.t, the number /l(S) E [0, 1] is the "mass" of S, relative to the whole space T). E3 The initial endowments: For every trader t E T, a commodity vector e(t) = (e 1 (t), e2 (t), . . . , eL (t)) E IR�, where ec (t) is the quantity of good .£ initially owned by trader t. E 4 The utility functions: For every trader t E T, a function u 1 : IR� � IR, where u 1 (x) is the utility of t from the commodity bundle x E IR�. We make the following assumptions: Gl (Measurability) e(t) is measurable in t and u 1 (x) is jointly measurable in (t, x) (relative to the Borel cr-field on IR� and the cr-field ([ on T). G 2 (Endowments) Every commodity is present in the market, i.e.,5 fr e df.l » 0. G3 (Utilities) The utility functions u 1 are all strictly monotonic (i.e., x � y implies u 1 (x) > u 1 (y)), continuous and uniformly bounded (i.e., there exists M such that lu 1 (x) l � M for all x and t). G4 (Standardness) (T, l.t, /1) is isomorphic to6 ([0, 1], �. v), the unit interval [0, 1] with the Borel cr-field � and the Lebesgue measure v; in particular, f.1 is a non atomic measure. All of these assumptions are standard; we will refer to G l -G4 (in addition, of course, to E l E4) as the general case. Some assumptions (like uniform boundedness of the utility functions) are made for simplicity of presentation only; the reader should consult the relevant papers for more general setups. Finally, we introduce an additional set of assumptions known as the diffe rentiable case: -
lR is the real line, and JR � is the non-negative orthant of the £-dimensional Euclidean space. I.e., J-i is non-negative and Jl,(T) 1 . 5 For vectors x, y E JRL , we take ;;:, y and x » y to mean, respectively, xe ;;:, l and xe l for all C . Next, x ;;; y means x ;;:, y and x =/= y (i.e., x f ;;:, l for all £, with at least one strict inequality). 6 There is little loss of generality here; see Aumann and Shapley (1974, Chapter VID).
3
4
=
x
>
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The utility functions u 1 are all concave and continuously differentiable on7 JR.;, and their gradients 'ilu1 are uniformly bounded and uniformly positive in bounded subsets of8 JR.; . D2 The initial endowments are uniformly bounded (i.e., there exists M such that ee (t) � M for all f and t). An allocation x is a feasible outcome of this exchange economy, i.e., a redistribution of the total initial endowment. That is, x : T --+ JR.; is an integrable function, where x(t) E JR.; is agent t's final commodity bundle, which satisfi e s: Dl
i x dp, i e dp,.
(2. 1 )
=
We now define the two concepts that will be the concern of this chapter: competitive equilibrium and value. An allocation x is competitive (or Walrasian) if there exists a price vector p E JR.;, p =/= 0 such that, for every agent t E T, the bundle x(t) is maximal with respect to u 1 in t's budget set B1 (p) := {x E JR.;: p x � p e(t)} (i.e., u1 (x(t)) � u1 (y) for all y E B1 (p)). We refer to e(t) as t's supply, and to x(t) as t's demand; equality (2. 1) then states that total demand equals total supply. An allocation x is a Shapley (1969) (NTU-)value allocation (see the chapter of McLean (2002) in this Handbook) if there exists a collection "A = ("A1 )t E T of nonnegative weights (formally, a measurable function "A : T --+ JR.+) such that9 ·
Is At U t (x(t)) dp,(t) = cpv;. (S),
·
(2.2)
for every S E
where v;. is the TU-game defined by v;. (S) := max
{ fs "At u1 (y(t)) dp, (t) : y(t) E lRi for all t E S and
Is y dp, Is e dp, } =
(2.3)
for all coalitions S E
Including the boundary of
lRt; that is, u1 is continuously differentiable in the interior of lRt, and all its lRt.
partial derivatives can be continuously extended to the boundary of
8
I.e., for every M < oo there exist c > 0 and C < oo such that c < 3u1 (x)jax£ < C for all t, e and x with ll x ll c( M. 9 Equivalently, in "density" terms, ).1u1 (x(t)) (d(
(which is the Radon-Nikodym derivative of
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the traders at the rates "A, then the value of the resulting TU-game (2.3) would in fact be achievable without transfers (2.2). 3. The Value Principle
In this section we will state the main results on the relations between value and com petitive equilibria. The following two theorems comprise what is known as the Value Principle.
THEOREM 3 . 1 (Value Equivalence). In the differentiable case: The set ofShapley value allocations coincides with the set of competitive allocations. THEOREM 3 . 2 (Value Inclusion). In the general case: The set of Shapley value allo cations is included in the set of competitive allocations (and there are cases where the inclusion is strict). In other words: First, value allocations are always competitive. And second, compet itive allocations are value allocations provided the economy is appropriately differen tiable (or smooth). In the framework of the previous section (markets with a continuum of agents), the "general case" refers to El-E4 together with Gl-G4; the "differentiable case", to El-E4, Gl-G4 and Dl-D2. Value Equivalence has been proved by Aumann (1975). We will provide an informal - and hopefully instructive - proof in the next section. Value Inclusion has been proved by Hart (1977b); see Subsections 5. 1 and 5.2 be low. As stated in the Introduction, parallel results have been obtained in other frame works (like limits of sequences, transferable utility, etc.; recall Table 1 and see Sec tion 5). All of these results use the concept of NTU-value due to Shapley ( 1969) (defined in the previous section). However, there are other notions of an NTU-value. A prominent one is the Harsanyi (1963) NTU-value. We have:
PROPOSITION 3 . 3 . In the differentiable case: The set of Harsanyi value allocations may be disjointfrom the set of competitive allocations. This is shown by Hart and Mas-Colell (1996a), who provide a robust example with a unique Harsanyi value which is different from the unique competitive equilib rium. This clearly raises doubts about either the Harsanyi value or the Value Principle. In view of the additional literature, Hart and Mas-Colell (1996a) suggest that, since the Harsanyi concept is no less appropriate than the Shapley one (and perhaps even more so), it follows that the Value Principle may not be valid in general - particu larly when the economy is far removed from the transferable utility case. See Subsec tion 5.4.
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4. Proof of Value Equivalence
We now provide an informal proof for the Value Equivalence Theorem 3 . 1 , in the frame work of Section 2 (under all the assumptions there, including differentiability); in par ticular, "value" stands for "Shapley NTU-value".
4.1. Value allocations are competitive allocations Let x be a value allocation with corresponding weights A . We ignore technical compli cations and assume that x(t) » 0 and A t > 0 for all t . Also, we will write "for all t" where in fact it should say "for tL-almost every t", and we regard a point t as if it were a "real agent". 10 First, we have
i At Ut (x(t)) dtL(t) =
V;.. (T)
(4.1)
by (2.2) for T together with cp v;.. (T) = v;.. (T) (since the TU-value is efficient). Thus the maximum in ( 2. 3 ) for T is achieved at x. Therefore 1 1
(4.2) (this is standard: if not, then a reallocation of goods between t and t' would increase the total utility v;.. (T)). Let p be the common value of12 A t Y'u t (x(t)) for all t:
Y'u t (x(t)) = (l/A t )P for all t E T. The utility functions U t are concave; therefore U t (y) from which it follows that
:S;
U t (x(t)) + Y'u 1 (x(t)) (y - x(t)), ·
if p y :S; p x(t) then u 1 (y) :S; u t (x(t)) . ·
(4.3)
·
Next, we claim that the asymptotic value cp v;.. of v;.. satisfies cp v;..
(t) = A t U t (x(t)) + p (e (t) - x(t)).
(4.4)
·
The intuition for this is as follows: The value of t is t's marginal contribution to a ran dom coalition Q (this holds in the case of finitely many players by Shapley's formula, 1 0 As in the case of finitely many agents. More appropriately, one should view dt as the "agent"; we prefer to use t since it may appear less intimidating for some readers. 1 1 We write \i'u(x) for the gradient of u at x, i.e., the vector of partial derivatives (au (x)jax i )f=l.. 12 Letting p (pe )t=I , one may interpret p£ as the Lagrange multiplier (or "shadow price") of the constraint J xi fT ...
=
T
...
=
et .
,L•
L·
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and it extends to the general case since the asymptotic value is the limit of values of finite approximations). Now a "random coalition" Q, when there is a continuum of players, is a perfect sample of the grand coalition T. (Assume for instance that we have a large finite population, consisting of two types of players: 1/3 of them are of type 1 , and 2/3 of them of type 2 . The coalition of the first half of the players in a random order will most probably contain, by the Law of Large Numbers, about 1 /3 players of type 1 and 2/3 of type 2. This holds for "one half" as well as for any other proportion strictly between 0 and 1.) Therefore the average marginal contribution of t to Q is essentially the same as t 's marginal contribution to the grand coalition 13 T, which we can write suggestively as
Since V). (T) is the result of a maximization problem, its derivative ("in the direc tion t") is obtained by keeping the optimal allocation fixed and multiplying the change in the constraints by the corresponding Lagrange multipliers. 1 4 The optimal allocation is x (by (4. 1)); rewriting JT x df-L= JT e df-L as JT \ { t J x df-L = JT \ {t } e df-L + (e(t) - x(t)) d{L(t) shows that the change in the constraints is 15 e(t) - x(t); and the Lagrange multipliers are p . Therefore indeed
In other words, the marginal contribution of t to the total utility V). (T) consists of t 's weighted utility contribution )..1 u 1 (x(t)), together with his net contribution of com modities e(t) - x(t), evaluated according to the "shadow prices" p (i.e., the marginal weighted utilities, which are common to everyone by (4.2)). We have thus shown (4.4). Comparing (2.2) (see Footnote 9) and (4.4) implies that
p (e(t) - x(t) ) = 0 for all t, ·
which together with (4.3) shows that x is a competitive allocation relative to the price vector p.
4.2. Competitive allocations are value allocations Let x be a competitive allocation with associated price vector p. From the fact that x(t) is the demand of t at the prices p (i.e., p y � p e(t) implies u 1 (y) � u 1 (x(t))), it ·
·
1 3 Technically, one uses here the homogeneity of degree 1 of hence the homogeneity of degree 0 of its derivatives. 14 Since the first order condition (that of vanishing derivatives) is satisfied at optimal allocations, the change in the optimal allocation is, in the first order, zero. This is the so-called "Envelope Theorem"; see, e.g., Mas Colell, Whinston and Green (1995, Section M.L). 15 In "density" terms. v,
2178
S. Hart
follows that the gradient 'Vu1 (x(t)) is proportional to p (again, we assume for simplicity that x(t) » 0 for all t). That is, there exist A t > 0 such that
This implies that the maximum in the definition of VJ.. (T) (for this collection A = (A1 )1) is attained at x. As we saw in the previous subsection (recall (4.4)), the asymptotic value CfJ VJ.. of VJ.. satisfies
CfJ VJ.. (t) = A1u1 (x(t) ) + p (e (t) ·
-
x(t) ) .
But x(t) is t's demand at p, so p x(t) p e(t) (by monotonicity), and therefore CfJ VJ.. (t) = A1u1 (x(t)), thus completing the proof that x is indeed a value allocation (by (2.2)). ·
=
·
5. Generalizations and extensions
5.1. The non-differentiable case In the general case (i.e., without the differentiability assumptions), the Value Inclusion Theorem says that every Shapley value allocation is competitive, and that the converse need no longer hold. In fact, in the non-differentiable case the asymptotic value of VJ.. may not exist (since different finite approximations lead to different limits), 16 and so there may be no value allocations at all. Moreover, even if value allocations do exist, they correspond only to some of the competitive allocations; roughly speaking, values "select" certain "centrally located" equilibria. The proof that every Shapley value is competitive is based on generalizing the value formula (4.4); see Hart (1977b) and the next subsection.
5.2. The geometry of non-atomic market games A (TU-)market game is a game that arises from an economy as in Section 2 (see (2.3)). To illustrate the geometrical structure that underlies the results of this chapter, sup pose for simplicity that there are finitely many types of agents, say n, with a continuum of mass 1 j n of each type. A coalition S is then characterized by its composition, or profile, s = (s 1 , s2 , . . . , s11) E JR.� , where Si is the mass of agents of type i in S. For mula (2.3) then defines a function v = VJ.. over profiles s; that is, 1 7 v : JR.� --+ R Such a 16 A try;
necessary condition for the existence of the asymptotic value is that the core possess a center of symme see Hart (1977a, 1977b). Note that the convex hull of k points in general position does not have a center of symmetry when k ? 3 (it is a triangle, a tetrahedron, etc.). 17 In fact, only s ,::; (1/n, lfn, . . . , 1/n) matter.
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function is clearly super-additive (i.e., v(s + s') � v(s) + v(s') for all s, s' E JR+) and homogeneous of degree 1 (i.e., v(a s ) a v (s ) for every s E JR+ and a � 0), and hence concave. 1 8 Let s = (1/n, 1/n, . . . , ljn) denote (the profile of) the grand coalition. A payoff vector1 9 n = (rr1 , rrz, . . . , nn ), where rr; is the payoff of each agent of type i, is in the core of v if :L7= 1 n;s; = v(s) and :L7= 1 rr;s; � v(s) for every s � s. Thus n is nothing other than a normal vector, or "supergradient" (recall that v is concave) to v at20 s. In particular, if v is differentiable, then the core has a unique element: the gradient Vv(s) of v at s. As for the TU-value of v, it is obtained as in Subsection 4. 1 (see the considera tions leading to (4.4)) as follows. Assume first that v is continuously differentiable. The profile q of a random coalition Q is (approximately) proportional to that of the grand coalition, i.e., q � as for some a E (0, 1) (by the Law of Large Numbers). The marginal contribution of an agent of type i to such a coalition Q is thus given by the partial deriva tive (&vj&s; ) (q) � (&vj&s; ) (as), which equals (& vj&s;)(s) by homogeneity. Therefore the value of type i is (&vj&s;)(s), and the value payoff vector is Vv(S) - identical to the core. 2 1 If v is not differentiable, then the partial derivatives are replaced by directional deriv atives, which correspond to supergradients, and are again independent of a by homo geneity. The set of supergradients is convex, therefore averaging (over random coali tions) implies that the value vector is also a supergradient of v at s - and so in this case too the value belongs to the core. Summarizing: In a non-atomic TU-market game, the value belongs to the core, and is the unique element in the core in the differentiable case. But the core and the set of competitive allocations coincide - this is the Core Equivalence Theorem (see the chapter of Anderson (1992) in this Handbook, and note that differentiability is not required) which yields the Value Principle in the TU-case. Moving now to the NTU-case, note that if n (rr1 , rr2 , . . . , nn ) is a Shapley NTU value with weights22 A = (AJ , A2 , . . . , An ), then the vector (A J1TJ , A2 1T2 , . . . , An nn ) is the TU-value of VJc (see (2.2)) and so, as we have seen above, it is a supergradient of VJc at s. Therefore :L;'=I A; rr;s; � VJc (s) for every s, implying that n cannot be improved upon by a coalition of profile s (recall the definition of VJc as a maximum). In other words, =
=
Such a function may be called "conical": its subgraph {(s, �) E IR'f_ JR: � :S; v(s)) is a convex cone. We only consider type-symmetric payoffs, where agents of the same type receive the same payoffs. 20 I.e., the graph of the linear function h (s) s is a supporting hyperplane to the subgraph of v (which is a convex cone; see Footnote 1 8) at the point (s, v(S)) (and thus also on the whole "diagonal" {(as, v(as)): 18
x
19
: = :n: ·
a ): 0}).
21 Another proof of this statement can be obtained using the potential approach of Hart and Mas-Colell (1989): Given v, there exists a unique function P P(v) : IR'f_ -+ lR with P (O) 0 and s '17 P (s) v(s) =
=
·
=
for all s the potential of v and moreover '17 P (S) is the value payoff vector. When v is homogeneous of degree 1 , the Euler equation implies s 'll v(s) v(s), and so P = v and '17 P (S) = 'll v (S) (or: value = core). 22 A.; is the weight corresponding to agents of type i (recall that, for simplicity, we assume type-symmetry). -
-
·
==
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belongs to the (NTU) core of the economy, and so it is a competitive allocation by the Core Equivalence Theorem. This establishes the Value Inclusion result of Theorem 3 .2. For precise presentations of these topics, the reader is referred to Shapley (1964), Hart (1977a, 1977b) and Hart and Mas-Colell (1996a, Section VII). 7T
5.3. Other non-atomic TU-values In the non-differentiable case, many of the relevant TU-games may not have an asymp totic value. 23 This happens when different sequences of finite games that approximate the given non-atomic game have values that converge to different limits. (The simplest such case is the so-called "3-gloves market"; see Aumann and Shapley (1974, Sec tion 19) and recall Footnote 16.) Therefore one looks for other TU-value concepts. One approach (first used by Aumann and Kurz (1977)) modifies the definition of the asymptotic value by considering only those finite approximations with players that are (approximately) equal in size, where "size" is determined by the underlying population measure p,. The resulting measure-based value (or p,-value) is shown by Hart (1980) to exist under wide conditions and, moreover, to yield a competitive allocation (i.e., the Value Inclusion result of Theorem 3.2 continues to hold for this value). An explicit formula, involving an appropriate normal distribution, is obtained for the resulting com petitive price; this price may then be viewed as an "expected equilibrium price", where the expectation is taken over random samples of the agents. Another approach, due to Mertens (1988a, 1988b), defines a value concept on a large space of non-atomic games, which includes all markets (whether differentiable or not); again, the Value Inclusion Theorem 3.2 holds for the Mertens value as well.
5.4. Other NTU-values Extending the concept of the Shapley value from the class of TU-games to the class of NTU-games is a conceptually challenging problem. One looks for an NTU-solution concept that, in particular: (i) extends the Shapley (1953) value for TU-games; (ii) ex tends the Nash (1950) bargaining solution for pure bargaining problems; (iii) is covari ant with independent rescalings of the utility functions; 24 and (iv) satisfies postulates like efficiency and symmetry. However, all this does not yet determine the solution, and additional constructs are needed. The two major NTU-value concepts are due to Harsanyi (1963) and to Shapley (1969). In fact, Shapley introduced his NTU-value originally as a simplification of the Harsanyi NTU-value. It then turned out to be of interest in its own right, and has been
23 See however Hart (1977b, Theorem C). 24 In the TU-case, where there is a common medium of utility transfer, one may multiply all utilities by the same constant without changing the problem. In contrast, in the NTU-case, each player's utility may be multiplied by a different constant.
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used in various models, economic and otherwise. However, further research has indi cated that the Harsanyi concept may well be the more appropriate one, at least in some cases; see Aumann (1985a, 1985b, 1986), Hart (1985a, 1985b), Roth (1980, 1986) and Shafer (1980) for some of the issues related to the interpretation and appropriateness of these concepts. The Harsanyi NTU-values of large differentiable economies25 were studied by Hart and Mas-Colell (1996a). It is shown, first, that the Value Inclusion result holds in some cases ?6 and, second, that in general the two (non-empty) sets of allocations the Harsanyi values and the competitive equilibria - may be disjoint. Moreover, this "non-equivalence" is a robust phenomenon, and a result of being far removed from the transferable utility case (i.e., the lack of substitutability among the agents' utilities). Another NTU-value that has recently been analyzed is the consistent NTU-value (see Maschler and Owen (1989, 1992), and also Hart and Mas-Colell (1996b)). For a first study on the connections between this value and the core, see Leviatan (1998).
5.5. Limits of sequences Rather than analyze the limit economy with a continuum of agents, one may instead consider sequences of finite economies. The simplest such approach, originally used in the study of the Core Equivalence Theorem, is that of "replicas": Each agent is replaced by r identical agents, and one looks at the limit as r increases to infinity. The results here are the same as in Theorems 3 . 1 and 3.2: Limits of Shapley value al locations are competitive, and the converse holds in the differentiable case. See Shapley (1964) (differentiable, TU); Champsaur (1975) (non-differentiable, TU and NTU); Mas Colell (1977) and Cheng (1981) (differentiable, NTU); and also Wooders and Zame (1987a, 1987b) for a "space of characteristics" framework (non-differentiable, TU and NTU).
5.6. Other approaches Among the other approaches to the Value Equivalence result, one should mention the axiomatic characterization of solutions of large economies due to Dubey and Neyman (1984, 1997), which captures many of the interesting concepts - competitive equilib rium, core, value - and thus implies their equivalence. Other works use non-standard analysis [Brown and Loeb (1977)] or fuzzy games [Butnariu (1987)].
25
The framework is that of a continuum of agents of finitely many types, as in Subsection 5.2. Moreover, as it is shown in Hart and Mas-Co1ell (1995a, 1995b), the analysis is borne out by limits of finite approximations. 26 Specifically, when the Harsanyi value is "tight".
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5. 7. Ordinal vs. cardinal The competitive allocations and the core are clearly "ordinal" concepts: They depend only on the preference relations of the agents and not on the particular utility represen tations. How about the value? The construction in Section 2 uses given utility functions u1• However, the relative weights At are obtained endogenously (as a "fixed-point" where (2.2) holds). Thus, if we were to apply linear transformations to the functions u 1 (with different coefficients for different t) the NTU-value allocations would not change. Moreover, a careful look at the proof of the Value Equivalence result in Section 3 shows that only "local" information is used (e.g., gradients or marginal rates of substitution), and so applying differentiable monotonic transformations will not matter either. Therefore, the NTU-value depends only on the preference orders, and is indeed an "ordinal" concept. In fact, Aumann ( 197 5) (and others) works directly with collections of utility representations (rather than with one utility representation and weights A, as we do here27 ). 5.8.
Impeifect competition
Perfect competition corresponds to all agents being individually insignificant. Imper fect competition results when there are "large" agents. This is modelled by population measures that are no longer non-atomic; the atoms are precisely the large agents. An interesting question is whether it is worthwhile for a coalition to become an atom, i.e., a "monopoly". For when this is measured by the value, see Guesnerie ( 1977) and Gardner (1977) (compare this with the results for the core; see the chapter of Gabszewicz and Shitovitz (1992) in this Handbook). For other economic applications, the reader is referred to the chapter of Mertens (2002) in this Handbook. References
Anderson, R.M. (1992), "The core in perfectly competitive economies", in: R.J. Aumann and S. Hart, eds., Handbook of Game Theory, Vol. 1 (North-Holland, Amsterdam) Chapter 14, 413-457. Aumann, R.J. (1964), "Markets with a continuum of traders", Econometrica 32:39-50. Aumann, R.J. (1975), "Values of markets with a continuum of traders", Econometrica 43:611-646. Aumann, R.J. (1985a), "An axiomatization of the non-transferable utility value", Econometrica 53:599-612. Aumann, R.J. (1985b), "On the non-transferable utility value: A comment on the Roth-Shafer examples", Econometrica 53:667-678. Aumann, R.J. (1986), "Rejoinder", Econometrica 54:985-989. Aumann, R.J., and M. Kurz (1977), "Power and taxes in a multi-commodity economy", Israel Journal of Mathematics 27:185-234. Aumann, R.J., and L.S. Shapley (1974), Values of Non-Atomic Games (Princeton University Press). 27
This "}.-approach" is commonly used, and we hope it will thus be easier on the reader.
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Brown, D., and P.A. Loeb (1977), "The values of non-standard exchange economies", Israel Journal of Math ematics 25:7 1-86. Butnariu, D. ( 1987), "Values and cores of fuzzy games with infinitely many players", International Journal of Game Theory 1 6:43-68. Champsaur, P. (1975), "Cooperation vs. competition", Journal of Economic Theory 1 1 :394-417. Cheng, H. (1981), "On dual regularity and value convergence theorems", Journal of Mathematical Economics 6:37-57. Dubey, P., and A. Neyman ( 1984), "Payoffs iu nonatomic economies: An axiomatic approach", Econometrica 52: 1 1 29-1 150. Dubey, P., and A. Neyman ( 1 997), "An equivalence principle for perfectly competitive economies", Journal of Economic Theory 75:3 14-344. Gabszewicz, J.J., and B. Shltovitz ( 1 992), "The core in imperfectly competitive economies", in: R.J. Aumann and S. Hart, eds., Handbook of Game Theory, Vol. 1 (North-Holland, Amsterdam) Chapter 1 5, 459-483. Gardner, R.J. ( 1 977), "Shapley value and disadvantageous monopolies", Journal of Economic Theory 16:5 135 17. Guesnerie, R. ( 1 977), "Monopoly, syndicate, and Shapley value: About some conjectures", Journal of Eco nomic Theory 1 5:235-251. Harsanyi, J.C. ( 1963), ' 'A simplified bargaining model for the n-person cooperative game", International Economic Review 4 : 194-220. Hart, S. (1977a), "Asymptotic value of games with a continuum of players", Journal of Mathematical Eco nomics 4:57-80. Hart, S. ( 1 977b), "Values of non-differentiable markets with a continuum of traders", Journal of Mathematical Economics 4: 103-1 16. Hart, S. ( 1980), "Measure-based values of market games", Mathematics of Operations Research 5: 1 97-228. Hart, S . (1985a), ''An axiomatization of Harsanyi's non-transferable utility solution", Econometrica 53: 1 2951313. Hart, S. ( 1985b), "Non-transferable utility games and markets: Some examples and the Harsanyi solution", Econometrica 53:1445-1450. Hart, S., and A. Mas-Colell ( 1 989), "Potential, value and consistency", Econometrica 57:589-614. Hart, S., and A. Mas-Colell ( 1995a), "Egalitarian solutions of large games: I. A continuum of players", Math ematics of Operations Research 20:959-1002. Hart, S., and A. Mas-Co1ell (1995b), "Egalitarian solutions of large games: II. The asymptotic approach", Mathematics of Operations Research 20: 1003-1022. Hart, S., and A. Mas-Colell (1996a), "Harsanyi values of large economies: Non-equivalence to competitive equilibria", Games and Economic Behavior 1 3 :74-99. Hart, S., and A. Mas-Colell (1996b), "Bargaining and value", Econometrica 64:357-380. Leviatan, S. ( 1 998), "Consistent values and the core in continuum market games with two types", The Hebrew University of Jerusalem, Center for Rationality DP- 1 7 1 . Maschler, M . , and G . Owen ( 1 989), "The consistent Shapley value for hyperplane games", International Journal of Game Theory 1 8 :389-407. Maschler, M., and G. Owen ( 1 992), "The consistent Shapley value for games without side payments", in: R. Selten, ed., Rational Interaction (Springer) 5-12. Mas-Colell, A. (1977), "Competitive and value allocations of large exchange economies", Journal of Eco nomic Theory 14:419-438. Mas-Colell, A., M.D. Whinston and J.R. Green (1995), Microeconomic Theory (Oxford University Press). McLean, R.P. (2002), "Values of non-transferable utility games", in: R.J. Aumann and S . Hart, eds., Handbook of Game Theory, Vol. 3 (North-Holland, Amsterdam) Chapter 55, 2077-21 20. Mertens, J.-F. (1988a), "The Shapley value in the non-differentiable case", International Journal of Game Theory 17: 1-65. Mertens, J.-F. ( 1988b), "Nondifferentiab1e TU markets: The value", in: A.E. Roth, ed., The Shapley Value (Cambridge University Press) 235-264.
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Mertens, J.-F. (2002), "Some other applications of the value", in: R.J. Aumann and S. Hart, eds., Handbook of Game Theory, Vol. 3 (North-Holland, Amsterdam) Chapter 58, 2185-2201. Nash, J . (1950), "The bargaining problem", Econometrica 18:155-162. Neyman, A. (2002), "Values of games with infinitely many players", in: R.J. Aumann and S. Hart, eds., Handbook of Game Theory, Vol. 3 (North-Holland, Amsterdam) Chapter 56, 2121-2167. Roth, A.E. (1980), "Values of games without side payments: Some difficulties with current concepts", Econometrica 48:457-465. Roth, A.E. (1986), "On the non-transferable utility value: A reply to Aumann", Econometrica 54:981-984. Shafer, W. (1980), "On the existence and interpretation of value allocations", Econometrica 48:467-477. Shapley, L.S. (1953), "A value for n-person games", in: H.W. Kuhn and A.W. Tucker, eds., Contributions to the Theory of Games II, Annals of Mathematics Studies 28 (Princeton University Press) 307-317. Shapley, L.S. (1964), "Values of large games VII: A general exchange economy with money", RM-4248-PR (The Rand Corporation, Santa Monica, CA). Shapley, L.S. (1969), "Utility comparison and the theory of games", La Decision (Editions du CNRS, Paris) 251-263. Shapley, L.S., and M. Shubik (1969), "Pure competition, coalitional power, and fair division", International Economic Review 24:1-39. Winter, E. (2002), "The Shapley value", in: R.J. Aumann and S. Hart, eds., Handbook of Game Theory, Vol. 3 (North-Holland, Amsterdam) Chapter 53, 2025-2054. Wooders, M., and W.R. Zame (1987a), "Large games: Fair and stable outcomes", Journal ofEconomic Theory 42:59-93. Wooders, M., and W.R. Zame (1987b), "NTU values oflarge games", Stanford University, I.M.S.S.S. TR-503 (mimeo).
Chapter 58 SOME OTHER ECONOM IC APP LICATIONS OF THE VALUE*
JEAN-FRAN<;;OIS MERTENS Centerfor Operations Research and Econometrics (CORE), Universite Catholique de Louvain, Louvain-la-Neuve, Belgium Contents
1. 2. 3. 4. 5.
Introduction The basic model Taxation and redistribution Public goods without exclusion Economies with fixed prices 5. 1 . 5.2. 5.3. 5.4. 5.5.
Introduction Dividend equilibria Value allocations The main result Concluding remarks
References
21 87 2187 2189 2192 2196 2196 2196 2197 2197 2200 2200
*Based on Chapter 10 of Game-Theoretic Methods in General Equilibrium Analysis, J.-F. Mertens and S. Sorin (editors), Vol. 77 of NATO AS! Series D, Kluwer Academic Publishers, 1994. Permission to use this material is gratefully acknowledged. Handbook of Game Theory, Volume 3, Edited by R.I. Aumann and S. Hart © 2002 Elsevier Science B. V. All rights reserved
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Abstract
Selected applications of the NTU Shapley value to settings other than that of general equilibrium in perfect competition. Three models are considered: Taxation and Redis tribution, Public Goods, and Fixed Prices. In each case, the value leads to significant conceptual insights which, while compelling once understood, are far from obvious at the outset. Keywords
NTU Shapley value, taxation, redistribution, public goods, fixed prices
JEL classification: H23, D45, D72, C7 1
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Ch. 58: Some Other Economic Applications ofthe Value 1. Introduction
The concept of value has proved to be a powerful tool in analyzing economic models. Indeed, since the Shapley value can be interpreted in terms of "marginal worth", it is closely related to traditional economic ideas (see Chapter 57 of this Handbook, which presents the Value Equivalence Theorem). Though other important applications exist, we focus here on three applications of values to economic models outside the general equilibrium model; each describes a different departure from the market environment. The first two are economic-political models dealing with taxation. Taxation has (at least) two purposes: redistribution and the financing of public goods. The classical literature assumes that a benevolent government aims to maximize some social utility function. The game approach is quite different: it assumes that the policy of the government is determined by the interests of those who elect it. Both with redistribution and with public goods, the value is a natural tool to analyze the corresponding games. The last section deals with fixed-price economies. Throughout this chapter, we will provide intuitions on why one would expect the results to hold (or not), rather than detailed proofs. For the latter, the reader may consult the original papers. 2. The basic model
Define a competitive economy as (T, £, e, (u 1 )1 ET ) where T [0, 1] stands for the set of traders, endowed with Lebesgue measure {t; ffi.� is the commodity space; e : T --+ ffi.�, integrable, is the initial allocation; u 1 : ffi.� --+ ffi. is the utility function of t. An allocation is a (measurable) map x : T --+ ffi.� such that fr Xt J-t(dt) = fr et J-t(dt) . An individual trader is best viewed as an "infinitesimal subset" dt of T. Thus e1 J-t (dt) is trader dt 's endowment, Xt J-t (dt) denotes what he gets under an allocation x, and u 1 (x1 )J-t (dt) is the utility he derives from it. We assume u 1 (0) 0. =
=
Value allocations. Consider a positive "weight" function A T --+ ffi.+ (integrable). The worth V)c (S) of a coalition S C T with respect to A is the maximum "weighted total util ity" it can get on its own, i.e., when the members of S reallocate their total endowment among each other. In the above setting, :
The value of a trader is his average marginal contribution to the coalitions to which he belongs, where "average" means expectation with respect to a distribution induced by a random order of the players. Thus, the value of trader dt is (1)
2 188
1.-F. Mertens
where Sct1 is the set of players "before dt" in a random order, and E stands for "expec tation". An allocation x is called a value allocation if there exists a weight function A. such that
for all t; in words, if it assigns to each trader - and hence to each coalition - precisely its (weighted) value.
Value equivalence. It is useful for the sequel to review here the argument showing that a value allocation x is also Walrasian. Note first that it is efficient - achieves the worth V"A (T) of T (i.e., JT A1u1 (x1)JJo(dt) = V"Jc (T)); this follows from the efficiency of the value (i.e., V"Jc (T) = (cp v"A )(T )) and the definition of value allocation. Denoting the gradient of u 1 at x by u; (x), we have
(otherwise, society could always profitably reallocate its endowment, contradicting x's efficiency). Call this common value p. Next, for any coalition S, define an S-allocation as a (measurable) map x5 : S ---* ffi.� with fs xf fJo (dt) = fs e1 fJo (dt); it is just like an allocation, except that it reallocates within S only. As with allocations, if such an x5 achieves v"A (S), then
in particular, this applies to Sct1• Now Sct1 is a "perfect sample" of T, in the sense that in Sct1 the distribution of players is the same as in T. From this it may be seen that the common value of the gradients for Sct1 is also p . Hence, dt's contribution to Sct1 is twofold: the weighted utility of dt itself, and the change in the other traders' aggregate utility due to dt's joining the coalition. Under the new optimal allocation of the endow ment of Sct1 U dt , dt gets X1JJo(dt) and therefore its weighted utility is A1u1 (x1)JJo(dt ) . So e1 fJo (dt ) - x1 fJo (dt) must be distributed among the traders in Sct1, so their increase in utility is p [e1 - xrJJJo(dt). So ·
so from the definition of "value allocation" it follows that p
·
[er - Xr] = 0.
Now (x1 , u1 (x1)) is on the boundary of the convex hull of the graph of u1. Otherwise, it would be possible to split trader dt (i.e., the bundle x1 dt allocated to him) into several
2189
Ch. 58: Some Other Economic Applications of the Value
players so that the weighted sum of their utilities would be greater than dt's utility, contradicting x's efficiency. Together with the equality of the gradients, this yields
Hence, x1 maximizes u 1 (x) under the constraint p x ( p x1 , and hence under the con straint p x ( p e1 • So x is a Walrasian allocation corresponding to the price system p . ·
·
·
·
3. Taxation and redistribution
Much of public economics regards the government as an exogenous benevolent eco nomic agent that aims to maximize some social utility, usually the sum of individ ual utilities [see Arrow and Kurz (1970)]. This ignores the workings of a democracy, in which government actions are determined by the pressures of competing interests, whose strength depends on political power (voting) and economic power (resources). This section, based on Aumann and Kurz (1977a, l977b), applies this idea to the is sues of taxation and redistribution. In our model, each agent has an endowment and a utility function, taxation and redistribution policy is decided by majority voting, and each agent can destroy part or all of his endowment, which we may think of, e.g., as labor. Thus while any majority can levy taxes even at 100% and redistribute the pro ceeds among its own members, anyone can decide not to work, so that the majority gets nothing by taxing the income from his labor. Though he will not feel better (no utility of leisure is assumed), he can use this as a threat to make the majority compromise. This influences the composition of the majority coalition and the tax policy it adopts. We start from the previous model but with a single commodity. Since we want to al low for threats and non-transferable utilities, we use the Harsanyi-Shapley NTU value. Suppose given a weight function A. . Then the worth of the all-player coalition T is
Suppose now that two complementary coalitions S and T \ S form. Think of VJc (S) as the aggregate utility of S if it bargains against T \ S. As in Nash (1953), suppose that the two parties can commit themselves to carrying out threat strategies if no satisfactory agreement is reached. If under these strategies S and T \ S get f and g respectively, then they are bargaining for VJc (T) - f - g, and, under the symmetry assumption, this is split evenly. Hence, S gets � (vJc(T) + f - g) and T \ S gets � (VJc(T) + g - f), so that the derived game between S and T \ S is a constant-sum game. The optimal threat strategy for the majority coalition is to tax at 100%, while the optimal threat strategy for the minority is to destroy all of its endowment; indeed, in this way the majority ensures its own endowment, and the minority ensures getting
2 190
J.-F Mertens
nothing, while each side is prevented from getting more. Thus the reduced game value is if JL(S) -!; if JL(S) < -!; >
and we have
it follows that A s in the previous section, Sct1 i s almost certainly a perfect sample of T. We can then use the self-explanatory notation Sctt = e T, where e E [0, 1] is the size of the sample. If we think of the players as entering a room at a uniform rate but in a random order, then e can be viewed as the random time when dt enters the room, and thus is uniformly distributed. Hence
Now there are three different cases: (1) both eT and eT u dt are majorities; (2) both are minorities; (3) eT is a minority, but eT U dt is a majority; i.e., dt is pivotal. Suppose qJ.. (T ) is achieved at the allocation x. Then dt 's expected contribution in the three cases is as follows, case 1 being as in the last section: (1) -!A.1 [u1 (x1) - u; (x1) • (x1 - er)]JL(dt) [= Jf [qJ.. (eT U dt) - qJ.. (eT )] de ]; (2) -! CO - O)[= fo�-f.IJctt ) (O) de ]; (3) -!qJ.. (T)JL(dt) [= ft (ctt [qJ.. ( -! T ) - O] dt]. ; -P, ) 2
I
All in all, therefore,
-
By the definition of value allocations, ({JVJ.. (dt) = A.1u1(x1)JL(dt) and, as in the previous section, A.1u ; (x1) is constant in t, say = p. It follows that
Ch. 58: Some Other Economic Applications of the Value
2191
p,-a.e. In the case of a single commodity (£ = 1) this comes to
(2) for all t, where C = q)JT) I p is a constant. Defining value allocation as in Section 2, we find that (2) is necessary and sufficient for x to be a value allocation. Moreover, (2) has a single solution (x, C), and C is positive. This constitutes the main result in Aumann and Kurz (1977a). Now to the economic interpretation. The left side of (2) is the (signed) tax on t. Note that when the utility functions Ut are increasing and concave, et is increasing in x1 or, more intuitively, x1 is increasing in e1 but with slope < ! . That is to say, marginal tax rates are between 50% and 100%. Though there is no explicit uncertainty in the model, define the fear of ruin as the ratio u1 (x1)lu; (x1). To explain why, consider its reciprocal u' lu. Suppose that some player is ready to play a game in which with probability 1 - p his initial fortune x is increased by a small amount c and with probability p he is ruined and his fortune is 0. If he is indifferent between playing the game or not, pIc is a measure of his boldness (at x) - and so, its reciprocal cl p a measure of his fear (of ruin). Indifference means that -
u (x ) = p 0 + (1 - p) u (x + c). ·
·
Hence, when c goes to zero, c I p goes to u (x) I u' (x) . Thus, the tax equals the fear of ruin at the net income, less a constant tax credit.
EXAMPLE. We end this section with an example of an explicit calculation of tax policy. Let all agents t have the same utilities u1(x) = u (x) = x"' , where 0 < a :( 1 . The fear of ruin is then u (x) u'(x)
x"' axa-l
x a
Integrating, we get
[ u (x) [x 1 [ C=J + O = J -;; = -;; J e , T T u ' (x) T since JT e = JT x. Hence the tax e1 - x1 is given by
this may be thought of as a uniform tax at the rate of 11(1 + a), followed by an equal distribution of the proceeds among the whole population. Recalling that the exponent a
2192
]. -F. Mertens
is an inverse measure of risk aversion, we conclude that the more risk averse the agents are, the larger the tax rate is. As a comment, recall that all estimates of coefficients of relative risk aversion are strongly negative - say ranging from a = -2 to a = - 15 (!) (cf., e.g., Dreze (198 1)), with ur (x) x"'a-e"' , , and note the limiting result for a = 0. This is clearly due to the enormous power attributed here to the majority: for a :( 0, it can - and will under the above "optimal threats" - put the minority effectively outside its consumption set. It might be interesting to investigate in this context (say a's distributed between - 15 and + 1) the implications of the traditional requirement that laws should be anonymous. For example, formulations of the sort: t's net tax T(y1, r) is a function only of his re ported (i.e., not destroyed) endowment y1 (0 :( y1 :( e1) and of the distribution r (in duality with the space of all Lipschitz functions of y) of those y1 's such that: (a) budget balance always holds, (b) T is jointly continuous, monotone and with slope :( 1 in y, (c) an increase in r in the sense of first-order stochastic dominance decreases r; for all y and T(O, ·), and (d) absence of "monetary illusion" - since the model satisfies it, - i.e., for A. > 0, denoting by r), the image of r under y A.y, one has T(A.y , rA.) = A.T ( y , r). One might even want to add progressivity: (e) r;' ;?: 0, or equivalently: consumption is concave in y; or still equivalently: tax is convex in consumption. And then the normal effect of an increased dispersion of r (second-order stochastic dominance) would also be to decrease r; for all y and T(O, ·): more tax receipts becom ing available, everyone and every marginal dollar should benefit from it. (Observe that the above tax policy, T (y, r ) = 1 �a (y - y) , satisfies all those requirements.) The aim here is clearly not to have to play explicitly with ideas like a higher utility for society of individuals with a lower risk aversion, and still to exhibit some similar effect. And hope fully less than all the above requirements would be sufficient to determine the solution. =
r+
4. Public goods without exclusion
The second purpose of taxation is the financing of public goods. As with redistribution, considering government as subject to the influence of its electors, rather than benev olently maximizing some social welfare function, sheds new light on the subject. As in Auman, Kurz and Neyman (1983, 1987), a Public Goods Economy is modeled by a continuum of agents, endowed with resources and voting rights, who take part in the production of non-exclusive public goods. More precisely, when a coalition forms, it chooses one strategy amongst the available, which together with the complementary coalition's choice determines the bundle of public goods that is produced. A natural question is the dependence of the outcome of the game on the distribution of voting rights, which should at first sight exert a major influence. But the Harsanyi-Shapley NTU Value leads to the surprising new and pessimistic result that the distribution of
Ch. 58: Some Other Economic Applications of the Value
2193
voting rights has (little or) nothing to do with the choice. Aside from its proof, this result is reinforced by an economic argument based on the implicit price of a vote. A nonatomic public goods economy is modeled by the set of agents T = [0, 1], the space JR.� of resources, and the public goods space JR.�. There is a correspondence G from JR.� to JR.�, representing the production function, which takes compact and nonempty values. Ut : JR.� --+ JR. is t's utility function. v is a nonatomic voting measure with v ( T ) 1 . Every agent has resources that can be used to produce public goods. The voting mea sure v is not necessarily identical to the population measure fl. For example, it is possi ble that noncitizens do not have the right to vote. A vote takes place and the majority decides which public good(s) will be produced. The minority is not entitled to produce public goods but, as in the previous model, it has the right to destroy its resources. The important point is that once the public good has been produced, the minority may not be excluded from consuming it. =
THEOREM measure
1.
In the voting game, the value outcomes are independent of the voting
v.
Consider an example with two public goods, TV and libraries. There are two kinds of agents in two equally weighted sets: one kind is fond of TV programs, the other of books. Assume further that TV fans possess all the voting rights. Then one might expect that after the voting, TV programs will be the only available leisure. But as it happens, both TV and books will be available in equal amounts ! Whatever the voting rights, the same bundle of public goods will be chosen. SKETCH OF PROOF. We first describe the Harsanyi-Shapley NTU value of a game, also used in Section 3. For a fixed weight function A., every coalition S announces a threat strategy zs that it will carry out if negotiations with the complementary coalition T \ S fail. Together with T\S's strategy, zs yields S a total payoff of V (S) = U(zs, ZT\S) (S) . After these announcements, the players are in a fixed-threat TU game, and its solu tion is taken to be the Shapley value. So each player wants the threats of the vari ous coalitions of which he is a member to be chosen so as to maximize his own final payoff according to this Shapley value. This can be shown to imply that all members of any given coalition S unanimously want to maximize H (S) = V (S) - V (T \ S). Since the same holds for T \ S, the optimal threat strategies z� and z h s are the sad dle points of the two-person zero-sum game H (S). Let qJc (S) = U(z�, z h5) (S) be the total payoff to S when both S and T\S carry out their optimal threats. Define WJc (S) = qJc(S) - qJc (T\S) = H(S)(z�, zh5); thus WJc (S) measures the "bargaining power" of S - its ability to threaten. Define also VJc (S) = � (WJc (S) + WJc(T)). As argued in the previous section, VJc (S) is the total utility S can expect to result from an efficient compromise with T \ S. Observe that f{JVJc = cpqJc (since the difference is a game where every coalition gets the same as its complement), but whereas qJc might depend on the particular choices of optimal threats z�, that is not so for VJc . The function VJc is the
2194
f. -F.
Mertens
Harsanyi coalitional form of the game with weight function A., and as before, we define a value allocation as a bundle y achieving the Harsanyi-Shapley NTU value, i.e., such that
We know (rp v)(dt)
=
fa\v(&T
u
dt) - v(&T) J de.
We want to compare this expression for two different voting measures v and � :
We call a coalition S even if S is either a majority under both v and � or a minority under both v and � . If S is even, so is T\S and their strategic options are the same under v and � , i.e., vv (S) - v; (S) 0. Every perfect sample of the whole population e T is even - it is determined by its size. For e T U dt, it is even if e > � or e < � - 8 [with 8 = max(v(dt), �(dt))]. The previous difference thus amounts to: =
If wv (e T U dt) is achieved at the outcome y , then, by additivity of the integral, and homogeneity: wv(& T U dt)
=
= =
HY (&T U dt) UY (&T) + 2UY (dt) - UY ( ( l - &)T ) (2& - l)UY (T) + 2A.t Ut (Yt ) P, (dt).
Now 2& - 1 as well as p,(dt) are infinitesimal. Independently of the voting measure v, we have shown that in the relevant range (& E [� - 8, �]), wv (&T U dt) is infinitesimal: the idea is that, under those circumstances, both the coalition and its complement re semble � T , thus are close to each other, and whatever the outcome, they enjoy the same utility derived from the common consumption of the same public good. Nobody enjoys a real bargaining advantage, and the efficient compromise induced by the Shapley value leads to equal treatment.
Ch. 58: Some Other Economic Applications of the Value
Going back to more technical arguments, assume all t. Then
2/8.
Given any coalition S, and 8 and n � Then we obtain
>
0, partition S into S1
Ut(Y ) U
2195 �
K for all feasible y and
· U S11, with (v + �)(Si ) � 8, · ·
cp v v (S) - cp vl; (S) = � �
� E h�8�2e - l ) (u
+ 2fs/'r(ut (Yn - u1 (yf)) f.L (dt)Jde � · 1 1 '2_ (28 - l) d8 1 · (2K) f A.1f.L(dt) + � · 4K8 1 A.1f.L(dt) 2 2-8 2 2K8 [i A.tf.L(dt) + is A.rf.L(dt) ] 4K8 i At f.L (dt). ·
(
Y
(T) - ul (T) )
n
s
�
This being true for all 8, we obtain cpvv (S) = cp v� (S). This being true for any weight function A., the result follows.
D
Though counterintuitive, this result might have been guessed from a similar analysis conducted in a transferable-utility (TU) context: we allow agents to trade their votes for money. The outcome of such a TU game consists of a public goods vector and a side payment vector. In the book vs. TV game, it can easily be derived from the following conditions: (1) It is Pareto-optimal. Under sensible assumptions, the only Pareto-optimal situation involves a produc tion of ( � , �) with the accompanying schedule of side-payments from book fans to TV fans. (3) To have this outcome approved, as book fans need only 50% of the vote and thus can play TV fans off against each other, they drive the price of a vote down to zero and can achieve the desired outcome without effective payments. In the case of non-exclusive public goods, every agent endowed with a voting right is potentially a free-rider: he believes his personal vote not to influence the final outcome, which he may like or dislike, and is thus ready to sell it even at a low price. This is wrong in the case of redistribution or more generally of exclusive public goods, where, as here, voting is assumed not to be secret, and the identity of the voters is crucial in eventually determining everyone's payoff. To conclude this section, we stress the connection between the TU and NTU situa tions: the TU games that we obtain corresponding to the latter are similarly public good economies, but in addition have a single desirable private good available for transfers.
(2)
2196
1. -F. Mertens
5. Economies with fixed prices
5.1. Introduction In economies with fixed prices all trading must take place at exogenously given prices, which determine (together with the positive orthant) a new consumption set - the net trade set - for each trader. This model has been used to describe market failures such as unemployment. In general, price rigidities prevent market clearance: a trader's con sumption set is exogenous and, under the standard assumptions, his utility function has an absolute maximum, a satiation point, generally in the interior of the consumption set; it may be the case that for any price vector, at least one of the traders' utility functions is satiated; this trader uses less than the maximum budget available to him, creating a total budget excess. Consider a fixed-price exchange economy with one commodity and two traders. Let their net trade sets be X 1 = X2 = [-5 , + 5] and their utility functions u 1 (x) = -(x - 1 ) 2 and uz (x) = - (x + 2) 2 . Since the satiation points are respectively 1 and - 2, if p > 0 then xi = 0 and x2 = -2. Hence, the market does not clear. Similarly if p < 0 or if p 0. EXAMPLE.
=
This suggests a generalization of the equilibrium concept in the general class of mar kets with satiation: the total budget excess is divided among all the traders, as dividends, so that supply matches demand. However, Dreze and Muller (1980) extended the First Theorem of Welfare Economics to this equilibrium concept, proving it to be too broad: with appropriate dividends, one can obtain any Pareto-optimum. In this respect, the Shapley value leads to more specific results (Aumann and Dreze, 1986): the income allocated to a trader depends only and monotonically on his trading opportunities and not on his utility function! This will be formally stated in Section 5.4, which includes a sketch of the proof. A formulation in the particular context of fixed price economies will then be presented.
5.2. Dividend equilibria Define a market with satiation as M 1 = (T; £; ( X1 ) t ET; (u 1 )t E T ), where T = { 1 , . . . , k} is a finite set of traders, JE.£ is the space of commodities, X1 C JE.£ is trader t's net trade set, supposed to be compact, convex, with nonempty interior and containing 0, and u1 is trader t 's utility function, assumed concave and continuous on X1 . A price vector is any element in JE.£ . Let B1 = {x E X1 I u 1 (x ) maxy E Xr U t (Y)} be the set of satiation points of trader t. B1 is nonempty, i.e., every trader has at least one satiation point. For simplicity, traders such that 0 E B1 may be taken out of the economy. They are fully satisfied with their initial endowment. Thus we will suppose that Vt, 0 ¢: B1 • An allocation is a vector x E fl t ET Xt such that Lt ET Xt = 0. =
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2197
As noted above, competitive equilibria may fail to exist since, whatever the price vector, a trader may well refuse to make use of his entire budget, thus preventing the market from clearing. The idea of dividends is to let the other traders use the excess budget. A dividend is a vector c E JR.k . A dividend equilibrium is a triple constituted of a price q , a dividend c and an allocation x such that, for all t, x1 maximizes u 1 (x) on X1 subject to q x � c1 • ·
5.3.
Value allocations
This is the "finite" version of the definition in Section 2. A comparison vector is a non-zero vector A E JR.�. For each A and each coalition S c T, the worth of S according to A is
{ I ES
V!c(S) = max L AtU1 (Xt) s.t. L Xt = 0 and Vt E S, Xt E X1
IES
};
Vic (S) is the maximum total utility that coalition S can get by internal redistribution when its members have weights A1 • An allocation is called a value allocation if there exists a comparison vector A such that A1u1(x1) = cp v!c (t ) for all t, where CfJV!c is the Shapley value of the game V)c . 5.4.
The main result
Mn, the n-fold replica of market M 1 , is the market with satiation where every agent of M 1 has n twins. Formally stated: Tn = ui E T Ijn (the Set Of nk traders). Vi E T, ! Tin I = n (there are n traders of type i) . Vt E r;n ' U t = Ui and Xr = xi . e
•
•
ELBOW ROOM ASSUMPTION. For all 1 C { l , . . . , k},
[
I:x1
O � bd L Bi + iEJ i!fj
In words, if it is possible to satiate all traders in some 1 simultaneously, then it is also possible to do so when they are restricted to the relative interior of their satiation sets, and the others to that of their net trade sets. Note that since the right-hand side is the boundary of a convex subset of JR.f, its dimension is at most (£ - 1). Since the possible 1 are finite in number, the assumption holds for all but an (£ - I)-dimensional set of total endowments. In that respect, it is generic. An allocation x of Mn is an equal treatment allocation if traders of the same type are assigned the same net trade. Trivially, there is then an associated allocation x of M 1 .
2198
J.-F. Mertens
THEOREM 2. Consider a sequence (xn ) n EN where xn is an allocation corresponding to an equal treatment value allocation in Mn . Let x00 be a limit of a subsequence of(xn ) n EN· Then, there is a dividend (vector) c and a price vector q such that (q , c, x00) is a dividend equilibrium where c is nonnegative, i.e., Vi, q � 0 c is monotonic, i.e., Vi, X; c Xj ==> c; � Cj . •
•
What gives substance to the theorem is the following existence result.
PROPOSITION 3 . There exists an equal treatment value allocation for every Mn . SKETCH OF PROOF OF THEOREM 2. This sketch is very informal. To make things simpler, suppose that: Li "A7 = 1 (normalization); Vi , x? and "A7 converge; Vi, x7 and xF are in int( Xi t ) ; Vi, u; is strictly concave and continuously differentiable on X; . Call those types i for which limn -+oo "A7 = 0 lightweight, the rest heavyweight. Sup pose that the weights of all the lightweight types converge to 0 at the same speed. We have • •
•
•
In the limit,
Now consider the contribution of a trader to a coalition S. If S is "large enough", it is very likely to be a good sample of the population Tn . Thus an optimal allocation for S is approximately the optimal xn for T n . The first term of the new trader's contribution is what he gets for himself. Since he does not change the optimal allocation by much, this is "A7 u; (x?). The second term is his influence on the other traders' utilities. Since the net trade must equal zero and all gradients are equal, this is approximately -q n x? . Thus, the contribution is approximately L1 = "A? u; (x?) - q n x? . If S is "small", the previous considerations do not hold. However, a new trader's contribution to a (small) coalition is uniformly bounded. This follows, e.g., from the continuous differentiability of the utilities on the compact net trade sets. Moreover, the probability p n of S being a small coalition goes to zero as n goes to infinity. Denote by 87 the expected contribution of a trader t of type i conditional on the coalition being small. Now we have (very roughly): ·
·
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2199
Hence, (3) Note that p n j(I- P n ) 0. Suppose there is no lightweight type. If simultaneous satiation of all traders is pos sible, then this is the Shapley value for all sufficiently large n, since then A-7 > 0 for all i . It is clear that the theorem holds then, for any price system q, and c; C suffi ciently large. Otherwise, u ; (xf") # 0 for at least some i . Hence, because of the equality of the gradients, uj (xj ) # 0 for all j . Hence q00 # 0 and for all i, the gradient of u; at x00 is in the direction of q00. Going to the limit in (3) gives q00 xf" 0. Hence, xf" maximizes u; (x) on X; subject to q 00 x ::::; 0. Hence, it is an ordinary competitive equilibrium and trivially a dividend equilibrium. Suppose now that type i is lightweight. We have q00 0. Hence uj (xj ) = 0 for any heavyweight type j ; that is to say, xj satiates j . Before letting n go to infinity, divide equality (3) by llqn /1 . We shall see that 8's order of magnitude is greater than that of "A?u; (x7) . Assume, for simplicity, that the sequence qn / llqn I converges to some point q . We get: �
=
•
=
•
=
q · X·'oo
=
pn · (8.n - "An. u; (x.n )) hm n --+oo (1 -Pn) l l q " l l l
'
z
.
Denote this quantity c; . If u ; (xf") = 0, then xf" maximizes u; over X; . Hence it maximizes u; over {x E X; , q x ::::; c; }. The same holds if u ; (xf) # 0, because then q is in the direction of u; (xf") . We claim that c; i s nonnegative, depends monotonically on X; , and not at all on u ; . A lightweight trader t's joining a "small" coalition S contributes to three utilities: (i) his own, (ii) that of other lightweight traders, and (iii) that of heavyweight traders. Roughly (again), since we are considering a small coalition, the heavyweight traders are probably not all simultaneously satiated. Since the weights in (i) and (ii) tend to zero when n becomes large, when trader t joins the coalition, his resources are best used if distributed to unsatiated heavyweight traders. His ability to give his resources depends only and monotonically on Xr, and not on Ut . Though the optimal redistribution of t's resources may involve several heavyweight traders, giving it all to only one trader gives a lower bound to (iii): 87 is of larger order than "A? . This establishes our claim about c; . ·
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1.-F.
Mertens
If trader t is heavyweight, contributions (i) and (ii) in 8? are at least cancelled out by A?ui (x;'), since A'f'ui (x'(') is the most t can get. Thus, the rest of the argument is as before with
(_!':__)
[component (iii) of 8] q .xr)Q � lim n -+ OO lf qn l l and Ci defined as the right side of this inequality. Since i is heavyweight, x'f' satiates. By the above inequality, it also satisfies the budget constraint. 0 5.5.
Concluding remarks
A question eluded until now is: "What about the core in economy that allows satia tion?" Remember that an allocation is in the core if it cannot be improved upon by any coalition. This can be understood either in a strong or a weak way. The strong version requires that no "weak improvement" is possible. That is to say that it is not possible that some members are strictly better off while others are not worse off. With this defin ition the Core Equivalence Theorem for replica economies applies. And since the set of competitive equilibria may be empty, the same holds for the core. On the other hand, the "weak" core may be too large and not even enjoy the equal treatment property. Either way, the core does not yield much insight into these economies. an
References
Arrow, K.J., and M. Kurz (1970), Public Investment, the Rate of Return, and Optimal Fiscal Policy (Johns Hopkins Press, Baltimore, MD). Aumann, R.J. (1964), "Markets with a continuum of traders", Econometrica 32:39-50. Aumann, R.J. ( 1975), "Values of markets with a continuum of traders", Econometrica 43:61 1-646. Aumann, R.J., and J.H. Dreze (1986), "Values of markets with satiation or fixed prices", Econometrica 54:1271-1318.
Aumann, R.J., and M. Kurz (1977a), "Power and taxes", Econometrica 45: 1 1 37-1 161. Aumann, R.J., and M. Kurz (1977b), "Power and taxes in a multi-commodity economy", Israel Journal of Mathematics 27: 186-234. Aumann, R.J., M. Kurz and A. Neyman (1983), "Voting for public goods", Review of Economic Studies 50:677-693.
Aumann, R.J., M. Kurz and A. Neyman (1987), "Power and public goods", Journal of Economic Theory 42: 108-127.
Champsaur, P. (1975), "Cooperation versus competition", Journal of Economic Theory 1 1 :394-417. Dreze, J.H. ( 1981 ), "Inferring risk tolerance from deductibles in insurance contracts", The Geneva Papers on Risk and Insurance 20:48-52. Dreze, J.H., and H. Miiller (1980), "Optimality properties of rationing schemes", Journal of Economic Theory 23: 150-159.
Harsanyi, J.C. (1959), "A bargaining model for the cooperative n-person game", in: A.W. Tucker and R.D. Luce, eds., Contributions to the Theory of Games IV, Ann. of Math. Studies 40 (Princeton University Press, Princeton, NJ) 325-355.
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Mertens, J.-F., and S. Sorin (eds.) (1994), Game-Theoretic Methods in General Equilibrium Analysis, Vol. 77 of NATO ASI Series - Series D: Behavioural and Social Sciences (Kluwer Academic Publishers, Dor drecht). Nash, J.F. (1953), "Two-person cooperative games", Econometrica 21: 128-140. Shapley, L.S. ( 1953), "A Value for n-person games", in: H.W. Kuhn and A.W. Tucker, eds., Contributions to the Theory of Games II, Ann. of Math. Studies 28 (Princeton University Press, Princeton, NJ) 307-317. Shapley, L.S. (1969), "Utility comparisons and the theory of games", in: La Decision (Editions du Centre National de Ia Recherche Scientifique (CNRS), Paris) 251-263.
Chapter 59 STRATEGIC ASPECTS OF POLITICAL SYSTEM S
JEFFREY S. BANKS* Division ofHumanities and Social Sciences, California Institute of Technology, Pasadena, CA, USA Contents 1 . Introduction 2. The cooperative approach 3. Sophisticated voting and agendas 4. The spatial environment 5. Incomplete information References
2205 2205 2207 2214 2220 2227
*Deceased December 21, 2000. The editors are grateful to David Austen-Smith and Richard McKelvey, who saw the manuscript through the final stages of the production process. It was evident from Jeff's marginal notes that he had intended to add material to the introductory section. However, since the paper seemed otherwise complete, it was decided not to alter the manuscript from the state that Jeff left it in. Handbook of Game Theory, Volume 3, Edited by R.I. Aumann and S. Hart © 2002 Elsevier Science B. V. All rights reserved
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Abstract
Early results on the emptiness of the core and the majority-rule-chaos results led to the recognition of the importance of modeling institutional details in political processes. A sample of the literature on game-theoretic models of political phenomena that ensued is presented. In the case of sophisticated voting over certain kinds of binary agendas, such as might occur in a legislative setting, equilibria exist and can be nicely charac terized. Endogenous choice of the agenda can sometimes yield "sophisticated sincer ity", where equilibrium voting behavior is indistinguishable from sincere voting. Under some conditions there exist agenda-independent outcomes. Various kinds of "structure induced equilibria" are also discussed. Finally, the effect of various types of incom plete information is considered. Incomplete information of how the voters will behave leads to probabilistic voting models that typically yield utilitarian outcomes. Uncer tainty among the voters over which is the preferred outcome yields the pivotal voting phenomenon, in which voters can glean information from the fact that they are pivotal. The implications of this phenomenon are illustrated by results on the Condorcet Jury problem, where voters have common interests but different information. Keywords
voting, political theory, social choice, asymmetric information
JEL classification: D7 1 , D72, D82, C70
Ch. 59: Strategic Aspects ofPolitical Systems
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1. Introduction
In this paper I review a small, selective sample of the existing literature on game theoretic models of political phenomena. Let X denote a set of outcomes, and N { 1 , . . . , n} a finite set of voters, with n � 2 and with each i E N having a binary preference relation Ri on X. In what follows we consider two separate environments: (1) the finite environment, where X is finite and, for all i E N, R; is a weak order; and (2) the spatial environment, where X c ffik is compact and convex and each R; is a continuous and strictly convex weak order. 1 In either case let Rn denote the set of admissible preference profiles, with common element p. =
2 . The cooperative approach
One popular approach to collective decision-making is to suppress any explicit behav ioral theory and model the interaction as a simple cooperative game with ordinal prefer ences. Thus let V c 2N denote the set of decisive or winning coalitions, required to be non-empty, monotonic ( C E V and C c C' implies C' E V) and proper ( C E V implies N\ C ¢:. V). The (strict) social preference relation of V at p is defined as x Pv (p)y iff 3C E V s.t. x P; y Vi E C, and the core of V at p is C(V, p)
=
{x E X: �y E X s.t. y Pv (p)x} .
Results on the possible emptiness of the majority-rule core have been known for some time. For instance, in the finite environment Condorcet (1785) constructed his famous three-person, three-alternative "paradox": x P1 yP1 z, y P2 z P2x , z P3 x P3y . In the spa tial environment, as convex preferences are single-peaked we know from Black (1958) that the majority-rule core will be non-empty in one dimension; however as Figure 1 demonstrates it is easy to construct three-person, two-dimensional examples where the core is empty. In this example individuals' preferences are "Euclidean": for all x , y E X, xR; y if and only if llx - x i II � II Y - x i I I , where x i is i 's ideal point. These results are generalized to arbitrary simple rules by reference to the rule's Nakamura number, 1J (V) , which is equal to if the rule is collegial, i.e., n e E D C # 0, and is otherwise equal to 00
{
min IV' I : V' c;;., V &
n
CED '
}
C=0 .
1 These assumptions on preferences are stronger than necessary for some of the results below; however they facilitate comparisons across a vatiety of models.
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Figure 1.
In words, the Nakamura number of a non-collegial simple rule D is the number of coali tions in the smallest non-collegial subfamily of D. For non-collegial rules this number ranges from a low of three (e.g., majority rule with n odd) to a high of n (e.g., a quota rule with q = n 1). Part (a) of the following is due to Nakamura (1979), while part (b) is due to Schofield (1983) and Strnad (1985): -
PROPOSITION 1 . (a) In the finite environment, C (D, p) -f. 0 for all p E R" if and only if I X I < 1J (D) . (b) In the spatial environment, C(D, p) -f. 0 for all p E R11 if and only if k < TJ (D) - 1 . Thus, for any simple rule one can identify, for both the finite and the spatial envi ronment, a critical number such that core non-emptiness is only guaranteed when the number of alternatives or the dimension of the policy space is below this number. As a corollary, we see that if the number of alternatives is at least n, or the dimension of the
Ch. 59: Strategic Aspects ofPolitical Systems
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policy space at least n 1 , then for any non-collegial rule the core will be empty for some profile of preferences. Furthermore, generalizing the logic of the Plott (1967) conditions for a majority-rule core point, one can identify a second critical number (necessarily larger than the first) for non-collegial simple rules in the spatial model with the following property: assum ing individual preferences are in addition representable by smooth utility functions, if the dimension of the policy space is above this second number the core will "almost al ways" be empty. Specifically, the set of utility profiles for which the core is empty will be open and dense in the Whitney C 00 topology on the space of smooth utility profiles [Schofield (1984), Banks (1995), Saari (1997)] . Therefore core non-emptiness for non collegial simple rules will be quite tenuous in high-dimensional policy spaces. Finally, McKelvey (1979) showed that when the core of a strong (C ¢:. V implies N\C E 'D) sim ple rule is empty, the strict social preference relation Pv(P) is extremely badly behaved, in the following sense: given any two alternatives x, y E X there exists a finite set of al ternatives {a 1 , . . , am } such that x Pv (p )a 1 Pv (p )a2 Pv (p) . . . Pv (p )am Pv (p) y. That is, one can get from any one alternative to any other, and back again, via the strict social preference relation.2 It is difficult to overstate the impact these "chaos" theorems have had on the for mal modeling of politics. Most importantly for this paper, the theorems gave rise to a newfound appreciation for political institutions such as committee systems, restrictive voting rules, etc. The theorems implied that additional structure was required on the col lective decision-making process so as to generate a well-posed, i.e., equilibrium, model. These institutional features then became the foundation of or the motivation for various non-cooperative game forms used to study specific political phenomena. In what fol lows we will examine a set of these game forms in depth, maintaining for the most part a focus on majority rule. Further, given the intrinsic appeal of majority-rule core points when they exist, we will inquire as to whether a game form is "Condorcet consistent"; that is, do the equilibrium outcomes correspond to majority-rule core points when the latter exist? -
.
3. Sophisticated voting and agendas
Consider the finite environment, where we make the additional assumptions that each individual's preference relation is a linear order, thereby ruling out individual indiffer ence; and that n is odd. Together these imply that the strict majority preference rela tion is complete (although of course not necessarily transitive), and therefore that any majority-rule core alternative is unique and is strictly majority-preferred to any other alternative, i.e., it is a "Condorcet winner". 2 An analogous result holds for non-strong rules when the strict social preference relation is replaced with the weak social preference relation [Austen-Smith and Banks (1999)]. One implication of this is that, when the core is empty, the transitive closure of the weak social preference relation exhibits universal indifference.
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X
X
z
(a)
W
Z
W
y
W
Z
W
(b)
Figure 2.
We consider sequential voting methods for selecting a unique outcome from X. Specifically, the voting procedures we examine can be characterized as occurring on a binary voting tree r (il., Q, ()) having many features in common with an extensive form game of perfect information: i1. is a finite set of nodes, and Q is a precedence relation on i1. generating a unique initial node A. 1 as well as a unique path between all A., A.' E il. for which A. Q A.' . The precedence relation Q partitions i1. into decision nodes il. d and terminal nodes i\. 1 , where i\. 1 = {A. E il. : A.' QA. for no A.' E il.} and il.d il. \il. 1 . The qualifier "binary" implies that each decision node A. E il. d has precisely two nodes immediately following it, l(A.) and r(A.) (for "left" and "right", respectively), and deci sion nodes characterize instances when a majority vote is taken over which of these two nodes to move. Finally, the function e i\. 1 X assigns to each terminal node exactly one of the alternatives; we require e to be onto, so that each element of X is the se lected outcome for some sequence of majority decisions. Figure 2 gives two examples of voting trees. Given a voting tree r, a strategy for any i E N is a decision rule cr; il. d {l, r} describing how i votes at each decision node. A profile of strategies cr = (cr 1 , . . . , ern ) then determines a unique sequence of majority-rule decisions through the voting tree, ultimately leading to a particular terminal node A.(cr) E i\. 1 , and hence an outcome e (A.(cr)) E X. Thus for any profile of preferences p we have a well-defined (ordinal) non-cooperative game. "Sophisticated" voting strategies are generated by iteratively deleting weakly dominated strategies [Farquharson (1969)], which can be characterized as follows [McKelvey and Niemi (1978), Gretlein (1983)] : (1) at all final decision nodes (i.e., those only followed by terminal nodes) individual i votes for her preferred alterna tive among those associated with the subsequent two terminal nodes (note that this con stitutes the unique undominated Nash equilibrium in the subgame); (2) at penultimate decision nodes i votes for her preferred alternative from those derived from (1), given common knowledge of preferences; and so on, back up the voting tree. This process, which is equivalent to simply applying the majority preference relation P (p) via back wards induction to r , can be characterized formally by associating with each node A. E A. its sophisticated equivalent s (A.) E X, which is the outcome that will ultimately =
=
:
--...
:
--...
Ch. 59: Strategic Aspects of Political Systems
2209
be selected if node A is reached. The mapping s(-) is defined inductively by: (a) for all A E A 1 , s(A) = 8(A), (b) for all A E A d , s (A) = s (l(A)) if s (l(A))P(p)s(r(A)) and s(A) = s(r(A)) otherwise. This process leads to a unique outcome s (AI), referred to as the sophisticated voting outcome (although the strategies generating s (A J ) need not be unique since for some A it may be that l(A) = r(A)). The outcome s (A J ) obviously de pends on the voting tree r; however it depends on the preference profile p only through the majority preference relation, P(p), and so we denote the sophisticated voting out come associated with r and P as s (r, P). Finally, we know from McGarvey (1953) that for any complete and asymmetric relation P on X there exists a value of n and a profile of linear orders p such that P is the majority preference relation associated with (n, p). Since these parameters are here unrestricted, spanning over all complete and asymmetric relations P on X is equivalent to spanning over all (n, p). The qualifier "sophisticated" is meant to contrast such behavior with "sincere" vot ing, in which branches are labeled with alternatives and at each decision node voters simply select the branch associated with the more preferred alternative. For instance, let X = {w, x , y, z} and n = 3, with preferences given by zPixPi y P1 w, wPzzPzxPzy, yP3wP3zP3x, and consider the subgame in Figure 2(b) beginning at node A. Sincere voting would require 1 to vote for z over y; however, since w will majority-defeat z if z is selected at A, y will defeat w if y is selected at A, and y P1 w, 1 's sophisticated strategy is to vote for y at A. On the other hand, both 2's and 3's sophisticated strategy prescribes voting sincerely at A. It is evident that the specifics of the voting tree play an important role in the determi nation of the sophisticated voting outcome; for instance, x is the outcome in Figure 2(a) but y is the outcome in Figure 2(b) given the above preference profile. For any majority preference relation P define V(P) = {x E X: :Jr s.t. x = s(r, P)} as the set of alter natives which are the sophisticated voting outcome for some binary voting game, and consider the following definition of the top cycle set:
T (P) = { x E X: 'v'y -j. x E X, :l{a1 , . . . , ar} s; X s.t. a 1 = x, a, = y, and 'v't E { 1 , . . . , r - 1}, a1 Pa1 + d · In words, x is in the top cycle set if and only if we can get from x to every other alternative via the majority preference relation P. If x is a Condorcet winner then it is easily seen that T(P) = {x}. Otherwise T (P) contains at least three elements, and can in general include Pareto-dominated alternatives; for instance, in the above example T (P) = X, but zP;x for all i E N. McKelvey and Niemi (1978) prove that for all P, V(P) s; T(P), while Moulin (1986) proves that for all P, T (P) s; V (P). Taken together we therefore have, PROPOSITION
2. For all P, V(P) = T (P).
Thus all binary voting procedures are Condorcet consistent, and so if P admits a Condorcet winner then the equilibrium outcome will be independent of the voting pro-
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cedure in place. Conversely, in the absence of a Condorcet winner the voting procedure itself will play some role in determining the collective outcome, where at times these outcomes can be inefficient. Much of the work on voting procedures and agendas has focused on the amendment voting procedure of which the game in Figure 2(b) is an example. Thus two alternatives are put to a vote; the winner is then paired with a new alternative, and so on. This pro cedure is attractive for a number of reasons, not the least of which is that it is consistent with the rules of the US Congress governing the perfecting of a bill through a series of amendments to the bill (i.e., Roberts' Rules of Order). An amendment procedure is characterized by an ordering of X or agenda a : { 1 , . . . m } X, where the first vote is between a(1) and a(2), the winner then faces a(3), etc. Let Ta denote the voting game associated with the agenda a, and let A (P) {x E X: 3a s.t. x s(Ta , P)} de note the set of sophisticated voting outcomes when restricting attention to amendment procedures. Building on the work of Miller (1980) and Shepsle and Weingast (1984), Banks (1985) provides the following characterization of the set A ( P) : let T ( P) = { Y � X: P is tran sitive on Y} be those subsets of alternatives for which the majority preference relation is transitive, and E(P) {Y � X: Vz ¢. Y 3y E Y s.t. yPz} be those subsets of alter natives for which every non-member of the set is majority-defeated by some member of the set. If P is transitive on the set Y then there will exist a unique P-maximal element of Y; label this alternative m(Y) and define B(P) {x E X: x = m (Y) for some Y E T(P) n E(P)}. Note that (i) for all P, B(P) � T(P); (ii) as with T (P), if a Condorcet winner exists B(P) will coincide with this alternative, and otherwise consist of at least three elements; and (iii) in contrast to T (P) any x E B(P) can not be Pareto-dominated by any other alternative. 3 For instance, in the above example B(P) = {y, z, w}, excluding the Pareto-dominated alternative x. ,
-+
=
=
=
=
PROPOSITION
3 . For all P, A (P) = B(P).
Therefore the amendment procedure always generates Pareto-efficient outcomes. On the other hand, in the absence of a Condorcet winner there remains a sensitivity of this outcome to the particular agenda employed. Two theoretical conclusions naturally arise at this level of the analysis. The first is that we should at times observe individuals voting against their "myopic" best interests, as voter 1 does in the above example. The second is that in the absence of a Condorcet winner there will be a diversity of induced preferences over the set of possible agendas; therefore to the extent the agenda is itself endogenously determined we have merely pushed the collective decision problem back one step (albeit limiting the relevant set of alternatives to A (P)). We consider each of these conclusions in turn.
zP;x for all i E N then zPy for all y such that xPy by the transitivity of individuals' m (Y) for some Y E T(P) then Y rfc E(P).
3 If
x
=
preferences; so if
221 1
Ch. 59: Strategic Aspects of Political Systems
With respect to "insincere" behavior, consider the spatial environment, with pref erence profile p for which the majority-rule core is empty, and suppose that m � n alternatives from X are selected to make up an amendment procedure in the fol lowing manner: initially voter m selects an alternative x E X to be a(m), i.e., the last alternative to be considered; then voter m 1 , with knowledge of a(m), selects a(m - 1), etc., with voter 1 selecting a(1). A proposal strategy for i � m is thus of the form rr; : x Cm -i) -+ X; given a profile of proposal strategies rr = (rr 1 , . . . , rrm ) let x(rr) = (a( l ; rr), . . . , a(m; rr)) denote the resulting amendment agenda. An equilib rium is then defined as (a) sophisticated voting strategies over any set of m alternatives from X, and (b) subgame perfect proposal strategies for all i � m. Finally, recall that for amendment procedures each node A. E Ad is a vote between alternatives a(j) and a (k), for some j, k � m, and hence sincere voting is well-defined. Austen-Smith (1987) proves the following. -
PROPOSITION 4. In any equilibrium the proposal strategies rr* are such that all indi viduals vote sincerely on x(rr*). Therefore when the alternatives on the agenda are endogenously chosen, we have "sophisticated sincerity" as the equilibrium voting behavior of the individuals will be indistinguishable from sincere voting. In particular, observing sincere voting is not in consistent with sophisticated voting; rather, the evidence of sophisticated voting will be much more indirect, with its influence felt through the proposals that make up the agenda. With respect to the endogenous choice of agenda for a fixed set of alternatives, sup pose there exists a "status quo" alternative which must be considered last in the agenda. For instance, after various amendments to a bill have been considered and either adopted or rejected, the bill in its final form is voted on against the status quo of "no change" in the policy. Let X0 E X denote this status quo policy, and let A(xo) denote the set of agendas with a(m) = X0• Finally, say that x E X is an agenda-independent outcome under the amendment procedure if x = s(rc, , P) for all a E A(xo). The presence of an agenda-independent outcome would obviously extinguish any conflict over the choice of agenda. One possibility of course is that X0 itself is an agenda-independent outcome, which only occurs when Xo is a Condorcet winner. Alternatively, consider the following con dition: Condition C: :Jx E X s.t. (i) x Px0 , and (ii) X Py Vy =f. X s.t. y P X0 • To see that this is sufficient for agenda independence, note that an amendment proce dure generates a particular form of assignment e of terminal nodes to alternatives; for instance, in Figure 2(b) each of x, y, and z are paired with the final alternative w at least once, and all final pairings involve w . This is a general feature of amendment
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procedures, so consider the voting game formed by replacing each final decision node ).. E Fr" with the majority-preferred alternative among its two immediate successors, s(fi.). Label this voting game r� (P), and note that the alternatives in this game, denoted X', are given by any y E X for which yPxa, along with possibly Xo itself (if Xa Pz for some z E X). Now if there exists an alternative x* E X' which is majority-preferred to all others in X' (i.e., x* is a Condorcet winner in X'), then applying Proposition 2 we have that x* = s (T� ( P ) , P) regardless of the structure of the voting game T� (P), in particular regardless of the ordering a E A(x0) . But then x* is an agenda-independent outcome. As an example of an environment where Condition C holds, consider the following classic model of distributive politics, which we label DP [Ferejohn et al. (1987)] : each i E N is the sole representative from district i in a legislature, and district i has asso ciated with it a project that would generate political benefits and costs of hi 0 and Ci 0, respectively. Assume Ci =f. Cj for all i, j E N, and assume without loss of gener ality that c1 < cz < < c11• The problem before the legislature is to decide which (if any) of the projects to fund; thus an outcome is a vector x = (x 1 , . . . , x11 ) E {0, 1 } 11 = X, where Xi = 1 denotes funding the ith project and Xi = 0 denotes rejection. For any x E X let F (x) s:; N denote the projects that are funded. If project i is funded the benefits accrue solely to legislator i , whereas the costs are divided equally among all the legislators; if project i is rejected, legislator i receives zero benefits but still must bear her share of the costs of any funded projects. Thus the utility for legislator i derived from any outcome x E X can be written >
>
· · ·
Ui (X) = biXi -
�n L.... j = l CjXj
n
.
Define x a E X by F(x a ) = { i E N: i � (n + 1)/2} and assume that for all i E F(x a ), Ui (x a ) 0; that is, the least-cost majority of projects generates a strictly positive utility for those legislators receiving projects. Finally, let the status quo alternative be Xo = (0, . . . , 0), i.e., no project is funded. >
PROPOSITION
procedure.
5. x a is the agenda-independent outcome in DP under the amendment
To see this, note first that for an alternative x to be majority-preferred to X0 it must be that F (x) > n /2, for otherwise a majority of legislators are receiving zero benefits while bearing positive costs, thereby generating a utility strictly less than zero. Second, among those alternatives with F(x) > n/2 there is one alternative that is majority-preferred to all others, namely x a . This follows since for any x =f. x a such that F(x) n /2 , all i E { 1 , . . . , (n + 1)/2} must be bearing a strictly higher cost (since project cost is increasing in the index) while receiving no higher benefit (either Xi = 1 and i receives the same benefit or else Xi = 0 and i receives a strictly lower benefit). Therefore the mqjority coalition of { 1 , . . . , (n + 1)/2} prefers xa to any other outcome which has a >
Ch. 59: Strategic Aspects ofPolitical Systems
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3
3,1,2
3,1
3,2
3
1,2
2
0
Figure 3.
majority of projects funded, and since U i (x a ) > 0 for all members of this coalition x a is majority-preferred to x0 as well. Therefore Condition C holds, with x a the agenda independent outcome. Notice that for this problem there exists another voting procedure which might ac tually seem more natural, namely, where each project is considered one at a time, and accepted or rejected according to a majority vote. Such a procedure would again gen erate an outcome x E {0, l }n , although the procedure would look quite different from the amendment procedure. For instance, suppose n = 3, and the projects are consid ered in the order 3 , 1 , 2; then the binary voting tree would look as in Figure 3 (where for simplicity we list F (x ) rather than x ) . Using the above utilities we can compute the sophisticated voting strategies: at each of the four final decision nodes the coalition { 1 , 3} prefers not to fund project 2, so this project is defeated regardless of the previous votes. Given this, the coalition {2, 3} prefers not to fund project 1 at the two penultimate nodes, and similarly the coalition { 1 , 2 } prefers not to fund project 3 at the initial node. Therefore the sophisticated voting outcome is that none of the three projects is funded. Furthermore, this logic is independent of the order in which the projects are considered, and immediately extends to the n-project environment. Therefore, the outcome under the "project-by-project" voting procedure is invariably the status quo Xo = (0, . . . , 0) . We can generalize from this example and create a different form of agenda inde pendence for the special case where the outcome set X is of the form X = {0, l } k , with the interpretation being that Xi = 1 signifies acceptance of issue i and Xi = 0 rejection. Equivalently, an outcome can be thought of as a (possibly empty) subset J � K = { 1 , . . , k} of accepted issues, with individual preference and hence majority preference then defined over these subsets. An issue-by-issue procedure considers each of the k issues one at a time, and is characterized by an agenda K K which orders .
t :
-+
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the k issues. Let f't be the issue-by-issue voting game associated with the agenda and say that E X is an agenda-independent outcome under the issue-by-issue procedure if = s(f't, P) for all Consider the following restriction on majority preference: t,
x
x
L
Condition S: For all 1 s; {1, . . . , k } and all j E 1, 1\{j} P 1. In words, slightly smaller (by inclusion) subsets of accepted issues are preferred by a majority to larger. To see that S is sufficient to guarantee agenda independence, note that any final decision node corresponds to a vote of the form 1 vs. 1\{j} (where ( j ) = k). If S holds then issue j will be rejected regardless of the earlier votes, i.e., regardless of the contents of 1. By backwards induction, then, the penultimate votes are of the form 1' vs. 11\{d} (where (d) = k - 1) and again d is defeated regardless of 1'. Continuing this logic, we see that if S holds then all issues are rejected for any ordering of the issues. Thus we have not only agenda independence but we have identified the sophisticated voting outcome as (0, . . . , 0) , all issues rejected. Returning to the above example, we have, t
t
PROPOSITION 6. procedure.
X0
is the agenda-independent outcome in DP under the issue-by-issue
Therefore, as with the amendment procedure, if attention is restricted to issue-by issue procedures there will not exist any conflict over the agenda to use. Finally, note that if it were put to a vote whether an amendment procedure or an issue-by-issue procedure should be adopted in DP, a majority (namely, the members of F (xa)) would prefer the former to the latter regardless of the agenda subsequently employed. 4. The spatial environment
It is evident that what drives the previous result is the decomposability of the collective choice problem into n smaller problems ("fund project i", "don't fund project i"), as well as the separability oflegislators' preferences across these n problems. This decom posability has a natural analogue in the spatial environment, when we think of each di mension of the policy space as a separate issue to be decided. To keep matters simple we make the simplifying assumption that X is rectangular, X L�:. 1 , x ! l x x [:!.k , xk], so that the feasible choices along any one dimension are independent of the choices along the others. We return to the notion of separable preferences below. As in the previous section, with n odd and each R; strictly convex the majority-rule where is strictly majority core, if non-empty, will consist of a single element, preferred to all other alternatives. On the other hand, Figure 1 demonstrated that, no mat ter how nice individual preferences are, in two or more dimensions a majority-rule core point rarely exists, i.e., for every alternative there is another that is majority-preferred =
x
c
,
· · ·
x
c
Ch. 59: Strategic Aspects ofPolitical Systems
2215
to it. From this perspective then no alternative is stable, in that for any alternative there exists a majority coalition with the willingness and ability to change the collective deci sion to some other alternative. Now suppose that we constrain the influence of majority coalitions by requiring that any such change from one alternative to another must be accomplished via a sequence of one-dimension-at-a-time changes. For any x , y E X, define I (x, y) = {j E { 1 , . . . , k}: XJ # Yi } as the set of issues along which x and y disagree. Then since not all of these changes will necessarily be accepted, in consider ing a move from x to y the set of possible outcomes are those points in X of the form x + (y - x) 1 , where 1 � I (x , y) and for any vector z E Dld and subset 1 � { 1 , . . . , k}, z ( = Z i if i E 1 and z( = 0 otherwise. For instance, in two dimensions the set of possi ble outcomes is given by {(xi, xz), (xi , yz), (YI , xz), (Y I , yz) } . Thus for any ordered pair of alternatives (x, y ) E X2 and any ordering of the el ements of I (x, y) we have a well-defined issue-by-issue voting procedure f:(x , y); and given individual and hence majority preferences P on X we can compute the sophisticated voting outcome s(f:(x , y), P).4 Say that an outcome x* E X is a so phisticated voting equilibrium if for any y # x* E X and any ordering of I (x* , y), s(f: (x*, y), P) = x*. That is, an alternative is a sophisticated voting equilibrium if no dimension-by-dimension change can be produced when all individuals vote sophisti catedly. Finally, say that individual i 's preferences are separable if for all (x, y) E X2 , 1 � I (x, y) and j E 1, [x + (y - x) 1 ]Ri [X + (y - x) 1\ Ul ] if and only if [x + (y x) Ul ]R; [x]. Kramer (1972) then proves the following: t
t
7 . If, for all i voting equilibrium.
PROPOSITION
E
N, R; is separable, then there exists a sophisticated
Since preferences are strictly convex, continuous and separable, and n is odd, along issue dimension j there will exist a unique point xT given by the median of the individ uals' ideal points. Letting x* = (x f , . . . , x;?), separability of preferences then implies that for any y E X Condition S above is satisfied: for any subset of changes 1 � I (x* , y) an individual's preferences over 1 and 1\{j} (where j E 1) are completely determined by their preferences along dimension j, and since xj = xT a majority prefers 1\{j} to 1. Kramer (1972) actually proves the existence of an issue-by-issue median, i.e., an al ternative for which no one-dimensional change could attract a majority, without requir ing separability. However he also shows by way of example that without separability one can construct situations in which two one-dimensional changes away from xm, to gether with sophisticated voting, will occur. Note that if a Condorcet winner xc exists, it must be that x c = xm, and so issue-by-issue voting is Condorcet consistent. On the Since individuals may be indifferent between accepting and rejecting a change on an issue we add the behavioral assumption that when indifferent an individual votes against the change. 4
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other hand, even when a Condorcet winner does not exist a sophisticated voting equilib rium will exist (with separable preferences), due to the constraints placed on movements through the policy space. Kramer (1972) is the first example of what became known as the structure-induced equilibrium approach to collective decision-making (as opposed to preference-induced equilibrium, i.e., the core). A second example is given by Shepsle (1979) where, rather than directly impeding a majority's influence through the policy space, the choice of collective outcome is decentralized into a set of k one-dimensional decisions by sub groups of individuals. Thus define a committee system as an onto function K : N K = { 1 , . . . , k}, with the interpretation that K (i) is the single issue upon which i E N is in fluential. For all j E K, let NK (j) {i E N: K (i) = j} denote the set of individuals, or "committee" assigned to issue j, and for simplicity assume INK (j) i is odd for all j E K . The committees play "Nash" against one another, taking the decisions on the other issues as given, and use majority rule to determine the outcome on their issue. Thus let F (x; j) = {y E X: yz xz for all / #- j} denote the set of feasible alternatives the committee NK (j) can generate, given that x summarizes the choices by the other committees, and similarly let CK (x; j) c F(x ; j ) denote the set of majority-rule core points for the committee. An outcome x* E X is a K-committee equilibrium if, for all j E K, x* E CK (x* ; j); that is, no majority of any committee can implement a preferred outcome. ---*
=
=
PROPOSITION 8 . For all K, a K -committee equilibrium exists. Since individual preferences are strictly convex and continuous on X, for each (x, j) E X x K the preferences of each i E NK (j) will be single-peaked on F(x; j) and hence CK (x ; j) will be equal to the unique median of the ideal points of members of NK (j) along F (x ; j). And since preferences are continuous and the median function is continuous, we have that CK ( ; j) : X X is a continuous function, as is the func tion !3 : X ---* X defined by B(x) = (Cr (x; 1), . . , Ck (x; k)) . By Brouwer's fixed-point theorem there exists x* E X such that x* B(x*). A few comments on this model are in order. The first is that if committees were to move sequentially as opposed to simultaneously, equilibria need not exist since the induced preferences for members of the committee moving first need not be single peaked along their issue when the latter committees' responses are taken into account [Denzau and Mackay ( 1981)]. On the other hand, if individual preferences are assumed to be separable it is easily seen that this will not be a concern, and in fact the equilibrium will be independent of the order in which the committees report their choices. Second, although as stated this model is a combination of cooperative (within com mittee) and non-cooperative (between committee) elements, we can easily make the entire process non-cooperative: let the strategy space for each i E NK (j) be given by [!_J , x:J ] , with the outcome function being the Cartesian product of the median choices along each dimension. If individuals' preferences are separable then i E NK (j) would have a dominant strategy to choose her ideal outcome from [!.J , xi], thereby imple·
---*
.
=
Ch. 59:
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menting the (now unique) K-committee equilibrium. If preferences are non-separable this ideal outcome may depend on the median choices along the other dimensions. However, taking these other choices as fixed, i 's best response is to choose her ideal outcome from L�.j , xj ], and therefore any K-committee equilibrium will constitute a Nash equilibrium of the above game (although there may be others). Finally, in contrast to Kramer (1972) here one does not necessarily get Condorcet consistency, even with separable preferences. This is so because the "core" voter can only be on a single committee, and even then there is no assurance she gets her way along the relevant dimension. To see this let k = 3 and n = 9, and let all individuals have Euclidean preferences, with ideal points x 1 = (0, 0, 0), x 2 = x 3 = (1, 0, 0), x4 = x5 = (- 1 , 0, 0), x6 (0, 1, 0), x7 = (0, - 1 , 0), x8 = (0, 0, 1), and x9 = (0, 0, -1). Then the majority-rule core i s at (0, 0, 0); but if the committee for the first dimension is given by { 1 , 2, 3} the first coordinate in any equilibrium is 1 . On the other hand, if K is such that NK (1) = { 1 , 2, 4}, NK (2) { 3, 6, 7}, and NK (3) {5, 8, 9}, then in fact the K-committee equilibrium is at the core. The above model was motivated by the committee system employed in the US Congress. Parliamentary democracies also typically display a certain structure in their collective decision-making, a structure which might again be leveraged to produce equi librium predictions [Laver and Shepsle ( 1990), Austen-Smith and Banks ( 1990)]. In par liaments the locus of control is much more at the party level as opposed to the individual level; however to avoid modeling intraparty behavior we imagine an n-party world with perfect party discipline so as to remain consistent with our n-individual analysis. As with the committee model each dimension of the policy space is associated with a dif ferent "ministry", the differences being that here a single party controls a ministry, and a party can control more than one ministry. Thus let P(K) denote the power set of K, and define an issue allocation as a mapping w: N --+ P(K) such that =
=
U w(i) = K
=
and, for all i, j E N, w(i) n w(j) = 0.
i EN
That is, each issue is assigned to precisely one party, with w(i) denoting the issues under the influence of party i . Let Q denote the (finite) set of such mappings. Parties for which w(·) =f. 0 constitute the governing coalition, and play Nash against one another to deter mine an outcome from X; thus the strategy sets are of the form S� = niEw(i J L!.j , Xj ] , and the outcome function is simply the admixture of the various choices. Let E (w) c X denote the set of Nash equilibrium outcomes for the allocation w. Given our stated as sumptions on individual/party preferences, a standard argument establishes that E (w) is non-empty for each allocation w in Q . Further, as in the committee model, if prefer ences are separable then each member of the governing coalition will have a dominant strategy, implying for each w E Q the Nash equilibrium will be unique. Hereafter we assume this uniqueness holds regardless of the separability of preferences. In the above model of Shepsle (1979) the committee system K was exogenous, thereby leaving the puzzle of collective choice somewhat unfinished. In contrast, here
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we endogenize the choice of allocation: let E = UwED E (w) C X denote the (finite) set of outcomes implementable by some allocation, and note that each i E N has well defined preferences defined over E and hence (by uniqueness) well-defined induced preferences over the set of allocations S? . Rather than modeling the choice of alloca tion and hence the government formation process explicitly [as in, e.g., Austen-Smith and Banks (1988) or Baron (1991)], Laver and Shepsle (1990) and Austen-Smith and Banks ( 1990) explore various core concepts associated with the set E of implementable outcomes. The most straightforward concept, although by no means the most reason able, is to identify the weighted majority core on the restricted set of outcomes, E . That is, each party has a certain number of seats and hence "weight" in the legislature; an alternative x E E is an "allocation" core if there does not exist y E E which a weighted majority prefers. Note that any such prediction gives not only a policy from X, but also the identity of the government and the distribution of policy influence within the gov ernment. Furthermore, since the set of implementable outcomes E is much smaller than the set of all possible outcomes X, these allocation core points can exist even when a core point in X does not. In fact, since E is finite the former need not lead the precar ious existence the latter often do in multiple dimensions with respect to small changes in preferences. Finally, since w (i) = K, i.e., allocating all issues to a single party, is al lowed, we have that the profile of ideal points {x 1 , . . . , xn } is necessarily a subset of E. Therefore, if we add the assumptions that the party weights are such that no coalition has precisely 1/2 the weight (i.e., weighted majority rule is strong) and that each R; is representable by a continuously differentiable utility function u; : X m, then when XC exists (and is interior to X) it must be that xc = xi for some i E N, and hence we have a weighted version of Condorcet consistency holding. This would give an instance of one-party government where that party does not itself hold a majority of the seats (i.e., a one-party, minority government). Austen-Smith and Banks (1990) prove the following: -7
PROPOSITION
is non-empty.
9. If n = 3 and, for all i E N, R; is Euclidean, then the allocation core
In fact, with n = 3 the allocation core is either equal to one of the ideal points, or else is equal to the issue-by-issue median (cf. Figure 1). Even without a general existence theorem, Laver and Shepsle (1996) take the model's predictions to the data on post World War II coalitional governments in parliamentary systems, and find an admirable degree of consistency. Each of the above three models employs some element of "real world" legislative decision-making to provide the analytical traction necessary to generate equilibrium predictions. The final model we consider in this section is relatively more generic, and is meant to capture individual and collective behavior in more unstructured situations. Specifically, we look at a discrete time, infinite horizon model of bargaining where, in contrast to Rubinstein-type bargaining with a deterministic proposer sequence, the proposer in each period is randomly recognized. Thus in period t = 1 , 2, . . . , indi-
Ch. 59: Strategic Aspects ofPolitical Systems
2219
vidual i E N is recognized with probability /-Li ;;::: 0 as the current proposer, where L 1 E N f-Li = 1 . Upon being recognized she selects some outcome x E X, and upon ob serving x each individual votes either to accept or reject. If the coalition of individuals voting to accept are in V, where V is some simple rule, the process ends and the out come is (x, t); otherwise the process moves to period t + 1 . Individual i 's preferences are represented by a continuous and strictly concave utility function u; : X ffi+ + and discount factor o; E [0, 1], with the utility to i from the outcome (x, t) then be ing o{ - 1 u; (x). This model, due to Banks and Duggan (2000), extends to the spatial environment the earlier work of Baron and Ferejohn (1989) in which this bargaining protocol was studied in a "divide-the-dollar" environment with "selfish" linear prefer ences, i.e., X = {x E [0, 1]": L i E N Xi = 1} and for all i E N and x E X, u ; (x) = x; , and where the focus was on majority rule. Banks and Duggan (2000) restrict attention to equilibria in stationary strategies, consisting of a proposal p; E X offered anytime i is recognized; and a voting rule v; : X ---+ {accept, reject} independent both of history and of the identity of the proposer. To prove existence they need to allow randomization in the proposal-making; thus each i E N selects a probability distribution li; E P(X) (with the latter endowed with the topology of weak convergence). However, each individual observes the realized policy proposal prior to voting. Finally, they characterize only "no-delay" equilibria in which the first proposal is accepted with probability one. When o; < 1 for all i E N it is easily seen that all stationary equilibria will involve no delay; however, since individuals here are allowed to be perfectly patient (i.e., o; = 1 ) this amounts to a selection from the set of all stationary equilibria. Let u = (u 1 , . . . , u 11 ), and as before let C(V, u) s; X denote the core of the simple rule V at the profile u. ---+
PROPOSITION 1 0 .
(a) There exist stationary no-delay equilibria; (b) If o; = 1 for all i E N and either V is collegial or X is one-dimensional, then (li { , . . . , li; ) is a stationary no-delay equilibrium if and only if li;* ({x*}) = 1 for all i E N andfor some x* E C(V, u). Comparing this result to Proposition 1, and cognizant of the fact that here we are assuming strict concavity rather than strict quasi-concavity (i.e., strictly convex pref erences), we see first that equilibria exist for all profiles u, all simple rules V, and all dimensions of the policy space k. Further, when individuals are perfectly patient we get a core equivalence result precisely when the core is non-empty for all profiles u and simple rules V (i.e., k = 1) and when the core is non-empty for all profiles u and all di mensions of the policy space k (i.e., V collegial). Banks and Duggan (2000) also show that the set of stationary no-delay equilibria is upper-hemicontinuous in (among other parameters) the discount factors, and hence if individuals are "almost" perfectly patient all equilibrium outcomes will be "close" to the core.
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Another interesting comparison is with the equilibrium predictions of the issue-by issue model and the parliamentary model described above. In Figure 1 both of these models predict the issue-by-issue median x111 as the unique equilibrium outcome. On the other hand, it is readily apparent that in the bargaining model under majority rule any equilibrium proposal lies on the contract curve between the proposer and one of the remaining individuals, with the exact location of the proposals (and the probability distribution over them) depending on the underlying parameters. Thus, the former two models predict a much more centrist outcome than does the bargaining model. 5. Incomplete information
All of the models surveyed in the previous two sections have been examples of politi cal processes in which the set of individuals N directly chooses the collective outcome from X. We begin this section with the classic model of representative democracy, namely, the Downsian model of electoral competition, and analyze it in the context of Proposition 1 . In the basic model two candidates, A and B ( i N) simultaneously choose policies Xa , Xb E X; upon observing these policies individuals simultaneously vote for either A or B (so no abstentions), with the majority-preferred candidate winning the election and implementing her announced policy. The candidates have no policy in terests themselves, and only care about winning the election: the winning candidate receives a utility payoff of 1 , while the loser receives - 1 . Given any (xa , xb)-pair each i E N has a weakly dominant strategy to vote for the candidate offering the preferred policy according to U i , with i voting for each with probability 0.5 when Ui (xa) = Ui (xb). Thus, given knowledge of voter preferences the two candidates are engaged in a sym metric, zero-sum game with common strategy space X, and so in any equilibrium each must receive an expected payoff of zero. It is easily seen that a pure strategy equilib rium to this game exists if and only if the majority-rule core is non-empty: if the core is empty then given any Xb there exists a policy which majority-defeats it, and hence if adopted by A would generate payoffs ( 1 , - 1 ) . Conversely, if xc is in the core candidate C guarantees herself an expected payoff of zero, and so if Xa and Xb are core policies (xa , Xb) constitutes an equilibrium. From Section 2 we know that the majority-rule core in the spatial environment is equal to the median voter's ideal point when the dimension of the policy space is one, thus giving the original prediction of Downs (1957) that both candidates would locate there. On the other hand, we also know that when this dimension is greater than one the core is almost always empty, and so pure strategy equilibria typically fail to ex ist. One possible escape route of course is to consider mixed strategies, as was done by Banks and Duggan (2000) in the bargaining context. However, while existence of mixed strategy equilibria is assured in the finite environment, 5 in the spatial environ-
See [Laffond et al. (1993)] for a characterization of the support ofthe mixed strategy equilibria in the finite environment.
5
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Ch. 59: Strategic Aspects ofPolitical Systems
ment it has proven to be somewhat more elusive given the inherent discontinuities in the majority preference relation and hence in the above zero-sum game [however see Kramer ( 1978)]. Alternatively, one can smooth over these discontinuities by positing a sufficient amount of uncertainty, from the candidates' perspective, about the intended behavior of the voters. To this end let the probability i E N votes for candidate A, given announced policy positions (xa , Xb), be Pi (u i (x0) , Ui (xb )), with 1 - Pi being the prob ability of voting for B (so again no abstentions). Candidates are assumed to maximize expected vote, which, in the absence of abstentions, is equivalent to maximizing ex pected plurality. Thus A solves
with a similar expression for B. As in the bargaining model, we assume that, for all i E N, Ui is strictly concave and maps into the positive real line.
PROPOSITION 1 1 . (a) If, for all i E N, Pi (·) is concave increasing in its first argument and convex decreasing in its second, then a pure strategy equilibrium exists. (b) If Pi (-) can be written as
where f3 is increasing and differentiable, then in equilibrium X xu solves
a
= Xb
= xu, where
max Z::> i (x) . i EN
(c) If Pi (-) can be written as
where j3 is increasing and differentiable, then in equilibrium Xa = Xb = x n , where x n solves max I )n (u i ( ) ) . x
i EN
Thus, while (a) [due to Hinich et al. (1972)] provides sufficient conditions for the existence of a pure strategy equilibrium in the candidate game, (b) and (c) demonstrate that, in certain situations, these equilibrium policies will coincide with either utilitar ian or Nash social welfare optima. Beyond any normative significance, this implies that the candidates' equilibrium policies are actually independent of their expected plurality
J.S. Banks
2222
functions, and, therefore, independent of the structure of the incomplete information in the model (other than the requirement that p and p only depend on i through u; ) . Part (b) extends the result of Lindbeck and Weibull (1987) for a model of income redistrib ution, while (c) is an extension of a result for the familiar binary Luce model employed by Coughlin and Nitzan (1981) and others, where Pi (u; (xa ), u;(xb )
)
=
u; (xa ) . ) U; (Xa + U; (Xb )
One weakness of the probabilistic voting model is that the analysis starts "in the middle", by positing some voter probability functions without generating them from primitives. This can be remedied by assuming that, from the candidates' perspective, there exist some additional considerations beyond a comparison of policies which in fluence the voters' decisions and which are unknown to the candidates. We can think of these as non-policy characteristics of the candidates, or (equivalently) as the candi dates' fixed positions on some other policy dimensions. For example, if i E N votes for candidate A if and only if u; (xa ) > u; (xb ) + {3, where f3 is distributed according to a smooth function G, then p; (u; (xa ), u; (xb )) is simply equal to G(u; (xa ) - u; (xb )) and (b) is satisfied. Additionally, if G is uniform on an interval [{3, /3] sufficiently large (and containing zero), then
and hence p will be linearly increasing in u; (xa ) and linearly decreasing in u; (xb ), and therefore (a) is satisfied. Alternatively, the "bias" term f3 could enter in multiplicatively rather than additively, thereby generating a form as in (c). One immediate consequence of moving back to this more primitive stage is to qualify the welfare statements above: it is not the sum of the individual utilities that is actually being maximized, but rather the sum of the known components of their utilities. 6 In the previous model incomplete information concerning voter behavior was present, in that the candidates could not perfectly predict how policy positions would be trans lated into votes. However, the decision problem on the part of the voters was straight forward: given only two alternatives, and with complete information about their own preferences, each voter's weakly dominant strategy was simply to vote for her pre ferred alternative. Suppose now that there exists some uncertainty among the voters about which is the preferred alternative [Austen-Smith and Banks (1996)]. Specifi cally, let X = {A, B}, and let there be two possible states of the world, S = {A, B}, with JT E (0, 1) the prior probability that the state is A . Individual i 's preferences over outcome-state pairs (x, s) are represented by u; : X x S � {0, 1 } , with u; (x, s) = 1 if
6 See [Banks and Duggan (1999)] for these and other extensions of results in probabilistic voting.
Ch. 59: Strategic Aspects ofPolitical Systems
2223
and only if = s. Hence, in contrast to the heterogeneous preferences assumed in all of the above models, here the individuals have a common interest in selecting the out come to match the state. Individuals simultaneously vote for either A or B, with B being the outcome if and only if h individuals vote for it, where h E { 1 , . . . , n}. Thus h = (n + 1)/2 means that majority rule is being employed, whereas h = n means that B is the outcome only if the individuals unanimously vote for B. While individuals share a common interest, they possess potentially different infor mation concerning the true state, and hence the optimal choice, in that prior to vot ing each i E N receives a private signal t; E T; correlated with the true state. Thus let v; T; � X denote a voting strategy for i , v = ( V J , . . . , Vn ) a profile of voting strategies, t = (t[ , . . . , tn ) E 0; E N T; a profile of signals, and d = (di , . . . , dn ) E Xn a profile of individual decisions. Given a common prior p ( ) over profiles of signals, then, we have a Bayesian game among the voters, with the expected utility of a pair (d, t) equal to the probability the true state is equal to the collective decision conditional on t. Defin ing c(d) E X by c(d) = B if and only if l {i E N: d; = B } l � h, we can write this as Pr[s = c(d); t], which (via Bayes' Rule) is equal to Pr[s = c(d) & t]/ p(t). Moreover, since p(L; ; t;) = p(t)/ LLi p(t; , L;), we can simplify the expression for the expected utility of i choosing d; , conditional on t; and v, from x
:
·
� Pr[s = c(d; , v(L;)) & t] EU(d; ; t; , v) = � p(L; ; t;) ------- p(t) L i
to �
EU(d; ; t; , v) = � T-i
Pr[s = c(d; , v(L;)) & t] . " L.... T . p(t; , L;) -r
Now in comparing the difference in expected utility from voting for A or B, we can obviously ignore the denominator above; further, this difference will only be non-zero when i is pivotal in determining the outcome. Therefore let
be the set of others' signals for which, according to v, i is pivotal. Finally, when i is pivotal her decision is equal to the collective decision, and hence we get that
EU(A; t; , v) - EU(B; t; , v) � 0 if and only if L [Pr[s = A & t] - Pr[s = B & t] ] � 0. T!i (v)
Thus, in identifying which is the better decision each individual conditions on being piv otal, which (through v) limits the set of possible inferences she can have about others' information.
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The presence of this "pivotal" information can have dramatic effects on individuals' voting behavior. The first thing to note is that, unlike when information is complete, sincere voting is no longer a weakly dominant strategy, where Vi is sincere if, for all ti E h vi (ti) = A if and only if E[u ; ( A , · ) ; t;] > E[u; (B · ) ; t;]. As a simple example, let n = 5, n = 0.5, h = (n + 1)/2, and for all i E N let T; = {a, b} with Pr[t; = a; s = A] = Pr[t; = b; s = B] = q > 0.5. Thus there are four pure strategies: "always vote A", "always vote B", "vote A if and only if ti = a", and "vote A if and only if ti = b", with the third strategy being sincere. Suppose voters 1 and 2 play "always A" while 3 and 4 play the sincere strategy; then 5 should play "always B" since the only time 5 is pivotal is when t3 = t4 b, in which case even if ts = a Bayes' Rule implies the more likely state is B . Therefore sincere voting is not a weakly dominant strategy. On the other hand, sincere voting does constitute an equilibrium: if the others are behaving sincerely and i is pivotal, she infers that two of the others have observed a and two have observed b. Given the symmetry of the situation these signals cancel out when applying Bayes' Rule, and so it is optimal for i to behave sincerely as well. The reason sincere voting constitutes an equilibrium is that majority rule is the "right" rule to use in symmetric situations such as this, in the sense that if all the signals were known choosing A if and only if a majority of the signals were a would maximize the likelihood of making the correct decision [Austen-Smith and Banks (1996)]. Con versely, suppose we keep all the underlying parameters the same but use unanimity rule, h = n [Feddersen and Pesendorfer (1998)]. Then it is easily seen that sincere vot ing does not constitute an equilibrium: under unanimity voter i is pivotal only when all others are voting for B . But if the others vote for B if and only if their signals are b, i prefers to vote for B when pivotal even if t; = a, as the (inferred) collection of n - 1 b's overwhelms her one a. On the other hand, everyone adopting the "always B" strategy is not an equilibrium either, since in this case i is always pivotal and hence infers nothing about the others' signals. Thus when t; = a i should vote for A. (Of course, everyone adopting the "always A" strategy is an equilibrium, since nobody is ever pivotal.) There fore, to find interesting symmetric equilibria Feddersen and Pesendorfer (1998) look to mixed strategies, letting (ti ) E [0, 1] be the probability a type-t; individual votes for B. Beyond symmetry, they require the strategies to be "responsive" to the private signals, so that u(a) =/= u (b). It is readily apparent that, given the symmetry of the environment, any such equilibrium under unanimity will be of the form 0 < u (a) < 1 = (b ) , i.e., vote for B upon observing the signal b, and randomize upon observing a. The goal is to compare unanimity rule to majority rule, as well as any other rule, in terms of the equilibrium probability of the collective making an incorrect decision. As they let the population size n vary, we can parametrize the rules by y E (0, 1) and set h = Iy n l where for any r E ffi+, Ir l is the smallest integer greater than or equal to r (e.g., y = 0.5 is majority rule). ,
=
u
u
12. (a) Under unanimity rule there exists a unique responsive symmetric equilibrium; as n .....,. oo u (a) .....,. 1, and the probabilities of choosing A when the state is B and choosing B when the state is A are bounded away from zero.
PROPOSITION
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(b) For any y E (0, 1) there exists n (y) such that for all n � n(y) there exists a responsive symmetric equilibrium; as n � oo any sequence of such equilibria has the probabilities of choosing A when the state is B and choosing B when the state is A converging to zero. Therefore from the perspective of error probabilities unanimity rule is unambiguously the worst rule for large collectives. Furthermore, Feddersen and Pesendorfer (1998) show by way of a twelve-person example that "large" need not be very large. They interpret this result as casting doubt on the appropriateness of using unanimity rule in, for instance, legal proceedings. 7 The preceding result showed how a model that Proposition l (a) would suggest is triv ial under complete information, namely, when there are only two alternatives, becomes not so trivial with the judicious introduction of incomplete information. Our last model shows the same phenomenon with respect to Proposition 1(b), and is due to Gilligan and Krehbiel (1987). Consider a one-dimensional policy space X = ffi within which two players, labelled the committee C and the floor F, interact to select a final policy. 8 Two different procedures are considered, yielding two different game forms: under a "closed" rule C makes a take-it-or-leave-it proposal p E ffi, where if F rejects the pro posal the status quo policy s E ffi is implemented. In contrast, under the open rule F can select any policy following C's proposal, thereby rendering the latter "cheap talk". As with Feddersen and Pesendorfer (1998), Gilligan and Krehbiel (1987) are inter ested in a comparison of these two rules with respect to the equilibrium outcomes they generate, with the motivation here being that F has the ability to choose ex ante which procedure to adopt. With complete information F would surely prefer the open rule, given its lack of constraints. However, suppose that under either rule, if z E ffi is the chosen policy the final outcome is not z but is rather x = z + t, where t is drawn uni formly from [0, 1] and whose value is known to C but unknown to F. Thus a strategy for C under either rule is a mapping from [0, 1] to ffi; however, to distinguish the two, let p(t) denote C's proposal under the closed rule and m (t) C's "message" under the open rule. Similarly, let a (p) denote the probability F accepts a proposal of p E ffi from C under the closed rule, and p(m) the policy F selects following the message m E ffi under the open rule. Players' utilities over final outcomes are quadratic: uc(x) = -(x - Xc) 2 , Xc > 0, and U f (x) -x 2 . Thus Xc measures the heterogeneity in the players' prefer ences. Given this specification, if F's updated belief about the value of t has mean t her best response under the closed rule is to accept p if liP - t i l ( lis - til and reject otherwise, i.e., select p or s depending on which is closer to t. Under the open rule, given mean t F's best response is simply to select p = -t. As with most signaling games there exist multiple equilibria under either rule. Under the open rule Gilligan and Krehbiel select the "most informative" equilibria, which have =
7
However see [Coughlan (2000)] for a rebuttal.
8 We can think ofF as the median member of the legislature, and C as the median member of a committee.
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J.S. Banks
the following partition structure [Crawford and Sobel (1982)]: let to = 0 and tN(xc) = 1, where N(xc) is the largest integer satisfying 12N(1 - N)xcl < 1 . For 1 < i < N(xc) define t; by
thus the equilibria will be parametrized by the choice of t1 . Given a distinct set {m 1 , . . , mN} = M of messages C's strategy is of the form m*(t) = m j if and only if t E [tj - l , ti ) . Given F's updated belief her optimal response to any m j E M is p* (m1) = -(t1 - tj- I )/2. Faced with this strategy C's strategy above is optimal if and only if for all i = 1 , . . . , N(xc), t; is indifferent between sending m* (t; - c) and m*(t; + c ) , which holds only when .
which is a second-order difference equation whose solution is given above. Of particular relevance is the fact that N(xc) is decreasing in Xc, and tends to infinity as Xc tends to zero. The equilibrium Gilligan and Krehbiel select under the closed rule has the following path of play: (1) for t E [0, S I ] U [s3 , 1], p* (t) = Xc - t, and a*(p* (t)) = 1 ; (2) for t E (SJ , s2 ], p*(t) = 4xc +s, and a*(p* (t)) = 1 ; (3) for t E (s2 , s3 ), p* (t) = p E (s, s +4xc) and a*(p* (t)) = 0. Thus, when the shift parameter t is sufficiently small or sufficiently large, C is able to separate and implement her optimal outcome. A third region of types pools together and has their proposal accepted, while a fourth has their proposal re jected, with the logic of the latter being that s + t, the outcome upon rejection, is already close to Xc (in fact, s3 = Xc - s ).
1 3 . There exists x 0 such that if Xc < x then the floor's ex ante equi librium utility under the closed rule is greater than that under the open rule.
PROPOSITION
>
Therefore, as long as the preferences of the committee and floor are not too dissim ilar, the floor prefers to grant the committee a certain amount of agenda control. The intuition behind this result is the following: under both the open and the closed rule the equilibrium expected utility of F is decreasing in Xc, so that as their preferences become less diverse F is better off under either rule. For the open rule this is true since the more homogeneous the preferences the greater is C's ability to transmit information, leading to a more informed decision by F. Under the closed rule the set of types that separate and subsequently implement their ideal outcome is itself increasing as Xc decreases, and as Xc decreases C's ideal outcome becomes more favorable from F's perspective. Now in selecting the closed rule over the open rule F faces a distributional loss, due to C's being able to bias the resulting outcome in her favor, as well as an informational gain, due to the increased amount of information transmitted and the assumed risk aversion of F. As Xc decreases this distributional loss tends to be outweighed by the informational gain, implying the closed rule is preferable.
Ch. 59: Strategic Aspects ofPolitical Systems
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References Austen-Smith, D. (1987), "Sophisticated sincerity: voting over endogenous agendas", American Political Science Review 81: 1323-1330. Austen-Smith, D., and J. Banks (1988), "Elections, coalitions, and legislative outcomes", American Political Science Review 82:409-422. Austen-Smith, D., and J. Banks (1990), "Stable governments and the allocation of policy portfolios", Ameri can Political Science Review 84:891-906. Austen-Smith, D., and J. Banks (1996), "Information, rationality, and the Condorcet jury theorem", American Political Science Review 90:34-45. Austen-Smith, D., and J. Banks (1999), "Cycling of simple rules in the spatial model", Social Choice and Welfare 16:663-672. Banks, J. (1985), "Sophisticated voting outcomes and agenda control", Social Choice and Welfare 1:295-306. Banks, J. (1995), "Singularity theory and core existence in the spatial model", Journal of Mathematical Eco nomics 24:523-536. Banks, J., and J. Duggan (1999), "The theory of probabilistic voting in the spatial model of elections", Work ing paper. Banks, J., and J. Duggan (2000), "A bargaining model of collective choice", American Political Science Review 94:73-88. Baron, D. (!991), "A spatial bargaining theory of government formation in parliamentary systems", American Political Science Review 85: 137-164. Baron, D., and J. Ferejohn (1989), "Bargaining in legislatures", American Political Science Review 83: 1 1 8 11206. Black, D. (1958), The Theory of Committees and Elections (Cambridge University Press, Cambridge). Condorcet, M. ( 1785/1994), Foundations of Social Choice and Political Theory, Trans!. I. McLean and F. He witt, eds. (Edward Elgar, Brookfield, VT). Coughlan, P. (2000), "In defense of unanimous jury verdicts: mistrials, communication, and strategic voting", American Political Science Review 94:375-393. Coughlin, P., and S. Nitzan (1981), "Electoral outcomes with probabilistic voting and Nash social welfare maxima", Journal of Public Economics 15:1 13-122. Coughlin, P. (1992), Probabilistic Voting Theory (Cambridge University Press, New York). Crawford, V., and J. Sobel (1982), "Strategic information transmission", Econometrica 50:1431-145 1 . Denzau, A., and R . Mackay (198 1), "Structure-induced equilibria and perfect foresight expectations", American Journal of Political Science 25:762-779. Downs, A. (1957), An Economic Theory of Democracy (Harper and Row, New York). Farquharson, R. (1969), The Theory of Voting (Yale University Press, New Haven). Feddersen, T., and W. Pesendorfer (1998), "Convicting the innocent: the inferiority of unanimous jury ver dicts", American Political Science Review 92:23-36. Ferejohn, J., M. Fiorina and R. McKelvey (1987), "Sophisticated voting and agenda independence in the distributive politics setting", American Journal of Political Science 3 1 : 169-193. Gilligan, T., and K. Krehbiel (1987), "Collective decision making and standing committees: an informational rationale for restrictive amendment procedures", Journal of Law, Economics and Organization 3:287-335. Gretlein, R. (1983), "Dominance elimination procedures on finite alternative games", International Journal of Game Theory 12:107-1 13. Hinich, M., J. Ledyard and P. Ordeshook (1972), "Nonvoting and the existence of equilibrium under majority rule", Journal of Economic Theory 4: 144-153. Kramer, G. (1972), "Sophisticated voting over multidimensional choice spaces", Journal of Mathematical Sociology 2: 1 65-180. Kramer, G. (1978), "Existence of electoral equilibrium", in: P. Ordeshook, ed., Game Theory and Political Science (NYU Press, New York).
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Laffond, G., J. Laslier and M. Le Breton (1993), "The bipartisan set of a tournament", Garnes and Economic Behavior 5:182-201. Laver, M., and K. Shepsle (1990), "Coalitions and cabinet government", American Political Science Review 84:873-890.
Laver, M., and K. Shepsle (1996), Making and Breaking Governments (Cambridge University Press, New York). Lindbeck, A., and J. Weibull ( 1987), "Balanced-budget redistributions as the outcome of political competi tion", Public Choice 52:273-297. McGarvey, D. (1953), "A theorem on the construction of voting paradoxes", Econometrica 21 :608-610. McKelvey, R. (1979), "Generalized conditions for global intransitivities in formal voting models", Journal of Economic Theory 47: 1085-1 1 12. McKelvey, R., and R. Niemi (1978), "A multistage game representation of sophisticated voting for binary procedures", Journal of Economic Theory 18: 1-22. Miller, N. (1980), "A new solution set for tournaments and majority voting: further graph-theoretical ap proaches to the theory of voting", American Journal of Political Science 24:68-96. Moulin, H. (1986), "Choosing from a tournament", Social Choice and Welfare 3:271-291 . Nakamura, K. (1979), "The vetoers in a simple game with ordinal preferences", International Journal of Game Theory 8:55-6 1 . Plott, C. (1967), "A notion of equilibrium and its possibility under majority rule", American Economic Review 57:787-806.
Saari, D. ( 1997), "The generic existence of a core for q-rules", Economic Theory 9:219-260. Schofield, N. (1983), "Generic instability of majority rule", Review of Economic Studies 50:695-705. Schofield, N. (1984), "Social equilibrium cycles on compact sets", Journal of Economic Theory 33:59-71. Shepsle, K. (1979), "Institutional arrangements and equilibrium in multidimensional voting models", American Journal of Political Science 23:37-59. Shepsle, K., and B. Weingast (1984), "Uncovered sets and sophisticated voting outcomes with implications for agenda institutions", American Journal of Political Science 28:49-74. Strnad, J. (1985), "The structure of continuous-valued neutral monotonic social functions", Social Choice and Welfare 2: 181-195.
Chapter 60 GAM E-THEORETIC ANALYSIS OF LEGAL RU LES AND INSTITUTIONS
JEAN-PIERRE BENOIT* and LEWIS A. KORNHAUSERt Department ofEconomics and School ofLaw, New York University, New York, USA Contents
1. Introduction 2. Economic analysis of the common law when contracting costs are infinite 2. 1. The basic model 2.2. Extensions within accident law 2.3. Other interpretations of the model 3. Is law important? Understanding the Coase Theorem 3.1. The Coase Theorem and complete information 3.2. The Coase Theorem and incomplete information 3.3. A cooperative game-theoretic approach to the Coase Theorem 4. Game-theoretic analyses of legal rulemaking 5. Cooperative game theory as a tool for analysis of legal concepts 5.1. Measures of voting power 5 .2. Fair allocation of losses and benefits 6. Concluding remarks References Articles Cases
2231 2233 2233 2235 2237 2241 2243 2245 2247 2249 2251 2252 225 9 2262 2263 2263 2269
*The support of the C.V. Starr Center for Applied Economics is acknowledged. Alfred and Gail Engelberg Professor of Law, New York University. The support of the Filomen D' Agostino and Max E. Greenberg Research Fund of the NYU School of Law is acknowledged. We thank John Ferejohn, Mark Geistfeld, Marcel Kahan, Martin Osborne and William Thomson for com ments. t
Handbook of Game Theory, Volume 3, Edited by R.J. Aumann and S. Hart © 2002 Elsevier Science B. V. All rights reserved
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J.-P. Benoft and L.A. Kornhauser
Abstract
We offer a selective survey of the uses of cooperative and non-cooperative game theory in the analysis of legal rules and institutions. In so doing, we illustrate some of the ways in which law influences behavior, analyze the mechanism design aspect of legal rules and institutions, and examine some of the difficulties in the use of game-theoretic concepts to clarify legal doctrine. Keywords
economic analysis of law, Coase Theorem, accidents, positive political theory, voting power
JEL classification: C7, D7, KO
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1. Introduction
Game-theoretic ideas and analyses have appeared explicitly in legal commentaries and judicial opinions for over thirty years. They first emerged in the immediate aftermath of early reapportionment cases that redrew districts in federal and state elections in the United States. As the U.S. courts struggled to articulate the meaning of equal represen tation and to give content to the slogan "one person, one vote", they considered, and for a time endorsed, a concept of voting power that has game-theoretic roots. At approxi mately the same time, the publication of Calabresi (1961) and Coase (1960) provoked an outpouring of economic analyses of legal rules and institutions that continues to grow. In the late 1960s, the first game-theoretic models of contract doctrine [Birming ham (1969a, 1969b)] appeared. Models of accident law in Brown (1973) and Diamond (1974a, 1974b) followed a few years later. In the mid-1980s, the explicit use of game theory in the economic analysis of law burgeoned. The early uses of game theory encompass the two different ways in which game the ory has been applied to legal problems. On the one hand, as in the reapportionment cases, courts and analysts have adopted game-theoretic tools to further the goals of tra ditional Anglo-American legal scholarship. In this tradition, both judge and commenta tor seek to rationalize a set of related judicial decisions by articulating the underlying normative framework that unifies and justifies the judicial doctrine. In this use, which we shall call doctrinal analysis, game-theoretic concepts elaborate the meaning of key, doctrinal ideas that define what the law seeks to achieve. On the other hand, as in the analyses of contract and tort, game theory has provided a theory of how individuals respond to legal rules. The analyst considers the games defined by a class of legal rules and studies how the equilibrium behavior of individuals varies with the choice of legal rule. In legal cultures that, as in the United States, regard law as an instrument for the guidance and regulation of social behavior, this analysis is a necessary step in the task facing the legal policymaker. It is worth emphasizing that nothing in this analytic structure ties the analyst, or the policymaker, to an "economic" objective such as efficiency or maximization of social welfare. In this essay, we offer a selective survey of the uses of game theory in the analy sis of law. The use of game theory as an adjunct of doctrinal analysis has been rel atively limited, and our survey of this area is correspondingly reasonably exhaustive. In contrast, the use of game theory in understanding how legal rules affect individual behavior has been pervasive. Game theory has been explicitly applied to analyses of contract, 1 torts between strangers, 2 bankruptcy and corporate reorganization, 3 corpo-
1 E.g., Birmingham (1969a, 1969b), Shavell (1980), Kornhauser (1983), Rogerson (1984), and Hermalin and Katz (1993). 2 E.g., Brown (1973), Diamond (1974a, 1974b), Diamond and Mirrlees (1975), and Shavell (1987). 3 E.g., Baird and Picker (1991), Bebchuk and Chang (1992), Gertner and Scharfstein (1991), Schwartz (1988, 1993), and White (1980, 1994).
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rate takeovers and other corporate problems,4 product liability, 5 and the decision to settle rather than litigate6 and the consequent effects on the likelihood that plaintiff will prevail at trial. 7 Kornhauser (1989) argues that virtually all of the economic analysis of law can be understood as an exercise in mechanism design in which a policymaker chooses among legal rules that define games played by citizens. Moreover, certain traditional subdis ciplines of economics and finance, such as industrial organization, public finance, and environmental economics, are inextricably tied to legal rules. Game-theoretic models in those fields invariably shed light on legal rules and institutions. Given the size and diversity of this latter literature, we cannot provide a compre hensive survey of game-theoretic models of legal rules and hence make no attempt to do so. Furthermore, in some areas at least, such literature surveys already exist: see, for example, Shaven (1987) on torts, Cooter and Rubinfeld (1989) on dispute resolu tion, Kornhauser (1986) on contract remedies, Pyle (1991) on criminal law, Hart and Holmstrom (1987) and Holmstrom and Tirole (1989) on the incomplete contracting lit erature, Symposium (1991) on corporate law generally and Harris and Raviv (1992) on the capital structure of firms. In addition, Baird, Gertner and Picker (1994) is an introduction to game theory that presents the concepts through a wide variety of legal applications. Having forsworn comprehensive coverage, we have three aims in this survey. Our first is to suggest the wide variety of legal rules and institutions on which game the ory as a behavioral theory sheds light. In particular, we illustrate the ways in which legal rules influence individual behavior. Our second is to emphasize the mechanism design aspect that underlies many game-theoretic analyses of the law. Our third is to examine some of the difficulties in the use of game-theoretic concepts to clarify legal doctrine. In Section 2, we examine the simple two-person game that underlies much of the economic analysis of accidents as well as the analysis of contract remedies and tort. In this model, the legal rule serves as a direct incentive mechanism. In Section 3, we examine the insights that game-theoretic models have provided into the "Coase Theorem" which often serves as an analytic starting point for economic analyses of law. In Section 4 we introduce a literature that extends the use of game theory as a behavioral predictor from private law to public law and the analysis of con stitutions. Section 5 discusses the small literature that has used game theory in doctrinal analysis. Section 6 offers some concluding remarks. 4 E.g., Bebchuk (1994), Bulow and Klemperer (1996), Cramton and Schwartz (1991), Grossman and Hart (1982), Kahan and Tuckman (1993) and Sercu and van Hulle (1995). 5 E.g., Oi (1973), Karnbhu (1982), Polinsky and Rogerson (1983), and Geistfeld (1995). 6 E.g., Bebchuk (1984), Daughety and Reinganum (1993), Hay (1995), Png (1982), Reinganum (1988), Reinganum and Wilde (1986), Shavell (1982), Spier (1992, 1994). 7 E.g., Priest and Klein (1984), Eisenberg (1990), Wittman (1988).
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2. Economic analysis of the common law when contracting costs are infinite
2.1. The basic model Brown (1973) offers a simple model of accident law. We discuss it extensively here for two reasons. First, it underlies many, if not most, economic analyses of law that examine how legal rules can solve externality problems. Second, and more importantly, this simple model of torts illustrates clearly the use of game theory as a theory of the behavior of agents in response to legal rules. There are three decisionmakers: the legal policymaker, generally thought of as the Court, and two agents, called the (potential) injurer 1 and the (potential) victim V. Agents 1 and V are each independently engaged in an activity that may result in harm to V. They simultaneously choose an amount x and y, respectively, to expend on care. These amounts determine the probability p (x, y) with which V may experience a loss and the value L (x, y) of such a loss. The functions p(x, y) and L(x , y) are common knowledge. The policymaker seeks to regulate the (risk-neutral) agents' choices of care levels. The policymaker selects a legal rule from a family of permissible rules of the form R(x, y; X, Y), where R(x, y; X, Y) is the proportion of the loss L(x, y) legally imposed on 1 , and the parameters X ) 0 and Y ) 0 will be interpreted as I 's and V 's standard of care, respectively. Although this framework is simple, it enables a comparison of many legal rules. Thus, a general regime of negligence with contributory negligence is characterized by a loss assignment function of the form:
R(x, y; X, Y) =
{�
if x < X and y ) Y, otherwise.
(la)
Under a pure negligence rule the injurer is responsible whenever she fails to take an "adequate" level of care; this corresponds to 0 < X < oo, Y = 0. With strict liability the injurer is responsible for damage regardless of the level of care she takes; this cor responds to X = oo, and Y = 0. 8 Under strict liability with contributory negligence the injurer is responsible whenever the victim takes an adequate level of care; thus, X = oo , 0 < y < 00 . 8 The legal literature generally distinguishes between "strict liability rules" and "negligence rules". As the text indicates, a strict liability rule (without contributory negligence) can be understood as a member of the general class of negligence with contributory negligence rules. A different distinction between strict liability and negligence rules is suggested by Equation (lb) below; namely, that under strict liability the default bearer of liability, i.e., the party who bears the loss when each party meets her standard of care, is the injurer rather than the victim.
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J.-P. Benoft and L.A. Kornhauser
Strict liability with dual contributory negligence has a different pattern of liability defined by: R(x , y; X, Y) =
{�
if x < X or y ;? Y, otherwise. 9
( lb)
Under a rule of comparative negligence, R may assume values strictly between 0 and 1 when each party fails to meet its standard of care. Any given rule R(x, y; X, Y) defines a two-person non-zero-sum game between I and V. Each agent seeks a level of care that minimizes her expected personal costs. Thus, I chooses to minimize: x
x + p (x , y)L(x , y)R(x , y; X, Y)
(2)
while V chooses y in order to minimize: y + p (x , y)L(x, y) [ l - R(x, y ; X, Y)] .
(3)
In principle, an economic analysis studies the equilibria of these various games. Once these equilibria have been identified, the policymaker can choose the rule that induces the best equilibrium, given the social objective function. The analysis is thus not in extricably linked to a search for an efficient or welfare maximizing rule. One can as easily search for the rule that minimizes the accident rate or equalizes care expendi tures or for any other social objective that the policymaker seeks to implement. One way to proceed would be to analyze the case law to identify the Court's objective func tion and then choose the legal rule that induces the equilibrium that is best in these terms. In practice, however, and perhaps unfortunately, the literature has to a large extent chosen as objective function the minimization of the expected social costs of acci dents. 10 That is, the minimization of: x + y + p(x , y)L(x , y).
(4)
Let (x*, y*) be the (assumed) unique minimizing care levels. Consider the common law regime of negligence with contributory negligence as characterized by ( l a). Clearly choosing standards X = x* and Y = y* induces the agents to take the optimal levels 9
In fact, if we rename the parties, the pattern of liability under this rule is identical to the pattern under negligence with contributory negligence. Put differently, in Brown's model, the loss is perfectly transferable between parties and the legal rule simply assigns the loss to one party or another. This is not an adequate model of personal injury, since a personal injury may alter V ' s utility function, and some injuries may not be compensable. See, for example, Arlen (1992) for a modification that addresses the problem of personal injury. Diamond (1974b), Kornhauser and Schotter ( 1992), and Endres and Querner (1995) study how the equi libria change as the standards of care change and are thus examples of exceptions to this general practice.
10
Ch. 60: Game-Theoretic Analysis ofLegal Rules and Institutions
2235
of care. However, these are not the only such standards. In fact, an infinite number of rules of the form {X = x*, Y E [0, y*]} and of the form {X E [M, oo] for sufficiently large M, Y = y*} induce an equilibrium in which the agents adopt care levels that are socially optimal. Consider first a rule with X = x* and Y :'( y*. When I chooses x*, V is responsible for the total loss, i.e., R = 0. From (3) V chooses y to minimize:
y + p (x*, y ) L (x*, y ) . But this is just (4) shifted down by x* (given that I chooses x*) so that (3) is also minimized at y*. Likewise, given V's choice of y*, I minimizes (2) by choosing x* . 1 1 A similar argument shows that (x*, y*) is the equilibrium for those games in which Y = y* and X is sufficiently large. Thus, efficiency can be obtained even if the court can only observe the level of care being taken by one of the parties. In particular, if the court can only observe x, then an appropriate pure negligence rule is efficient - the court sets X = x*, Y = 0 and lets V, whose action it cannot observe, optimize against /'s choice. Similarly, when the court only observes y, the strict liability with contributory negligence rule (X = oo, Y = y*) is efficient. Notice that these payoff functions, which accurately capture the structure of the neg ligence rule, are discontinuous at the standard of care. 12 This discontinuity plays an im portant role in generating the efficient outcome, because it allows both agents to "see" the full cost of their decisions. For example with the rule X = x* and Y = 0, I bears the full cost of the accident in the event he chooses x < x*, and none of the cost at x = x*. Therefore, at a choice of x = x*, / 's marginal cost of lowering x is the entire accident cost. At the same time, when I chooses x = x*, V bears the entire loss and sees the full cost of her actions.
2.2. Extensions within accident law Several modifications of this basic framework have been studied. Shaven (1987) ana lyzes a model in which each party chooses both a level of care and a level of activity, but the allocation of the loss still depends only on the parties' choice of care levels. Unsurprisingly, no legal rule can induce both parties to choose efficient activity levels and efficient care levels; after all, the court has only one instrument (for each party) to 1 1 (x * , y*) is the only pure strategy equilibrium for games with the standards of care as indicated. Sup pose I chooses x ' < x*. If V chooses y ' < Y, then V's cost is y' + p(x' , y ')L(x ' , y') > x* - x' + y* + p(x*, y*)L(x*, y*) > y* > Y. Therefore V would choose Y and / 's best response to Y is X = x* # x ' . If I chooses x > x*, she makes unnecessary expenditures on care. 1 2 Kahan ( 1989), however, argues that these payoff functions do not correctly model the measure of damages. He contends that, in an appropriate model, the payoffs are continuous functions. These functions still show the agent the full marginal cost of her decisions.
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J. -P. Benoft and L.A. Kornhauser
control the two decisions that each party makes. One can still identify the rule that is best given the social objective function. Shavell (1987) also investigates the impact of an insurance market on care decisions. (See also Winter (1991) for a discussion of the dynamics of the market for liability insurance.) A number of other variants of this model have also been analyzed. Green (1976) and Emons and Sobel (199 1), for example, study the equilibria of the basic model of negligence with contributory negligence when injurers differ in the cost of care. Beard (1990) and Kornhauser and Revesz (1990) consider the effects of the potential insol vency of the parties on care choice. Shaven (1983) and Wittman (1981) analyze games with a sequential structure. Leong (1989) and Arlen (1990) study negligence with con tributory negligence rules when both the victim and the injurer suffer damage. Arlen (1992) studies these same rules when the agents suffer personal (or non-pecuniary) in jury. Several authors have integrated the analysis of the incentives to take care into a fuller description of the litigation process that enforces liability rules. Ordover (1978) shows that when litigation costs are positive some injurers will be negligent in equilibrium. Craswell and Calfee (1986) studies how uncertainty concerning legal standards affects the equilibria. Polinsky and Shaven (1989) studies the effects of errors in fact-finding on the probability that a victim will bring suit and hence on the incentives to comply with the law. Hylton (1990, 1995) introduces costly litigation and legal error. Png (1987) and Polinsky and Rubinfeld (1988) study the effect of the rate of settlement on compliance. Png (1987) and Hylton (1993) consider how the rule allocating the costs of litigation between the parties affects the incentives to take care. Cooter, Kornhauser, and Lane (1979) assumes that the court only observes "local" information about the private costs of care and the costs of accidents. In their model the courts adjust the standards of care over time and converge to the optimum. Polinsky (1987b) compares strict liability to negligence when the injurer is uncertain about the size of the loss the victim will suffer. Many authors have compared the efficiency of negligence with contributory negli gence rules to comparative negligence rules. Under a rule of comparative negligence injurer and victim share the loss when both fail to meet the standard of care. That is, the injurer pays the share
R(x ,
y; X, Y)
f(x ,
�
{ �(x ,
xx (x
for < X and y ): Y, y; X, Y) for < X and y < Y, otherwise ): 0),
where 0 ::( y ; X, Y) ::( 1 . Landes and Posner (1980) argues that both rules can induce efficient caretaking; Haddock and Curran (1985) and Cooter and Ulen (1986) argue for the superiority of comparative negligence when actual care levels are observed with error; Edlin (1994) argues that with the appropriate choice of standards of care the two rules perform equally well even when such error is present. Rubinfeld (1987)
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Ch. 60: Game-Theoretic Analysis ofLegal Rules and Institutions
considers the performance of comparative negligence when the court cannot observe the private costs of care. 2.3.
Other interpretations of the model
This model can be recast to yield insights into other areas of law. Indeed, some [Cooter (1985)] have suggested that the pervasiveness of this model shows the inherent unity of the common law. While this claim seems too strong - common law rules govern a wider set of interactions than those described here - the model does illuminate a wide variety of legal institutions. To begin, we reinterpret the problem in terms of remedies for breach of contract. Con sider, for example, two parties, a buyer B and a seller S, contemplating an exchange. 1 3 B requires a part produced by S. The value to B of the part depends upon the level of investment y that B makes in advance of delivery. Denote this value by v (y) . y is known as the "reliance" level, as it indicates the extent to which B is relying upon receiving the part. S agrees to deliver the part and chooses an investment level x . This results in a probability [ 1 - p(x) ] that S will successfully deliver the part and a probability p (x) that he will not perform the contract. B pays S an amount k up front for the part. A legal rule R(x , y) determines the amount R(x, y)v(y) that S must pay B if S fails to deliver the part. S then chooses to maximize: x
k - x - p(x)v(y)R(x, y) ,
(5)
while B simultaneously chooses y to maximize:
-k - y + [ 1 - p(x) ] v(y) + p(x)R(x, y)v(y).
(6)
This model is perfectly congruent to Brown's basic model of Section 2.1 . To see this, first subtract a term g(y) from the victim's objective function (3). We interpret g(y) as V 's (added) payoff from her investment of y, should no accident occur. In Brown's basic model g(y) is zero. This corresponds to a potential victim who, by taking care, can reduce the value of her assets that are at risk, say by putting some of them in a fireproof safe, but whose actions have no effect on the overall value of the assets. On the other hand, consider a potential victim whose investment determines the value of some assets should no accident occur, but which assets will be completely lost in the event of an accident. Then g(y) = L(y) ( L(x , y)) in (3). If this invest ment affects only the value of the assets, and not the probability that they are lost, then p(x , y) = p(x). Now the accident model of Equations (2) and (3) is identical to the breach of contract model (5) and (6) (with v(y) L(y) and k = 0). =
=
1 3 The model in the text is similar to ones presented in Shavell (1980), Kornhauser (1983) and Rogerson (1984).
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Thus, the accident interpretation and the contract interpretation differ largely in the characterization of the net benefits of the victim's action. Despite this congruence, courts, and analysts, have treated these situations differently. Rather than rules setting standards of care X and Y to determine liability for breach of contract, courts have com monly used a "reliance measure", R(x, y) = R(y) yjv(y) (so that R(y)v(y) = y),1 4 and an expectation measure, R (x, y) 1 . These measures of contract damages are rules of "strict liability" as the amount of the seller's liability is not contingent on its own deci sions; they are also "actual damage" measures as they depend on the actual, as opposed to any hypothetical, choice of the Buyer. It is easy to see that neither of these two damage measures will generally induce an efficient level of care and reliance. The total expected social welfare of the contract is =
=
[1 - p(x)]v(y) - x - y. Let (x*, y*) maximize this social welfare function. 15 Thus,
- 1/v(y*), or p(x* ) 0 and p' (x*) < 1/v(y*), 1/(1 - p(x*)) .
p' (x*) v' (y*)
=
When R(y) = 1 , B's maximizing choice, Ye. is independent of S's action and is implicitly defined by
v' (Ye) 1 . =
Given this, S's equilibrium action is defined by:
Note that (xe, Ye) = (x* , y*) only when p(x*) 0. When p(x*) # 0, this expectation measure leads to too much reliance on the part of B ; S, however, adopts the optimal amount of care given B's reliance. Under the reliance measure R(y) yjv(y), B again maximizes independently of S by setting: =
=
v' (yr) = 1 . 14
R(y)v(y) = y is one interpretation of a reliance measure. Some legal commentators understand reliance as a measure that places the promisee in the same position he would have been in had the contract not been made. Reliance then includes the opportunity cost of entering the contract. On this interpretation, reliance damages would equal expectation damages in a competitive market. We assume sufficient conditions for a unique nonzero solution, and that all our functions are differentiable.
15
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2239
S's equilibrium action is now defined by:
Again Yr is efficient only if p(xr) 0. In this case, S adopts the efficient level of care too. Otherwise, B under-relies and S breaches too frequently given B's reliance. We may also use this example to suggest how non-efficiency concerns might enter into the legal analysis. Consider a technology in which, by taking a reasonable amount of care, S can get p close to 0; further reductions in p are very costly without much gain. Indeed, there may be a residual uncertainty which S cannot avoid. Then the so lutions to the three programs above may well all result in a very small probability of nonperformance. The solutions will all be close to each other, so that fairness consider ations may easily outweigh efficiency considerations. Under the reliance measure, for instance, when S fails to deliver despite his "best efforts" he must only repay B for B 's out-of-pocket (or reliance) losses, rather than the loss of B's potential gain, which may be quite high. 1 6 Brown's model of accident law is easily converted to a model of the taking of prop erty. The Fifth Amendment to the Constitution of the United States requires that "prop erty not be taken for a public use without compensation". How does the level of required compensation affect the extent to which the landowner develops her land and the nature of the public takings? We follow the model in Hermalin (1995). Let the landowner choose an amount y to expend on land development, resulting in a property value v(y). v(y) is an increas ing function of her investment. The value of the land to the government is a random value whose realization is unknown at the time the landowner chooses his investment level. Let q (a) denote the probability that the value of the land in public hands will be greater than a, and let p(y) q (v(y)). Assume that the state appropriates the land if and only if it is worth more in public hands than in private hands. As before, we =
=
16
One might convert this fairness argument into an efficiency argument by extending the analysis to consider why B does not choose to integrate vertically with S. A number of other aspects of contract law have also been modeled. Hadfield ( 1994) considers complications created by the inability of courts to observe actions. Hermalin and Katz (1993) examines why contracts do not cover all contingencies. Katz ( l990a, 1990b, 1993) has studied doctrines surrounding offer and acceptance, which determine when a contract has been formed. Various authors - e.g., Posner and Rosenfield (1977), White (1988), and Sykes (1992) - have considered the related doctrines of impossibility and impracticability which are, in a sense, the converse of breach as they state conditions under which contractual obligation is discharged. Rasmusen and Ayres (1993) examine the doctrines governing unilateral and bilateral mistake. Polinsky (1983, 1987a) addresses some more general questions of the risk-bearing features of contract that are raised by the mistake and impossibility doctrines. The rule of Hadley v. Baxendale has also received much attention as in Perloff (1981), Ayres and Gertner (1989, 1992), Bebchuk and Shavell (1991), and Johnston (1990). The latter four articles are particularly concerned with the role of the legal rule in inducing parties to reveal information.
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consider compensation rules of the form R(y)v(y). Then the landowner chooses y to maximize:
-y + (1 - p(y))v(y) + p(y)R(y)v(y).
(7)
Note that (7) is identical to (6), when k = 0 and p is written more generally as p(x, y). Suppose that the state is benevolent so that its objective function is net social
value:
- y + [ 1 - p(y)]v(y) + E[a l a > v(y)] .
(8)
The government chooses R(y) to maximize (8), given the landowner's program (7). 1 7 As a simple example, consider a rule R (y) = a and suppose that the land is either worth more to the state than max v(y) or worth less than v(O) (perhaps a new rail line must be situated). Thus p' (y) 0, E' ( v(y)) = 0 and efficiency demands that: =
v' (y) = 1/[1 - p(y)]. The landowner maximizes by setting:
v' (y) = 1/ [ 1 - p (y) ( 1 - a)] . Efficiency calls for no compensation, i.e., a = 0. However, if p(y) is small, then setting a = 1 may be fairer while causing only a small loss in total welfare. It is striking that radically different legal institutions are captured by models that are structurally so similar. Why do legal institutions vary so much across what seem to be structurally similar strategic situations? The literature does not offer a clear answer; indeed, it has not explicitly addressed the issue. Three suggestions follow. First, the nature of the externality differs slightly across legal institutions. In the ac cident case, the likelihood of an adverse outcome depends on the care choices of each party. In the contract case, the seller determines the likelihood of breach while the buyer determines the size of the loss. In the takings context, the landowner determines the mar ket value of the land. Perhaps the shifting judicial approaches reflect important differ ences in the court's ability to monitor the differing transactions. Second, the differences in institutions may reflect different values that the courts seek to further in the differing contexts. Finally, the judges who developed these common law solutions may not have 1 7 Analyses of takings as in Blume et al. (1984) and Fischel and Perry (1989) are more complex because they are set in a broader framework in which the government must also set taxes in order to finance the takings. In this context one is able to consider a non-benevolent government as well. This government acts in the interest of the majority; see Fischel and Perry (1989) and Hermalin (1995). Hermalin, whose analysis is explicitly normative, examines a set of non-standard compensation rules as well. We have ignored these complexities because we are interested in showing the doctrinal reach of the simple model.
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perceived the common strategic features of these situations and thus may have generated a multiplicity of responses. 18 3. Is law important? Understanding the Coase Theorem
The analysis of accident law in the prior section assumed that the parties could not negotiate or enforce agreements concerning the level of care that each should adopt. The interpretation of the model as one of accidents between strangers presents a situation in which such a bar to negotiation is plausible; the pedestrian contemplating a stroll rarely has the opportunity to negotiate with all drivers who might strike her during her ramble. Under other interpretations, however, the barriers to negotiation are less. In these contexts, one must confront a seductive, though ill-specified, argument for the irrelevance of certain legal rules. This irrelevance argument is generally attributed to Ronald Coase (1960) who ar gued through two examples that the assignment of liability would not, in the absence of transaction costs, affect the economic efficiency of the behavior of agents because an "incorrect" assignment of liability could always be corrected through private, en forceable agreements. This insight has come to be known as the Coase Theorem. In the context of contract remedies, for example, the parties could negotiate a contingent claims contract specifying the level of reliance that the buyer should undertake. Coase also emphasized that the assignment of liability might indeed matter when transaction costs were present. Coase never actually stated a "theorem" and subsequent scholars have had difficulty formulating a precise statement. Precision is difficult because Coase argued from example and did not clearly define several key concepts in the argument. In particular, "transaction costs" are never defined and the legal background in which the parties act is underspecified. Game theorists have generally treated the theorem as a claim about bargaining and, as we shall see, have used both the tools of cooperative and non-cooperative game theory to illuminate our understanding of Coase's examples. The Coase Theorem, however, is also a claim about the importance of legal rules to bargaining outcomes. At the outset, we must thus understand precisely which legal rules are in question so that they may be modeled appropriately. The Coase Theorem implicates two sets of legal rules: the rules of contract and the assignment of entitlements. The rules of contract determine which agreements are legally enforceable and what the consequences of non-performance of a contract are. For the Coase Theorem to hold, the courts should be willing to enforce every con tract and the consequences of non-performance should be sufficiently grave to deter non-performance. Absent this possibility, repetition and reputation effects might ensure compliance with contracts in certain situations. 18
We thank a referee for suggesting this third possibility.
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The entitlement defines the initial package of legal relations over which the parties negotiate. 1 9 This package identifies the set of actions which the entitlement holder is free to undertake and the consequences to others for interfering with the entitlement holder's enjoyment of her entitlement. The law and economics literature, drawing on Calabresi and Melamed (1 972), has generally focused on the two forms of entitlement protection called property rules and liability rules. Under a property rule, all transfers must be consensual; consequently the transfer price is set by the entitlement holder herself. Under a liability rule, an agent may force the transfer of the entitlement; if the parties fail to agree on a price, a court sets the price at which the entitlement transfers. Consider, for example, two adjoining landowners A and B. B wishes to erect a build ing which will obstruct A's coastal view. Suppose A has an entitlement to her view. Property rule protection of A 's entitlement means that A may enjoin B 's interfering use of his land and thus subject B to severe fines or imprisonment for contempt should B erect the building. Put differently, A may set any price, including infinity, for the sale of his view, without regard to the "reasonableness" of this price. Under a liability rule, A's recourse against B is limited to an action for damages. Put differently, should B erect a building against A's wishes, the courts will determine a reasonable "sale" price for the loss of A ' s view. The Coase Theorem is commonly understood as a claim that, in a world of "zero transaction costs", the nature and assignment of the entitlement does not affect the fi nal allocation of real resources; in this sense, then, legal rules are irrelevant. One might ask two very different questions about the asserted irrelevance of legal rules. First, one might assume a perfect law of contract and specify more precisely what "zero transac tion costs" means in order to assure the irrelevance of the assignment of entitlements. Alternatively, one might ask how "imperfect" contract law could be without disturbing the irrelevance of the assignment of the entitlements. Following most of the literature, we investigate the first question. Much controversy revolves around the concept of transaction costs. At the most ba sic level, transaction costs involve such things as the time and money needed to bring the various parties together and the legal fees incurred in signing a contract. One may
I9
An entitlement is actually a complex bundle of various types of legal relations. A classification of legal relations offered by Hohfeld (1913) is useful. He identified eight fundamental legal relations which can be presented as four pairs of jural opposites: right duty
privilege no-right
power disability
immunity liability
When an individual has a right to do X, she may invoke state power to prevent interference with her doing X. By contrast, i f she has the privilege t o do X, no one else may invoke the state to interfere (though they may be able to "interfere" in other ways). An individual may hold a privilege without holding a right and conversely. When an individual has a power, she has the authority to alter the rights or privileges of another; if the individual holds an immunity (with respect to a particular right or privilege), no one has a power to alter it. Kennedy and Michelman (1980) apply this classification to illuminate different regimes of property and contract.
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Game- Theoretic Analysis of Legal Rules and Institutions
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go further, however, and, somewhat tautologically, consider a transaction cost to con stitute anything which prevents the consummation of efficient trades. Game theory has clarified the different impediments to trade by focusing on two types of barriers: those that arise from strategic interaction per se and those that arise from incomplete infor mation. We might then understand the Coase Theorem as one of two claims about bargain ing. 2° First, we might understand the Coase Theorem as a claim that bargains are ef ficient regardless of the assignment and nature of the entitlement. Call this the Coase Strong Efficient Bargaining Claim. Second, we might understand the Coase Theorem as a claim that, although bargaining is not always efficient, the degree of efficiency is unaffected by the nature and assignment of the entitlement. Call this the Coase Weak Efficient Bargaining Claim. The game-theoretic investigations do provide some support for Coase's valuable in sight. 2 1 However, the bulk of the analysis suggests that both Coase Claims are, in general, false. One set of arguments focuses on the strategic structure of games of complete information. A second set considers the problems created by bargaining un der incomplete information. A third set circumvents the bargaining problem using co operative game theory. The next three subsections discuss each set of arguments in tum. First, it is worthwhile emphasizing one point. Discussions of the Coase Theorem often characterize it as a claim that, under appropriate conditions, the legal regime is economically irrelevant. This loose characterization obscures an important fact. At most, the Coase Theorem asserts that efficiency may be unaffected by the assignment of the entitlements. There are, of course, other economic considerations. In particular, Coase never asserted that the final distribution of assets would be unaffected by the le gal regime. On the contrary, every "proof" of the Coase Theorem relies on the parties negotiating transfers, the direction of which depends on the assignment of the entitle ment. Thus, even in circumstances in which the Coase Theorem holds, the choice of a legal rule will certainly matter to the parties affected by its announcement and it may also matter to a policymaker.
3.1. The Coase Theorem and complete information Reconsider the injurer/victim model of Section 2. 1 . Suppose that I engages in an ac tivity that may cause a fixed amount of damage, L(x, y) = d, to agent V. Agent I can avoid causing this damage at a cost of c. That is, p(x, y) = 0 if x � c, p(x, y) = 1 if x < c. Both c and d lie in the interval (0, 1 ) . We assume that the benefit to I of the 20 The literature on the Coase Theorem is vast and offers a variety of characterizations of the claim. For overviews of the literature that offer accounts consistent with that in the text see Cooter ( 1987) and Veljanovski (1982). 21 Hoffman and Spitzer (1982. 1985. 1 986) and Harrison and McAfee (1 986) provide experimental evidence in support of the Coase Claims.
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activity is greater than 1, so that it is always socially desirable for I to undertake the potentially harmful activity. We can view V as being entitled to be free from damage. As previously noted, an entitlement can be protected with a property rule or a liability rule. Consider a compen sation rule that specifies a lump sum M that I must pay V for damage. We interpret M � 1 as a property rule, since I will only fail to take care if V consents. 22 V 's en titlement is protected by a liability rule when 0 < M < 1 . Clearly, the larger M the greater the protection afforded V . When M = 0, V has no protection; this can be in terpreted as giving I an entitlement to cause damage that is protected by a property rule. A surface analysis suggests that if M = 0, I will not take care since the damage I causes is external to I, whereas if M = 1 , I will take care since c < 1 . In particular, when M = 0 and c < d, I will cause harm to V, and when M = 1 and c > d I will take care, even though both these outcomes are inefficient. The Coase Theorem holds that this conclusion is unduly negative. For instance, when M = 0 and c < d, V could induce I to take care by offering I a side payment of, say, c + (d - c)/2. This is possible provided that "transaction costs" are not too high. If making the side payment involved a separate cost greater than (d - c), no mutually beneficial payment could be made. When transaction costs are negligible then, for any values of c and d, and for any value of M, we might expect the firms to negotiate a mutually beneficial transfer which yields the efficient level of care. Note that even if this expectation is correct, it does not imply that the legal regime M is irrelevant, as M affects the direction and level of the transfer. While the insight that firms will have an incentive to "bargain around" any legal rule is important, the blunt assertion that they will successfully do so is hasty. Exactly how are the firms to settle on an appropriate transfer? What if, with c < d, V offers to pay c to I, while I insists upon receiving d ? Negotiations could break down, resulting in the inefficient outcome of no care being taken. In any case, Coase did not specify the bargaining game which would permit us to rule out this possibility. Suppose that firms always seek to reach mutually beneficial arrangements and that they can costlessly implement any contract upon which they agree. Consider the polar cases of M = 0, and M = 1 . A simple dominance argument shows that in the former case I will never take care when c > d, since it is dominated for V to pay I more than d to take care. Thus, there will never be more than an efficient level of care taken. Similarly, when M (x, d) = 1 there will never be less than an efficient level of care. Without additional assumptions no more than this can be guaranteed. In particular, the legal regime may affect the level of care taken and the Coase Theorem fails in both its
22 This characterization of a property rule is not wholly satisfactory because the court might enforce a prop erty rule by injunction (and by contempt proceedings should the injunction be violated). Thus this character ization might differ from the operation of such a legal rule should I act irrationally.
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forms. To establish a Coase Theorem one requires additional structure. Notice that the material facts do not in themselves determine this additional structure. To evaluate the Coase Theorem we must posit a bargaining game and examine how its solution changes with legal regimes. Consider any legal rule M and a simple game in which V makes a take-it-or-leave-it offer to I . Then, in the unique subgame perfect equilibrium, when c < d V offers I a side-payment of max[c - M, 0] to take care and I takes care. Hence I takes care regardless of the legal rule. When c > d, V offers to accept a side-payment of min[ c, M] from I if I does not take care, and under every legal rule I does not take care. In both cases, the efficient outcome is reached regardless of the legal rule M ( c, d). This is a particularly simple model of bargaining. Rubinstein (1982) offers a more sophisticated model in which two parties altematingly propose divisions of a surplus until one of the parties accepts the other's offer. In our present situation, if c < d there is a surplus when care is taken and if c > d there is a surplus when care is not taken, regardless of the rule M (c d). Suppose that when c < d, in period 1 V offers I a payment to take care. If I rejects the payment, in period 2 I makes a counterdemand of a payment If V rejects I 's proposal he makes a counteroffer in period 3 and so forth. When c > d the game is the same except that the payments are for not taking care. Each party attaches a subjective time cost to delay and the game ends when an offer is accepted or when I decides to act unilaterally. Rubinstein's results imply that for all M (c , d) the parties instantly agree upon an efficient transfer. Thus, we have some support for the Coase Strong Efficient Bargaining Claim. How ever, the result that bargaining will be efficient is delicate. Shaked has shown that with three or more agents bargaining in Rubinstein's model may be inefficient. 23 Av ery and Zemsky (1994) present a unifying discussion of a variety of bargaining mod els that result in inefficient bargaining. 24 Perhaps most importantly, we have been as suming complete information. We consider incomplete information in the next sec tion. ,
3.2. The Coase Theorem and incomplete information Generally, we might expect a firm to possess more information about its costs than out side agents possess. We now modify the model of the previous section in this direction. Specifically, following Kaplow and Shavell (1996), suppose that while I knows the cost c of taking care, V knows only that c is distributed with continuous positive den sity f(c) on the unit intervaL Similarly, while V knows the level of damage d which I would cause, I knows only that d is distributed with density g(d) on the unit intervaL
23 Shaked's model is discussed in Osborne and Rubinstein (1990). 24 These models, which include Fernandez and Glazer (1991) and Haller and Holden (1990), all have com
plete information.
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We focus on the two rules M = 0 and M = 1 . Call these rules Mo and MJ , respec tively. Notice that to implement Mo and MJ the court only needs to know whether or not damage has occurred. Efficiency requires that I take care whenever c < d. Let S; = { ( c, d) I I takes care under rule Mi }. The strong efficient bargaining claim asserts that So = SJ = {(c , d) l c � d}.25 Call this set E. The weak efficient bargaining claim could be taken to mean one of two things; that So = SJ =f. E or that, although So =f. SJ , Mo and MJ are equally inefficient (given a measure of the degree of inefficiency of rules). Again consider the simple bargaining procedure in which V makes a take-it-or-leave it offer to I . Consider first the situation when Mo prevails. Then V may seek to bribe I to take care. We calculate how V 's offer to buy I's right to cause harm varies with V 's damage cost d. Suppose V of type d makes an offer of o(d) to I. I accepts if c � o(d). Thus, when V makes an offer of o(d), he incurs expected costs of F(o(d))o(d) + (1 F(o(d))d. These costs are minimized by that o(d) which satisfies
o(d) = d - F(o(d))/f (o(d)) . Call this value oo(d). Thus:
So = { (c, d) I c � oo(d) } . Since oo(d) < d efficiency fails in the presence of incomplete information, the strong efficiency claim is false. 26 To check the Weak Efficiency Claim we examine the effect of a change in the legal regime. Consider the situation when MJ prevails. V may now offer to sell I the right to cause the harm. Suppose that V offers to forego his damage payment of 1 at a price o(d) . I accepts if c > o(d) as it would be cheaper for I to buy the right than to avoid the harm. Thus, when V makes an offer of o(d), he expects profits of (1 - F(o(d))(o(d) - d) . These profits are maximized by that o(d) which satisfies
o(d) = d + [ 1 - F(o(d))]/f(o(d)) . Call this value OJ (d) . Thus:
SJ = { (c, d) I c � OJ (d) } . Since o1 (d) > d, again efficiency fails. There is an important difference in regimes Mo and MJ , however. Under Mo , there are instances when efficiency demands
25 Of course, if c = d efficiency permits that care either be taken or not taken. Throughout, we will be a little loose as to what happens at degenerate "equality" points. 26 Indeed, Myerson and Satterthwaite (1983) show that, quite generally, bargaining under incomplete infor mation will be inefficient.
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that I take care but he does not. In no case does I take care when he should not. In contrast, under M 1 , there may be too much care taken, but never too little. 27 From this comparison, we may conclude that the Weak Efficiency Claim is false in both its forms. For instance, if F places more weight in the interval [oo (d) d] than in the interval [d, OJ (d)] then Mo results in an inefficient level of care more often than M1 does. 28 ,
3.3. A
cooperative game-theoretic approach to the Coase Theorem
In the example of the previous section the firms were in a situation of asymmetric infor mation. One might interpret the difficulties imposed by this asymmetric information as an informational transaction cost, thereby rescuing the Coase Theorem. Similarly, one might interpret the inefficiencies that arise with three or more agents in a Rubinstein type bargaining model as resulting from strategic transaction costs. On this view, co operative game theory would seem to provide the best hope for the Coase Strong Ef ficiency Claim. After all, the characteristic function form of a game simply posits that firms can reach any agreement they choose, effectively assuming away any "bargaining costs". Consider the classic externality problem in which air pollution generated by two fac tories interferes with the operation of a nearby laundry. There are two possible assign ments of the entitlement: ( 1 ) the factories pay for damage caused to the laundry and (2) the factories do not pay for such damage. Each of these assignments yields a game in characteristic function form that derives from the underlying technological interaction between factories and laundry. In this context, one might formulate the Coase Theorem as a claim either that the (relevant) solution to each of the games is the same or that each solution contains only equally efficient allocations. Although there is no general theory that studies (or even formulates) this problem, it is easy to provide examples that con tradict each claim. We present here a modification of the examples underlying Aivazian and Callen ( 1 9 8 1 ) and Aivazian, Callen, and Lipnowski ( 1987). There are three firms. Firms 1 and 2 are industrial factories, and firm 3 is a laundry. Each firm can operate at one level or shut down. Firm 3 operating on its own earns a profit of 9. If either firm 1 or 2 operates, however, a pollution cost of 2 is imposed on firm 3. If both firms 1 and 2 operate, independently or jointly, a cost of 8 is imposed on firm 3. Firms 1 and 2 operating independently each earn a profit of 1 , regardless of the actions of other firms. Firms 1 and 2 operating together earn joint profits of 7.
27 This i s in keeping with a claim from the previous section. 28 Several authors have considered the differential effects of legal rules on the revelation of private informa tion. Ayres and Gertner (1989) argues that this consideration should inform judicial choice of contract default rules. The response to their claim [Johnston (1990)], their reply [Ayres and Gertner (1992)], and a related analysis [Bebchuk and Shaven (199 1)] provide insight into the force of Coase Theorem arguments in contract law. See also Ayres and Talley (1995) and Kaplow and Shaven (1996). Johnston's (1995) study of the effects of blurriness in the entitlement in situations of incomplete information provides related insights.
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We must define two characteristic function form games: one in which the laundry legally bears the pollution cost, and one in which the factories legally bear the cost. In general, defining a characteristic function game from underlying data is not altogether obvious. For instance, in defining the value of a firm operating on its own, should it be assumed that the other firms operate independently or jointly? Should it be assumed that the other firms maximize their profits or that they minimize the profits of the firm under consideration? Our example has been chosen to circumvent these difficulties. Here, the same game is defined regardless of how these questions are answered. We first derive the characteristic function of the game in which the laundry bears the pollution costs. We have v (3) = 1 ; this value of 1 is obtained by taking the 9 that firm 1 earns by operating and subtracting the cost of 8 which is imposed upon it by firms 1 and 2 operating separately or jointly. v (23) = 7 since the best that these firms can do jointly is have firm 2 shut down and firm 3 operate. Their profits are then 9 minus the cost of 2 imposed by firm 1 . v ( 123) = 9 ; firm 3 alone operates, since having one of the other firms also operate would yield the grand coalition a total of 8 and having all three firms operate would yield it 3 . Similar reasoning yields the following characteristic function: v(1)
=
v (2)
=
v (3) = 1 ;
v ( 12) = v ( 1 3) = v (23) = 7;
v ( 123) = 9.
Consider now a liability regime in which the court awards damages to the laundry based upon the maximal possible return to any coalition containing the laundry. Under such a regime we have: w ( 1 ) = w (2)
=
w ( 12) = 0;
w (3) = w ( 13)
=
w(23)
=
w ( 123)
=
9.
The Coase Theorem relates the solutions of the two games v and w . Possibly the most important solution concept for cooperative games is the core. The core of the liability game w consists of the unique point (0, 0, 9) . On the other hand, the no-liability game v has an empty core. Informal Coase-type reasoning would hold that in the absence of a liability rule, firm 3 will simply bribe firms 1 and 2 not to produce, say by offering each of them 3.5. However, such an arrangement is unstable. Firm 1 , for instance, could suggest a sepa rate agreement just between itself and firm 3. Since the core is empty, there is in fact no arrangement among the three firms which cannot be blocked by a subset of them. Al though there are no (obvious) transaction costs, it is quite possible that the firms will fail to settle on an efficient outcome. On the other hand, in the liability regime of game w, the efficient outcome in which only firm 3 produces is the obvious stable outcome. Given that the core of v is empty we might want to consider other solution concepts. The Shapley value and nucleolus are both the point (0, 0, 9) for w and the point (3, 3, 3) for v . These solution concepts always exist and are always efficient for superadditive
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games so that this Coasean result is not terribly revealing about the validity of the Strong Efficiency Claim.29 Now consider the bargaining set. 3° For the game w the bargaining set still consists of the unique point (0, 0, 9) . For v, however, while the efficient outcome (3, 3, 3) with coalitional structure { (123)} is an element of the bargaining set, the inefficient point ( 1 , 3.5, 3 .5) with coalitional structure { 1 , (23)} is also an element of the bargaining set. 3 1 Note that this inefficient outcome in which firms 2 and 3 form a coalition is better for firms 2 and 3 than the efficient element of the bargaining set (and better than the Shapley value and nucleolus). The complete information, no transaction cost world of cooperative game theory might seem to be the perfect setting for the Coase Theorem. However, even here the theorem is problematic; the choice of entitlement might very well matter to a legal pol icymaker concerned solely with efficiency. Clearly, a policymaker with other concerns or faced with a more recalcitrant environment or a policymaker designing the contract regime rather than assigning the entitlement will also need to evaluate the solutions of games defined by different legal rules.
4. Game-theoretic analyses of legal rulemaking
Legal controversies arise over broad questions concerning the structure of adjudica tion and the consequences of specific legal doctrines. Game theory has played an in creasingly important role in illuminating these structural questions. Schwartz ( 1 992), Cameron ( 1 993), Kornhauser (1 992, 1 996) and Rasmusen ( 1 994), for instance, have analyzed the practice of precedent both within a court and with respect to lower court obedience to higher court decisions. Others have addressed the claim that judicial re view of legislative action is countermajoritarian (see, e.g., Ferejohn and Shipan (1990) and the effects of specific decisions of the United States Supreme Court [e.g., Cohen and Spitzer ( 1995)]. Eskridge and Ferejohn ( 1992) employs a now standard model to illuminate difficult questions of the interrelation of Congress, administrative agencies and the courts. In principle, under the U.S. Constitution policy is promulgated through Congress, with the policy views of the President accommodated to some extent because of the ex istence of the presidential veto power. The recent rise of lawmaking promulgated by administrative agencies under the control of the President has shifted additional power
29 The difference in the solutions for the two games, however, may be of great significance to a legal policy maker whose social objective extends beyond efficiency to fairness or other concerns. 30 See Aumann and Maschler (1964) for a definition of the bargaining set. We are using the definition of M** . 31 Other elements of the bargaining set are ( ! , 1 , 1) for coalitional structure { ( ! ) , (2), ( 3 )), (3.5, 3.5, 1) for coalitional structure {(1, 2) , (3)}, and (3.5, I , 3.5) for coalitional structure {(1, 3)}. See Aivasian, Callen, and Lipnowski ( 1 9 87) for a derivation (in an equivalent game).
J.-P. Benoit and L.A. Kornhauser
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p
h
s
H
s
SQ
Diagram 1 .
to the President. I n response, Congress has attempted to redress the balance b y as suming the power to veto administrative action. In Immigration and Nationalization Service v. Chadha, the Supreme Court ruled such legislative vetoes unconstitutional. Eskridge and Ferejohn analyze the consequences of these decisions in the following simple model. There are four actors: The House of Representatives, the Senate, the President, and an executive Agency. They play a sequential game which results in some policy which can be represented as a point along a one-dimensional continuum. Diagram 1 presents a possible distribution of preferences. P is the ideal point of the President; H is the ideal point of the median voter in the House; h is the ideal point of the "2/3rds" house voter; 32 S is the ideal point of the median voter in the Senate; s is the ideal point of the 2j3rds Senate voter; SQ is the status quo. Preferences are spatial so that a person with ideal policy Xi prefers x to y if lx - Xi ! < I Y - Xi ! . Consider the Legislation Game which involves the House, the Senate, and the Pres ident. This game determines whether the United States will shift away from the pre vailing status quo policy, SQ. In stage 1 of the Legislation Game, the House chooses some point x in the policy space. If x = SQ the game ends and the policy SQ is main tained. If x =f. SQ the game continues. In stage 2, the Senate approves or rejects the proposed point x . If the Senate rejects x , the game ends with SQ. If the Senate ap proves x , the game proceeds to stage 3 . In stage 3, the President either signs the bill x or vetoes it. If the President signs the bill the game ends and the bill x becomes law. If the President vetoes the bill, the game proceeds to stage 4, in which the Senate and the House move simultaneously. If two thirds of each house approves x , the bill is passed and becomes law. If x lacks a two-thirds majority in either house, then SQ pre vails. For any specified distribution of ideal points, this game is easily solved. Consider the pattern displayed in Diagram 1. Since the status quo is to the right of all ideal policies, clearly some new policy will be enacted. ts in Diagram 1 is a point such that Its - Sl = I S - SQI . The new policy will be no further to the left of S than ts, since the Senate would reject any such proposal in favor of SQ. In fact, in the unique subgame perfect equilibrium for the configuration of Diagram 1, the House proposes ts and this is approved by both the Senate and the President. Next consider the Delegation Game in which Congress does not formulate policy, but instead delegates this task to an executive agency (such as the Environmental Protection 32 The 2j3rds voter is the voter such that she and all voters to her right just constitute a 2/3 majority.
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Agency). In the Delegation Game, the Agency chooses a policy y in stage 0 and then the above Legislation Game is played with y serving as the new status quo. Since the executive agency is staffed by the President we will assume that its ideal point is that of the President (i.e., P). With the configuration of Diagram 1, the unique subgame perfect equilibrium policy outcome is now h. The delegation of power to the Agency dramatically shifts the equilibrium policy from ts to h. How might Congress moderate Agency's exercise of its discretion? In the Two-House Veto Game and the One-House Veto Game, Congress intervenes directly. These two games model the effects of Chadha, where the Supreme Court ruled such supervision unconstitutional. In the Two-House Veto Game, Congress delegates decision-making authority to an executive agency but reserves the right to overturn any policy proposal by a majority vote of each house. Specifically, in stage 1 of the Two-House Veto Game, the Agency chooses a policy y . In stage 2, the House votes on the policy. If a majority approves the policy it is adopted. Otherwise, the game proceeds to stage 3 . If a majority of the Senate also approves y, it is adopted. Otherwise, the game ends with the original status quo SQ remaining in place. Reconsider the ideal point configuration Diagram 1. The point tH is such that ltH - H I = I H - SQ I . Clearly, fH is the equilibrium of the Two-House Veto Game as any point to the left of tH will be vetoed by both Houses. The Two-House Veto Game thus moderates the action of the Agency by bringing the equilibrium from h to tH , which is closer to the policy ts that would be enacted directly by Congress (with consent of the President). In the One-House Veto Game a specific house of Congress has the power to approve or disapprove of the Agency's policy choice. Clearly the outcome will be tH if the House has the veto power and the policy ts if the Senate has the veto power. Thus, Chadha's announcement of a constitutional prohibition on the power of Congress to veto agency decisions dramatically restricts Congress' ability to control power that it statutorily delegates to an executive agency. The previous analysis of the Legislation, Delegation and Veto games suggests how one might analyze the way that different institutional structures determine which legal policies prevail. Of course, a complete analysis would compare equilibria across all possible distributions of prefer ences of the relevant institutional actors.
5. Cooperative game theory as a tool for analysis of legal concepts
The prior discussion has examined the use of game theory to understand the behav ioral consequences of legal rules. As we have seen, the structure of much legal doc trine can be illuminated by models of simple games. These models have had an im portant impact on the legal commentary of cases and legislation. Moreover, courts
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have occasionally cited such game-theoretic analyses in support of particular deci sions. 33 Game-theoretic tools have also played another role in legal analysis, a role that may have had a more direct impact on legal development. Game theory, particularly cooper ative game theory, permits a conceptual analysis of ideas central to some areas of legal doctrine. This doctrinal approach may explicate the social objective underlying legal rules. 34 In this section, we discuss two doctrinal analyses. We first examine the treatment of an index of voting power by the United States' judicial system. We then consider a case law problem of fair division.
5.1. Measures of voting power In 1 962, the Supreme Court of the United States began a voting rights revolution when it held in Baker v. Carr (1 962) that the apportionment of state legislatures is subject to
33 The Supreme Court of the United States has referred to game-theoretic analyses of settlement in at least three of its opinions. Two of these cases concern settlement under joint and several liability. McDermott, Inc. v. Amclyde et al. (1994) cites Easterbrook, Landes and Posner (1981), Polinsky and Shavell ( 1980) and Ko rnhauser and Revesz (1993); Texas Industries v. Radcliff Materials (1981) cites Cirace ( 1980), Easterbrook, Landes and Posner ( 1981) and Polinsky and Shavell ( 1 980). The dissent in Marek v. Chesny ( 1985) cites Shaven (1982) and Katz ( 1987) in a discussion of fee-shifting. The intermediate appellate courts in the federal system and the state courts of the United States also oc casionally refer to game-theoretic analyses of settlement or of joint and several liability (or of both). Judge Easterbrook, himself a noted scholar in the field of economic analysis of law, has cited Shavell ( 1982) and Katz ( 1987) in Packer Engineering v. Kratville (1 992), Easterbrook, Landes and Posner (1981), Polinsky and Shavell ( 1980) and Kornhauser and Revesz ( 1989) in In re Oil Spill of Amoco Cadiz, 954 F.2d 1279 (7th Cir. 1991) (per curiam), Kornhauser and Revesz ( 1989) in Akzo Coatings v. Aigner Corp. (1 994), and Shavell ( 1982) in Premier Electric Construction Company v. National Electrical Contractors Assn. (1987). In addi tion, Rosenberg and Shavell ( 1985) were cited by the First Circuit in Weinberger v. Great Northern Nekoosa Corp. ( 1 991) and Easterbrook, Landes and Posner (198 1), Polinsky and Shaven ( 1980) and Kornhauser and Revesz ( 1993) were cited by the District of Columbia Court of Appeals in Berg v. Footer (1996). In re Larsen (1992) cites Bebchuk (1988) and Rosenberg and Shaven ( 1 985). Again, two economically sophisticated judges, Judge Calabresi and Judge Easterbrook, have cited Ayres and Gertner (1989) on the problems of information revelation in American National Fire Insurance Co. v. Ke nealy and Kenealy (1995) and Harnischfeger Corp. v. Harbor Insurance Co. ( 1991), respectively. Similarly, both Judges Easterbrook (in United States v. Herrera (1995)) and Posner (in In re Hoskins (!996)) have cited Baird, Gertner and Picker ( 1994) on game theory generally. It is worth noting that Coase (1960), the most cited article in the legal academic literature, is only cited 42 times by state and federal courts of the United States. Fifteen of these citations are from United States Court of Appeals for the Seventh Circuit. (LEXIS search done on 24 April 2000.) 34 The distinction between the use of game theory to predict behavior and game theory to explicate doctrine is sometimes fuzzy. For example, one might rationalize the structure of product liability doctrine as a set of legal rules that induce efficient behavior on the part of manufacturers and consumers. This rationalization might use game theory both as a predictive tool (to show that some particular rule in fact induces efficient conduct) and as a conceptual tool (to illuminate the meaning of efficiency, e.g., to distinguish between ex ante, interim, and ex post efficiency).
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judicial review under the 14th amendment of the U.S. Constitution. In the cases that im mediately followed, the Court announced that the U.S. Constitution guarantees each cit izen "fair and effective representation" (Reynolds v. Sims ( 1964) ). It enunciated a maxim of "one person, one vote" to guide states in legislative reapportionment. In the context of legislative elections from single-member districts, this slogan had a clear meaning and an immediate impact: it required redistricting when the population discrepancy be tween the largest and smallest districts was too great. 35 As the Court widened the reach of jurisdictions and governmental bodies subject to judicial oversight it gradually con fronted a wider and wider range of election practices which required it to elaborate a more complete theory of fair representation. In particular, the Court encountered sys tems that weight the votes of legislators to counterbalance population disparities as well as systems in which larger districts elect proportionately more legislators. Federal and state courts then confronted the question: When do these systems satisfy the require ments of fair and effective representation? In 1 954, Shapley and Shubik (1954) offered an interpretation of the Shapley value as an index of voting power. They interpreted the Shapley value as measuring the chance a voter would be critical to the passage of a bill. These ideas influenced John Banzhaf III who, as a law student, proposed a related index in Banzhaf (1965). 36 Banzhaf also in terpreted his index as a measure of the chance a voter would be critical. Banzhaf ( 1965) used this index to analyze weighted voting schemes and Banzhaf (1968) applied the index to the problem of multi-member districts. In each article, Banzhaf acknowledged the influence of the Shapley value on his thinking. 37 Banzhaf ( 1965) had a substantial impact on litigation; even while in draft, it was introduced into the deliberations of the state and federal courts considering a plan for the temporary reapportionment of the New York Legislature in which weighted vot ing would be used. (For a discussion of the plan see Glinski v. Lomenzo ( 1 965) and WMCA, Inc. v. Lomenzo ( 1 965).) After he published his articles, Banzhaf continued to intervene in reapportionment litigation; the courts thus frequently considered his view. Before we examine Banzhaf's influence on the case law, we define the two power in dices mentioned and briefly discuss the relation between them. Straffin (1994) provides a comprehensive survey of the technical literature. Consider a simple, proper voting game (N, v) . (v (S) = 0 or 1 for all coalitions S; v (S) = 1 implies v (N IS) = 0.) Let ei be the number of coalitions S in which i is a
35
Of course, as the Court was soon forced to learn, there are many ways to divide a population into roughly equal districts.
36
The Banzhaf index is equivalent to measures independently proposed by Coleman ( 1 973), Dahl (1957), and Rae (1969). On this equivalence, see Dubey and Shapley ( 1979).
37
Footnote 32 to Banzhaf (1965) begins, "Although the definition presented in this article is based in part on the ideas of Shapley-Shubik, their definition is rejected because . . . [it] has not been shown that the order in which votes are cast is significant; . . . Likewise it seems unreasonable to credit a legislator with different amounts of voting power depending on when or for what reasons he joins a particular voting coalition". Footnote 55 of Banzhaf (1968) thanks Shapley.
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swing voter; that is, the number of coalitions S such that v(S) The unnormalized Banzhaf power index {3; is given by:
=
1 and v(S - {i })
=
0.
while the normalized Banzhaf index is:
The Shapley-Shubik power index is defined as: CJi =
S is a swing for i
(s - 1)!(n - s)!jn!
where s = l S I .
An individual's (unnormalized) Banzhaf index gives the probability that she will be a swing voter under the assumption that all coalitions are equally likely to form. An individual's Shapley-Shubik index gives the probability that she will be a swing voter under the assumption that it is equally likely that a coalition of any size will form. 38 Although these two indices often give similar measures of power, they need not. Indeed, they may differ radically both in the relative weights they assign and in the rank order they place players. (See Straffin ( 1 988) for examples.) Straffin (1988) derives the indices within a probabilistic model in which there are n voters and each voter i has probability Pi of voting "yes" on a given issue, and a prob ability ( 1 Pi ) of voting "no" on the issue. Voter i 's power is given by the probability that her vote will affect the outcome of the election. Thus, a voter's power depends on the joint distribution from which the Pi 's are drawn. Straffin shows that (a) CJ; measures i 's power if p; = p for all i , and p is drawn from the uniform distribution on [0, 1] while (b) {3; measures i 's power if, for each i , P i i s drawn independently from the uni form distribution on [0, 1]. Equivalently, f3i measures i 's power on the assumption that each voter independently has a 50% chance of voting "yes" on the issue. 39 We tum now to the judicial reception of these ideas. The maxim of "one person, one vote" was articulated in the context of a political structure in which legislators are elected from single-member districts and each legislator has one vote within the leg islature. The early reapportionment cases therefore identified one political structure equal-sized, single-member districts - that satisfies the constitutional requirement that -
38 More precisely, the Shapley-Shubik index assumes that it is equally likely that a coalition of any size will form, and that all coalitions of a given size are equally like to form. An equivalent characterization is that the Shapley-Shubik index gives the probability that as an individual joins a coalition she will be a swing under the assumption that all orders of coalition formation are equally likely. For an axiomatic characterization of the two indices see Dubey and Shapley (1979).
39
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each citizen's vote have equal influence on political outcomes.40 Other political struc tures, however, were soon challenged. These challenges required the judiciary to elabo rate a conception of "equal voting power". Two political structures were of particular interest. Suppose that districts vary in size. In systems of weighted voting, each district elects one legislator but the elected rep resentatives of larger districts are given more votes within the legislature. What set of voting weights, if any, yield equal influence of each citizen's vote? In the second polit ical structure, larger districts elect more representatives. How should representatives be allocated among the districts to satisfy the constitutional mandate? The Banzhaf index (and Shapley-Shubik index) suggest answers to these questions. Citizens in a representative democracy play a compound voting game. They elect rep resentatives, and these representatives then pass legislation. There are three ways that the Banzhaf index could be used in this context. First, it could be used to measure the power of the elected representatives voting in the legislature. Second, it could be used to measure the power of citizens when voting for representatives. Finally, it could be used to measure the net power of citizens in the compound voting game. Since the probability that a citizen is a swing voter for the final passage of legislation equals the probability that her vote is crucial to getting her legislator elected times the probability that the legislator's vote is crucial, this last measure of Banzhaf power is simply the product of the first two. From roughly 1 967 through 1 990 the Banzhaf index served as a judicially accepted criterion for the determination of the power of representatives at the legislative level in weighted voting schemes. The Banzhaf index had particular importance in New York State where many county boards elected supervisors from towns with widely varying populations.4 1 Indeed, Banzhaf's first article analyzed the striking case of the Nassau County Board of Supervisors in 1958 and 1 964. Voters in different municipalities elected a total of six representatives. Since the municipalities differed in size, the elected representatives were given different voting weights, presumably to equalize the power of the citizens in the different municipalities. Incredibly, the weights were distributed among the rep resentatives in such a way that three representatives had no swings - their votes could never influence the outcome (see Table 1). The weighting schemes masked the fact that in both 1 958 and 1 964 the six representatives were actually playing a game of three player majority rule; three of the representatives had power 1 /3 and three had power 0 under both the Shapley and the normalized Banzhaf index.
40
The case law focuses on the formulation of the conception of right articulated in Reynolds v. Sims (1964). There the U.S. Supreme Court stated that "each and every citizen has an inalienable right to full and effective participation in the political processes of his State's legislative bodies" (at 565). Moreover, "full and effective participation by all citizens in state government requires . . . that each citizen have an equally effective voice in the election of members of his state legislature" (at 565). Imrie (1973) and Lucas (1983) provide brief accounts of the use of the Banzhaf index in New York State.
41
J.-P. Benoit and L.A. Kornhauser
2256 Table 1 Board of Supervisors - Nassau County 1958 Municipality
Hempstead! Hempstead2 N. Hempstead Oyster Bay Glen Cove Long Beach
Population
Weight of representative
Unnormalized power
Normalized power
618,065
9 9 7 3
1/2 1/2 1/2 0 0 0
1/3 1/3 1/3 0 0 0
1/2 1/2 0 1/2 0 0
1/3 1/3 0 1/3 0 0
184,060 164,716 19,296 17,999
1964 Hempstead! Hempstead2 N. Hempstead Oyster Bay Glen Cove Long Beach
728,625 213,225 285 ,545 22,752 25,654
31 31 21 28 2 2
Subsequently, in Iannucci v. Board of Supervisors of Washington County (1 967) the New York Court of Appeals, the highest court in New York State, endorsed the Banzhaf index as the appropriate standard for determining the weights accorded to representa tives of districts of varying sizes. The Court stated that the principle of "one person, one vote" was satisfied as long as the "[Banzhaf] power of a representative to affect the passage of legislation by his vote . . . roughly corresponds to the proportion of the population of his constituency" (at 252). Interestingly, the Court explicitly endorsed the abstract nature of the power calculus, ruling that "the sole criterion [of the consti tutionality of weighted voting schemes] is the mathematical voting power which each legislator possesses in theory (emphasis added) . . . and not the actual voting power he possesses in fact . . ." (at 252). In Franklin v. Kraus ( 1973), the New York Court of Ap peals again approved a weighted voting system that made the Banzhaf power index of each representative proportional to his district's size. In both these cases the court used the Banzhaf index to determine the voting power of the legislators. The courts, however, did not carry the Banzhaf reasoning through to the citizen level. It can be shown that the probability that an individual is a swing voter in the election of her legislator is (approximately) inversely proportional to the square root of the population of her district. Thus, equalizing each citizen's net Banzhaf power in the compound voting game would require that a legislator be given power proportional to this square root. Nonetheless, the courts decided that a legislator's power should be proportional to the population itself. How do we account for the failure of the courts to consider the citizen's power in the compound game? Apparently, the courts adopted a different view of the relation
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between representative and citizen than that implicit in Banzhaf. 42 The swing voter analysis assumes that citizens disagree about which candidate would best represent the jurisdiction and that the prevailing candidate acts in the interests of those citizens who votedfor her. On a different conception of representation, however, once elected a can didate represents the interests of all citizens within her constituency. Each citizen in a district then exerts an equal influence upon the district representative, so that a citizen's influence is inversely proportional to the district population. Hence, under this concep tion, a representative's power should be proportional to the number of constituents in her district, as the courts ruled. We now consider judicial treatment of multi-member districts. In Abate v. Mundt (1969), the New York Court of Appeals upheld a system of multi-member districts that allocated supervisors to election districts in proportion to district populations. The court again ignored the compound game analysis. It should be noted, however, that unlike Banzhaf's analysis of weighted voting schemes, his analysis of multi-member districts is seriously flawed. His assertion that voters in districts should be given representatives in proportion to the square root of the size of the population is unwarranted. First of all, it is clear that no recommendation on the number of representatives in a multi-member district can be made independently of the voting method being used to elect these representatives. Is each voter being given one vote or the same number of votes as there are representatives or is some other method in use? Second, if individuals are given many votes, then what is the proper assumption to make on the way these votes are cast? Clearly, the assumption that an individual is casting her own votes in dependently of each other is absurd. Finally, what assumption is being made about the correlation of the voting behavior of representatives voted for by a single individual? If the correlation is perfect, then, in effect, the multi-member districting scheme results in a weighted voting game in the legislature. Of course, the ratios of voting powers in such a game are not necessarily equal to the ratios of voting weights. On the other hand, if these representatives are voting differently from each other, in what sense are they all representing the same individual? There may well be such a sense, but it is ill-captured by the Banzhaf conception, which considers the probability that a voter will be decisive on an issue. A few years after Abate, the United States Supreme Court (in Whitcomb v. Chavis (1973)) rejected the Banzhaf index in the judicial evaluation of multi-member districts. The Court argued that the index offered only a theoretical measure of the voter's power while constitutional considerations had to rest on disparities in actual voting power.43 Thus, the theoretical character of the power index which had pleased the New York Court of Appeals, moved the United States Supreme Court to reject the index. 42 Banzhaf (1965) is noncommittal about the relationship between a citizen and his representative. Banzhaf ( 1968) explicitly endorses the compound voting computation suggested above. Banzhaf was an expert witness for plaintiffs at trial and signed the plaintiff-appellee's brief to the United States Supreme Court. Interestingly, plaintiffs sought single-member districts as a remedy in Whitcomb v.
43
Chavis.
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In 1 989 the U.S. Supreme Court explicitly carried out a compound voting game analysis when considering the weighted voting scheme used by the Board of Estimate of New York City. 44 The court rejected the Banzhaf index, again citing its "theoretical" nature. On the basis of this decision, one federal district court struck down as uncon stitutional the weighted voting scheme that then prevailed in Nassau County.45 Two federal district courts have approved a weighted voting system in the New York coun ties of Roxbury and Schoharie respectively which assigned weights proportionally to population. 46 Thus, for the time being at least, the Banzhaf index appears to have been discredited for all United States legislative apportionment purposes. This brief history illustrates both the potential significance and difficulties of judicial use of game-theoretic (or other formal) concepts in the articulation of legal rules. The Banzhaf index governed the shape of weighted voting systems for roughly twenty years and partially shaped the judicial treatment of voting systems that consisted of unequal sized districts. Unfortunately, however, the courts and the community of political sci entists and game theorists failed to engage in a constructive dialogue that would have enabled the courts to formulate a coherent conception of equal and effective represen tation for each citizen. The courts never clearly articulated or resolved three important questions. First, as we remarked earlier, the courts had to choose whether to focus atten tion on the game constituted by voters electing representatives, the game played by the elected representatives in the legislature, or the compound game. The role of citizens in the compound game seems normatively the correct one for the courts to consider, but they were never clear on this point. Second, and as a consequence of their failure to consider the compound game explicitly, the courts did not articulate a theory of repre sentation that would allow analysis of the compound game. Finally, the courts did not question whether power was the appropriate concept for expressing equality of repre sentation. The indices of voting power measure the probability that an individual will be a swing voter under various assumptions ; the related concept of satisfaction measures the probability that the outcome will be the one that the voter desires. In a town with like-minded citizens each one will have zero voting power since decisions will always be unanimous and no individual voter will be crucial. On the other hand, each citizen will have a satisfaction of one. The courts should have at least considered whether sat isfaction was a more appropriate concept than power on which to construct the right to fair and effective representation. The analytic community on the other hand bears some responsibility for the judicial failure to confront these three questions. Neither Banzhaf himself nor other commenta tors on the judicial efforts clearly framed the question in terms of the compound game.
44 Morris 45 Jackson
v.
Board of Estimate ofNew York City ( 1989). Nassau County Board of Supervisors (1993).
The weights at issue in the litigation had been allocated so that the Banzhaf power of each representative was proportional to the population of the represen tative's district. 46 Roxbury Taxpayers Alliance v. Delaware County Board of Supervisors (1995) and Reform of Schoharie County v. Schoharie County Bd. ofSupervisors (1997). v.
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Moreover, discussions of the importance of the concept of representation to the analysis of the game were sparse. Finally, although a voter's satisfaction is arguably more impor tant than a voter's power, the former concept has received comparatively little attention from game theorists.47
5.2. Fair allocation of losses and benefits Adjudication resolves disputes among parties. Courts are thus often more concerned, at least superficially, with the fair resolution of the dispute between the parties before the court than with the ex ante behavior that may avoid similar disputes in the future (or reshape them). Many classes of disputes, in fact, seem explicitly directed at the fair division of property among disputing claimants. The most obvious area of law in which questions of this type arise is bankruptcy where the court must allocate the assets of the bankrupt among its creditors in a way that distributes the loss - i.e., the difference between the (valid) claims of creditors and the value of the estate - fairly.48 Other areas include: the interpretations of wills that divide more assets among the heirs than are actually available; the rules of contribution when the share of an insolvent tortfeasor must be allocated among the remaining joint tortfeasors; rate proceedings for common carriers such as telecommunications or electricity providers when the common costs of service provision must be allocated among different classes of customers; and the determination of fee awards to class action attorneys when the common costs of repre sentation must be allocated among different classes of plaintiffs. One might reasonably ask what conception of fairness is embodied in legal rules gov erning these allocations? What explains variation, if any, in these rules? In the case of the common law, a judge confronted with a dispute consults the decisions in analogous, previously decided cases and attempts to articulate a principle or set of principles that rationalizes these outcomes. Here, answering the above questions is particularly impor tant and difficult. Talmudic law, like Anglo-American law, is a form of common law adjudication. Au mann and Maschler (1985) and O'Neill ( 1982) have deployed game-theoretic tools to explicate in part the rules implicit in two areas of Talmudic law that concern the fair allocation of property among disputing claimants. Aumann and Maschler consider the following bankruptcy problem from the Talmud. There are three creditors on an estate. Creditor 1 is owed 100; creditor 2 is owed 200, and creditor 3 is owed 300. The estate is worth less than the 600 total claim. The Talmud states that when the estate is valued at 100, the creditors should each receive 33 �; when 47 On satisfaction see Brarns and Lake ( 1 978) and Straffin, Davis, and Brarns (1982). 48 In fact bankruptcy law must balance other concerns with fairness. In particular, the rules of bankruptcy may
affect the pre-bankruptcy behavior of both debtor and creditors. Moreover, even after bankruptcy, the value of a debtor corporation is generally greater as an ongoing concern than liquidated; capturing this surplus may entail creating incentives to some parties such as managers or specific creditors that otherwise seem unfair. Finally, we have ignored concerns about priority of claims of different types of creditors.
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the estate is valued at 200, the creditors should receive, respectively, 50, 75, and 75; when the estate is valued at 300, the allocation should be (50, 1 00, 1 50) . What principle underlies the prescribed allocations? How should these three examples be extended to other bankruptcy cases? As a preliminary, consider the "contested-garment problem" from the Mishnah: one individual claims an entire garment; a second individual claims one-half of the same garment. The Talmud allocates the garment (3/4, 1/4) to the two claimants respec tively. Here, the underlying principle is well understood by Talmudic scholars. The sec ond claimant implicitly concedes half the garment to the first, while the first claimant concedes nothing. Each claimant is given the amount that is conceded to him, and the remainder is divided equally. However, this principle is articulated only for the case of two claimants. How can this principle for the two-claimant case be extended to the n-claimant case? Define a general claim problem P = (A; CJ , cz, . . . , c,) , where A is the total amount of assets to be distributed and Ci is i 's claim on A . Aumann and Maschler call a dis tribution x of the amount A "contested-garment-consistent" if (x i xi) is the contested garment solution to the claim problem (xi + xi ; Ci , c i) for all claimants i and j . They show that each claim problem has a unique contested-garment-consistent solution. Moreover, this solution gives the allocations prescribed by the Talmud for the above three bankruptcy problems. Note that while the notion of contested-garment consistency does not appear else where in the game theory literature, this "consistency" approach is certainly game theoretic in spirit. For instance, Harsanyi has characterized the product solution to the n-person Nash bargaining problem as the unique solution which is Nash-consistent for all pairs of bargainers. As in O'Neill ( 1 982), Aumann and Maschler turn the claim problem into a coopera tive game by associating the following characteristic function game with it:
,
(
v(S) = max A - L ci , o iEN - S
)
for S c;
N.
This characteristic function is constructed on the assumption that claims are satisfied in full as they arrive and that a coalition can only guarantee itself what it would receive if it were the last to make a claim on the assets A. Alternatively, a coalition's worth is what it obtains when it concedes the claims of the other claimants. Aumann and Maschler demonstrate that the contested-garment-consistent solution (and hence the Talmudic division to the three bankruptcy problems) is the nucleolus of this coalitional game. 49 While this result is remarkable and Aumann and Maschler's discussion is insightful and revealing, various objections can be raised. These objections pertain to the limita tions of the game-theoretic approach in general.
49 See Benoit (1998) for
a
simple proof of this result.
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A difficulty in applying game theory is that "real world" situations are not presented as games. Rather a game must be defined. While the characteristic function Aumann and Maschler associate with the claim problem is a reasonable one, other specifications are also reasonable. For instance, we could define a coalition's worth to be its expected payment assuming that the other claimants do not form coalitions, that all claimants are paid off as they arrive, and that all orders of arrival are equally likely. Similarly, while consistency notions are interesting, several can be defined. Thus, call a solution X to a claim problem "CG2-consistent" if for all s c N, o:=s X; ' L N !S x;) is the contested-garment solution to (A; L s Ci , L N ;s ci ) . As the claim problem
(500; 100, 200, 300) shows, there is no CG2-consistent solution. 5° This non-existence might be taken as evidence that CG2-consistency is the wrong notion of consistency, but it might just as well indicate that the contested-garment example should not be extended to more than two people.5 1 As noted earlier, two questions arise in the analysis of a body of common law cases. What is the general rule that is to be derived from the disposition of particular cases, and what is the conception of fairness embodied in this law? These two questions are related because a rule that rationalizes a set of cases must both "fit" those cases and offer an attractive rationale for them. After all, many rules will be consistent with the decisions in the set of cases. Aumann and Maschler explicitly address the first question. Implicitly, they address the second as well. The nucleolus of a bankruptcy problem minimizes the maximum loss suffered by any coalition. Thus, it has an egalitarian spirit, but one in which coalitions and not just individuals are taken as objects of concern. O'Neill considers a problem in which a testator bequeaths shares in his estate, val ued at 120, that total to more than one. The entire estate is bequeathed to heir 1 , one half the estate to heir 2, one-third the estate to heir 3 and one-quarter the estate to heir 4. According to O'Neill, the 1 2th-century scholar Ibn Ezra claimed that the divi sion (97 / 144, 25/ 1 44, 13/ 144, 9/ 144) is the appropriate one. Ibn Ezra reasoned that heir 4 lays claim to a fourth of the estate, but that all four heirs make equal claim to this quarter and so this quarter is split evenly. Next heir 3 claims one third of the estate. He has already received his share of 1 j4 of the estate, so now he and the three larger claimants receive an additional 1 / 3 ( 1 /3-1 j4) for a total of 13/144. Proceeding in this manner yields the above numbers. In contrast to the earlier bankruptcy problem, here the reasoning underlying the di vision is given explicitly. O'Neill is interested in both extending this reasoning and proposing alternative, albeit related, solutions. Notice that while each heir only makes a claim for himself, this claim can be taken as implicitly imputing a division of the remain der of the assets to the other heirs. Indeed, given a specific heir's claim and an allocation 50 Solving for x; by taking S = [i} for each claimant one at a time yields (50, !50, 250) as the putative CG2-solution, but this does not sum to 500.
51
Aumann and Maschler consider a principle similar to CG2-consistency. They show how a modification of this principle can be used to again derive the CO-consistent solution.
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rule, use the rule to impute a division of the remainder of the assets to each of the other heirs. Since no heir has any greater legitimacy than any other, compute the allocation that results from averaging the various divisions obtained in this manner. O'Neill calls a rule consistent if the allocation thus obtained is the same as the allocation the rule prescribed for the initial problem. Ibn Ezra's solution is not consistent in this sense. Again associate the characteristic function v(S) = max(A - L iEN � S Ci , 0) with this claim problem. The Shapley value yields an allocation which is consistent in O'Neill's sense. Furthermore, the Shapley allocation affords an interesting interpretation. Sup pose that each heir is simply paid off in full as he presents his claim until the estate is exhausted. The allocation which results from averaging over all possible orders in which the claims could be presented is the Shapley value. When making allocations, courts might want to balance practical considerations with ethical ones. For instance, courts may not want to give claimants the incentive to combine or split their claims. Thus, they might require that (xi , . . . , Xi, , . . . , xi , . . . , Xn) is the solution to (A ; C ] , . . . , Ci , . . . , Cj , . . . , en ) if and only if (XJ , . . . , Xi + Xj , . . . , Xn) is the solution to (A; C J , . . . , Ci + cj , . . . , en ) . The only such rule is proportional allocation. 52 One should note that U.S. law generally does not resolve claims problems similarly to the Talmudic solutions that inspired O'Neill and Aumann and Maschler. In bankruptcy law, the assets A are, in fact, allocated among claimants proportionally to their claims. U.S. law presumably adopts this approach because creditors may freely transfer claims on the debtor-in-bankruptcy. With inheritances the problem is considered one of inter preting the intent of the testator. If no clear intent can be inferred, the will is invalid, the individual dies intestate, and the state's default rules of inheritance govern. Determina tion of intent depends critically on the precise wording of the document. See, e.g., Estate ofHeisserer v. Loos (1985), In re Vzsmar's Estate ( 1921), and Rodisch v. Moore (1913). The shares of insolvent defendants among joint tortfeasors are generally allocated using one of two rules. Under legal contribution, the tortfeasor against whom the plain tiff chooses to execute the judgment must bear the entire share of the insolvent parties. Under equitable contribution, these insolvent shares are divided among the solvent tort feasors either equally or proportionally to their own shares. (Kornhauser and Revesz (1990) summarizes the law in this area.) 6. Concluding remarks
Law pervades daily life. Legal rules structure relations among individuals, institutions, and government. The theory of games provides a framework within which to study the effects these legal rules have on the behavior of these agents. In a legal culture that considers law a means to guide behavior to specified goals, this framework will interest
52 See Chun (1988), de Frutos ( 1994) and O'Neill (198 2). Thomson (1997) provides a survey of axiomatic models of bankruptcy.
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not only legal scholars but also lawmakers directly as they attempt to design rules that forward their goals. In the last decade, analysts have deployed increasingly sophisticated models to an ever-widening range of legal rules. Our discussion has merely scratched the surface of this literature. Many analyses of legal rules focus not on the behavioral effects of these rules but on their conceptual and normative foundations. The tools of game theory have also offered insights into these areas. References
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Rubinstein, A. ( 1982), "Perfect equilibrium in a bargaining model", Econometrica 50: 1 1 5 1-1 172. Samuelson, W. (1 985), "A comment on the Coase Theorem", in: A.E. Roth, ed., Game-Theoretic Models of Bargaining, 321-339. Schwartz, A. ( 1989), "A theory of loan priorities", Journal of Legal Studies 18:209-261 . Schwartz, A . ( 1 993), "Bankruptcy workouts and debt contracts", Journal of Law and Economics 36:595-632. Schwartz, E.P. (1992), "Policy, precedent, and power: A positive theory of supreme court decision-making", Journal of Law, Economics & Organization 8:219-252. Sercu, P., and C. van Hulle (1994), "Financing instruments, security design, and the efficiency of takeovers: A note", International Review of Law and Economics 15:373-393. Shapley, L.S., and M. Shubik (1954), "A method for evaluating the distribution of power in a committee system", American Political Science Review 48:787-792. Shavell, S. ( 1980), "Damage measures for breach of contract", Bell Journal of Economics and Management Science 1 1 :466-490. Shavell, S. ( 1982), "Suit settlement and trial: A theoretical analysis under alternative methods for the allocation of legal costs", Journal of Legal Studies 1 1 :55-82. Shavell, S. (1983), "Torts in which victims and injurers move sequentially", Journal of Law & Economics 26:589-612. Shavell, S. ( 1987), Economic Analysis of Accident Law (Harvard University Press, Cambridge, MA). Shavell, S. ( 1995), "The appeals process as a means of error correction", Journal of Legal Studies 24:379-426. Sobel, J. (1985), "Disclosure of evidence and resolution of disputes: Who should bear the burden of proof?", in: A.E. Roth, ed., Game-Theoretic Models of Bargaining, 341-361. Spier, K. ( 1992), "The dynamics of pretrial negotiation", Review of Economic Studies 59:93-108. Spier, K. (1994), "Pretrial bargaining and the design of fee shifting rules", RAND Journal of Economics 25: 1 97-214. Straffin, P.D., Jr. (1988), "The Shapley-Shubik and Banzhaf power indices as probabilities", in: A.E. Roth, ed., The Shapley Value: Essays in Honor of Lloyd Shapley, 71-8 1 . Straffin, P.D., Jr. ( 1 994), "Power and stability i n politics", in: R.J. Aumann and S. Hart, eds., Handbook of Game Theory, Vol. 2 (North-Holland, Amsterdam) Chapter 32, 1 127-1 152. Straffin, P.D., Jr., M.D. Davis and S.J. Brams ( 1982), "Power and satisfaction in an ideologically divided voting body", in: M.J. Holler, ed., Power, Voting, and Voting Power, 239-255. Sykes, A.O. (1990), "The doctrine of commercial impracticability in a second-best world", Journal of Legal Studies 19:43-94. Symposium ( 1 99 1 ), "Just winners and losers: The application of game theory to corporate law and practice", University of Cincinnati Law Review 60:2-448. Thomson, W. (1997), "Axiomatic analyses of bankruptcy and taxation problems: A survey", mimeo. Veljanovski, C.J. ( 1982), "The Coase theorems and the economic theory of markets and law", Kyklos 35:53-74. Wang, G.H. (1994), "Litigation and pretrial negotiations under incomplete information", Journal of Law, Economics & Organization 10: 1 87-200. Weingast, B., and M. Moran (1 983), "Bureaucratic discretion of congressional control? Regulatory policy-making by the Federal Trade Commission", Journal of Political Economy 91 :756-800. White, M.J. ( 1980), "Public policy toward bankruptcy: Me-first and other priority rules", Bell Journal of Economics and Management Science 1 1 :550-564. White, M.J. ( 1 988), "Contract breach and contract discharge due to impossibility: A unified theory", Journal of Legal Studies 17:353-376. White, M.J. ( 1994), "Corporate bankruptcy as a filtering device: Chapter 1 1 . Reorganization and out-of-court debt restructurings", Journal of Law, Economics & Organization 10:268-295. Winter, R. (1991), "Liability insurance", Journal of Economic Perspectives 5 : 1 15-136. Wittman, D. (1981), "Optimal pricing of sequential inputs: Last clear chance, mitigation of damages, and related doctrines in the law", Journal of Legal Studies 10:65-92.
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Wittman, D. (1988), "Dispute resolution, bargaining, and the selection of cases for trial: A study of the generation of biased and unbiased data'', Journal of Legal Studies 17:313-352. Young, H.P. ( 1 994), Equity: In Theory and Practice (Princeton University Press, Princeton, NJ).
Cases Abate v. Mundt, 25 N.Y.2d 309 (1969) Abate v. Mundt, 403 U.S. 182 (1971) Akzo Coatings, Inc v. Aigner Corp., 30 F.3d 761 (7th Cir. 1994) American National Fire Insurance Co. v. Kenealy and Kenealy, 72 F.3d 264 (2d Cir. 1995) Baker v. Carr, 369 U.S. 186 (1962) Bechtle v. Board of Supervisors of County ofNassau, 441 N.Y.S. 2d 403 (2d Dept 1981) Berg v. Footer, 673 A. 2d 1244 (D.C. App., 1996) Board of Estimate ofNew York City v. Morris, 489 U.S. 688 ( 1 989) Chapman v. Meier, 420 U.S. 1 (1975) Chevron, U.S.A., Inc v. Natural Resources Defense Council Inc, 467 U.S. 837 ( 1 984) Estate ofHeisserer v. Laos, 698 S.S. 2d 6 (Mo. Ct. of Appeals 1985) Franklin v. Kraus, 72 Misc.2d (Sup.Ct., Nassau Cty 1972) reversed 32 NY 2d 234 ( 1 973) reargument denied, motion to amend remittitur granted 33 N.Y. 2d 646 ( 1 973) appeal dismissed for want of a substantial federal question 415 U.S. 904 ( 1974) Franklin v. Mandeville, 57 Misc.2d 1 072 (Sup.Ct. Nassau Cty. l 968) aff'd 299 N.Y.S.2d 953 (2d Dept 1969) modified 26 NY2d 65 (1970) Glinski v. Lomenzo, 16 N.Y.2d 27 (1 965) Harnischefeger Corp. v. Harbor Insurance Co. , 927 F.2d 974 (7th Cir 1 991) Holder v. Hall, 1 14 S.Ct. 2581 ( 1 994) Iannucci v. Board of Supervisors of Washington County, 20 N.Y.2d 244 (1967) Immigration and Nationalization Service v. Chadha, 462 U.S. 919 ( 1983) In re Hoskins, 102 F.3d 3 1 1 (7th Cir. 1996) In re Larsen, 616 A.2nd 529 (Pa. S. Ct 1 992) In re Oil Spill ofAmoco Cadiz, 954 F.2d 1279 (7th Cir 1991) In re Vismar's Estate, 191 NYS 752 (Surrogates Ct 1921) Jackson v. Nassau County Board of Supervisors, 818 F. Supp. 509 (EDNY 1993) Kilgarlin v. Hill, 386 U.S. 120 ( 1966) League of Women Voters v. Nassau County Board of Supervisors, 737 F.2d 155 (2d Cir. 1984) cert denied sub nom. Schmertz v. Nassau County Board of Supervisors, 469 U.S. 1 108 (1985) Marek v. Chesny, 473 U.S. 1 (1985) McDermott Inc. v. Amclyde et al. , 5 1 1 S. Ct. 202 at notes 14, 20, and 24 (1994) Packer Engineering v. Kratville, 965 F.2d 174 (7th Cir. 1992) Premier Electric Construction Company v. National Electrical Contractors Assn., 814 F.d 358 (7th Cir. 1987) Reynolds v. Sims, 377 U.S. 533 (1964) Rodisch v. Moore, 101 N.E. 206 (TIL Sup. Ct 1913) Roxbury Taxpayers Alliance v. Delaware County Board ofSupervisors, 886 F. Supp. 242 (NDNY 1995) aff'd, 80 F. 3d 42 (2d Cir.), cert. denied sub nom. MacDonald v. Delaware County Board of Supervisors, 5 1 9 U.S. 872 ( 1996) Reform of Schoharie County v. Schoharie County Board of Supervisors, 975 F. Supp. 191 (NDNY, 1997) Texas Industries v. Radcliff Materials, 451 U.S. 630 ( 1 981) United States v. Herrera, 70 F. 3d 444 (7th Cir, 1995) Weinberger v. Great Northern Nekoosa Corp. , 925 F.2d 5 1 8 ( 1 st Cir., 1991) Whitcomb v. Chavis, 403 U.S. 124 ( 1 971) WMCA Inc. v. Lomenzo, 246 F. Supp. 953 (S.D.N.Y. 1965) affirmed per curiam sub nom Travia v. Lomenzo, 382 U.S. 287 (1965)
Chapter 61 IM P LEMENTATION THEORY THOMAS R. PALFREY*
Division ofHumanities and Social Sciences, California Institute of Technology, Pasadena, CA, USA Contents
1 . Introduction 1 . 1 . The basic structure of the implementation problem 1 .2. Informational issues 1 .3. Equilibrium: Incentive compatibility and uniqueness 1 .4. The organization of this chapter
2. Implementation under conditions of complete information 2. 1 . Nash implementation 2. 1 . 1 . Incentive compatibility 2.1 .2. Uniqueness 2.1 .3. Example 1 2.1 .4. Example 2 2. 1 .5. Necessary and sufficient conditions 2.2. Refined Nash implementation 2.2. 1 . Subgame perfect implementation 2.2.2. Example 3 2.2.3. Characterization results 2.2.4. Implementation by backward induction and voting trees 2.2.5. Normal form refinements 2.2.6. Example 4 2.2.7. Constraints on mechanisms 2.3. Virtual implementation 2.3. 1 . Virtual Nash implementation 2.3.2. Virtual implementation in iterated removal of dominated strategies
3 . Implementation with incomplete information 3 . 1 . The type representation 3.2. Bayesian Nash implementation 3.2. 1 . Incentive compatibility and the Bayesian revelation principle
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*I thank John Duggan, Matthew Jackson, and Sanjay Srivastava for helpful comments and many enlightening discussions on the subject of implementation theory. Suggestions from Robert Aumann, Sergiu Hart and two anonymous readers are also gratefully acknowledged. The financial support of the National Science Foundation is gratefully acknowledged.
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2272 3.2.2. Uniqueness 3.2.3. Example 5 3.2.4. Bayesian monotonicity 3.2.5. Necessary and sufficient conditions 3.3. Implementation using refinements of Bayesian equilibrium 3.3. 1 . Undominated Bayesian equilibrium 3.3.2. Example 6 3.3.3. Example 7 3.3.4. Virtual implementation with incomplete information 3.3.5. Implementation via sequential equilibrium
4. Open issues 4. 1 . Implementation using perfect information extensive forms and voting trees 4.2. Renegotiation and information leakage 4.3. Individual rationality and participation constraints 4.4. Implementation in dynamic environments 4.5. Robustness of the mechanism
References
2304 2304 2305 2307 23 10 23 10 23 1 1 23 1 1 23 1 2 23 12 23 14 23 14 23 14 23 1 6 23 17 23 1 8 2320
Abstract
This chapter surveys the branch of implementation theory initiated by Maskin ( 1 999) . Results for both complete and incomplete information environments are covered.
Keywords
implementation theory, mechanism design, game theory, social choice
JEL classification: 025, 026
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1. Introduction
Implementation theory is an area of research in economic theory that rigorously investi gates the correspondence between normative goals and institutions designed to achieve (implement) those goals. More precisely, given a normative goal or welfare criterion for a particular class of allocation problems, or domain of environments, it formally char acterizes organizational mechanisms that will guarantee outcomes consistent with that goal, assuming the outcomes of any such mechanism arise from some specification of equilibrium behavior. The approaches to this problem to date lie in the general domain of game theory because, as a matter of definition in the implementation theory literature, an institution is modelled as a mechanism, which is essentially a non-cooperative game form. Moreover, the specific models of equilibrium behavior are usually borrowed from game theory. Consequently, many of the same issues that intrigue game theorists are the focus of attention in implementation theory: How do results change if informational assumptions change? How do results depend on the equilibrium concept governing sta ble behavior in the game? How do results depend on the distribution of preferences of the players or the number of players? Also, many of the same issues that arise in social choice theory and welfare economics are also at the heart of implementation theory: Are some first-best welfare criteria unachievable? What is the constrained second-best solu tion? What is the general correspondence between normative axioms on social choice functions and the possibility of strategic manipulation, and how does this correspon dence depend on the domain of environments? What is the correspondence between social choice functions and voting rules? In order to limit this chapter to manageable proportions, attention will be mainly fo cused on the part of implementation theory using non-cooperative equilibrium concepts that follows the seminal paper by Maskin ( 1 999) . 1 This body of research has its roots in the foundational work on decentralization and economic design of Hayek, Koopmans, Hurwicz, Reiter, Marschak, Radner, Vickrey and others that dates back more than half a century. The limitation to this somewhat restricted subset of the enormous literature of im plementation theory and mechanism design excludes four basic categories of results. The first category is implementation via dominant strategy equilibrium, perhaps more familiarly known as the characterization of strategyproof mechanisms. This research, mainly following the seminal work of Gibbard ( 1973) and Satterthwaite ( 1 975), has close connections with social choice theory, and for that reason has already been treated in some depth in Chapter 3 1 of this Handbook (Vol. II). The second category excluded is implementation via solution concepts that allow for coalitional behavior, most notably, strong equilibrium [Dutta and Sen ( 1 99 1c)] and coalition-proof equilibrium [Bernheim and Whinston ( 1 987)] . The third category involves practical issues in the design of mechanisms. There is a vast literature identifying specific classes of mechanisms such
1
The published article is a revision of a paper that was circulated in 1977.
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as divide-and-choose, sequential majority voting, auctions, and so forth, and study ing/characterizing the range of social choice rules that are implementable using such mechanisms. The fourth category is the many applications of the revelation principle to design mechanisms in Bayesian environments. 2 This includes much work in auc tion theory, bargaining, contracting, and regulation. These other categories are already covered in some detail in several other chapters of this Handbook. Later in this chapter we discuss practical aspects of implementation, as it relates to the specific mechanisms used in the constructive proofs of the main theorems. But a few introductory remarks about this are probably in order. In contrast to the literature devoted to studying classes of "natural" mechanisms, the most general characterizations of implementable social choice rules often resort to highly abstract mechanisms, with little or no concern for practical application. The reason for this is that the mechanisms constructed in these theorems are supposed to apply to arbitrary implementable social choice rules. A typical characterization result in implementation theory takes as given an equilibrium concept (say, subgame perfect Nash equilibrium) and tries to identify necessary and sufficient conditions for a social choice rule to be implementable, un der minimal domain restrictions on the environment. The method of proof (at least for sufficiency) is to construct a mechanism that will work for any implementable social choice rule. That is, a standard game form is identified which will implement all such rules with only minor variation across rules. It should come as no surprise that this often involves the construction of highly abstract mechanisms. A premise of the research covered in this chapter is that to create a general founda tion for implementation theory it makes sense to begin by identifying general conditions on social choice rules (and domains and equilibrium concepts) under which there ex ists some implementing mechanism. In this sense it is appropriate to view the highly abstract mechanisms more as vehicles for proving existence theorems than as specific suggestions for the nitty gritty details of organizational design. This is not to say that they provide no insights into the principles of such design, but rather to say that for most practical applications one could (hopefully) strip away a lot of the complexity of the abstract mechanisms. Thus a natural next direction to pursue, particularly for those interested in specific applications, is the identification of practical restrictions on mechanisms and the characterization of social choice rules that can be implemented by mechanisms satisfying such restrictions. Some research in implementation theory is beginning to move in this direction, and, by doing so, is beginning to bridge the gap between this literature and the aforementioned research which focuses on implementa tion using very special classes of procedures such as binary voting trees and bargaining rules.
2
The approach based on the revelation principle studies the truthful implementation [Dasgupta, Hammond and Maskin (1979)] of allocation rules [referred to as weak implementation in Palfrey and Srivastava ( 1 993, p. 15)]. That is, an allocation rule is truthfully implementable if there is some equilibrium of some mechanism that leads to the desired allocation rule. This chapter only deals with full implementation, where the imple menting mechanism must have the property that all equilibria of the mechanism lead to the desired allocation rule.
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F
X
M Figure 1. Mount-Reiter diagram.
Finally, there are other surveys that interested readers would find useful. These in clude Allen ( 1 997), Jackson (2002), Maskin (1 985), Moore ( 1992), Palfrey ( 1 992), Pal frey and Srivastava (1 993), and Postlewaite ( 1 985).
1.1. The basic structure of the implementation problem The simple structure of the general implementation problem provides a convenient way to organize the research that has been conducted to date in this area, and at the same time organize the possibilities for future research. Before presenting this classification scheme, it is useful to present one of the most concise representations of the implemen tation problem, called a Mount-Reiter diagram, a version of which originally appeared in Mount and Reiter ( 1 974). Figure 1 contains the basic elements of an implementation problem. The notation is as follows: £: The domain of environments. Each environment, e E £, consists of a set offeasible outcomes, A (e), a set of individuals, I (e) , and a preference profile R(e), where R; (e) is a weak order on A (e). 3 X: The outcome space. Note: A (e) � X for all e E £. F � {f : £ --;. X} : The welfare criterion (or social choice set) which specifies the set of acceptable mappings from environments to outcomes. An element, f, of F is called a social choice function. M = Mr x x MN : The message space. g : M --;. X: The outcome function. f.L = (M, g): The mechanism. IJ = The equilibrium concept that maps each f.L into IJ�"' � { u : £ --;. M } As an example to illustrate what these different abstract concepts might be, consider the domain of pure exchange economies. Each element of the domain would consist of a · · ·
.
3 Except where noted, we will assume that the set of feasible outcomes is a constant A that is independent of e, and the planner knows A. Furthermore, we will typically take the set of individuals I = { 1 , 2, . . . , N ) as fixed.
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set of traders, each of whom has an initial endowment, and the set of feasible outcomes would just be the set of all reallocations of the initial endowment. Many implementa tion results rely on domain restrictions, which in this illustration would involve standard assumptions such as strictly increasing, convex preferences, and so forth. The welfare criterion, or social choice set, might consist of all social choice functions satisfying a list of conditions such as individual rationality, Pareto optimality, interiority, envy freeness, and so forth. One common social choice set is the set of all selections from the Walrasian equilibrium correspondence. For this case the message space might be either an agent's entire preference mapping or perhaps his demand correspondence, and the "planner" would take on the role of the auctioneer. An example of an outcome function would be the allocations implied by a deterministic pricing rule (such as some selection from the set of market clearing prices), given the reported demands. Common equilib rium concepts employed in these settings are Nash equilibrium or dominant strategy equilibrium. The arrows of the Mount-Reiter diagram indicate that the diagram has the commu tative property that, under the equilibrium concept IJ, the set of desirable social choice functions defined by F correspond exactly to the outcomes that arise under the mech anism f.L. That is, F = g o IJ!L . When this happens, we say that " f.L implements F via IJ in £.". Whenever there exists some mechanism such that that statement is true, we say "F is implementable via IJ in £.". Implementation theory then looks at the rela tionship between domains, equilibrium concepts, welfare criteria, and implementing mechanisms, and the various questions that may arise about this relationship. The re mainder of this chapter summarizes a small part of what is known about this. Because this has been the main focus of the literature, the discussion here will concentrate pri marily on the existence question: Under what conditions on F, IJ, and £. does there exist a mechanism f.L such that f.L implements F via IJ in £. ?
1.2. Informational issues To this point, nothing has been said about what information can be used in the construc tion of a mechanism nor about what information the individuals have about £. . A com mon interpretation given to the implementation problem is that there is a mythical agent, called "the planner", who has normative goals and very limited information about the environment (typically one assumes that the planner only knows £. and X). The planner then must elicit enough information from the individuals to implement outcomes in a manner consistent with those normative goals. This requires creating incentives for such information to be voluntarily and accurately provided by the individuals. Nearly all of the literature in implementation theory assumes that the details of the environment are not directly verifiable by the planner, even ex post.4 Thus, implementation theory char acterizes the limits of a planner's power in a society with decentralized information.
4
There are some results on auditing and other ex post verification procedures. See, for example, Townsend (1979), Chander and Wilde ( 1 998) and the references they cite.
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The information that the individuals have about E is also an important consideration. This information is best thought of as part of the description of the domain. The main information distinction that is usually made in the literature is between complete in formation and incomplete information. Complete information models assume that e is common knowledge among the individuals (but, of course, unknown to the planner). Incomplete information assumes that individuals have some private information. This is usually modeled in the Harsanyi ( 1 967, 1 968) tradition, by defining an environment as a profile of types, one for each individual, where a type indexes an individual's informa tion about other individuals' types. In this manner, an environment (preference profile, set of feasible allocations, etc.) is uniquely defined for each possible type profile. One branch of implementation theory addresses a somewhat different informational issue. Given a social choice correspondence and a domain of environments, how much information about the environment is minimally needed to determine which outcome should be selected, and how close to this level of minimal information gatheting (infor mationally efficient) do different "natural" mechanisms come? In particular, this ques tion has been asked in the domain of neoclassical pure exchange environments, where the answer is that the Walrasian market mechanism is informationally efficient [Hurwicz ( 1 977); Mount and Reiter ( 1 97 4)] . With few exceptions that branch of implementation theory does not directly address questions of incentive compatibility of mechanisms. This chapter will not cover the contributions in that area.
1.3. Equilibrium: Incentive compatibility and uniqueness In principle, the equilibrium concept E could be almost anything. It simply defines a systematic rule for mapping environments into messages for arbitrary mechanisms. However, nearly all work in implementation theory and mechanism design restricts attention to equilibrium concepts borrowed from noncooperative game theory, all of which require the rational response property in one form or another. That is, each indi vidual, given their information, preferences, and a set of assumptions about how other individuals are behaving, and given a set of rules (M, g ) , adopts a rational response, where rationality is usually based on maximization of expected utility. 5 The requirement of implementation can be broken down into two components. The first component is incentive compatibility: 6 This is most transparent for the special case of social choice functions (i.e., F is a singleton). If a mechanism (M, g) implements a social choice function f , it must be the case that there is an equilibrium strategy profile
5
There are some exceptions, notably the use of maximin strategies [Thomson (1979)] and undominated strategies [Jackson (1992)], which do not require players to adopt expected utility maximizing responses to the strategies of the other players. 6 This is sometimes referred to as Truthful Implementability [Dasgupta, Hammond, and Maskin (1979)] because, in the framework where individual preferences and information are represented as "types", if the mechanism is direct in the sense that each individual is required to report their type (i.e., M = T), then the truthful strategy u (t) = t is an equilibrium of the direct mechanism f-l = (T, f).
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£ --+ M such that g o a = f. The second component is uniqueness: If a mechanism (M, g) implements a social choice function f, it must be the case, for all social choice functions h =/= f, that there is not an equilibrium strategy profile a ' such that g o a ' = h . For the more general case of the implementation of a social choice set, F , these two components extend in the natural way. If a mechanism (M, g) implements a social choice set F, it must be the case that, for each f E F there is an equilibrium strategy profile a such that g o a = f. If a mechanism (M, g) implements a social choice set F, it must be the case, for all social choice functions h rf. F, that there is not an equilibrium strategy profile a such that g o a = h . a :
1.4. The organization of this chapter The remainder of this paper is divided into three sections. Section 2 presents character izations of implementation under conditions of complete information for several differ ent equilibrium concepts. In particular, the relatively comprehensive characterization for Nash implementation (i.e., implementation in complete information domains via Nash equilibrium) is set out in considerable detail. The partial characterizations for refined Nash implementation (subgame perfect equilibrium, undominated Nash equilibrium, dominance solvable equilibrium, etc.) are then discussed. This part also describes the problem of implementation by games of perfect information, and a few results in the area, particularly with regard to "voting trees", are briefly discussed. Finally results for virtual implementation (both in Nash equilibrium and using refinements) are described, where social choice functions are only approximated as the equilibrium outcomes of mechanisms. Section 3 explains how the results for complete information are extended to incom plete information "Bayesian" domains, where environments are represented as collec tions of Harsanyi type-profiles, and players are assumed to have well-defined common knowledge priors about the distribution of types. Section 4 discusses some "difficult" problems in the area of implementation theory that have either been ignored or studied in only the simplest settings. This includes dy namic issues such as renegotiation of mechanisms and dynamic allocation problems, considerations of simplicity, robustness, and bounded rationality, and issues of incom plete control by the planner over the mechanism (side games played by the agents, or preplay communication).
2. Implementation under conditions of complete information
By complete information, we mean that individual preferences and feasible alterna tives are common knowledge among all the individuals. This does not mean that the planner knows these preferences. The planner is assumed to know only the set of pos-
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sible individual preference profiles and the set of feasible allocations.? For this reason, we simplify the notation considerably for this section of the chapter. First, we rep resent the domain by R, the set of possible preference profiles, with typical element R = (Rt , R2 , . . . , RN ), and the set of feasible alternatives is A. A social choice set can, without loss of generality, be represented as a correspondence F mapping R into subsets of A. We denote the image of F at R by F(R).
2.1. Nash implementation Consider a mechanism fL = (M, g) and a profile R. The pair (/L, R) defines an N -player noncooperative game. D EFINITION 1 . A message profile m* E M is called a Nash equilibrium of fL at R if, for all i E /, and for all m; E M;
g (m* ) R;g (m; , m"_ ; ) . Therefore, the definition of Nash implementability is: 2 . A social choice correspondence F is Nash implementable in R if there exists a mechanism fL = (M, g) such that: (a) For every R E R and for every x E F(R), there exists m* E M such that m* is a Nash equilibrium of fL at R and g(m*) = x . (b) For every R E R and for every y tj. F(R), there does not exist m* E M such that m* is a Nash equilibrium of fL at R and g(m*) = y .
D EFINITION
Alternatively, writing a* (R) as the set of Nash equilibria of fL at R , we can state (a) and (b) as: (a') F(R) s; g o a* (R) for all R E R, (b') g o a* (R) s; F(R) for all R E R. Condition (a) corresponds to what we have referred to as incentive compatibility and condition (b) is what we have referred to as uniqueness. We proceed from here by characterizing the implications of (a) and (b).
2.1.1. Incentive compatibility Suppose a social choice function f : R � A is implementable via Nash equilibrium. Then there exists a mechanism fL that implements f via 2J in £. What exactly does this 7
Nearly always, the set of feasible allocations is taken as fixed in implementation theory. A notable ex ception to this is Hurwicz, Maskin, Postlewaite (1995). In this paper, the mechanism has individuals report endowments as well as preferences to the planner, but it is assumed that it is impossible for an individual to overstate his endowment (although understatements are possible). See Hong (1996, 1998) and Tian (1 994) for extensions to Bayesian environments.
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mean? First, it means that there is a Nash equilibrium of fL that yields f as the equilib rium outcome function. With complete information this turns out to have very little bite. That is, examples of social choice functions that are not "incentive compatible" when the individuals have complete information are rather special. How special, you might ask. First, the examples must have only two individuals. This fact is quickly established below as Proposition 3 . P ROPOSITION 3 . In complete information domains with N 2, every social choice function f is incentive compatible (i.e., there exists a mechanism such that (a) is satisfied). >
Consider the following mechanism, which we call the agreement mechanism. M; = R for all i E /. That is, individuals report profiles. 8 Arbitrarily pick some ao E A. Partition the message space into two parts, Ma (called the agreement region) and Md (called the disagreement region). P ROOF.
Let
Ma = {m E M I 3j E /, R E R such that m; = R for all i =/= j}; Md = {m E M i m tf: Ma } . In other words, the agreement region consists of all message profiles where either every individual reports the same preference profile, or all except one individual report the same preference profile. The outcome function is then defined as follows.
{
m E Ma, g(m) = f(R) if m if E Md . Go
It is easy to see that if the profile of preferences is R, then it is a Nash equilibrium for all individuals to report m; = R, since unilateral deviations from unanimous agreement D will not affect the outcome. Therefore, (a) is satisfied. Therefore, incentive compatibility9 is not an issue when information is complete and N > 2. Of course the problem with this mechanism is that any unanimous message profile is a Nash equilibrium at any preference profile. Therefore, this mechanism does not satisfy the uniqueness requirement (b) of implementation. We return to this problem shortly, after addressing the question of incentive compatibility for the N = 2 case. 8 This mechanism could be simplified further by having each agent report a feasible outcome. 9 The reader should not confuse this with a number of negative results on implementation of social choice functions via Nash equilibrium when mechanisms are restricted to being "direct" mechanisms (M; = R j ) . If individuals do not report profiles, but only report their own component o f the profile (sometimes called privacy-preserving mechanisms), then clearly incentive compatibility can be a problem. This kind of incentive compatibility is usually called strategyproofness, and is closely related to the problem of implementation under the much stronger equilibrium concept of dominant strategy equilibrium. See Dasgupta, Hammond, and Maskin (1979).
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When N = 2 the outcome function of the mechanism used in Proposition 3 is not well defined, since a unilateral deviation from unanimous agreement is not well defined. If m 1 = R and m 2 = R', then it is unclear whether g(m) = f(R) or g(m) = f(R'). There are some simple cases where incentive compatibility is assured when N = 2. First, if there exists a uniformly bad outcome, w, with the property that, for all a E A, and for all R E R, aRi w, i = 1 , 2. In that case, the mechanism above can be modified so that Ma requires unanimous agreement, and a0 = w . Clearly any unanimous report of a profile is a Nash equilibrium regardless of the actual preferences of the individuals, so this modified mechanism satisfies (a) but fails to satisfy (b). A considerably weaker assumption, called nonempty lower intersection, is due to Dutta and Sen (199lb) and Moore and Repullo (1990). We state a slightly weaker ver sion below, which is sufficient for the incentive compatibility requirement (a) when N = 2. They define a slightly stronger version that is needed to satisfy the uniqueness requirement (b). 4. A social choice function f satisfies Weak Nonempty Lower Intersec tion 1 0 if, for all R, R' E R, such that R =I= R', :Jc E A such that f(R)Rtc and f(R')R�c. D EFINITION
The definition of social choice correspondences is similar: D EFINITION 5 . A social choice correspondence F satisfies Weak Nonempty Lower Intersection if for all R, R' E R, such that R =!= R', and for all a E f(R) and b E f(R') , :Jc E A such that aR 1 c and bR�c.
To see that this is a sufficient condition for (a), consider implementing the social choice function f. From Definition 4, we can define a function c(R, R') for R =I= R' with the property that j(R)R 1 c(R, R') and f(R')R�c(R, R'). We can then modify the mechanism above by:
{
g(m) = f(R) c(R', R)
if m 1 = m 2 = R, if m 1 = R and m 2 = R'.
This mechanism is illustrated in Figure 2. It is easy to check that weak nonempty lower intersection guarantees m = (R, R) is a Nash equilibrium when the actual profile is R. There are two interesting special cases where Nonempty Lower Intersection holds. The first is when there exists a universally "bad" outcome [Moore and Repullo ( 1 990)] with the property that it is strictly less preferred than all outcomes in the range of the social choice rule, for all agents, at all profiles in the domain. 1 1 This is satisfied by any 1 0 The stronger version, called Nonempty Lower Intersection, requires f(R)P c and 1 1 1 Notice that this is a joint restriction on the domain and the social choice function.
f(R 1) P2 c.
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R
2
R
R'
f (R)
C(R', R)
Player 1
R' C(R, R')
Figure
2.
j(R')
Nonempty Lower Intersection: f(R)R r C(R, R'), f(R')R;C(R, R ' ) f(R1)Rj C(R', R), j(R)R2 C(R', R) =? f(R) E N E(R), f(R') E N E(R').
(Weak)
and
nonwasteful social choice rule in exchange economies with free disposal and strictly in creasing preferences, since destruction of the endowment is a bad outcome. The second special case is any Pareto efficient and individually rational interior social choice corre spondence in exchange economies (with or without free disposal) with strictly convex and strictly increasing preferences and fixed initial endowments [Dutta and Sen (1991 b); Moore and Repullo ( 1 990)].
2.1.2. Uniqueness Clearly, incentive compatibility places few restrictions on Nash implementable social choice functions (and correspondences) with complete information. The second require ment of uniqueness is more difficult, and the major breakthrough in characterizing this was the classic paper of Maskin (1 999). In that paper, he introduces two conditions, which are jointly sufficient for Nash implementation when N ;:?: 3 . These conditions are called Mono tonicity and No Veto Power (NVP). D EFINITION
6. A social choice correspondence F is Monotonic if, for all
R, R' E R
(x E F(R), x � F(R') ) =} 3i E 1, a E A such that xRiaPfx. The agent i and the alternative a are called, respectively, the test agent and the test alternative. Stated in the contrapositive, this says simply that if is a socially desired alternative at R, and does not strictly fall in preference for anyone when the profile is changed to R', then must be a socially desired alternative at R'. Thus monotonic social x
x
x
choice correspondences must satisfy a version of a nonnegative responsiveness criterion with respect to individual preferences. In fact, this is a remarkably strong requirement
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for a social choice correspondence. For example, it rules out nearly any scoring rule, such as the Borda count or Plurality voting. Several other examples of nonmonotonic so cial choice functions in applications to bilateral contracting are given in Moore and Re pullo ( 1988). One very nice illustration of a nonmonotonic social choice correspondence is given in the following example. It is a variation on the "King Solomon's Dilemma" example of Glazer and Ma ( 1 989) and Moore ( 1 992), where the problem is to allocate a baby to its true mother. 2. 1.3.
Example 1
There are two individuals in the game (Ms. a and Ms.
/3), and four possible alternatives:
a = give the baby to Ms. a, b = give the baby to Ms. f3, c
= divide the baby into two equal halves and give each mother one half,
d = execute both mothers and the child. Assume the domain consists of only two possible preference profiles depending on whether a or f3 is the real mother, and we will call these profiles R and R ' respectively. They are given below: Ra = a
>- b >- c >- d, Rf3 = b >- c >- a >- d,
R� = a
>- c >- b >- d, R{; = b >- a >- c >- d.
The social choice function King Solomon wishes to implement is f (R) = a and f (R ' ) = b. This is not monotonic. Consider the change from R to R ' . Alternative a does not fall in either player's preference order as a result of this change. However, f ( R ') = b =/=- a, a contradiction of monotonicity. Notice however that this social choice function is incentive compatible since there is a universally bad outcome, d, which is ranked last by both players in both of their preference orders. A second example, from a neoclassical 2-person pure exchange environment illus trates the geometry of monotonicity. 2. 1.4.
Example 2
Consider allocation x in Figure 3 . Suppose x E f (R) where the indifference curves through x o f the two individuals are labelled R t and Rz respectively in that figure. Now consider some other profile R ' where Rz = R� , and R i is such that the lower contour set of x for individual ! has ex panded. Monotonicity would require x E f (R ' ) . Put another way (formally stated in the definition), if f is monotonic and x ¢ f ( R ' ) for some R ' -=f. R , then one of the two indi viduals must have an indifference curve through x that either crosses the R-indifference
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Agent 2
,-------,
Agent
I Figure 3. Illustration of monotonicity:
x E F(R) =? x E F(R ').
curve through x or bounds a strictly smaller lower contour set. Figure 4 illustrates the (generic) case in which the R ' -indifference curve of one of the individuals (individ ual 1 , in the figure) crosses the R -indifference curve through x. Thus, in this example agent 1 is the test agent. One possible test alternative a E A (an alternative required in the definition of monotonicity which has the property that x R; a P(x) is marked in that figure. 2. 1.5.
Necessary and sufficient conditions
Maskin (1999) proved that monotonicity is a necessary condition for Nash implemen tation.
THEOREM 7.
If F
is Nash implementable then F is monotonic.
Consider any mechanism f.L that Nash implements F and consider some x E F (R) and some Nash equilibrium message, m*, at profile R, such that g(m*) = x . Define the "option set" 12 for i at m * as
PROOF.
0; (m * ; f.L) = {a E A I 3m; E M; such that g (m; , m :..; ) = a } . 12 This is similar to the role of option sets in the strategyproofness literature.
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Agent 2
,-------------------------------------------------�
Agent
1 Figure 4. lllustration of tests agent and test allocation:
x R 1 a P{x .
That is, fixing the messages o f the other players at m*__ i , the range o f possible out comes that can result for some message by i under the mechanism fL is Oi (m* ; fL) . By the definition of Nash equilibrium, x( Ri a for all i and for all a E Oi (m*; fL). Now consider some new profile R' where x fj. F(R'). Since fL Nash implements F, it must be that m is not a Nash equilibrium at R'. Thus there exist some i and some alternative a E O i (m*; /L) such that a P(x . Thus a is the test alternative and i is the test agent as D required in Definition 6, with the property that x Ria P(x . The second theorem in Maskin (1999) was proved in papers by Williams (1984, Saijo (1988), McKelvey (1989), and Repullo (1987). 13 It provides an elegant and simple sufficient condition for Nash implementation for the case of three or more agents. This is a condition of near unanimity, called No Veto Power (NVP).
1986),
8. A social choice correspondence F satisfies No Veto Power (NVP) if, R E R and for all x E A, and i E I,
D EFINITION
for all
[xRjy for all j =/= i, 13
for all y
E
Y] =} x E F(R).
Groves ( 1979) credits Karl Vind for some o f the ideas underlying the now-standard constructive proofs of sufficiency, and writes down a version of the theorem that uses these ideas in the proof. That version of the theorem, as well as the version of the theorem proved by Williams (1984, 1 986), imposes stronger assumptions than in Maskin (1999).
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THEOREM 9 . If N ;?: 3 and F is Monotonic and satisfies NVP, then F is Nash imple mentable. PROOF.
The proof given here is constructive and is similar to one in Repullo ( 1 987). A very general mechanism is defined, and then the rest of the proof consists of demon strating that the mechanism implements any social choice function that satisfies the hypotheses of the theorem. This is usually how characterization theorems are proved in implementation theory. Consider the following generic mechanism, which we call the
agreement/integer mechanism: Mi = R
X
A
X
{0,
1 , 2, . . . }.
That is, each individual reports a profile, an allocation, and an integer. The outcome function is similar to the agreement mechanism, except the disagreement region is a bit more complicated, and agreement must be with respect to an allocation and a profile.
Ma
=
Md
=
E R , a E F (R) such that m i = (R, a, Zi) , where Z i = 0 for each i # j } ; {m E M I m r:f. Ma }. {m E M I jj,
R
The outcome function i s defined as follows. The outcome function i s constructed so that, if the message profile is in Ma , then the outcome is either a or the allocation an nounced by individual j, which we will denote ai . If the outcome is in Md , then the outcome is ak where k is the individual announcing the highest integer (ties broken by a predetermined rule). This feature of the mechanism has become commonly known as an integer game (although in actuality, it is only a piece of the original game form). Formally,
if m E Ma and aj Pj a , if m E Ma and a Rj aj , if m E Md and k = max{i E I I Z i ;?: ZJ for all j E /}. Recall that we must show that F (R) � NEJL (R) and NEJL (R) � F(R) for all R , where NEJL (R) = {a E A I jm E M such that a = g(m)Ri g(m; , m-i ) for all i E I, m; E Mt } is the set of Nash equilibrium outcomes to JL at R . ( 1 ) F (R) � NEJL (R). At any R , and for any a E F (R), there is a Nash equilibrium in which all in dividuals report m i = (R, a, 0) . Such a message lies in Ma and any unilateral deviation also lies in Ma . The only unilateral deviation that could change the outcome is a deviation in which some player j reports an alternative ai such that a Rj aj . Therefore, a is a Rrmaximal element of 0i (m; JL) for all j E I, so m = (R, a, 0) is a Nash equilibrium.
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(2)
NEJL (R) s; F(R).
This is the more delicate part of the proof, and is the part that exploits Monotonic ity and NVP. (Notice that part ( 1 ) of the proof above exploited only the assump tion that N ;? 3.) Suppose that m E NEJL(R) and g(m) = a rl. F(R). First no tice that it cannot be the case that all individuals are reporting ( R', a, 0) where a E F(R') for some R' E R . This would put the outcome in Ma and Monotonicity guarantees the existence of some j E I, b E A such that a Rj bPja, so player j is better off changing to a message ( b, ) which changes the outcome from a to b. Thus m; # m i for some i, j . Whenever this is the case, the option set for at least N - 1 of the agents is the entire alternative space, A. Since a rl. F (R) and F sat isfies NVP, it must be that there is at least one of these N - 1 agents, k, and some element c E A such that cPka. Since the option set for k is the entire alternative space, A, individual k is better off changing his message to (-, c, Zk) # m i where Zk > zi , j # k, which will change the outcome from a to c. This contradicts the hypothesis that m is a Nash equilibrium. 0 · ,
·
Since these two results, improvements have been developed to make the characteriza tion of Nash implementation complete and/or to reduce the size of the message space of the general implementing mechanism. These improvements are in Moore and Repullo ( 1 990), Dutta and Sen (199 l b), Danilov (1 992), 14 Saijo ( 1 988), Sjostrom (199 1 ) and McKelvey ( 1 989) and the references they cite. The last part of this section on Nash implementation is devoted to a simple application to pure exchange economies. It turns out the Walrasian correspondence satisfies both Monotonicity and NVP under some mild domain restrictions. First notice that in private good economies with strictly increasing preferences and three or more agents, NVP is satisfied vacuously. Next suppose that indifference curves are strictly quasi-concave and twice continuously differentiable, endowments for all individuals are strictly positive in every good, and indifference curves never touch the axis. It is well known that these conditions are sufficient to guarantee the existence of a Walrasian equilibrium and to further guarantee that all Walrasian equilibrium allocations in these environments are interior points, with every individual consuming a positive amount of every good in every competitive equilibrium. Finally, assume that "the planner" knows everyone's endowment. 15 1 4 O f these, Danilov (1992) establishes a particularly elegant necessary and sufficient condition (with three or more players), which is a generalization of the notion of monotonicity, called essential monotonicity. However, these results are limited somewhat by this assumption of universal domain. Nash implementable social choice correspondences need not satisfy essential monotonicity under domain restrictions. Yamato (1992) gives necessary and sufficient conditions on restricted domains.
15
This assumption can be relaxed. See Hurwicz, Maskin, and Postlewaite (1995). The Walrasian corre spondence can also be modified somewhat to the "constrained Walrasian correspondence" which constrains individual demands in a particular way. This modified competitive equilibrium can be shown to be imple mentable in more general economic domains in which Walrasian equilibrium allocations are not guaranteed to be locally unique and interior. See the survey by Postlewaite (1985), or Hurwicz (1986).
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Palfrey
Since the planner knows the endowments, a different mechanism can be constructed for each endowment profile. Thus, to check for monotonicity it suffices to show that the Walrasian correspondence, with endowments fixed and only preferences changing, is monotonic. If a is a Walrasian equilibrium allocation at R and not a Walrasian equilib rium allocation at R', then there exists some individual for whom the supporting price line for the equilibrium at R is not tangent to the R; indifference curve through a. But this is just the same as the illustration in Figure 4, and we have labelled allocation b as a "test allocation" as required by the monotonicity definition. The key is that for a to be a Walrasian equilibrium allocation at R and not a Walrasian equilibrium allocation at R' implies that the indifference curves through x at R and R' cross at x . A s mentioned briefly above, there are many environments and "nice" (from a nor mative standpoint) allocation rules that violate Monotonicity, and in the N = 2 case ("bilateral contracting" environments) NVP is simply too strong a condition to im pose on a social choice function. There are two possible responses to this problem. One possibility, and the main direction implementation theory has pursued, is that Nash equilibrium places insufficient restrictions on the behavior of individuals. This leads to consideration of implementation using refinements of Nash equilibrium, or refined Nash implementation. A second possibility is that implementability places very strong restrictions on what kinds of social choice functions a planner can hope to enforce in a decentralized way. If not all social choice functions can be implemented, then we need to ask "how close" we can get to implementing a desired social choice function. This has led to the work in virtual implementation. These two directions are discussed next.
16
2.2.
Refined Nash implementation
More social choice correspondences can be implemented using refinements of Nash equilibrium. The reason for this is straightforward, and is easiest to grasp in the case of N � 3. In that case, the incentive compatibility problem does not arise (Proposition 3), so the only issue is (ii) uniqueness. Thus the problem with Nash implementation is that Nash equilibrium is too permissive an equilibrium concept. A nonmonotonic so cial choice function fails to be implementable simply because there are too many Nash equilibria. It is impossible to have f ( R) be a Nash equilibrium outcome at R and at the same time avoid having a =/::. f ( R) also be a Nash equilibrium outcome at R. But of 1 6 One might argue to the contrary that in other ways Nash equilibrium places too strong a restriction on individual behavior. Both directions are undoubtedly true. Experimental evidence has shown that both of these are defensible. On the one hand, some refinements of Nash equilibrium have received experimental support indicating that additional restrictions beyond mutual best response have predictive value [Banks, Camerer, and Porter ( 1994)] . On the other hand, many experiments indicate that players are at best imperfectly rational, and even violate simple basic axioms such as transitivity and dominance. Thus, from a practical standpoint, it is very important to explore the implementation question under assumptions other than the simple mutual best response criterion of Nash equilibrium.
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course this is exactly the kind of problem that refinements of Nash equilibrium can be used for. The trick in implementation theory with refinements is to exploit the refine ment by constructing a mechanism so that precisely the "bad" equilibria (the equilibria whose outcomes lie outside of F) are refined away, while the other equilibria survive the refinement. 1 7 2.2.1.
Subgame perfect implementation
The first systematic approach to extending the Maskin characterization beyond Nash equilibrium in complete information environments was to look at implementation via subgame perfect Nash equilibrium [Moore and Repullo ( 1 988); Abreu and Sen ( 1 990)] . They find that more social choice functions can be implemented via subgame perfect Nash equilibrium than via Nash equilibrium. The idea is that sequential rationality can be exploited to eliminate certain bad equilibria. The following simple example in the "voting/social choice" tradition illustrates the point. 2.2.2.
Example 3
There are three players on a committee who are to decide between three alternatives, A = {a, b, c}. There are two profiles in the domain, denoted R and R'. Individuals 1 and 2 have the same preferences in both profiles. Only player 3 has a different preference order under R than under R'. These are listed below:
Rt = Ri = a >- b >- c, R3 = c >- a >- b,
R2 = R� = b >- c >- a, R; = a >- c >- b.
The following social choice function is Pareto optimal and satisfies the Condorcet criterion that an alternative should be selected if it is preferred by a majority to any other alternative:
f(R) = b,
f(R' ) = a.
This social choice function violates monotonicity since b does not go down in player 3 's rankings when moving from profile R to R' (and no one else's preferences change). Therefore it is not Nash implementab1e. However, the following trivial mech anism (extensive form game form) implements it in subgame perfect Nash equilib rium:
17
Earlier work by Farquharson ( 1957/1969), Moulin (1979), Crawford (1 979) and Demange ( 1984) in spe cific applications of multistage games to voting theory, bargaining theory, and exchange economies fore shadows the more abstract formulation in the relatively more recent work in implementation theory with refinements.
T.R. Palfrey
2290
1
b
"Pass" 2
Outcome = b a
Outcome =
a
c
Outcome =
c
Figure 5. Game tree for Example 2.
Stage 1 : Player 1 either chooses alternative b, or passes. The game ends if b is chosen. The game proceeds to stage 2 if player 1 passes. Stage 2: Player 3 chooses between a and c. The game ends at this point. The voting tree is illustrated in Figure 5 . To see that this game implements f, work back from the final stage. In stage 2 , player 3 would choose c in profile R and a i n profile R'. Therefore, player 1 's best response is to choose b in profile R and to pass in profile R'. Notice that there is another Nash equilibrium under profile R', where player 2 adopts the strategy of choosing c if player 1 passes, and thus player 1 chooses b in stage 1 . But of course this is not sequentially rational and is therefore ruled out by subgame perfection.
2.2.3. Characterization results Abreu and Sen (1990) provide a nearly complete characterization of social choice cor respondences that are implementable via subgame perfect Nash equilibrium, by giving a general necessary condition, which is also sufficient if N � 3 for social choice func tions satisfying NVP. This condition is strictly weaker than Monotonicity, in the follow ing way. Recall that monotonicity requires, for any R, R' and a E A, with a = f(R), a =!= f(R'), the existence of a test agent i and a test allocation b such that aRibR;a. That there is some player and some allocation that produces a preference switch with f (R) when moving from R to R'. The weakening of this resulting from the sequential rationality refinement is that the preference switch does not have to involve f (R) di rectly. Any preference switch between two alternatives, say b and c, will do, as long as these alternatives can be indirectly linked to f (R) in a particular fashion. We formally state this necessary condition and call it indirect monotonicity, 1 8 to contrast it with the 1 8 Abreu and Sen (1990) call it Condition a.
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direct linkage to ity.
f(R)
of the test alternative in the original definition of monotonic
1 0 . A social choice correspondence F satisfies indirect monotonicity if s; A such that F ( R) s; B for all R E R, and if for all R, R' and a E A , with a E F(R), a rt. F(R'), 3L < and 3 a sequence of agents {jo, . . . , h } and 3 sequences of alternatives {ao, . . . , aL+d , {bo, . . . , h } belonging to B such that: (i) akRjkak+ t , k = 0, 1 , . . . , L ; (ii) aL +t P} L aL ; (iii) bk Pjk ab k = 0, 1 , . . . , L; (iv) (aL+t R} L b Vb E B) ::::} (L = 0 or h - 1 = h ) . D EFINITION
there 3B
oo,
The key parts of the definition of indirect monotonicity are (i) and (ii). A less restric tive version of indirect monotonicity consisting of only parts (i) and (ii) was used first by Moore and Repullo ( 1988) as a weaker necessary condition for implementation in subgame perfect equilibrium with multistage games. The main two general theorems about implementation via subgame perfect imple mentation are the following. The proofs [Abreu and Sen ( 1 990)] are long and tedious and are omitted, although an outline of the proof for the sufficiency result is given. Similar results, but slightly less general, can be found in Moore and Repullo (1988). 1 1 (Necessity). If a social choice correspondence F is implementable via subgame peifect Nash equilibrium, then F satisfies indirect monotonicity. T HEOREM
THEOREM 1 2 (Sufficiency). If N � 3 and F satisfies NVP and indirect monotonicity, then F is implementable via subgame peifect Nash equilibrium.
This is an outline of the sufficiency proof. Since F satisfies indirect monotonic ity, there exists the required set B and for any (R, R', a) such that a E F(R) and a rt. F(R') there exists an integer L and the required sequences {}k(R, R', a)h=O, l , ... ,L and {ak(R, R', a ) h=O, i , . .. ,L+i that satisfy (i)-(iv) of Definition 6. In the first stage of the mechanism, all agents announce a triple of the form (m ; 1 , m i 2 , m;3 ) where m ; 1 E R , m ; 2 E A, and m i 3 E {0, 1 , . . . } . The first stage of the game then conforms fairly closely to the agreement/integer mechanism, with a minor exception. If there is too much dis agreement (there exist three or more agents whose reports are different) the outcome is determined by m; 2 of the agent who announced the largest integer. If there is unan imous agreement in the first two components of the message, so all agents send some (R , a , z;) and a E F(R), then the game is over and the outcome is a. The same is true if there is only one disagreeing report in the first two components, unless the dissenting report is sent by io(R, mw 1 , a), in which case the first of a sequence of at most L "bi nary" agreement/integer games is triggered in which either some agent gets to choose his most preferred element of B or the next in the sequence of binary agreement/integer PROOF.
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games is triggered. If the game ever gets to the (L + l ) st stage, then the outcome is aL+l and the game ends. The rest of the proof follows the usual order. First, one shows that for all R E R and for all a E F(R) there is a subgame perfect Nash equilibrium at R with a as the equilibrium outcome. Second, one shows that for all R E R and for all a r:J. F(R) there is no subgame perfect Nash equilibrium at R with a as the equilibrium out come. D In spite of its formidable complexity, some progress has been made tracing out the implications of indirect monotonicity for two well-known classes of implementation problems: exchange economies and voting. Moore and Repullo ( 1 988) show that any selection from the Walrasian equilibrium correspondence satisfies indirect monotonic ity, in spite of being nonmonotonic. There are also some results for the N = 2 case that can be found in Moore ( 1 992) and Moore and Repullo ( 1988) which rely on sidepay ments of a divisible private good. The case of voting-based social choice rules con trasts sharply with this. Abreu and Sen (1990), Palfrey and Srivastava ( 1 99 l a) and Sen (1 987) show that many voting rules fail to satisfy indirect monotonicity, as do most runoff procedures and "scoring rules" (such as the famous Borda rule). How ever, a class of voting-based social choice correspondences, including the Copeland rule, is implementable via subgame perfect Nash equilibrium [Sen (1987)] . Some re lated findings are in Moulin (1979), Dutta and Sen ( 1993), and the references they cite. There are a number of applications that exploit the combined power of sidepayments and sequential mechanisms. See Glazer and Ma ( 1 989), Varian ( 1 994), and Jackson and Moulin ( 1 990). Moore ( 1 992) also gives some additional examples.
2.2.4. Implementation by backward induction and voting trees In general, it is not possible to implement a social choice function via subgame perfect Nash equilibrium without resorting to games of imperfect information. At some point, it is necessary to have a stage with simultaneous moves. Others have investigated the im plementation question when mechanisms are restricted to games of perfect information. In that case, the refinement implied by solving the game in its last stage and working back to earlier moves, generates similar behavior as subgame perfect equilibrium. 19 Example 3 above illustrates how it is possible for nonmonotonic social choice func tions to be implemented via backward induction. The work of Glazer and Ma ( 1 989) illustrates how economic contracting and allocation problems similar in structure to Example 1 (King Solomon's Dilemma) can be solved with backward induction imple mentation if sidepayments are possible. Crawford's work (1977, 1979) on bargaining 1 9 In fact, it is exactly the same if players are assumed to have strict preferences. Much of the work in this area has evolved as a branch of social choice theory, where it is common to work with environments where A is finite and preferences are linear orders on A (i.e., strict).
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mechanisms20 proves in fairly general bargaining settings that games of perfect infor mation can be used to implement nonmonotonic social choice functions that are fair. The general problem of implementation by backward induction has been studied by Hererro and Srivastava (1 992) and Trick and Srivastava ( 1 996). The characterizations, unfortunately, are quite cumbersome to deal with, and the necessary conditions for im plementation via backward induction are virtually impossible to check in most settings. But some useful results have been found for certain domains. Closely related to the problem of implementation by backward induction is imple mentation by voting trees, using the solution concept of sophisticated voting as devel oped by Farquharson ( 1 957/1969). Sophisticated voting works in the following way. First, a binary voting tree is defined, which consists of an initial pair of alternatives, which the individuals vote between. Depending on which outcome wins a majority of votes, 21 the process either ends or moves on to another, predetermined pair and another vote is taken. Usually one of the alternatives in this new vote is the winner of the pre vious vote, but this is not a requirement of voting trees. The tree is finite, so at some point the process ends regardless of which alternative wins. Sophisticated voting means that one starts at the end of the voting tree, and, for each "final" vote, determines who will win if everyone votes sincerely at that last stage. Then one moves back one step to examine every penultimate vote, and voters vote taking account of how the final votes will be determined. Thus, as in subgame perfect Nash equilibrium, voters have perfect foresight about the outcomes of future votes, and vote accordingly. The problem of implementation by voting trees was first studied in depth by Moulin ( 1979), using the concept of dominance solvability, which reduces to sophisticated vot ing [McKelvey and Niemi ( 1 978)] in binary voting trees. There are two distinct types of sequential voting procedures that have been investigated in detail. The first type consists of binary amendment procedures. In a binary amendment procedure, all the alternatives (assumed to be finite) are placed in a fixed order, say, (a 1 , a2 , . . . , al A I). At stage 1, the first two alternatives are voted between. Then the winner goes against the next alterna tive in the list, and so forth. A major question in social choice theory, and for that matter, in implementation theory, is how to characterize the set of social choice functions that are implementable by binary amendment procedures via sophisticated voting. This work is closely related to work by Miller ( 1 977), Banks (1985), and others, which explores general properties of the majority-rule dominance relation, and following in the foot steps of Condorcet, looks at the implementability of social choice correspondences that satisfy certain normative properties. Several results appear in Moulin (1 986), who iden tifies an "adj acency condition" that is necessary for implementation via binary voting trees. For more details, the reader is referred to the chapter on Social Choice Theory by Moulin ( 1 994) in Volume II of this Handbook.
20
This includes the divide-and-choose method and generalizations of it. 21 It is common to assume an odd number of voters for obvious reasons. Extensions to permit even numbers of voters are usually possible, but the occurrence of ties clutters up the analysis.
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More recent results on implementation via binary voting trees are found in Dutta and Sen (1993). First, they show that implementability by sophisticated voting in binary voting trees implies implementability in backward induction using games of perfect information. They also show that several well-known selections from the top-cycle set22 are implementable, but that certain selections that have appealing normative properties are not implementable.
2.2.5. Normalform refinements There are some other refinements of Nash equilibrium that have been investigated for mechani sms in the normal form. These fall into two categories. The first category relies on dominance (either strict or weak) to eliminate outcomes that are unwanted Nash equi libria. This was first explored in Palfrey and Srivastava ( 1 99 1a) where implementation via undominated Nash equilibrium is characterized. Subsequent work that explores this and other variations of dominance-based implementation in the normal form includes Jackson ( 1 99 2), Jackson, Palfrey, and Srivastava ( 1 994), Sjostrom (199 1 ), Tatamitani ( 1 993) and Yamato ( 1 993). Using a different approach, Abreu and Matsushima ( 1 990) obtain results for implementation in iteratively weakly undominated strategies, if ran domized mechanisms can be used and small fines can be imposed out of equilibrium. The work by Abreu and Matsushima ( 1992a, 1992b), Glazer and Rosenthal ( 1 992), and Duggan (1993) extends this line of exploiting strategic dominance relations to refine equilibria by looking at iterated elimination of strictly dominated strategies and also in vestigating the use of these dominance arguments to design mechanisms that virtually implement (see Section 2.3 below) social choice functions. The second category of refinements looks at implementation via trembling hand per fect Nash equilibrium. The main contribution here is the work of Sjostrom ( 1 993). The central finding of the work in implementation theory using normal form refine ments is that essentially anything can be implemented. In particular, it is the case that dominance-based refinements are more powerful than refinements based on sequential rationality, at least in the context of implementation theory. A simple result is in Palfrey and Srivastava ( 1 99 1 a), for the case of undominated Nash equilibrium. Consider a mechanism t-t == (M, g). A message profile m* E M is undominated Nash equilibrium of f-t at R if, for all i E I, for all m; E M; , g(m*)R;g(m; , m':_) and there does not exist i E I and m; E M such that DEFINITION 1 3 .
called an
g(m ; , m �; ) R;g (m7, m � ;) g(m; , m �; ) P; g (m 7 , m �;)
for all m�; E M�; and for some m �; E M�i ·
22 The top-cycle set at R is the minimal subset, TC, of A, with the property that for all a , b such that and b ¢c TC, a majority strictly prefers committees.
a
to
b.
a E
TC
This set has a very prominent role in the theory of voting and
Ch. 61: Implementation Theory
In other words, m* is an undominated Nash equilibrium at rium and, for all i , m; is not weakly dominated.
2295
R if it is a Nash equilib
THEOREM 1 4 . Suppose R contains no profile where some agent is indifferent between all elements of A. If N � 3 and F satisfies NVP, then F is implementable in undomi nated Nash equilibrium. PROOF. The proof of this theorem is quite involved. It uses a variation on the agree ment/integer game, but the general construction of the mechanism uses an unusual tech nique, called tailchasing. Consider the standard agreement/integer game, JL, used in the proof of Theorem 9. If m* is a Nash equilibrium of JL at R, but g(m*) rf- f (R ) , then one can make m * dominated at R by amending the game in a simple way. Take some player i and two alternatives x, y such that x P; y. Add a message m; for a player i and a message for each of the other players j -=f. i, m j , such that
g (m r , m -; ) = g(m; , m - ; ) for all m _; -=f. m'_ ; , g(m; , m'_ ; ) = x. g(m7, m'_ ; ) = y, Now strategy m * is dominated at R. Of course, this is not the end of the story, since it is now possible that (m; , m*_ ; ) is a new undominated Nash equilibrium which still produces the undesired outcome a rf- f ( R) . To avoid this, we can add another message for i , m;' and another message for the other players j -=f. i, m 'J and do the same thing again. If we repeat this an infinite number of times, we have created an infinite sequence of strategies for i, each one of which is dominated by the next one in the sequence. The complication in the proof is to show that in the process of doing this, we have not disturbed the "good" undominated Nash equilibria at R and have not inadvertently 0 added some new undominated Nash equilibria. This kind of construction is illustrated in the following example.
2.2.6. Example 4 This example is from Palfrey and Srivastava ( 1 99 1 a, pp. 488-489).
A = {a , b, c, d}, N = 2, R = {R, R'}: a b cd
b c d F( R) = {a, b}, F (R') = {a}.
T.R. Palfrey
2296
It is easy to show that there is no implementation with a finite mechanism, and any implementation must involve an infinite chain of dominated strategies for one player in profile R'. One such mechanism is:
Player
2.2. 7.
M2l m l1 a m 21 c m 31 c m41 c
Player 2
M22 c b b b
M23 c d c c
M24 c d d c
Constraints on mechanisms
Few would argue that mechanisms of the sort used in the previous example solve the implementation problem in a satisfactory manner. 23 This concern motivated the work of Jackson (1992) who raised the issue of bounded mechanisms.
1 5 . A mechanism is bounded relative to R if, for all R E R, and m; E M; , m; is weakly dominated at R, then there exists an undominated (at R) message m ; E M; that weakly dominates m; at R. D EFINITION
if
In other words, mechanisms that exploit infinite chains of dominated strategies, as occurs in tailchasing constructions, are ruled out. Note that, like the best response crite rion, it is not just a property of the mechanism, but a joint restriction on the mechanism and the domain. Jackson (1992)24 shows that a weaker equilibrium notion than Nash equilibrium, called "undominated strategies", has a similar property to undominated Nash implementation, namely that essentially all social choice correspondences are im plementable. He shows that if mechanisms are required to be bounded, then very re strictive results reminiscent of the Gibbard-Satterthwaite theorem hold, so that almost no social choice function is implementable via undominated strategies with bounded mechanisms. However, these negative results do not carry over to undominated Nash implementation.
23
In fact, few would argue that any of the mechanisms used in the most general sufficiency theorems are particularly appealing. That is also the first paper to seriously raise the issue of mixed strategies. All of the results that have been described so far in this paper are for pure strategy implementation. Only very recently have results been appearing that explicitly address the mixed strategy problem. See for example the work by Abreu and Matsushima ( 1992a) on virtual implementation.
24
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2297
Following the work of Jackson ( 1 992), Jackson, Palfrey, and Srivastava ( 1 994) pro vide a characterization of undominated Nash implementation using bounded mecha nisms and requiring the best response property. They find that the boundedness restric tion, while ruling out some social choice correspondences, is actually quite permissive. First of all, all social choice correspondences that are Nash implementable are imple mentable by bounded mechanisms [see also Tatamitani (1 993) and Yamato ( 1993)]. Second, in economic environments with free disposal, any interior allocation rule is implementable. Furthermore, there are many allocation rules that fail to be subgame perfect Nash implementable that are implementable via undominated Nash equilibrium using bounded mechanisms. Roundedness is not the only special property of mechanisms that has been inves tigated. Hurwicz ( 1960) suggests a number of criteria for judging the adequacy of a mechanism. Saijo ( 1 988), McKelvey ( 1989), Chakravarti (1991), Dutta, Sen, and Vohra ( 1994 ), Reichelstein and Reiter ( 1 988), Tian ( 1 990) and others have argued that message spaces should be as small as possible and have given results about how small the mes sage spaces of implementing mechanisms can be. Related work focuses on "natural" mechanisms, such as restrictions to price-quantity mechanisms in economic environ ments [Saijo et al. ( 1996); Sjostrom ( 1995); Corchon and Wilkie (1995)] . Abreu and Sen ( 1 990) argue that mechanisms should have a best response property relative to the domain for which they are designed. Continuity of outcome functions as a property of implementing mechanisms has been the focus of several papers [Reichelstein ( 1 984); Postlewaite and Wettstein ( 1 989); Tian ( 1989); Wettstein (1990, 1992) ; Peleg (1996a, 1 996b) and Mas-Colell and Vives ( 1993)] .
2.3. Virtual implementation 2.3.1. Virtual Nash implementation A mechanism virtually implements a social choice function25 if it can (exactly) imple ment arbitrarily close approximations of that social choice function. The concept was first introduced by Matsushima ( 1988). 1t is immediately obvious that, regardless of the domain and regardless of the equilibrium concept, the set of virtually implementable social choice functions contains the set of all implementable social choice functions. What is less obvious is how much more is virtually implementable compared with what is exactly implementable. It turns out that it makes a big difference. One way to see why so much more is virtually implementable can be seen by refer ring back to Figure 3 . That figure shows how the preferences R and R' must line up in order for monotonicity to have any bite in pure exchange economies. As can readily
25 The work on virtual implementation limits attention to single-valued social choice correspondences. Since the results in this area are so permissive (i.e., few social choice functions fail to be virtually implementable), this does not seem to be an important restriction.
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be seen, this is not a generic picture. Rather, Figure 4 shows the generic case, where monotonicity places no restrictions on the social choice at R' if a = f(R). Virtual im plementation exploits the nongenericity of situations where monotonicity is binding. 26 It does so by implementing lotteries that produce, in equilibrium at R, f(R) with very high probability, and some other outcomes with very low probability. In finite or countable economic environments, every social choice function is virtually implementable if individuals have preferences over lotteries that admit a von Neumann Morgenstern representation and if there are at least three agents. 27 The result is proved in Abreu and Sen ( 1991) for the case of strict preferences and under a domain restriction that excludes unanimity among the preferences of the agents over pure alternatives. They also address the 2-agent case, where a nonempty lower intersection property is needed. A key difference between the virtual implementation construction and the Nash im plementation construction has to do with the use of lotteries instead of pure alternatives in the test pairs. In particular, virtual implementation allows test pairs involving lotter ies in the neighborhood (in lottery space) of f(R) rather than requiring the test pairs to exactly involve f(R). It turns out that by expanding the first allocation of the test pair to any neighborhood of f ( R), one can always find a test pair of the sort required in the definition of monotonicity. There are several ways to illustrate why this is so. Perhaps the simplest is to con sider the case of von Neumann-Morgenstern preferences for lotteries. If an individual maximizes expected utility, then his indifference surfaces in lottery space are parallel hyperplanes. For the case of three pure alternatives, this is illustrated in Figure 6. For this three alternative case, consider two preference profiles, R and R', which dif fer in some individual's von Neumann-Morgenstern utility function. This means that the slope of the indifference lines for this individual has changed. Accordingly, in every neighborhood of every interior lottery in Figure 6, there exists a test pair of lotteries such that this agent has a preference switch over the test pair of lotteries. Now consider a social choice function that assigns a pure allocation to each preference profile, but that fails to satisfy monotonicity. In other words, the social choice function assigns one of the vertices of the triangle in Figure 6 to each profile. We can perturb this social choice function ever so slightly so that instead of assigning a vertex, it assigns an interior lot tery, x, arbitrarily close to the vertex. This "approximation" of the social choice function satisfies monotonicity because there exists agent i (whose von Neumann-Morgenstern utilities have changed), and a lottery y such that x R; y R;x. In this way, every (interior)
26
This fact that monotonicity holds generically is proved formally in Bergin and Sen (1996). They show for classical pure exchange environments with continuous, strictly monotone (but not necessarily convex) preferences there exists a dense subset of utility functions that always "cross" (i.e., there are never tangencies of the sort depicted in Figure 2). In fact, more general lottery preferences can be used, as long as they satisfy a condition that guarantees individuals prefer lotteries that place more probability weight on more-preferred pure alternatives.
27
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2299 c
a
b
Figure 6. Vertices a, b, and c represent pure alternatives, with all other points representing lotteries over those alternatives. The indifference curves passing through lottery x for agent i under two von Neumann Morgenstern utility functions are labelled R; and R; , with the direction of preference marked with arrows. Lottery y satisfies x R; y R; x .
approximation of every pure social choice function in this simple example is monotonic and hence (if veto power problems are avoided) implementable. Abreu and Sen (199 1 ) prove that this simple construction outlined above for the case of von Neumann-Morgenstem preferences and lA I = 3 is very general. The upshot of this is that moving from exact to virtual implementation completely eliminates the necessity of monotonicity. 2.3.2.
Virtual implementation in iterated removal of dominated strategies
An even more powerful result is established in Abreu and Matsushima (1992a). They show that if only virtual implementation is required, then in finite-profile environments one can find mechanisms such that not only is there a unique Nash equilibrium that approximately yields the social choice function, but the Nash equilibrium is strictly dominance solvable. They exploit the fact that for each agent there is a function h from his possible preferences to lottery space, h, such that if R; =/= R;, then h(R;)R; h(R;) and h(R;>R;h (R;). The message space of agent i consists of a single report of i 's own preferences and multiple (ordered) reports of the entire preference profile, with the final outcome patching together pieces of a lottery, each piece of which is determined by some portion of the reported profiles. The payoff function is then constructed so that falsely reporting one's own type as R' at R will lead to individual i receiving the g(R') lottery instead of the g(R) lottery with some probability, so this false report is a strictly
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T.R. Palfrey
dominated strategy. Incentives are provided so that subsequent2 8 reports of the profile will "agree" with earlier reports in a particular way. The first defection from agreement is punished severely enough to make it unprofitable to defect from the truthful self reports of the first component of the message space. The degree of approximation can then be made as fine as one wishes simply by requiring a very large number of reported profiles. Formally, a message for i is a K + 1 vector m i = (mf, mf , . . . , mf ) , where the first component is an element of i 's set of possible preferences and the other K components are each elements of the set of possible preference profiles. The outcome function is then pieced together in the following way. Let 8 be some small positive number. With probability 81 I (where I is the number of players), the outcome is based only on mf, and equals h(mf), so i is strictly better off reporting mf honestly. With probability 8 2 I I agent i is rewarded if, for all k = 1 , . . . , K, m7 = m0 whenever m 7 = m0 for all
j J = i for all h < k. That is, i gets a small reward (in expected terms) for honestly revealing his preference, and then gets an order of magnitude smaller reward for always agreeing with the vector of first reports (including his own). These are the only pieces of the outcome function that are affected by m0. Clearly, for 8 small enough the first order loss overwhelms any possible second order gain from falsely reporting mf. Thus messages involving false reports of mf are strictly dominated. The remaining pieces of the outcome function (each of which is used with probabil 2 ity (1 - 8 - 8 ) I K) correspond to the final K components of the messages, where each agent is reporting a preference profile. If everyone agrees on the kth profile, then that kth piece of the outcome function is simply the social choice function at that commonly reported profile. For K large enough, the gain one can obtain from deviating and report ing m7 f. m0 in the kth piece can be made arbitrarily small. But the penalty for being the first to report m7 f. m0 is constant with respect to K , so this penalty will exceed any gain from deviating when K is large. Thus deviating at h = k + 1 can be shown to be dominated once all strategies involving deviations at h < k + 1 have been elim inated. Variations on this "piecewise" approximation technique also appear in Abreu and Matsushima (1990) where the results are extended to incomplete information (see below) and Abreu and Matsushima (1994) where a similar technique is applied to exact implementation via iterated elimination of weakly dominated strategies. 29 This kind of construction is quite a bit different from the usual Maskin type of con struction used elsewhere in the proofs of implementation theorems. It has a number of attractive features, one of which is the avoidance of any mixed strategy equilibria. 28 The term "subsequent" should not be interpreted as meaning that the profiles are reported sequentially, since the game is simultaneous-move. Rather, the vector of reported profiles is ordered, so subsequent refers to the reported profile with the next index number. Glazer and Rubinstein ( 1996) show that there is a similar sequential game that can be constructed which is dominance solvable following similar logic. Glazer and Perry ( 1996) show that this mechanism can be reconstructed as a multistage mechanism which can be solved by backward induction. Glazer and Rubinstein (1996) propose that this reduces the computa tional burden on the players.
29
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2301
In other constructions, mixed strategies are usually just ignored. This can be problem atic as an example of Jackson ( 1 992) shows that there are some Nash implementable social choice correspondences that are impossible to implement by a finite mechanism without introducing other mixed strategy equilibria. A second feature is that in finite do mains one can implement using finite message spaces. While this is also true for Nash implementation when the environment is finite, there are several examples that illus trate the impossibility of finite implementation in other settings. Palfrey and Srivastava ( 1 99 1 a) show that sometimes infinite constructions are needed for undominated Nash implementation, and Dutta and Sen ( 1994b) show that Bayesian Nash implementation in finite environments can require infinite message spaces. Glazer and Rosenthal (1 992) raise the issue that in spite of the obvious virtues of the implementing mechanism used in the Abreu and Matsushima (1 992a) proof, there are other drawbacks. In particular, Glazer and Rosenthal ( 1 992) argue that the kind of game that is implied by the mechanism is precisely the same kind of game that game theorists have argued as being fragile, in the sense that the predictions of Nash equilibrium are not a priori plausible. Abreu and Matsushima (1992b) respond that they believe iterated strict dominance is a good solution concept for predictive30 purposes, especially in the context of their construction. However, preliminary experimental findings [Sefton and Yavas (1996)] indicate that in some environments the Abreu-Matsushima mechanisms perform poorly. This is part of an ongoing debate in implementation theory about the "desirability" of mechanisms and/or solution concepts in the constructive existence proofs that are used to establish implementation results. The arguments by critics are based on two premises: ( 1 ) equilibrium concepts, or at least the ones that have been explored, do not predict equally well for all mechanisms; and (2) the quality of the existence result is diminished if the construction uses a mechanism that seems unattractive. Both premises suggest interesting avenues of future research. An initial response to ( 1 ) is that these are empirical issues that require serious study, not mere introspection. The obvious implication is that experimental3 1 work in game theory will be crucial to generating useful predictive models of behavior in games. This in turn may require a redirection of effort in implementation theory. For example, from the game theory experiments that have been conducted to date, it is clear that limited rationality considerations will need to be incorporated into the equilibrium concepts, as will statistical (as opposed to deterministic) theories of behavior. 32
30
I n implementation theory, it i s the predictive value o f the solution concept that matters. One can think of the solution concept as the planner's model for predicting outcomes that will arise under different mechanisms and in different environments. If the model predicts inaccurately, then a mechanism will fail to implement the planner's targeted social choice function. The use of controlled experimentation in settling these empirical questions is urged in Abreu and Mat sushima's (l992b) response to Glazer and Rosenthal (1992). See, for example, McKelvey and Palfrey (1995, 1996, 1998).
31
32
2302
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Palfrey
Possible responses to (2) are more complicated. The cheap response is that the im plementing mechanisms used in the proofs are not meant to be representative of mecha nisms that would actually be used in "real" situations that have a lot of structure. These are merely mathematical techniques, and any mechanism used in a "real" situation should exploit the special structure of the situation. Since the class of environments to which the theorems apply is usually very broad, the implementing mechanisms used in the constructive proofs must work for almost any imaginable setting. The question this response begs is: for a specific problem of interest, can a "reasonable " mechanism be found? The existence theorems do not answer this question, nor are they intended to. That is a question of specific application. So far, even with the alternative mech anisms of Abreu-Matsushima, the mechanisms used in general constructive existence theorems are impractical. However, some nice results for familiar environments exist [e.g., Crawford (1979); Moulin (1984); Jackson and Moulin (1990)] that suggest we can be optimistic about finding practical mechanisms for implementation in some com mon economic settings.
3. Implementation with incomplete information
This section looks at the extension of the results of Section 2 to the case of incomplete information. Just as most of the results above are organized around Nash equilibrium and refinements, the same is done in this section, except the baseline equilibrium con cept is Harsanyi's (1967, 1968) Bayesian equilibrium. 33 3.1.
The type representation
The main difference in the model structure with incomplete information is that a domain specifies not only the set of possible preference profiles, but also the information each agent has about the preference profile and about the other agents' information. We adopt the "type representation" that is familiar to literature on Bayesian mechanism design [see, e.g., Myerson (1985)]. An incomplete information domain34 consists of a set, I, of n agents, a set, A, of fea sible alternatives, a set of types, T;, for each agent i E I, a von Neumann-Morgenstern utility function for each agent, u; : T x A --+ n, and a collection of conditional proba bility distributions {q; (L; It;)}, for each i E I and for each t; E T; . There are a variety of familiar domain restrictions that will be referred to, when necessary, as follows:
(1) Finite types: I Ti I < oo . (2) Diffuse priors: q ; (L; I t;)
>
0 for all i E I, for all t; E T; , and for all L; E L; .
33
This should come as no surprise to the reader, since Bayesian equilibrium is simply a version of Nash equilibrium, adapted to deal with asymmetries of information.
34 Myerson (1985) calls this a Bayesian Collective Decision Problem.
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Private values: 35 u; (t; , L; , a) = u; (t; , t'_ 1 , a) for all i , ti , Li , t'_1 , a . Independent types: qi (L i \ti ) = qi (L i lt[) for all i, ti , t'_ i L i , a . (5) Value-distinguished types: For all i, t1 , t[ E T; , t1 i: t[, 3a, b such that ui (ti , L 1 , a) > Ui (ti , Li , b) and Ui (t[, L i , b) > Ui (t;, Li , a) for all Li E L 1 . A social choice function (or allocation rule) f : T � A assigns a unique outcome to each type profile. A social choice correspondence, F, is a collection of social choice (3) (4)
,
functions. The set of all allocation rules in the domain is denoted by X, so in general, we have f E F s; X. A mechanism fJ = (M, g) is defined as before. A strategy for i is a function mapping T; into M1 , denoted a; : Ti � M1 . We also denote type t; of player i 's interim utility of an allocation rule x E X by:
where E1 is the expectation over t. Similarly, given a strategy profile a in a mecha nism f-L, we denote type t1 of player i 's interim utility of strategy a in fJ by:
Ui (a, ti ) = Et { Ui (g (a (t) ) , t ) iti } .
3.2. Bayesian Nash implementation The Bayesian approach to full implementation with incomplete information was initi ated by Postlewaite and Schmeidler ( 1 986). Bayesian Nash implementation, like Nash implementation, has two components, incentive compatibility and uniqueness. The main difference is that incentive compatibility imposes genuine restrictions on social choice functions, unlike the case of complete information. When players have private infor mation, the planner must provide the individual with incentives to reveal that informa tion, in contrast to the complete information case, where an individual's report of his information could be checked against another individual's report of that information. Thus, while the constructions with complete information rely heavily on mutual audit ing schemes that we called "agreement mechanisms", the constructions with incomplete information do not. 3 6 DEFINITION 1 6 . A strategy
t1 E T;
a
U; (a, ti) � U1 (a;, a-i , t;)
is a Bayesian
equilibrium of fJ if, for all i
and for all
for all a; : T; � Mi .
35 In this case, we simply write ui (ti , a), since i 's utility depends only on his own type. 36 There are special exceptions where mutual auditing schemes can be used, which include domains in which there is enough redundancy of information in the group so that an individual's report of the state may be checked against the joint report of the other individuals. This requires a condition called Non-Exclusive Infor mation (NEI). See Postlewaite and Schmeidler (1986) or Palfrey and Srivastava (1986). Complete information is the extreme form of NEI.
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DEFINITION 1 7 . A social choice function f : T -+ A (or allocation rule x : T -+ A) is Bayesian implementable if there is a mechanism f.L = (M, g) such that there exists a Bayesian equilibrium of f.L and, for every Bayesian equilibrium, u, of M, f (t) = g(u (t)) for all t E T .
Implementable social choice sets are defined analogously. For the rest of this section, we restrict attention to the simpler case of diffuse types, defined above. Later in the chapter, the extension of these results to more general infor mation structures will be explained.
3.2.1. Incentive compatibility and the Bayesian revelation principle Paralleling the definition for complete information, a social choice function (or allo cation rule) is called (Bayesian) incentive compatible if and only if it can arise as a Bayesian equilibrium of some mechanism. The revelation principle [Myerson (1979); Harris and Townsend ( 1 9 8 1 )] is the simple proposition that an allocation rule x can arise as the Bayesian equilibrium to some mechanism if and only if truth is a Bayesian equilibrium of the direct37 mechanism, f.L = (T, x). Thus, we state the following. DEFINITION 1 8 . An allocation rule
ti , t; E Ti
x
is
incentive compatible if, for all i
and for all
3.2.2. Uniqueness Just as the multiple equilibrium problem can arise with complete information, the same can happen with incomplete information. In particular, direct mechanisms often have this problem (as was the case with the "agreement" direct mechanism in the complete information case). Consider the following example.
3.2.3. Example 5 This is based on an allocation rule investigated in Holmstrom and Myerson ( 1 983). There are two agents, each of whom has two types. Types are equally likely and statistically independent and individuals have private values. The alternative set is A = {a, b, c}. Utility functions are given by (u ij denotes the utility to type j of player i):
u 1 1 (a) = lu 1 1 (b) = l u 1 1 (c) = 0, u 21 (a) = l u 21 (b) = luzt (c) = 0, 3? A mechanism i s direct if M;
=
T1 for all i E I .
u 12 (a) = Ou !2 (b) = 4u !2 (c) = 9 , un(a) = 2un(b) = l uzz(c) = -8.
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2305
The following social choice function, f, is incentive compatible and efficient (where denotes the outcome when player 1 is type i and player 2 is type j):
fij
f1 1 = a,
f12 = b,
h 1 = c,
f22 = b.
It is easy to check that for the direct revelation mechanism (T, f), there is a "truthful" Bayesian equilibrium where both players adopt strategies of reporting their actual type, i.e., f is incentive compatible. However, there is another equilibrium of (T, f), where both players always report type 2 and the outcome is always b. We call such strategies in the direct mechanism deceptions, since such strategies involve falsely reported types. Denoting this deceptive strategy profile as a, it defines a new social choice function, which we call fa , defined by fa (t) = f(a(t))t. This illustrates that this particular al location rule is not Bayesian Nash implementable by the direct mechanism. However, it turns out to be possible to add messages, augmenting38 the direct mechanism into an "indirect" mechanism that implements f. One way to do this is by giving player 1 another pair of messages, call them "truth" and "lie", one of which must be sent along with the report of his type. The outcome function is then defined so that g(m) = f(t) if the vector of reported types is t and player one says "truth". If player 1 says "lie", then g(m) = f (ti , t�) where t1 is player 1 's reported type and t� is the opposite of player 2's reported type. This is illustrated in Figure 7. It is easy to check that if the players use the a deception above, then player 1 will announce "lie", which is not an equilibrium since player 2 would be better off always responding by announcing type 1 . In fact, simple inspection shows that there are no longer any Bayesian equilibria that lead to social choice functions different from f, and (truth, "truth") is a Bayesian equilibrium39 that leads to f. 3.2.4.
Bayesian monotonicity
Given that the incentive compatibility condition holds, the implementation problem boils down to determining for which social choice functions it is possible to augment the direct mechanism as in the example above, to eliminate unwanted Bayesian equilib ria. This is the so-called method of selective elimination [Mookherjee and Reichelstein ( 1990)] that is used in most of the constructive sufficiency proofs in implementation theory. Again paralleling the complete information case, there is a simple necessary condition for this to be possible, which is an "interim" version of Maskin' s monotonic ity condition (Definition 6), called Bayesian monotonicity. DEFINITION 1 9 . A social choice correspondence F is Bayesian monotonic if, for every f E F and for every joint deception a : T -+ T such that fa rj: F, 3i E I, ti E Ti ,
38 The terminology "augmented" mechanism is due to Mookheljee and Reichelstein ( 1990). 39 There is another Bayesian equilibrium that also leads to f . See Palfrey and Srivastava ( 1 993).
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T.R. Palfrey
Player
(t 1 , "truth" )
a
b
(t2, "truth" )
c
b
b
a
b
c
1
Figure 7. Implementing mechanism for Holmstrom-Myerson example.
and an allocation rule
Ui (f, t[) � Ui (y, t[).
y:T � A
such that
Ui Cfa , ti )
<
Ui (Ya , ti )
and, for all
t[ E Ti ,
The intuition behind this condition is simpler than i t looks. In particular, think of the relationship between f and fa being roughly the same in the above definition as the relationship between R and R ' , the difference being that with asymmetric information, we need to consider changes in the entire social choice function f, rather than limiting attention to the particular change in type profile from R to R' (or t to t', in the type notation). So, if fa ¢:. F (analogous to a ¢:. F(R') in the complete information formula tion), we need a test agent, i , and a test allocation rule y (analogous to test allocation, in the monotonicity definition), such that i 's (interim) preference between f and y is the reverse of his preference between fa and Ya (with the appropriate quantifiers and qualifiers included). Thus the basic idea is the same, and involves a test agent and a test allocation rule.
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3.2.5. Necessary and sufficient conditions We are now ready to state the main result regarding necessary conditions40 for Bayesian implementation. THEOREM 20. If F is Bayesian Nash implementable, then F is Bayesian monotonic, and every f E F is incentive compatible. PROOF. The necessity of incentive compatibility is obvious. The proof for necessity of Bayesian monotonicity follows the same logib as the prooffor necessity of monotonicity D with complete information (see Theorem 7). As with complete information, sufficient conditions generally require an allocation rule to be in the social choice correspondence if there is nearly unanimous agreement among the individuals about the "best" allocation rule. This is the role of NVP in the original Maskin sufficiency theorem. There are two ways to guarantee this. The first way (and by far the simplest) is to make a domain assumption that avoids the problem by rul ing out preference profiles where there is nearly unanimous agreement. The prototypical example of such a domain is a pure exchange economy. In that case, there is a great deal of conflict across agents, as each agent's most preferred outcome is to be allocated the entire societal endowment, and this most preferred outcome is the same regardless of the agent's type. Another related example includes the class of environments with side payments using a divisible private good that everyone values positively, the best-known case being quasi-linear utility. We present two sufficiency results, one for the case of pure exchange economies, and the second for a generalization. We assume throughout that information is diffuse and n ;): 3 . Consider a pure exchange economy with asymmetric information, E, with L goods and n individuals, where the societal endowment4 1 is given by w = (w 1 , . . . , wL ) . The alternative set, A, is the set of all nonnegative allocations of w across the n agents.42 The set of feasible allocation rules mapping T into A are denoted X. THEOREM 2 1 . Assume n ;): 3 and information is diffuse. A X is Bayesian Nash implementable if and only if it satisfies
x E
and Bayesian monotonicity.
social choice function incentive compatibility
40 There are other necessary conditions. For example, F must be closed with respect to common knowledge
concatenations. See Postlewaite and Schmeidler (1986) or Palfrey and Srivastava (1993) for details.
4 1 We will not be addressing questions of individual rationality, so the initial allocation of the endowment is
left unspecified.
42 One could permit free disposal as well, but this is not needed for the implementation result. The construc
tions by Postlewaite and Schmeidler (1986) and Palfrey and Srivastava (1989a) assume free disposal. We do not assume it here, but do assume diffuse information. Free disposability simplifies the constructions when information is not diffuse, by permitting destruction of the entire endowment (i.e., all agents receive 0) when the joint reported type profile is not consistent with any type profile in T .
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PROOF. Only if follows from Theorem 20. It is only slightly more difficult. Once again, we use a variation on the agreement/integer game, adapted to the incomplete informa tion framework. Notice, however, that there is always "agreement" with diffuse types, since each player is submitting a different component of the type profile, and all re ported type profiles are possible. Each player is asked to report a type and either an allocation rule that is constant in his own type (but can depend on other players' types) or a nonnegative integer. Thus:
X-i denotes the set of allocation rules that are constant with respect to t; . The agreement region is the set of message profiles where each player sends a reported type and "0". The unilateral disagreement region is the set of message profiles where exactly one agent reports a type and something other than "0". Finally, the disagreement region is the set of all message profiles with at least two agents failing to report "0". In the agreement region, the outcome is just x (t) , where t is the reported type profile. In the unilateral disagreement region the outcome is also just x (t), unless the disagreeing agent, i, sends y E X - i with the property that Ui (x, t;) ? Ui (y, t[ ) for all t[ E Ti . In that case, the outcome is y(t). In the disagreement region, the agent who submits the highest
where
integer43 is allocated w and everyone else is allocated 0. Notice how the mechanism parallels very closely the complete information mecha nism. The structure of the unilateral disagreement region is such that if all individuals are reporting truthfully, no player can unilaterally falsely report and/or disagree and be better off. By incentive compatibility it does not pay to announce a false type. The fact that y does not depend on the disagreer's types implies that it doesn't pay to report y and a false (or true) type. Therefore, there is a Bayesian equilibrium in the agreement re gion, where all players truthfully report their types. There can be no equilibrium outside the agreement region, because there would be at least two agents each of whom could unilaterally change his message and receive w . Thus the only possible other equilibria that might arise would be in the agreement region, where agents are using a joint de ception a . But the Bayesian monotonicity condition (which f satisfies by assumption) says that either Xa = x or there exists a y, and i, and a li , such that Ui (x, t[ ) ? Ui (y, t[ ) for all t[ E T; but Ui (Ya , ti) > Ui (Xa , ti ). Since it is easy to project y onto X -i [see Palfrey and Srivastava ( 1 993)] and preserve these inequalities, it follows that i is better D off deviating unilaterally and reporting y instead of "0". The extension of the above result to more general environments is simple, as long as individuals have different "best elements" that do not depend on their type. For each i , suppose that there exists an alternative bi such that U; (hi , t) ? Ui (a, t) for all a E A
43
In this region, if a player sends an allocation instead of an integer, this is counted as in favor of the agent with the lowest index.
"0". Ties are broken
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and t E for all t
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T , and further suppose that for all i , j it is the case that U; (b; , t) > U; (bJ , t) T. If this condition holds, we say players have distinct best elements.
E
THEOREM 22. lfn � 3, information is diffuse andplayers have distinct best elements, then f is Bayesian implementable if and only iff is incentive compatible and Bayesian monotonic. PROOF. Identical to the proof of Theorem 2 1 , except in the disagreement region the outcome is b; , where i is winner of the integer game. D Jackson ( 199 1) shows that the condition of distinct best elements can be further weak ened, and their result is summarized in Palfrey and Srivastava ( 1 993, p. 35). An even more general version that considers nondiffuse as well as diffuse information structures is in Jackson (199 1 ) . That paper identifies a condition that is a hybrid between Bayesian monotonicity and an interim version of NVP, called monotonicity-no-veto (MNV). The earlier papers by Postlewaite and Schmeidler ( 1986) and Palfrey and Srivastava (1987, 1 989a) also consider nondiffuse information structures. Dutta and Sen ( 199 1b) provide a sufficient condition for Bayesian implementation, when n � 3 and information is diffuse, that is even weaker than the MNV condition of Jackson (199 1 ) . They call this condition extended unanimity, and it, like MNV, incor porates Bayesian monotonicity. They also prove that when this condition holds and T is finite, then any incentive compatible social choice function can be implemented using a finite mechanism. They do this using a variation on the integer game, called a modulo game,44 which accomplishes the same thing as an integer game but only requires using the first n positive integers. Dutta and Sen ( 1 994b) raise an interesting point about the size of the message space that may be required for implementation of a social choice function. They present an example of a social choice function that fails their sufficiency condition (it violates una nimity and there are only two agents), but is nonetheless implementable via Bayesian equilibrium. But they are able to show that any implementing mechanisms must use an infinite message space, in spite of the fact that both A and T are finite. Dutta and Sen ( 1994a) extend their general characterization of Bayesian imple mentable social choice correspondences when n � 3 to the n = 2 case, using an interim version of the nonempty lower intersection property that they used in their n = 2 charac terization with complete information [Dutta and Sen (1991a)]. This complements some earlier work on characterizing implementable social choice functions by Mookherjee and Reichelstein ( 1 990). Dutta and Sen (1 994a) extend these results to characterize
44 The modulo game is due to Saijo (1988) and is also used in McKelvey (1989) and elsewhere. A potential weakness of a modulo game is that it typically introduces unwanted mixed strategy equilibria that could be avoided by the familiar greatest-integer game.
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Bayesian implementable social choice correspondences for the n = 2 case, for "eco nomic environments". 45 All of the results described above are restricted (either explicitly or implicitly) to fi nite sets of player types. Obviously for many applications in economics this is a strong requirement. Duggan ( 1994b) provides a rigorous treatment of the many difficult tech nical problems that can arise when the space of types is uncountable. He extends the re sults of Jackson (199 1 ) to very general environments, and identifies some new, more in clusive conditions that replace previous assumptions about best elements, private values, and economic environments. The key assumption he uses is called interiority, which is satisfied in most applications. 3.3.
Implementation using refinements ofBayesian equilibrium
Just as in the case of complete information, refinements permit a wider class of social choice functions to be implemented. These fall into two classes: dominance-based re finements using simultaneous-move mechanisms, and sequential rationality refinements using sequential mechanisms. In both cases, the results and proof techniques have sim ilarities to the complete information case. 3.3.1.
Undominated Bayesian equilibrium
The results for implementation using dominance refinements in the normal form are limited to undominated Bayesian implementation, where a nearly complete character ization is given in Palfrey and Srivastava (1 989b). An undominated Bayesian equilib rium is a Bayesian equilibrium where no player is using a weakly dominated strategy. There are several results, some positive and some negative. First, in private value en vironments with diffuse types and value-distinguished types, any incentive compatible allocation rule satisfying no veto power is implementable via undominated Bayesian equilibrium. The proof assumes the existence of best and worst elements46 for each type of each agent, but does not require No Veto Power. They also show that with non private values, some additional very strong restrictions are needed, and, moreover, the assumption of value distinction is critical. 47 45 The term economic is vague. "Informally speaking, an economic environment is one where it is possible to make some individual strictly better off from any given allocation in a manner which is independent of her type. This hypothesis, while strong, will be satisfied if there is a transferable private good in which the utilities of both individuals are strictly increasing". [Dutta and Sen (1994a, p. 52).] 46 Notice that if A is finite or more generally if A is compact and preferences are continuous, then best and worst elements exist. The proof can be extended to cover some special environments where best elements do not exist, such as the quasi-linear utility case. The assumption of value distinction is stronger than might appear. It rules out environments where two types of an agent differ only in their beliefs about the other agents. One can imagine some natural environ ments where value distinction might be violated, such as financial trading environments, where a key feature of the information structure involves what agents know about what other agents know.
47
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Two simple voting public goods examples illustrate both the power and the limitations (with common values) of the undominated refinement. 3.3.2.
Example 6
There are three agents, two feasible outcomes, A = {a, b}, private values, independent types, and each player can be one of two types. Type ct strictly prefers a to b and type f3 strictly prefers b to a, and the probability of being type ct is q , with q 2 � ! . The "best solution" according to almost any set of reasonable normative criteria is to choose a if and only if at least two agents are type ct. Surprisingly, this "majoritarian" solution, while incentive compatible, is not implementable via Bayesian equilibrium. It is fairly easy to show that for any mechanism that produces the majoritarian solution as a Bayesian equilibrium that mechanism will have another equilibrium in which outcome b is produced at every type profile. However, it is easy to see that the majoritarian solution is implementable via undominated Bayesian equilibrium, since it is the unique undominated Bayesian equilibrium48 outcome in the direct mechanism. 3.3.3.
Example 7
This is the same as Example 6 (two feasible outcomes, three agents, two independent types and type ct occurs with probability q 2 > !), except there are common values.49 The common preferences are such that if a majority of agents are type ct, then everyone prefers a to b, and if a majority of agents are type /3 , then everyone prefers b to a . We call these "majoritarian preferences". Obviously, there is a unique best social choice function for essentially any non-malevolent welfare criterion, which is the majoritar ian (and unanimous, as well ! ) solution: choose a if and only if at least two agents are type ct . First observe that because o f the common values feature of this example, players no longer have a dominant strategy in the direct game for agents to honestly report their true type. (Of course, truth is still a Bayesian equilibrium of the direct game.) One can show [Palfrey and Srivastava ( 1989b)] that this social choice function is not even im plementable in undominated Bayesian equilibrium. In particular, any mechanism which produces the majoritarian solution as an undominated Bayesian equilibrium always has another undominated B ayesian equilibrium where the outcome is always b. The point of Example 7 i s to illustrate that with common values, using refinements may have only limited usefulness in a Bayesian framework. We know from the work in complete information that implementation requires the existence of test agents and test
48 Notice that it is actually a dominant strategy equilibrium of the direct mechanism. This example illustrates how it is possible for an allocation rule to be dominant strategy implementable (and strategy-proof), but not Bayesian Nash implementable.
49
By common values we mean that every agent has the same type-contingent preferences. A related mecha nism design problem is explored in more depth by Glazer and Rubinstein (1998).
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pairs of allocations that involve (often delicate) patterns of preference reversal between preference profiles. Analogously, in the Bayesian setting such preference reversals must occur across type profiles. With private values, such preference reversals are easy to find. With common values and/or non-value-distinguished types, such preference reversals often simply do not exist, even in very natural examples of social choice functions that satisfy No Veto Power and N > 2.
3.3.4. Virtual implementation with incomplete information For virtual implementation, Abreu and Matsushima have an extension of their complete information paper on the use of iterated elimination of strictly dominated strategies [Abreu and Matsushima ( 1990)] for implementation in incomplete information environ ments. They show that, under a condition they call measurability and some additional minor restrictions on the domain, any incentive compatible social choice function de fined on finite domains can be virtually implemented by iterated elimination of strictly dominated strategies. Abreu and Matsushima ( 1994) conjecture that with some addi tional assumptions (such as the ability to use small monetary transfers) one may be able to obtain exact implementation via iterated elimination of weakly dominated strategies in finite incomplete information domains. Duggan (1997) looks at the related issue of virtual implementation in Bayesian equi librium (rather than iterated elimination of dominated strategies). He shows that the measurability of Abreu and Matsushima (1990) is not necessary for virtual Bayesian implementation, and uses a condition called incentive consistency. Serrano and Vohra ( 1999) have recently shown that both measurability and incen tive consistency are stronger conditions than originally implied in these papers. Results and examples in that paper provide insight into the kinds of domain restrictions that are required for the permissive virtual implementation results. In a sequel to that pa per, Serrano and Vohra (2000) identify a useful and easy-to-check domain restriction, called type diversity. They show that any incentive-compatible social choice function is virtually Bayesian implementable as long as the environments satisfy this condition. Type diversity requires that different types of an agent have different interim preferences over pure alternatives. In private-values models, it reduces to the familiar condition of value-distinguished types [Palfrey and Srivastava (1989b)] .We turn next to the question of implementation using sequential rationality refinements, where results parallel (to an extent) the results for subgame perfect implementation.
3.3.5. Implementation via sequential equilibrium There are some papers that partially characterize the set of implementable social choice functions for incomplete information environments using the equilibrium refinement of sequential equilibrium. The main idea behind these characterizations is the same as the ideas behind the results for subgame perfect equilibrium implementation under condi tions of complete information. Instead of requiring a test pair involving the social choice
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function, x, a s i s required in Bayesian monotonicity, all that i s needed i s some (interim) preference reversal between some pair of allocation rules, plus an appropriate sequence of allocation rules that indirectly connect x with the test pair of allocation rules. The details of the conditions analogous to indirect monotonicity for incomplete in formation are messy to state, because of quantifiers and qualifiers that relate to the posterior beliefs an agent could have at different stages of an extensive form game in which different players are adopting different deceptions. However, the intuition behind the condition is similar to the intuition behind Condition a in Abreu and Sen (1 990). As with the necessary and sufficient results for Bayesian implementation, results are easiest to state and prove for the special case of economic environments, where No Veto Power problems are assumed away and where there are at least three agents. Bergin and Sen ( 1 998) have some results for this case. They identify a condition which is sufficient for implementation by sequential equilibrium in a two-stage game. That paper also makes the point that with incomplete information there exist social choice functions that are implementable sequentially, but are not implementable via un dominated Bayesian equilibrium (or via iterated dominance), and are not even virtually implementable. This contrasts with the complete information case, where any social choice function in economic environments that is implementable via subgame perfect Nash equilibrium is also implementable via undominated Nash equilibrium. They are able to obtain these very strong results by showing that the "consistent beliefs" condi tion of sequential equilibrium can be exploited to place restrictions on equilibrium play in the second stage of the mechanism. Baliga ( 1999) also looks at implementation via sequential equilibrium, limited to finite stage extensive games. His paper makes additional restrictions of private values and independent types, which lead to a significant simplification of the analysis. A more general approach is taken in Brusco (1995) who does not limit himself to stage games nor to economic environments. He looks at implementation via per fect Bayesian equilibrium and obtains the incomplete information equivalent to indi rect monotonicity which he calls "Condition {3+". This condition is then combined with No Veto Power in a manner similar to Jackson's ( 1 99 1 ) monotonicity-no-veto condition to produce sequential monotonicity-no-veto. His main theorem50 is that any incentive-compatible social choice function satisfying SMNV is implementable in per fect Bayesian equilibrium. He also identifies a weaker condition than fJ + (called Con dition fJ), which he proves is necessary for implementation in perfect Bayesian equilib rium. However, Brusco's (1995) results are weaker than Bergin and Sen (1998) because his conditions on the requisite sequence of test allocations include a universal quanti fier on beliefs that makes it much more difficult to guarantee existence of the sequence. Bergin and Sen show that the very tight condition of belief consistency can replace the universal quantifier. Loosely speaking, Brusco's results exploit only the sequential
50 The main theorem is stated more generally. In particular he allows for social choice correspondences, which means that the additional restriction of closure under the common knowledge concatenation is required.
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rationality part of sequential equilibrium, while Bergin and Sen exploit both sequen tial rationality and belief consistency. This seemingly minor distinction actually makes quite a difference in proving what can be implemented. Duggan ( 1 995) focuses on sequentially rational implementation5 1 in quasi-linear en vironments where the outcomes are lotteries over a finite set of public projects (and transfers). He shows that any incentive-compatible social choice function is imple mentable in private-values environments with diffuse priors over a finite type space if there are three or more agents. He also shows that these results can be extended, with some modifications, in a number of directions: two agents; the domain of exchange economies; infinite type spaces; bounded transfers; nondiffuse priors; and social choice correspondences. He also shows how a "belief revelation mechanism" can be used if the planner does not know the agents' prior beliefs, as long as these prior beliefs are common knowledge among the agents.
4. Open issues
Open problems in implementation theory abound. Several issues have been explored at only a superficial level, and others have not been studied at all. Some of these have been mentioned in passing in the previous sections. There are also numerous untied loose ends having to do with completing full characterizations of implementability under the solution concepts discussed above.
4. 1. Implementation using perfect information extensive forms and voting trees Among these uncompleted problems is implementation via backward induction and the closely related problem of implementing using voting trees. If this class of implementa tion problems has a shortcoming, it is that extensions to the more challenging problem of implementation in incomplete information environments are limited. The structure of the arguments for backward induction implementation fails to extend nicely to incom plete information environments, as we know from the literature on sophisticated voting with incomplete information [e.g., Ordeshook and Palfrey ( 1988)] .
4.2. Renegotiation and information leakage Many of the above constructions have the feature that undesirable (e.g., Pareto ineffi cient, grossly inequitable, or individually irrational) allocations are used in the mech anism to break unwanted equilibria. The simplest examples arise when there exists a universally bad outcome that is reverted to in the event of disagreement. This is not nec essarily a problem in some settings, where the planner's objective function may conflict 5 1 Duggan (1995) defines sequentially rational implementation as implementation simultaneously in Perfect Bayesian Equilibrium and Sequential Equilibrium.
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with Pareto optimality from the point of view of the agents (as in many principal-agent problems). However, in some settings, most obviously exchange environments, one usu ally thinks of the mechanism as being something that the players themselves construct in order to achieve efficient allocations. In this case, one would expect agents to renego tiate outcomes that are commonly known among themselves to be Pareto dominated. 5 2 Maskin and Moore ( 1 999) examine the implications of requiring that the outcomes al ways be Pareto optimal. This approach has the virtue of avoiding mixed strategy equi libria and also avoiding implausibly bad outcomes off the equilibrium path. It has the defect of implicitly permitting the outcome function of the mechanism to depend on the preference profile, which makes the specification of renegotiation somewhat arbitrary. Ideally, one would wish to specify a bargaining game that would arise in the event that an outcome is reached which is inefficient. 5 3 But then the bargaining game itself should be considered part of the mechanism, which leads to an infinite regress problem. More generally, the planner may be viewed directly as a player, who has state contingent preferences over outcomes, just as the other players do. The planner has prior beliefs over the states or preference profiles (even if the players themselves have complete information). Given these priors, the planner has induced preferences over the allocations. This presents a commitment problem for the planner, since the outcome function must adhere to these preferences. This places restrictions both on the mecha nism that can be used and also on the social choice functions that can be implemented. Several papers have been written recently on this subject, which vary in the assumptions they make about the extent to which the planner can commit to a mechanism, the extent to which the planner may update his priors after observing the reported messages, and the extent to which the planner participates directly in the mechanism. The first paper on this subject is by Chakravarti, Corchon, and Wilkie ( 1 994) and assumes that the so cial choice function must be consistent with some prior the planner might have over the states, which implies that the outcome function is restricted to the range of the social choice function. The planner is not an active participant and does not update beliefs based on the messages of the players, nor does he choose an outcome that is optimal given his beliefs. Baliga, Corchon, and Sjostrom ( 1 997) obtain results with an actively participating planner,5 4 who acts optimally, given the messages of the players and can not commit ex ante to an outcome function. Thus, the mechanism consists of a message space for the players and a planner's strategy of how to assign messages to outcomes. This strategy replaces the familiar outcome function, but is required to be sequentially rational. Thus, the mechanism is really a two-stage (signalling) game, and an equilib rium of the mechanism must satisfy the conditions of Perfect Bayesian Equilibrium. B aliga and Sjostrom ( 1 999) obtain further results on interactive implementation with an
5 2 One doesn't have to look very hard to find counterexamples to this in the real world. Institutional structures (such as courts) are widely used to enforce ex post inefficient allocations in order to provide salient incentives. Some forms of criminal punishments, such as incarceration and physical abuse, fall into this category. See, for example, Aghion, Dewatripont, and Rey ( 1994) or Rubinstein and Wolinsky (1992). 54 That is, the planner also submits messages. They call this "interactive implementation".
53
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uninformed planner who can commit to an outcome function, and who also participates in the message-sending stage. Another sort of renegotiation arises in incomplete information settings if the social choice function calls for allocations that are known by at least some of the players to be inefficient. In particular, for certain type realizations, some players may be able to propose to replace the mechanism with a new one that all other types of all other players would unanimously prefer to the outcome of the social choice function. This problem of lack of durability55 [Holmstrom and Myerson ( 1983)] opens up another kind of renego tiation problem, which may involve the potential leakage of information between agents in the renegotiation process. Related issues are also addressed in Sjostrom (1 996), and in principal-agent settings by Maskin and Tirole (1992) and in exchange economies by Palfrey and Srivastava ( 1993). Also related to this kind of renegotiation is the problem of preplay communication among the agents. It is well known that preplay communication can expand the set of equilibria of a game, and a similar thing can happen in mechanism design. This can occur because information can be transmitted by preplay communication and be cause communication opens up possibilities for coordination that were impossible to achieve with independent actions. In nearly all of the implementation theory research, it is assumed that preplay communication is impossible (i.e., the message space of the mechanism specifies all possible communication). An exception is Palfrey and Srivas tava (199 1 b), which explicitly looks at designing mechanisms that are "communication proof", in the sense that the equilibria with arbitrary kinds of preplay communication are interim-payoff-equivalent to the equilibria without prep lay communication. They show that for a wide range of economic environments one can construct communication proof mechanisms to implement any interim-efficient, incentive-compatible allocation rule. Chakrovorti ( 1 993) also looks at implementation with preplay communication. 4.3.
Individual rationality and participation constraints
A close relative to the renegotiation problem is individual rationality. Participation con straints pose similar restrictions on feasible mechanisms as do renegotiation constraints, and have similar consequences for implementation theory. In particular, individual ra tionality gives every player a right to veto the final allocation, which restricts the use of unreasonable outcomes off the equilibrium path, which are often used in constructive proofs to knock out unwanted equilibria. Jackson and Palfrey (200 1) show that individ ual rationality and renegotiation can be treated in a unified structure, which they call
voluntary implementation.
Ma, Moore, and Turnbull ( 1 988) investigate the effect of participation constraints on the implementability of efficient allocations in the context of an agency problem of
55
Closely related to this are the notions of ratifiability and secure allocations [Cramton and Palfrey ( 1 995)] and stable allocations [Legros (1990)].
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Demski and Sappington ( 1984). 56 I n the mechanism originally proposed b y Demski and Sappington (1984), there are multiple equilibria. Ma, Moore, and Turnbull ( 1 988) show that a more complicated mechanism eliminates the unwanted equilibria. Glover ( 1 994) provides a simplification of their result.
4.4. Implementation in dynamic environments Many mechanism design problems and allocation problems involve intertemporal allo cations. One obvious example is bargaining when delay is costly. In that case both the split of the pie and the time of agreement are economically important components of the final allocation. Recently, Rubinstein and Wolinsky ( 1 992) looked at the renegotiation proof problem in implementation theory by appending an infinite horizon bargaining game with discounting to the end of each inefficient terminal node. This is an alterna tive approach to the same renegotiation problem that Maskin and Moore (1999) were concerned about. However, like the rest of implementation theory, their interest is in im plementing static allocation rules (i.e., no delay) in environments that are (except for the final bargaining stages) static. This is true for all the other sequential game constructions in implementation theory: time stands still while the mechanism is played out. Intertemporal implementation raises additional issues. Consider, for example, a set ting in which every day the same set of agents is confronted with the next in a series of connected allocation problems, and there is discounting. A preference profile is not an infinite sequence of "one shot" profiles corresponding with each time period. A social choice function is a mapping from the set of these profile sequences into allocation se quences. Renegotiation-proofness would impose a natural time consistency constraint that the social choice function would have to satisfy from time t onward, for each t. With this kind of structure one could begin to look at a broader set of economic issues related to growth, savings, intertemporal consumption, and so forth. Jackson and Palfrey (1998) follow such an approach in the context of a dynamic bar gaining problem with randomly matched buyers and sellers. Many buyers and sellers are randomly matched in each period, and bargain under conditions of complete in formation. Those pairs who reach agreements leave the market, and the unsuccessful pairs are rematched in the following period. This continues for a finite number of peri ods. The matching process is taken as exogenous, and the implementation problem is to design a single bargaining mechanism to implement the efficiently dynamic allocation rule (which pairs of agents should trade, as a function of the time period), subject to the constraints of the matching process and individual rationality. They identify necessary and sufficient conditions for an allocation rule to be implemented and show that it is often the case that constrained efficient allocations are not implementable under any bargaining game. 56 There is a large and growing literature on implementation theoretic applications to agency problems. The techniques used there apply many of the ideas surveyed in this chapter. See, for example, Arya and Glover (1995), Arya, Glover, and Rajan (2000), Ma (1988), and Duggan (1998).
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There are some very simple intertemporal allocation problems that could be investi gated as a first step. One example is the one-sector growth model of Boylan et al. (1996) which compares different political mechanisms for deciding on investments. As a sec ond example, Bliss and Nalebuff (1984) look at an intertemporal public goods problem. There is a single indivisible public good which can be produced once and for all at any date t = 1 , 2, 3 , . . . , and preferences are quasilinear with discounting. The production technology requires a unit of private good for the public good to be provided. Thus, an allocation is a time at which the public good is produced and an infinite stream of taxes for each individual, as a function of the profile of preferences for the public good. Bliss and Nalebuff (1984) look at the equilibrium of a specific mechanism, the volun tary contribution mechanism. At each point in time an individual must decide whether or not to privately pay for the public good, depending on their type. The unique equi librium is for types that prefer the public good more strongly to pay earlier. Thus the public good is always produced by having the individual with the strongest preference for the public good pay for it, and the time of delay before production depends on what the highest valuation is and on the distribution of types. One could generalize this as a dynamic implementation problem, which would raise some interesting questions: What other allocation rules are implementable in this setting? Is the Bliss and Nalebuff ( 1984) equilibrium allocation rule interim incentive efficient?57 4.5.
Robustness of the mechanism
The approach of double implementation looks at robustness with respect to the equi librium concept. For example, Tatamitani ( 1993) and Yamato ( 1993) both consider im plementation simultaneously in Nash equilibrium and undominated Nash equilibrium. The idea is that if we are not fully confident about which equilibrium concept is more reasonable for a certain mechanism, then it is better if the mechanism implements a social choice function via two different equilibrium concepts (say, one weak and one strong concept). Other papers adopting this approach include Peleg ( 1996a, 1996b) and Corchon and Wilkie ( 1995). The constructive proofs used by Jackson, Palfrey, and Sri vastava ( 1 994) and Sjostrom ( 1994) also have double implementation properties. A second approach to robustness is to require continuity of the implementing mech anism. This was discussed briefly earlier in the chapter. Continuity will protect against local mis-specification of behavior by the agents. Duggan and Roberts (1997) provide a formal justification along these lines. 58 On the other hand, the preponderance of dis continuous mechanisms in real applications (e.g., auctions) suggests that the issue of mechanism continuity is of little practical concern in many settings. Related to the con tinuity requirement is dynamical stability, under various assumptions of the adjustment
57 Notice that it is not ex post efficient since there is always delay in producing the public good. 58 Cotchon and Ortuno-Ortin (1995) consider the question of robustness with respect to the agents' prior
beliefs.
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path. See Hurwicz ( 1994), Cabrales ( 1 999), Kim ( 1 993). Chen (2002) and Chen and Tang ( 1 998) have experimental results demonstrating that stability properties of mech anisms can have significant consequences. There is also the issue of robustness with respect to coalitional deviations. Implemen tation via strong Nash equilibrium [Mask:in ( 1 979); Dutta and Sen ( 199 1c); Suh (1997)] and coalition-proof equilibrium [Boylan ( 1 998)] are one way to address these issues. The issue of collusion among the agents is related to this [Baliga and Brusco (2000); Sjostrom ( 1 999)] . These approaches look at robustness with respect to profitable devi ations by coalitions. Eliaz (1999) has looked at implementation via mechanisms where the outcome function is robust to arbitrary deviations by subsets of players, a much stronger requirement. An alternative approach to robustness, which is related to virtual implementation, involves considerations of equilibrium models based on stochastic choice, 59 so that so cial choice functions may be approximated "on average" rather than precisely. Im· plementation theory (even the relaxed problem of virtual implementation) so far has investigated special deterministic models of individual behavior. The key assumption for obtaining results is that the equilibrium model that is assumed to govern individual behavior under any mechanism is exactly correct. Many of the mechanisms have no room for error. One would generally think of such fragile mechanisms as being non robust. Similarly (especially in the Bayesian environments), the details of the environ ment, such as the common knowledge priors of the players and the distribution of types, are known to the planner precisely. Often mechanisms rely on this exact knowledge. It should be the case that if the model of behavior or the model of the environment is not completely accurate, the equilibrium behavior of the agents does not lead to outcomes too far from the social choice function one is trying to implement. This problem suggests a need to investigate mechanisms that either do not make spe· cial use of detailed information about the environment (such as the distribution of types) or else look at models that permit statistical deviation from the behavior that is predicted under the equilibrium model. In the latter case, it may be more natural to think of social choice functions as type-contingent random variables rather than as deterministic func tions of the type profile. Related to the problem of robustness of the mechanisms and the possible use of statistical notions of equilibrium is bounded rationality. The usual defense for assuming in economic models that all actors are fully rational, is that it is a good first cut on the problem and often captures much of the reality of a given economic situation. In any case, most economists regard the fully rational equilibrium as an ap propriate benchmark in most situations. Unfortunately, since implementation theorists and mechanism designers get to choose the economic game in very special ways, this
60
59 One such approach is quantal response equilibrium [McKelvey and Palfrey (1995, 1996, 1998)]. 60 Note the subtle difference between this and virtual implementation. In the virtual implementation ap proach, something is virtually implementable if a nearby social choice function can be exactly implemented. The players do not make choices stochastically, although the outcome functions of the mechanisms may use lotteries.
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rationale loses much of its punch. It may well be that the games where rational models predict poorly are precisely those games that implementation theorists are prone to de signing. Integer games, modulo games, "grand lottery" games like those used in virtual implementation proofs, and the enormous message spaces endemic to all the general constructions would seem for the most part to be games that would challenge the limits of even the most brilliant and experienced game player. If such constructions are un avoidable we really need to start to look beyond models of perfectly rational behavior. Even if such constructions are avoidable, we have to be asking more questions about the match (or mismatch) between equilibrium concepts as predictive tools and limitations on the rationality of the players.
References Abreu, D., and H. Matsushima (1990), "Virtual implementation in iteratively undominated strategies: Incom plete information", mimeo. Abreu, D., and H. Matsushima ( l 992a), "Virtual implementation in iteratively undominated strategies: Complete information", Econometrica 60:993-1008. Abreu, D., and H. Matsushima ( l992b), "A response to Glazer and Rosenthal", Econometrica 60: 1439-1442. Abreu, D., and H. Matsushima ( 1994), "Exact implementation", Journal of Economic Theory 64:1-20. Abreu, D., and A. Sen (1990), "Subgame perfect implementation: A necessary and almost sufficient condition", Journal of Economic Theory 50:285-299. Abreu, D., and A. Sen (1991), "Virtual implementation in Nash equilibrium", Econometrica 59:997-102 1 . Aghion, P., M. Dewatripont and P. Rey (1 994), "Renegotiation design with unverifiable information", Econo metrica 62:257-282. Allen, B. (1997), "Implementation theory with incomplete information", in: S. Hart and A. Mas-Colell, eds., Cooperation: Game Theoretic Approaches (Springer, Heidelberg). Arya, A., and J. Glover (1995), "A simple forecasting mechanism for moral hazard settings", Journal of Economic Theory 66:507-521. Arya, A., J. Glover and U. Rajan (2000), "Implementation in principal-agent models of adverse selection", Journal of Economic Theory 93:87-109. Baliga, S. (1999), "Implementation in economic environments with incomplete information: The use of multi stage games", Games and Economic Behavior 27:173-183. Baliga, S., and S. Brusco (2000), "Collusion, renegotiation, and implementation", Social Choice and Welfare 17:69-83. Baliga, S., L. Corchon and T. Sjostrom ( 1997), "The theory of implementation when the planner is a player", Journal of Economic Theory 77: 15-33. Baliga, S., and T. Sjostrom (1999), "Interactive implementation", Games and Economic Behavior 27:38-63. Banks, J. (1 985), "Sophisticated voting outcomes and agenda control", Social Choice and Welfare 1 : 295-306. Banks, J., C. Camerer and D. Porter (1994), "An experimental analysis of Nash refinements in signalling games", Games and Economic Behavior 6:1-3 1 . Bergin, J., and A . Sen (1996), "Implementation i n generic environments", Social Choice and Welfare 13 :467448. Bergin, J., and A. Sen (1998), "Extensive form implementation in incomplete information environments", Journal of Economic Theory 80:222-256. Bernheim, B.D., and M. Whinston ( 1987), "Coalition-proof Nash equilibrium, II: Applications", Journal of Economic Theory 42: 1 3-29. Bliss, C., and B. Nalebuff (1 984), "Dragon-slaying and ballroom dancing: The private supply of a public good", Journal of Public Economics 25:1-12.
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Boylan, R. ( 1 998), "Coalition-proof implementation", Journal of Economic Theory 82: 1 32-143. Boylan, R., J. Ledyard and R. McKelvey ( 1996), "Political competition in a model of economic growth: Some theoretical results", Economic Theory 7:191-205. Brusco, S. (1995), "Perfect Bayesian implementation", Economic Theory 5:419-444. Cabrales, A. (1999), "Adaptive dynamics and the implementation problem with complete information", Jour nal of Economic Theory 86: 1 59-184. Chakravarti, B. (1991), "Strategy space reductions for feasible implementation of Walrasian performance", Social Choice and Welfare 8:235-246. Chakravarti, B. ( 1 993), "Sequential rationality, implementation and pre-play communication", Journal of Mathematical Economics 22:265-294. Chakravarti, B., L. Corchon and S. Wilkie (1994), "Credible implementation", mimeo. Chander, P., and L. Wilde (1998) "A general characterization of optimal income taxation and enforcement", Review of Economic Studies 65: 165-183. Chen, Y. (2002), "Dynamic stability of Nash-efficient public goods mechanisms: Reconciling theory and experiments", Games and Economic Behavior, forthcoming. Chen, Y., and F.-F. Tang (1998), "Learning and incentive-compatible mechanisms for public goods provision: An experimental study", Journal of Political Economy 106:633-662. Corchon, L., and I. Ortufio-Ortin ( 1 995), "Robust implementation under alternative information structures", Economic Design 1 : 1 57-172. Corchon, L., and S. Wilkie (1 995), "Double implementation of the ratio correspondence by a market mecha nism", Economic Design 2:325-337. Cramton, P., and T. Palfrey ( 1995), "Ratifiable mechanisms: Learning from disagreement", Games and Eco nomic Behavior 10:255-283. Crawford, V. (1977), "A game of fair division", Review of Economic Studies 44:235-247. Crawford, V. (1979), "A procedure for generating Pareto-efficient egalitarian equivalent allocations", Econo metrica 47:49-60. Danilov, V. ( 1992), "Implementation via Nash equilibrium", Econometrica 60:43-56. Dasgupta, P., P. Hammond and E. Maskin (1979), "The implementation of social choice rules: Some general results on incentive compatibility", Review of Economic Studies 46: 1 85-216. Demange, G. ( 1984), "Implementing efficient egalitarian equivalent allocations", Econometrica 52:1 1671 1 77. Demski, J., and D. Sappington (1984), "Optimal incentive contracts with multiple agents", Journal of Eco nomic Theory 33: 152-171. Duggan, J. (1993), "Bayesian implementation with infinite types", mimeo. Duggan, J. ( 1994a), "Virtual implementation in Bayesian equilibrium with infinite sets of types", mimeo (California Institute of Technology, CA). Duggan, J. ( 1994b), "Bayesian implementation", Ph.D. dissertation (California Institute of Technology, CA). Duggan, J. (1995), "Sequentially rational implementation with incomplete information", mimeo (Queens Uni versity). Duggan, J. ( 1997), "Virtual Bayesian implementation", Econometrica 65 : 1 175-1 199. Duggan, J. ( 1998), "An extensive form solution to the adverse selection problem in principal/multi-agent environments", Review ofEconomic Design 3: 167-1 9 1 . Duggan, J . , and J. Roberts ( 1997), "Robust implementation", Working paper (University o f Rochester, Rochester). Dutta, B., and A. Sen ( 1 99 1 a), "A necessary and sufficient condition for two-person Nash implementation", Review of Economic Studies 58: 1 21-128. Dutta, B., and A. Sen ( 1 991b), "Further results on Bayesian implementation", mimeo. Dutta, B., and A. Sen ( 1 99 l c), "Implementation under strong equilibrium: A complete characterization", Journal of Mathematical Economics 20:49-67. Dutta, B., and A. Sen ( 1 993), "Implementing generalized Condorcet social choice functions via backward induction", Social Choice and Welfare 10: 149-160.
T.R.
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Dutta, B., and A. Sen (1 994a), "Two-person Bayesian implementation", Economic Design 1 :41-54. Dutta, B., and A. Sen (1 994b), "Bayesian implementation: The necessity of infinite mechanisms", Journal of Economic Theory 64: 1 30-14 1 . Dutta, B., A. Sen and R . Vohra ( 1994), "Nash implementation through elementary mechanisms i n exchange economies", Economic Design 1 : 173-203. Eliaz, K. (1 999), "Fault tolerant implementation", Working paper (Tel Aviv University, Tel Aviv). Farquharson, R. ( 1957/1969), Theory of Voting (Yale University Press, New Haven). Gibbard, A. (1973), "Manipulation of voting schemes", Econometrica 41 :587-601 . Glazer, J., and C.-T. M a (1989), "Efficient allocation of a 'prize' - King Solomon's dilemma", Games and Economic Behavior 1:222-233. Glazer, J., and M. Perry (1 996), "Virtual implementation by backwards induction", Games and Economic Behavior 15:27-32. Glazer, J., and R. Rosenthal (1992), "A note on the Abreu-Matsushima mechanism", Econometrica 60: 14351438. Glazer, J., and A. Rubinstein (1 996), "An extensive game as a guide for solving a normal game", Journal of Economic Theory 70:32-42. Glazer, J., and A. Rubinstein (1998), "Motives and implementation: On the design of mechanisms to elicit opinions", Journal of Economic Theory 79:157-173. Glover, J. (1994), 62:221-229.
''A simpler mechanism that stops agents from cheating", Journal of Economic Theory
Groves, T. (1979), "Efficient collective choice with compensation", in: J.-J. Laffont, ed., Aggregation and Revelation of Preferences (North-Holland, Amsterdam) 37-59. Harris, M., and R. Townsend (1981), "Resource allocation with asymmetric information", Econometrica 49:33-64. Harsanyi, J. ( 1 967, 1 968), "Games with incomplete information played by Bayesian players", Management Science 14: 159-182, 320-334, 486-502. Hererro, M., and S. Srivastava (1992), "Implementation via backward induction", Journal of Economic The ory 56:70-88. Holmstrom, B., and R. Myerson (1983), "Efficient and durable decision rules with incomplete information", Econometrica 5 1 : 1799-1819. Hong, L. ( 1996), "Bayesian implementation in exchange economies with state dependent feasible sets and private information", Social Choice and Welfare 13:433-444. Hong, L. (1998), "Feasible Bayesian implementation with state dependent feasible sets", Journal of Economic Theory 80:201-22 1 . Hong, L., and S. Page ( 1994), "Reducing informational costs i n endowment mechanisms", Economic Design 1 : 103-1 17. Hurwicz, L. ( 1960), "Optimality and information efficiency in resource allocation processes", in: K. Arrow, S. Karlin and P. Suppes, eds., Mathematical Methods in the Social Sciences (Stanford University Press, Stanford) 27-46. Hurwicz, L. (1972), "On informationally decentralized systems", in: C.B. McGuire and R. Radner, eds., Decision and Organization (North-Holland, Amsterdam). Hurwicz, L. ( 1973), "The design of mechanisms for resource allocation", American Economic Review 6 1 : 130. Hurwicz, L. ( 1 977), "Optimality and informational efficiency in resource allocation problems", in: K. Arrow, S. Karlin and P. Suppes, eds., Mathematical Methods in the Social Sciences (Stanford University Press, Stanford) 27-48. Hurwicz, L. ( 1986), "Incentive aspects of decentralization", in: K. Arrow and M. Intriligator, eds., Handbook of Mathematical Economics, Vol. III (North-Holland, Amsterdam) 1441-1482. Hurwicz, L. (1994), "Economic design, adjustment processes, mechanisms, and institutions", Economic De sign 1 : 1-14.
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Hurwicz, L., E. Maskin and A. Postlewaite (1995), "Feasible Nash implementation of social choice corre spondences when the designer does not know endowments or production sets", in: J. Ledyard, ed., The Economics of Information Decentralization: Complexity, Efficiency, and Stability (Kluwer, Amsterdam). Jackson, M. (1991), "Bayesian implementation", Econometrica 59:461-477. Jackson, M. (1992), "Implementation in undominated strategies: A look at bounded mechanisms", Review of Economic Studies 59:757-775. Jackson, M. (2002), "A crash course in implementation theory", in: W. Thomson, ed., The Axiomatic Method: Principles and Applications to Game Theory and Resource Allocation, forthcoming. Jackson, M., and H. Moulin (1990), "Implementing a public project and distributing its cost", Journal of Economic Theory 57: 124-140. Jackson, M., and T. Palfrey (1998), "Efficiency and voluntary implementation in markets with repeated pair wise bargaining", Econometrica 66: 1353- 1388. Jackson, M., and T. Palfrey (2001), "Voluntary implementation", Journal of Economic Theory 98: 1-25. Jackson, M., T. Palfrey and S. Srivastava ( 1994), "Undominated Nash implementation in bounded mecha nisms", Games and Economic Behavior 6:474-501 . Kalai, E., and J . Ledyard ( 1998), "Repeated implementation", Journal of Economic Theory 83:308-3 17. Kim, T. (1993), "A stable Nash mechanism implementing Lindahl allocations for quasi-linear environments", Journal of Mathematical Economics 22:359-371 . Legros, P. (1990), "Strongly durable allocations", CAE Working Paper No. 90-05 (Cornell University). Ma, C. (1988), "Unique implementation of incentive contracts with many agents", Review of Economic Stud ies 55:555-572. Ma, C., J. Moore and S. Turnbull ( 1988), "Stopping agents from cheating", Journal of Economic Theory
46:355-372. Mas-Colell, A., and X. Vives ( 1993), "Implementation in economies with a continuum of agents", Review of Economic Studies 10:61 3-629. Maskin, E. (1999), "Nash implementation and welfare optimality", Review of Economic Studies 66:23-38. Maskin, E. (1979), "Implementation and strong Nash equilibrium", in: J.-J. Laffont, ed., Aggregation and Revelation of Preferences (North-Holland, Amsterdam). Maskin, E. ( 1985) "The theory of implementation in Nash equilibrium", in: L. Hurwicz, D. Schmeidler and H. Sonnenschein, eds., Social Organization: Essays in Memory of Elisha Pazner (Cambridge University Press, Cambridge). Maskin, E., and J. Moore (1999), "Implementation with renegotiation", Review of Economic Studies 66:39-
56. Maskin, E., and J. Tirole (1992), "The principal-agent relationship with an informed principal, II: Common values", Econometrica 60: 1-42. Matsushima, H. (1988), "A new approach to the implementation problem", Journal of Economic Theory
45: 128-144. Matsushima, H.
(1993), "Bayesian monotonicity with side payments", Journal of Economic Theory 59: 107-
121. McKelvey, R . (1989), "Game forms for Nash implementation o f general social choice correspondences", Social Choice and Welfare 6: 139-156. McKelvey, R., and R. Niemi ( 1978), "A multistage game representation of sophisticated voting for binary procedures", Journal of Economic Theory 8 1 : 1-22. McKelvey, R., and T. Palfrey ( 1995), "Quantal response equilibria for normal form games", Games and Eco nomic Behavior 10:6-38. McKelvey, R., and T. Palfrey (1996), "A statistical theory of equilibrium in games", Japanese Economic Review 47: 186-209. McKelvey, R., and T. Palfrey ( 1998), "Quantal response equilibria for extensive form games", Experimental Economics 1 :9-41 . Miller, N . (1977), "Graph-theoretic approaches to the theory of voting", American Journal of Political Science
21:769-803.
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Mookherjee, D., and S. Reichelstein ( 1990), "Implementation via augmented revelation mechanisms", Review of Economic Studies 57:453-475. Moore, J. ( 1992), "Implementation, contracts, and renegotiation in environments with complete information", in: J.-J. Laffont, ed., Advances in Economic Theory, Vol. 1 (Cambridge University Press, Cambridge). Moore, J., and R. Repullo (1988), "Subgame perfect implementation", Econometrica 46: 1 191-220. Moore, J., and R. Repnllo (1990), "Nash implementation: A full characterization", Econometrica 58:10831099. Moulin, H. (1979), "Dominance-solvable voting schemes", Econometrica 47: 1 337-1351. Moulin, H. ( 1 984), "Implementing the Kalai-Smorodinsky bargaining solution", Journal of Economic Theory 33:32-45. Moulin, H. (1986) "Choosing from a tournament", Social Choice and Welfare 3:27 1-29 1 . Moulin, H . (1994), "Social choice", in: R.J. Aumann and S . Hart, eds., Handbook of Game Theory, Vol. 2 (North-Holland, Amsterdam) Chapter 3 1 , 1091-1 1 25. Mount, K., and S. Reiter (1974), "The informational size of message spaces", Journal of Economic Theory 8 : 1 6 1-192. Myerson, R. ( 1979), "Incentive compatibility and the bargaining problem", Econometrica 47:6 1-74. Myerson, R. (1985), "Bayesian equilibrium and incentive compatibility: An introduction", in: L. Hurwicz, D. Schmeidler and H. Sonnenschein, eds., Social Goals and Organization: Essays in Memory of Elisha Pazner (Cambridge University Press, Cambridge) 229-259. Ordeshook, P., and T. Palfrey ( 1988), "Agendas, strategic voting, and signaling with incomplete information", American Journal of Political Science 32:441-466. Palfrey, T. (1992), "Implementation in Bayesian equilibrium: The multiple equilibrium problem", in: J.-J. Laf font, ed., Advances in Economic Theory, Vol. 1 (Cambridge University Press, Cambridge). Palfrey, T., and S. Srivastava ( 1986), "Private information in large economies", Journal of Economic Theory 39: 34-58. Palfrey, T., and S. Srivastava ( 1987), "On Bayesian implementable allocations", Review of Economic Studies 54:193-208. Palfrey, T., and S. Srivastava ( 1989a), "Implementation with incomplete information in exchange economies", Econometrica 57 : 1 15-134. Palfrey, T., and S. Srivastava (1989b), "Mechanism design with incomplete information: A solution to the implementation problem", Journal of Political Economy 97:668-691 . Palfrey, T., and S . Srivastava (l99 l a), "Nash implementation using undominated strategies", Econometrica 59:479-501. Palfrey, T., and S. Srivastava (l991b), "Efficient trading mechanisms with preplay communication", Journal of Economic Theory 55:17-40. Palfrey, T., and S. Srivastava (1993), Bayesian Implementation (Harwood Academic Publishers, Reading). Peleg, B. ( 1996a), "A continuous double implementation of the constrained Walrasian equilibrium", Eco nomic Design 2:89-97. Peleg, B. ( l996b), "Double implementation of the Lindahl equilibrium by a continuous mechanism", Eco nomic Design 2:31 1-324. Postlewaite, A. (1 985), "Implementation via Nash equilibria in economic environments", in: L. Hurwicz, D. Schmeidler and H. Sonnenschein, eds., Social Goals and Organization: Essays in Memory of Elisha Pazner (Cambridge University Press, Cambridge) 205-228. Postlewaite, A., and D. Schmeidler ( 1 986), "Implementation in differential information economies", Journal of Economic Theory 39: 14-33. Postlewaite, A., and D. Wettstein ( 1989), "Feasible and continuous implementation", Review of Economic Studies 56:603-6 1 1 . Reichelstein, S. (1 984), "Incentive compatibility and informational requirements", Journal of Economic The ory 34:32-5 1 . Reichelstein, S . , and S. Reiter (1988), "Game forms with minimal strategy spaces", Econometrica 56:661692.
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Repullo, R. ( 1987), "A simple proof of Maskin's theorem on Nash implementation", Social Choice and Wel fare 4:39-41. Rubinstein, A., and A. Wolinsky (1992), "Renegotiation-proof implementation and time preferences", Amer ican Economic Review 82:600�614. Saijo, T. (1988), "Strategy space reductions in Maskin's theorem: Sufficient conditions for Nash implementa tion", Econometrica 56:693�700. Saijo, T., Y. Tatamitani and T. Yamato (1996), "Toward natural implementation", International Economic Review 37:949�980. Satterthwaite, M. (1975), "Strategy-proofness and Arrow's conditions: Existence and correspondence theo rems for voting procedure and social welfare functions", Journal of Economic Theory 10: 187�217. Sefton, M., and A. Yavas (1996), "Abreu�Matsushima mechanisms: Experimental evidence", Games and Economic Behavior 16:280�302. Sen, A. ( 1987), "Two essays on the theory of implementation", Ph.D. dissertation (Princeton University). Serrano, R., and R. Vohra (1999), "On the impossibility of implementation under incomplete information", Working Paper #99-10 (Brown University, Department of Economics, Providence) (forthcoming in Econo metrica). Serrano, R., and R. Vohra (2000), "Type diversity and virtual Bayesian implementation", Working Paper #00-16 (Brown University, Department of Economics, Providence). Sjostrom, T. (199 1), "On the necessary and sufficient conditions for Nash implementation", Social Choice and Welfare 8:333�340. Sjostrom, T. (1993), "Implementation in perfect equilibria", Social Choice and Welfare 10:97�106. Sjostrom, T. (1994), "Implementation in undominated Nash equilibrium without integer games", Games and Economic Behavior 6:502-5 1 1. Sjostrom, T. (1995), "Implementation by demand mechanisms", Economic Design 1:343-354. Sjostrom, T. (1996), "Credibility and renegotiation of outcome functions in implementation", Japanese Eco nomic Review 47:157-169. Sjostrom, T. (1999), "Undominated Nash implementation with collusion and renegotiation", Games and Eco nomic Behavior 26:337-352. Suh, S.-C. ( 1997), "An algorithm checking strong Nash implementability", Journal of Mathematical Eco nomics 25: 109-122. Tatamitani, Y. (1993), "Double implementation in Nash and undominated Nash equilibrium in social choice environments", Economic Theory 3: 109-1 17. Thomson, W. (1979), "Maximin strategies and elicitation of preferences", in: J.-J. Laffont, ed., Aggregation and Revelation of Preferences (North-Holland, Amsterdam) 245-268. Tian, G. ( 1989), "Implementation of the Lindahl correspondence by a single-valued, feasible, and continuous mechanism". Review of Economic Studies 56:613-621. Tian, G. ( 1990), "Completely feasible continuous implementation of the Lindahl correspondence with a mes sage space of minimal dimension", Journal of Economic Theory 5 1 :443-452. Tian, G. (1994), "Bayesian implementation in exchange economies with state dependent preferences and feasible sets", Social Choice and Welfare 16:99-120. Townsend, R. (1979), "Optimal contracts and competitive markets with costly state verification", Journal of Economic Theory 21:265-293. Trick, M., and S. Srivastava (1996), "Sophisticated voting rules: The two tournaments case", Social Choice and Welfare 13:275-289. Varian, H. (1994), "A solution to the problem of externalities and public goods when agents are well informed", American Economic Review 84: 1278-1293. Wettstein, D. (1990), "Continuous implementation of constrained rational expectations equilibria", Journal of Economic Theory 52:208-222. Wettstein, D. (1992), "Continuous implementation in economies with incomplete information", Games and Economic Behavior 4:463-483. Williams, S. (1984), "Sufficient conditions for Nash implementation", mimeo (University of Minnesota).
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Williams, S. ( 1986), "Realization and Nash implementation: Two aspects of mechanism design", Economet rica 54: 1 39-151. Yamato, T. (1992), "On Nash implementation of social choice correspondences", Games and Economic Be havior 4:484--492. Yamato, T. (1993), "Double implementation in Nash and undominated Nash equilibria", Journal of Economic Theory 59:31 1-323.
Chapter 62 GAM E THEORY AND EXPERIMENTAL GAM ING MARTIN SHUBIK*
Cowles Foundation, Yale University, New Haven, CT, USA Contents
1 . Scope 2. Game theory and gaming 2. 1. The testable elements of game theory 2.2. Experimentation and representation of the game 3. Abstract matrix games 3 . 1 . Matrix games 3.2. Games in coalitional form 3.3. Other games 4. Experimental economics 5. Experimentation and operational advice 6. Experimental social psychology, political science and law 6. 1 . Social psychology 6.2. Political science 6.3. Law 7 . Game theory and military gaming 8. Where to with experimental gaming? References
*I would like to thank the Pew Charitable Trust for its generous support.
Handbook of Game Theory, Volume 3, Edited by R.J. Aumann and S. Hart © 2002 Elsevier Science B. V All rights reserved
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Abstract
This is a survey and discussion of work covering both formal game theory and experi mental gaming prior to 1 99 1 . It is a useful preliminary introduction to the considerable change and emphasis which has taken place since that time where dynamics, learning, and local optimization have challenged the concept of noncooperative equilibria.
Keywords
experimental gaming, game theory, context, minimax, and coalitional form
JEL classification: C7 1 , C72, C73, C90
Ch. 62: Game Theory and Experimental Gaming
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1. Scope
This article deals with experimental games as they pertain to game theory. As such there is a natural distinction between experimentation with abstract games devoted to testing a specific hypothesis in game theory and games with a scenario from a discipline such as economics or political science where the game is presented in the context of some particular activity, even though the same hypothesis might be tested. 1 John Kennedy, professor of psychology at Princeton and one of the earliest experi menters with games, suggested2 that if you could tell him the results you wanted and give him control of the briefing he could get you the results. Context appears to be critical in the study of behavior in games. R. Simon, in his Ph.D. thesis ( 1 967), controlled for context. He used the same two-person zero-sum game with three different scenarios. One was abstract, one described the game in a business context and the other in a military context. The players were selected from ma jors in business and in military science. As the game was a relatively simple two-person zero-sum game a reasonably strong case could be made out for the maxmin solution concept. The performance of all students was best in the abstract scenario. Some of the military science students complained about the simplicity of the military scenario and the business students complained about the simplicity of the business scenario. Many experiments in game theory with abstract games are to "see what happens" and then possibly to look for consistency of the outcomes with various solution concepts. In teaching the basic concepts of game theory, for several years I have used several simple games. In the first lecture when most of the students have had no exposure to game theory I present them with three matrices, the Prisoner's Dilemma, Chicken and the Battle of the Sexes (see Figures 1 a, b and c). The students are asked if they have had any experience with these games. If not, they are asked to associate the name with the game. Table 1 shows the data for 1988. There appears to be no significant ability to match the words with the matrices. 3 There are several thousand articles on experimental matrix games and several hun dred on experimental economics alone, most of which can be interpreted as pertaining to game theory as well as to economics. Many of the business games are multipurpose and contain a component which involves game theory. War gaming, especially at the tactical level and to some extent at the strategic level, has been influenced by game the ory [see Brewer and Shubik (1 979)]. No attempt is made here to provide an exhaustive review of the vast and differentiated body of literature pertaining to blends of opera tional, experimental and didactic gaming. Instead a selection is made concentrating on
1
For example, one might hypothesize that the percentage of points selected in an abstract game which lie in the core will be greater than the points selected in an experiment repeated with the only difference being a scenario supplied to the game. Personal communication, 1957.
2
3 The rows and columns do not all add to the same number because of some omissions among 30 students.
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M. Shubik A
2
B
2 5,5
-9, 10
10,-9
0,0
2
2 -10,-10
5,-5
-5,5
0,0
Figure
c
2
2 2,1
0,0
0,0
1 ,2
1.
Table I Guessed name
Actual game Chicken
Chicken Battle of the Sexes Prisoner's Dilemma
Battle of the Sexes
7
II
9 10
12 7
Prisoner's Dilemma 8 8
9
the modeling problems, formal structures and selection of solution concepts in game theory. Roth (1 986), in a provocative discussion of laboratory experimentation in economics, has suggested three kinds of activities for those engaged in economic experimental gam ing. They are "speaking to theorists", "searching for facts" and "whispering in the ears of princes". The third of these activities, from the viewpoint of society, is possibly the most immediately useful. In the United States the princes tend to be corporate exec utives, politicians and generals or admirals. However, the topic of operational gaming merits a separate treatment, and is not treated here.
2. Game theory and gaming
Before discussing experimental games which might lead to the confirmation or rejection of theory, or to the discovery of "facts" or regularities in human behavior, it is desirable to consider those aspects of game theory which might benefit from experimental gam ing. Perhaps the most important item to keep in mind is the fundamental distinction be tween the game theory and the approach of social psychology to the same problems. Underlying a considerable amount of game theory is the concept of external symmetry. Unless stated otherwise, all players in game theory are treated as though they are in trinsically symmetric in all nonspecified attributes. They have the same perceptions, the same abilities to calculate and the same personalities. Much of game theory is devoted to deriving normative or behavioral solution concepts to games played by personality-free players in context-free environments. In particular, one of the impressive achievements
Ch. 62: Game Theory and Experimental Gaming
233 1
of formal game theory has been to produce a menu of different solution concepts which suggest ingeniously reasoned sets of outcomes to nonsymmetric games played by ex ternally symmetric players. In bold contrast, the approach of the social psychologists has for the most part been to take symmetric games and observe the broad differences in play which can be attributed to nonsymmetric players. Personality, passion and perception count and the social psy chologist is interested to see how they count. A completely different view from either of the above underlies much of the approach of the economists interested in studying mass markets. The power of the mass market is that it turns the social science problem of competition or cooperation among the few into a nonsocial science problem. The message of the mass market is not that personality differences do not exist, but that in the impersonal mass market they do not matter. The large market has the individual player pitted against an aggregate of other players. When the large market becomes still larger, all concern for the others is attenuated and the individual can regard himself as playing against a given inanimate object known as the market. 2.1.
The testable elements of game theory
Before asking who has been testing what, a question which needs to be asked is what is there to be tested concerning game theory. A brief listing is given and discussed in the subsequent sections. Context: Does context matter? Or do we believe that strategic structure in vitro will be treated the same way by rational players regardless of context? There has been much abstract experimental gaming with no scenarios supplied to the players. But there also is a growing body of experimental literature where the context is explicitly economic, political, military or other. External symmetry and limited rationality: The social psychologists are concerned with studying decision-makers as they are. A central simplification of game theory has been to build upon the abstract simplification of rational man. Not only are the play ers in game theory devoid of personality, they are endowed with unlimited rationality, unless specified otherwise. What is learned from Maschler's schoolchildren, Smith's or Roth's undergraduates or Battaglia's rats [Maschler (1978), Smith ( 1 986), Roth ( 1 983), Battaglia et al. (1985)] requires considerable interpretation to relate the experimental results with the game abstraction. Rules of the game and the role of time: In much game theory time plays little role. In the extensive form the sequencing of moves is frequently critical, but the cardinal aspects of the length of elapsed time between moves is rarely important (exceptions are in dueling and search and evasion models). In particular, one of the paradoxes in attempting to experiment with games in coalitional form is that the key representation - the characteristic functions (and variants thereof) - implicitly assumes that the bar gaining, communication, deal-making and tentative formation of coalitions is costless and timeless. Whereas in actual experiments the amount of time taken in bargaining and discussion is often a critical limiting factor in determining the outcome.
2332
M. Shubik
Utility and preference: Do we carry around utility functions? The utility function is an extremely handy fiction for the mathematical economist and the game theorist applying game theory to the study of mass markets, but in many of the experiments with matrix games or market games the payoffs have tended to be money (on a performance or hourly basis or both), points or grades [see Rapoport, Guyer and Gordon ( 1 976)] . For example, Mosteller and Nogee ( 1 95 1 ) attempted to measure the utility function of individuals with little success. In most experimental work little attempt is made to obtain individual measures before the experiment. The design difficulties and the costs make this type of pre-testing prohibitively difficult to perform. Depending on the specifics of the experiment it can be argued that in some experiments all one needs to know is that more money is preferred to less; in others the existence of a linear measure of utility is important; in still others there are questions concerning interpersonal comparisons. See however Roth and Malouf ( 1979) for an experimental design which attempts to account for these difficulties.4 Even with individual testing there is the possibility that the goals of the isolated indi vidual change when placed in a multiperson context. The maximization of relative score or status [Shubik ( 1 97 1b)] may dominate the seeking of absolute point score. Solution concepts: Given that we have an adequate characterization of the game and the individuals about to play, what do we want to test for? Over the course of the years various game theorists have proposed many solution concepts offering both normative and behavioral justifications for the solutions. Much of the activity in experimental gam ing has been devoted to considering different solution concepts.
2.2. Experimentation and representation of the game The two fundamental representations of games which have been used for experimenta tion have been the strategic and cooperative form. Rarely has the formal extensive form been used in experimentation. This appears to be due to several fundamental reasons. In the study of bargaining and negotiation in general we have no easy simple description of the game in extensive form. It is difficult to describe talk as moves. There is a lack of structure and a flexibility in sequencing which weakens the analogy with games such as chess where the extensive form is clearly defined. In experimenting with the game in matrix form, frequently the experimenter provides the same matrix to be played several times; but clearly this is different from giving the individual a new and considerably larger matrix representing the strategic form of a mul tistage game. Shubik, Wolf and Poon ( 1 974) experimented with a game in matrix form played twice and a much larger matrix representing the strategic form of the two-stage game to be played once. Strikingly different results were obtained with considerable emphasis given in the first version to the behavior of the opponent on his first trial.
4
Unfortunately, given the virtual duality between probability aud utility, they have to introduce the assump tion that the subjective probabilities are the same as those presented in the experiment. Thus one difficulty may have been traded for auother.
2333
Ch. 62: Game Theory and Experimental Gaming
How big a matrix can an experimental subject handle? The answer appears to be a 2 x 2 unless there is some form of special structure on the entries; thus for example Fouraker, Shubik and Siegel ( 1 96 1 ) used a 57 x 25 matrix in the study of triopoly, but this matrix had considerable regularities (being generated from continuous payoff functions from a Cournot triopoly model).
3. Abstract matrix games
3.1. Matrix games The 2 x 2 matrix game has been a major source for the provision of didactic examples and experimental games for social scientists with interests in game theory. Limiting ourselves to strict orderings there are 4! x 4! = 576 ways in which two payoff entries can be placed in each of four cells of a 2 x 2 matrix. When symmetries are accounted for there are 78 strategically different games which remain [see Rapoport, Guyer and Gordon ( 1 976)] . If ties are considered in the payoffs then the number of strategically different games becomes considerably larger. Guyer and Hamburger (1968) counted the strategically different weakly ordinal games and Powers ( 1986) made some corrections and provided a taxonomy for these games. There are 726 strategically different games. O'Neill, in unpublished notes dating from 1987, has estimated that the lower bound on the number of different 2 x 2 x 2 games is (8!) 3 /(8 6) = 1 ,365,590,01 6,000. For the 3 x 3 game we have (9 !) 2 / ((3 !) 2 2) = 1 ,828,915,200. Thus we see that the reason why the 2 x 2 x 2 and the 3 x 3 games have not been studied, classified and analyzed in any generality is that the number of different cases is astronomical. Before considering the general nonconstant-sum game, the first and most elementary question to ask concerns the playing of two-person zero- and constant-sum games. This splits into two questions. The first is how good a predictor is the saddlepoint in two person zero-sum games with a pure strategy saddlepoint. The second is how good a predictor is the maxmin mixed strategy of a player's behavior. Possibly the two earliest published works on the zero-sum game were those of Morin (1 960) and Lieberman ( 1 960) who considered a 3 x 3 game with a saddlepoint. He has 15 pairs of subjects play a 3 x 3 for 200 times each at a penny a point. He used the last 10 trials as his statistic. In these trials approximately 90% selected their optimal strategy. One may query the approach of using the first 1 90 trials for learning. Lieberman ( 1962) also considered a zero-sum 2 x 2 game played against a stooge using a minimax mixed strategy. Rather than approach an optimal strategy the live player appeared to follow or imitate the stooge. Rapoport, Guyer and Gordon [( 1976, Chapter 2 1 )] note three ordinally different games of pure opposition. Using 4 to stand for the most desired outcome and 1 for the least desired, Game 1 below has a saddlepoint with dominated strategies for each player, Game 2 has a saddlepoint but only one player has a dominated strategy and Game 3 has no saddlepoint. ·
·
2334
M. Shubik 2 2
2
2
4 3
4
3 2
2 2 2·
2
Figure
4
3
2.
Column Player
1
2
3
0 Row Player
0 0
2 3 J
0 0
0
Figure
J
0
0
0
3.
They report on one-shot experiments by Frenkel (1973) with a large population of players with reasonably good support for the saddlepoints but not for the mixed strat egy. The saddlepoint for small games appears to provide a good prediction of behavior. Shubik has used a 2 x 2 zero-sum game with a saddlepoint as a "rationality test" prior to having students play in nonzero-sum games and one can bet with safety that over 90% of the students will select their saddlepoint strategy. The mixed strategy results are by no means that clear. More recently, O'Neill (1987) has considered a repeated zero-sum game with a mixed strategy solution. The O'Neill experiment has 25 pairs of subjects play a card-matching game 105 times. Figure 3 shows the matrix. The + is a win for Row and the - a loss. There is a unique equilibrium with both players randomizing using probabilities of 0.2, 0.2, 0.2, 0.4, each. The value of the game is 0.4. The students were initially given $ 2.50 each and at each stage the winner received 5 cents from the loser. The aggregate data involving 2625 joint moves showed high consistency with the predictions of minimax theory. The proportion of wins for the row player was 40.9% rather than 40% as predicted by theory. Brown and Rosenthal ( 1987) have challenged O'Neill's analysis but his results are of note [see also O'Neill ( 1 991)]. One might argue that minimax is a normative theory which is an extension of the concept of rational behavior and that there is little to actually test beyond observing that naive players do not necessarily do the right thing. In saddlepoint games there is reasonable evidence that they tend to do the right thing [see, for example, Lieberman (1 960)] and as the results depend only on ordinal properties of the matrices this is dou bly comforting. The mixed strategies however in general require the full philosophical
2335
Ch. 62: Game Theory and Experimental Gaming
mysteries of the measurement of utility and choice under uncertainty. One of the attrac tive features of the O'Neill design is that there are only two payoff values and hence the utility structure is kept at its simplest. It would be of interest to see the O' Neill experi ment replicated and possibly also performed with a larger matrix (say a 6 x 6) with the same structure. It should be noted, taking a cue from Miller's ( 1 956) classical article, that unless a matrix has special structure it may not be reasonable to experiment with any size larger than 2 x 2. A 4 x 4 with general entries would require too much memory, cognition and calculation for nontrained subjects. One of the most comprehensive research projects devoted to studying matrix games has been that of Rapoport, Guyer and Gordon (1976). In their book entitled The 2 x 2 Game, not only do they count all of the different games, but they develop a taxonomy for all 78 of them. They begin by observing that there are 12 ordinally symmetric games and 66 nonsymmetric games. They classify them into games of complete, partial and no opposition. Games of complete opposition are the ordinal variants of two-person constant-sum games. Games of no opposition are classified by them as those which have the best outcome to each in the same cell. Space limitations do not permit a full discussion of this taxonomy, yet it is my belief that the problem of constructing an adequate taxonomy for matrix games is valuable both from the viewpoint of theory and experimentation. In particular, although much stress has been laid on the noncooperative equilibrium for games in strategic form, there appears to be a host of considerations which can all be present simultaneously and contribute towards driving a solution to a predictable outcome. These may be reflected in a taxonomy. A partial list is noted here: ( 1 ) uniqueness of equilibria; (2) symmetry of the game; (3) safety levels of equilibria; (4) existence of dominant strategies; (5) Pareto optimality; (6) interpersonal comparison and sidepayment conditions; (7) information conditions. 5 Of all of the 78 strategically different 2 x 2 games I suggest that the following one is structurally the most conducive to the selection of its equilibrium point solution. 6 The cell ( 1 , 1 ) will be selected not merely because it is an equilibrium point but also because it is unique, Pareto-optimal, results from the selection of dominant strategies, 2
2
5
4,4
3,2
2,3
1 ,1
In an experiment with several 2 x 2 games played repeatedly with low information where each of the players was given only his own payoff matrix and information about the move selected by his competitor after each of a sequence of plays, the ability to predict the outcome depended not merely on the noncooperative equilibrium but on the coincidence of an outcome having several other properties [see Shubik (1962)]. Where the convention is that 4 is high and I low.
6
2336
M. Shubik
is symmetric and coincides with maxmin (Pt - P2 ). Rapoport et al. ( 1976) have this tied for first place with a closely related no-conflict game, with the off-diagonal entries re versed. This new game does not have the coincidence of the maxmin difference solution with the equilibrium. The work of Rapoport, Guyer and Gordon ( 197 6) to this date is the most exhaustive attempt to ring the changes on conditions of play as well as structure of the 2 x 2 game. 3.2.
Games in coalitional form
If one wishes to be a purist, a game in coalitional or characteristic function form cannot actually be played. It is an abstraction with playing time and all other transaction costs set at zero. Thus any attempt to experiment with games in characteristic function form must take into account the distinction between the representation of the game and the differences introduced by the control and the time utilized in play. With allowances made for the difficulties in linking the actual play with the character istic function, there have been several experiments utilizing the characteristic function especially for the three-person game and to some extent for larger games. In particular, the invention of the bargaining set by Aumann and Maschler ( 1 964) was motivated by Maschler's desire to explain experimental evidence [see Maschler (1963, 1 978)]. The book of Kahan and Amnon Rapoport (1 984) has possibly the most exhaustive discussion and report on games in coalitional form. They have eleven chapters on the theory development pertaining to games in coalitional form, a chapter on experimental results with three-person games, a chapter7 on games for n ) 4, followed by concluding remarks which support the general thesis that the experimental results are not indepen dent of normalization and of other aspects of context; that no normative game-theoretic solution emerges as generally verifiable (often this does not rule out their use in special context-rich domains). It is my belief that a fruitful way to utilize a game in characteristic function form for experimental purposes is to explore beliefs or norms of students and others concerning how the proceeds from cooperation in a three-person game with sidepayments should be split. Any play by three individuals of a three-person game in characteristic func tion form is no longer represented by the form but has to be adjusted for the time and resource costs of the process of play as well as controlled for personality. In a normative inquiry Shubik ( 1 975a, 1 978) has used three characteristic functions for many years. They were selected so that the first game has a rather large core, the sec ond a single-point core that is highly nonsymmetric and the third no core. For the most part the subjects were students attending lectures on game theory who gave their opin ions on at least the first game without knowledge of any formal cooperative solution. Game 1 :
7
v ( l ) = v (2) = v (3) v ( l 23) = 4.
=
0;
v ( l 2)
=
1, v ( l 3)
=
2, v (23)
=
3;
These two chapters provide the most comprehensive summary of experimental results with games in char acteristic form up until the early 1980s. These include Apex games, simple games, Selten's work on the equal division core and market games.
2337
Ch. 62: Game Theory and Experimental Gaming Table 2 Percentage even split
Percentage in core Game
1980 1981 1983 1984 1985 1988
1
Game 3
Game 2
86 86.8 87 89.5 93 92
36 32.5 7.5
28.3 5 21.8 19 20 11
5
Game 2:
v ( l ) = v (2) v ( l 23) = 4.
v(3)
=
0;
v ( l 2)
=
0, v ( 1 3)
Game 3 :
v ( l ) = v (2) = v (3) v ( l 23) = 4 .
=
0;
v ( l 2)
=
2.5, v ( 1 3)
=
=
0, v (23) =
=
3, v (23)
4; =
3.5;
Table 2 shows the percentage of individuals selecting a point in the core for games 1 and 2. The smallest sample size was 17 and the largest was 50. The last column shows the choice of an even split in the game without a core. There was no discernible pattern for the game without a core except for a selection of the even split, possibly as a focal point. There was little evidence supporting the value or nucleolus. When the core is "flat" and not too nonsymmetric (game 1 ), it is a good predictor. When it is one point but highly skewed, its attractiveness is highly diminished (game 2). Three-person games have been of considerable interest to social psychologists. Caplow (1956) suggested a theory of coalitions which was experimented with by Gam son (1961) and later the three-person simple majority game was used in experiments by Byron Roth ( 1 975). He introduced a scenario involving a task performed by a scout troop with three variables: the size of the groups considered as a single player, the effort expended, and the economic value added to any coalition of players. He compared both questionnaire response and actual play. His experimental results showed an important role for equity considerations. Selten, since 1972, has been concerned with equity considerations in coalitional bar gaining. His views are summarized in a detailed survey [Selten (1987)]. In essence, he suggests that limited rationality must be taken into consideration and that normaliza tion of payoffs and then equity considerations in division provide important structure to bargaining. He considers and compares both the work by Maschler (1978) on the bargaining set and his theory of equal division payoff bounds [Selten ( 1 982)] . Selten defines "satisficing" as the player obtaining his aspiration level which in the context of a characteristic function game is a lower bound on what a player is willing to accept in a genuine coalition. The equal division payoff bounds provide an easy way to obtain as piration levels, especially in a zero-normalized three-person game. The specifics for the calculation of the bounds are lengthy and are not reproduced here. There are however
2338
M. Shubik
three points to be made. ( 1 ) They are chosen specifically to reflect limited rationality. (2) The criterion of success suggested is an "area computation", i.e., no point predic tion is used as a test of success, but a range or area is considered. (3) The examination of data generated from four sets of experiments done by others "strongly suggests that equity considerations have an important influence on the behavior of experimental sub jects in zero-normalized three-person games". This also comes through in the work of W. Giith, R. Schmittberger and B. Schwartz (1 982) who considered one-period ultima tum bargaining games where one side proposes the division of a sum of money M into two parts, M x and x . The other side then has the choice of accepting or rejecting the split with both sides obtaining 0. The perfect equilibrium is unique and has the side that moves first take all. This is not what happened experimentally. The approach in these experiments is towards developing a descriptive theory. As such it appears to this writer that the structure of the game and both psychological and socio-psychological factors must be taken into account explicitly. The only way that process considerations can be avoided is by not actually playing the game but asking individuals to state how they think they should or they would play. The characteristic function is an idealization. The coalitions are not coalitions, they are costless coalitions in-being. Attempts to experiment with playing games in characteristic function form go against the original conception of the characteristic form. As the results of Byron Roth indicated both the questionnaire and actual play of the game are worth utilizing and they give different results. Much is still to be learned in comparing both approaches. -
3.3.
Other games
One important but somewhat overlooked aspect of experimental gaming is the construc tion and use of simple paradoxical games to illustrate problems in game theory and to find out what happens when these games are played. Three examples are given. Haus ner, Nash, Shapley and Shubik (1964) in the early 1 950s constructed a game called "so long sucker" in which the winner is the survivor; in order to win it is necessary, but not sufficient, to form coalitions. At some point it will pay someone to "doublecross" his partner. However, to a game theorist the definition of doublecross is difficult to make precise. When John McCarthy decided to get revenge for being doublecrossed by Nash, Nash pointed out that McCarthy was "being irrational" as he could have easily calcu lated that Nash would have to do what he did given the opportunity. The dollar auction game [see Shubik (197 1 a)] is another example of an extremely simple yet paradoxical game. The highest bidder pays to win a dollar, but the second highest bidder must also pay but receives nothing in return. Experiments with this game belong primarily to the domain of social psychology and a recent book by Tegler ( 1980) covers the experimental work. O'Neill (1987) however has suggested a game-theoretic solution to the auction with limited resources. The third game is a simple game attributed to Rosenthal. The subjects are confronted with a game in extensive form as indicated in the game tree in Figure 4. They are told that they are playing an unknown opponent and that they are player 2 and have been given the move. They are required to select their move and justify the choice.
2339
Ch. 62: Game Theory and Experimental Gaming
177
1 2,3
3 ,4
3,6
2,3
Figure 4.
The paradox comes in the nature of the expectations that player 2 must form about player 1 's behavior. If player 1 were "rational" he would have ended the game: hence the move should never have come to player 2. In an informal8 classroom exercise in 1985 17 out of 2 1 acting as player 2 chose to continue the game and 4 terminated giving reasons for the most part involving the stupidity or the erratic behavior of the competition.
4. Experimental economics
The first adequately documented running of an experimental game to study competition among firms was that of Chamberlain (1948). It was a report of a relatively informal oligopolistic market run in a class in economic theory. This was a relatively isolated event [occurring before the publication of the Nash equilibrium ( 195 1 )] and the gen eral idea of experimentation in the economics of competition did not spread for around another decade. Perhaps the main impetus to the idea of experimentation on economic competition came via the construction of the first computer-based business game by Bellman et al. ( 1 957). Within a few years business games were widely used both in business schools and in many corporations. Hoggatt ( 1 959, 1969) was possibly the ear liest researcher to recognize the value of the computer-based game as an experimental tool. In the past two decades there has been considerable work by Vernon Smith primarily using a computer-based laboratory to study price formation in various market structures. Many of the results appear to offer comforting support for the virtues of a competitive price system. However, it is important to consider that possibly a key feature in the functioning of an anonymous mass market is that it is designed implicitly or explicitly to minimize both the socio-psychological and the more complex game-theoretic aspects of strategic human behavior. The crowning joy of economic theory as a social science is its theory of markets and competition. But paradoxically markets appear to work the best
8
The meaning of "informal" here is that the students were not paid for their choice. I supplied them with the game as shown in Figure 4 and asked them to write down what they would do, if as player 2 they were called on to move. They were asked to explain their choice.
2340
M. Shubik
when the individuals are converted into nonsocial anonymous entities and competition is attenuated, leaving a set of one-person maximization problems. 9 This observation is meant not to detract from the value of the experimental work of Smith and many others who have been concerned with showing the efficiency of markets but to contrast the experimental program of Smith, Plott and others concerned with the design of economic mechanisms with the more general problems of game theory. 1 0 Are economic and game-theoretic principles sufficiently basic that one might have cleaner experiments using rats and pigeons rather than humans? Hopefully, animal pas sions and neuroses are simpler than those of humans. Kagel et al. ( 1 975) began to in vestigate consumer demand in rats. The investigations have been primarily aimed at isolated individual economic behavior in animals, including an investigation of risky choice [Battaglia et al. ( 1985)] . Kagel ( 1 987) offers a justification of the use of animals, noting problems of species extrapolation and cognition. He points out that (he believes that) animals unlike humans are not selected from a background of political, economic and cultural contexts, which at least to the game theorist concerned with external sym metry is helpful. The Kagel paper is noted (although it is concerned only with individual decision making) to call attention to the relationship between individual decision-making and game-playing behavior. In few, if any, game experiments is there extensive pre-testing of one-person decision-making under exogenous uncertainty. Yet there is an implicit assumption made that the subjects satisfy the theory for one-person behavior. Is the jump from one- to two-person behavior the same for humans, sharks, wolves and rats? Especially given the growth of interest in game theory applications to animal be havior, it would be possible and of interest to extend experimentation with animals to two-person zero- and nonconstant-sum games. In much of the experimental work in game theory anonymity is a key control factor. It should be reasonably straightforward to have animals (not even of the same species) in separate cages, each with two choices leading to four prizes from a 2 x 2 matrix. There is the problem as to how or if the animals are ever able to deduce that they are in a two-animal competitive situation. Fur thermore, at least if the analogy with economics is to be carried further, can animals be taught the meaning and value of money, as money figures so prominently in business games, much of experimental economics, experimental game theory and in the actual economy? I suspect that the answer is no. But how far can animal analogies be made seems to be a reasonable question. Another approach to the study of competition has been via the use of robots in com puterized oligopoly games. Hoggatt ( 1969) in an imaginative design had a live player play two artificial players separately in the same type of game. One artificial player
9
But even here the dynamics of social psychology appears in mass markets. Shubik (1970) managed to run a simulated stock market with over I 00 individuals using their own money to buy and sell shares in four corporations involved in a business game. Trailing was at four professional broker-operated trading posts. Both a boom and a panic were encountered. IO See the research series in experimental economics starting with Smith (1979).
Ch. 62: Game Theory and Experimental Gaming
2341
was designed to be cooperative and the other competitive. The live players tended to cooperate with the former and to compete with the latter. Shubik, Wolf and Eisenberg ( 1 972), Shubik and Riese (1 972), Liu (1973), and others have over the years run a series of experiments with an oligopolistic market game model with a real player versus either other real players or artificial players [Shubik, Wolf and Lockhart (197 1 )] . In the Systems Development Corporation experiments markets with 1 , 2, 3, 5, 7, and 10 live players were run to examine the predictive value of the noncooperative equilibrium and its relationship with the competitive equilibrium. The average of each of the last few periods tended to lie between the beat-the-average and noncooperative solution. A money prize was offered in the Shubik and Riese ( 1 972) experiment with 10 duopoly pairs to the firm whose profit relative to its opponent was the highest. This converted a nonzero-sum game into a zero-sum game and the predicted outcome was reasonably good. In a series of experiments run with students in game theory courses the students each first played in a monopoly game and then played in a duopoly game against an artificial player. The monopoly game posed a problem in simple maximization. There was no significant correlation in performance (measured in terms of profit ranking) between those who performed well in monopoly and those who performed well in duopoly. Business games have been used for the most part in business school education and corporate training courses with little or no experimental concern. Business game models have tended to be large, complex and ad hoc, but games to study oligopoly have been kept relatively simple in order to maintain experimental control as can be seen by the work of Sauermann and Selten (1959); Hoggatt (1959); Fouraker, Shubik and Siegel ( 1 961); Fouraker and Siegel ( 1963); Friedman (1967, 1 969); and Friedman and Hoggatt ( 1 980). A difficulty with the simplification is that it becomes so radical that the words (such as firm production cost, advertising, etc.) bear so little resemblance to the com plex phenomena they are meant to evoke that experience without considerable ability to abstract may actually hamper the players. 1 1 The first work in cooperative game experimental economics was the prize-winning book of Siegel and Fouraker (1 960) which contained a series of elegant simple experi ments in bilateral monopoly exploring the theories of Edgeworth ( 1 8 8 1 ), Bowley ( 1928) and Zeuthen ( 1 930). This was interdisciplinary work at its best with the senior author being a quantitatively oriented psychologist. The book begins with an explicit statement of the nature of the economic context, the forms of negotiation and the information con ditions. In game-theoretic terms the Edgeworth model leads to the no-sidepayment core, the Zeuthen model calls for a Nash-Harsanyi value and the Bowley solution is the out come of a two-stage sequential game with perfect information concerning the news.
1 1 Two examples where the same abstract game can be given different scenarios are the Ph.D. thesis of R. Simon, already noted, and a student paper by Bulow where the same business game was run calling the same control variable advertising in one run and product development in another.
2342
M. Shubik
In the Siegel-Fouraker experiments subjects were paid, instructed to maximize profits and anonymous to each other. Their moves were relayed through an intermediary. In their first experiments they obtained evidence that the selection of a Pareto-optimal outcome increased with increasing information about each others' profits. The simple economic context of the Siegel-Fouraker experiments combined with care to maintain anonymity gave interesting results consistent with economic theory. In contrast the imaginative and rich teaching games of Raiffa ( 1983) involving face-to-face bargaining and complex information conditions illustrate how far we are from a broadly acceptable theory of bargaining backed with experimental evidence. Roth ( 1983), recog nizing especially the difficulties in observing bargainers' preferences, has devised sev eral insightful experiments on bargaining. In particular, he has concentrated on two person bargaining where failure to agree gives the players nothing. This is a constant threat game and the various cooperative fair division theories such as those of Nash, Shapley, Harsanyi, Maschler and Pedes, Raiffa, and Zeuthen can all be considered. Both in his book and in a perceptive article Roth ( 1 979, 1 983) has stressed the prob lems of the game-theoretic approach to bargaining and the importance of and difficul ties in linking theory and experimentation. The value and fair division theories noted are predicated on the two individuals knowing their own and each others' preferences. Abstractly, a bargaining game is defined by a convex compact set of outcomes S and a point t in that set which represents the disagreement payoff. A solution to a bargaining game (S, t) is a rule applied to (S, t) which selects a point in S. The game theorist, by assuming external symmetry concerning the personalities and other particular fea tures of the players and by making the solution depend just upon the payoff set and the no-agreement point, has thrown away most if not all of the social psychology. Roth's experiments have been designed to allow the expected utility of the partic ipants to be determined. Participants bargained over the probability that they would receive a monetary prize. If they did not agree within a specified time both received nothing. The experiment of Roth and Malouf ( 1979) was designed to see if the outcome of a binary lottery game 1 2 would depend on whether the players are explicitly informed of their opponent's prize. They experimented with a partial and a complete information condition and found that under incomplete information there was a tendency towards equal division of the lottery tickets, whereas with complete information the tendency was towards equal expected payoffs. The experiment of Roth and Murnighan (1982) had one player with a $20 prize and the other with a $5 prize. A 4 x 2 factorial design of information conditions and common knowledge was used: ( 1 ) neither player knows his competitor's prize; (2) and (3) one does; (4) both do. In one set of instances the state of information is common knowledge, in the other it is not common knowledge. The results suggested that the effect of information is primarily a function of whether the player with the smaller prize is informed about both. If this is common knowledge 12 If a player obtained 40% of the lottery tickets he would have a 40% chance of winning his personal prize and a 60% chance of obtaining nothing.
2343
Ch. 62: Game Theory and Experimental Gaming
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I
Figure 5 .
there is less disagreement than if it is not. Roth and Schoumaker ( 1 983) considered the proposition that if the players were Bayesian utility maximizers then the previous results could indicate that the effect of information was to change players' subjective beliefs. The efforts of Roth and colleagues complement the observations of Raiffa ( 1 983) on the complexities faced in experimenting with bargaining situations. They clearly stress the importance of information. They avoid face-to-face complications and they bring new emphasis to the distinction that must be made between behavioral and normative approaches. Perhaps one of the key overlooked factors in the understanding of compe tition and bargaining is that they are almost always viewed by economists and game theorists as resolution mechanisms for individuals who know what things are worth to themselves and others. An alternative view is that the mechanisms are devices which help to clarify and to force individuals to value items where previously they had no clear scheme of evaluation. In many of the experiments money is used for payoffs possibly because it is just about the only item which is clearly man-made and artificial that exists in the economy in such a form that it is normatively expected that most individuals will have a conscious value for it even though they have hardly worked out valuations for most other items. Schelling (1961) suggested the possible importance of "natural" focal points in games. Stone ( 1958) performed a simple experiment in which two players were required to select a vertical and a horizontal line respectively on a payoff diagram as shown in Figure 5 . If the two lines intersect within or on the boundary (indicated by ABC) the players receive payoffs determined by the intersection of the lines (X provides an ex ample). If the lines intersect outside of ABC then both players receive nothing. Stone's results show a bimodal distribution with the two points selected being B , a "prominent point" and E, an equal-split point. In class I have run a one-sided version of the Stone experiment six times. The results are shown in Table 3 and are bimodal. A more detailed discussion of some of the more recent work 1 3 on experimental gam ing in economics is to be found in the survey by Roth ( 1988), which includes the consid-
13
A bibliography on earlier work together with commentary can be found in Shubik (1 975b). A highly useful source for game experiment literature is the volumes edited by Sauermann starting with Sauermann (1967-78).
2344
M. Shubik Table 3
1980 198 1 1983 1984 1985 1988
E
B
9 10 12 8 8 24
16 11 8 6 10 15
Other offers*
13 II 4 5 7 7
* These are scattered on several other points.
erable work on bidding and auction mechanisms [a good example of which is provided by Radner and Schotter (1989)] . 5. Experimentation and operational advice
Possibly the closest that experimental economics comes to military gaming is in the testing of specific mechanisms for economic allocation and control of items involving some aspects of public goods. Plott (1 987) discusses policy applications of experimen tal methods. This is akin to Roth's category of "whispering into the ears of princes", or at least bureaucrats. He notes agenda manipulation for a flying club [Levine and Plott (1977)], rate-filing policies for inland water transportation [Hong and Plott ( 1 982)] and several other examples of institution design and experimentation. Alger ( 1 986) consid ers the use of oligopoly games for policy purposes in the control of industry. Although this work is related to the concerns of the game theorist and indicates how much behav ior may be guided by structure, it is more directly in the domain of economic application than aimed at answering the basic questions of game theory. 6. Experimental social psychology, political science and law
6.1. Social psychology It must be stressed that the same experimental game can be approached from highly dif ferent viewpoints. Although interdisciplinary work is often highly desirable, much of experimental gaming has been carried out with emphasis on a single discipline. In par ticular, the vast literature on the Prisoner's Dilemma contains many experiments strictly in social psychology where the questions investigated include how play is influenced by sex differences or differences in nationality.
6.2. Political science We may divide much of gaming experiments in political science into two parts. One is concerned with simple matrix experiments used for the most part for analogies to situations involving threats and international bargaining. The second is devoted to ex perimental work on voting and committees.
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The gaming section of Conflict Resolution contains a large selection of articles heav ily oriented towards 2 x 2 matrix games with the emphasis on the social psychology and political science interpretations. Much of the political science gaming activities are not of prime concern to the game theorist. The experiments dealing with voting and committees are more directly related to strictly game-theoretic concerns. An up-to-date survey is provided by McKelvey and Ordeshook ( 1987). They note two different ways of viewing much of the work. One is a test of basic propositions and the other an attempt to gain insights into poorly under stood aspects of political process. In the development of game theory at the different levels of theory, application and experimentation, a recognition of the two different as pects of gaming is critical. The cooperative form is a convenient fiction. In the actual playing of games, mechanisms and institutions cannot be avoided. Frequently the act of constructing the playable game is a means of making relevant case distinctions and clarifying ill-specified models used in theory.
6.3. Law There is little of methodological interest to the game theorist in experimental law and economics. A recent lengthy survey article by Hoffman and Spitzer (1985) provides an extensive bibliography on auctions, sealed bids, and experiments with the provision of public goods (a topic on which Plott and colleagues have done considerable work). The work on both auctions and public goods is of substantive interest to those interested in mechanism design. But essentially as yet there is no indication that the intersection between law, game theory and experimentation is more than some applied industrial organization where the legal content is negligible beyond a relatively simplistic misin terpretation of the contextual setting of threats. The cultural and academic gap between game theorists and the laissez-faire school of law and economics is typified by virtually totemistic references to "The Coase The orem". This theorem, to the best of this writer's understanding, amounts to a casual comment by Coase ( 1 960) to the effect that two parties who can harm each other, but who can negotiate, will arrive at an efficient outcome. Hoffman and Spitzer (1 982) offer some experimental tests of the Coase so-called theorem, but as there is no well-defined context-free theorem, it is somewhat difficult to know what they are testing beyond the proposition that if people can threaten each other but also can make mutually beneficial deals they may do so.
7. Game theory and military gaming
The earliest use of formal gaming appears to have been by the military. A history and discussion of war gaming has been given elsewhere [Brewer and Shubik ( 1979)]. The expenditures of the military on games and simulations for training and planning pur poses are orders of magnitude larger than the expenditures of all of the social sciences on
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experimental gaming. Although the games devoted to doctrine, i.e., the actual testing of overall planning procedures, can in some sense be regarded as research on the relevance and viability of overall strategies, at least in the open literature there is surprisingly little connection between the considerable activity in war gaming and the academic research community concerned with both game theory and gaming. War games such as the Global War Game at the U.S. Naval War College may involve up to around 800 players and staff and take many weeks spread over several years to play. The compression of game time as compared to clock time is not far from one to one. In the course of play threats and counterthreats are made. Real fears are manifested as to whether the simulated war will go nuclear. Yet there is little if any connection between these activities and the scholarly game-theoretic literature on threats.
8. Where to with experimental gaming?
Is there such a thing as context-free experimental gaming? At the most basic level the answer must be no. The act of running an experimental game involves process. Games that are run take time and require the specification of process rules. ( 1 ) In spite of the popularity of the 2 x 2 matrix game and the profusion of cases for the 2 x 2 x 2 and 3 x 3 , several basic theoretical and experimental questions remain to be properly formulated and answered. In particular with the 2 x 2 game there are many special cases, games with names which have attracted considerable attention. What is the class of special games for the larger examples? Are there basic new phenomena which appear with larger matrix games? What, if any, are the special games for the 2 x 2 x 2 and the 3 x 3 ? (2) Setting aside game theory, there are basic problems with the theory of choice under risk. The critique of Kahneman and Tversky (1979, 1984) and various explana tions of the Allais paradox indicate that our understanding of one-person reaction to risk is still open to new observations, theory and experimentation. It is probable that different professions and societies adopt different ways to cope with individual risk. This suggests that studies of fighter pilots, scuba divers, high rise construction work ers, demolition squads, mountaineers and other vocations or avocations with high-risk characteristics is called for. The social psychologists indicate that group pressures may influence corporate risk taking [Janis (1982)] . In the economics literature it seems to have been overlooked that probably over 80% of the economic decisions made in a society are fiduciary decisions with someone acting with someone else's money or life at stake. Yet no experimentation and little theory exists to account for fiduciary risk behavior. A socialized individual has a far better perception of the value of money than the value of thousands of goods. Yet little direct consideration has been given to the important role of money as an intermediary between goods and their valuation. (3) The experimental evidence is overwhelming that one-point predictions are rarely if ever confirmed in few-player games. Thus it appears that solutions such as the value
Ch. 62: Game Theory and Experimental Gaming
2347
are best considered as normative or as benchmarks for the behavior of abstract players. In mass markets or voting much (but not all) of the social psychology may be wiped out: hence chances for prediction are improved. For few-person games more explicitly interdisciplinary work is called for to consider how much of the observations are being explained by game theory, personal or social factors. (4) Game-theoretic solutions proposed as normative procedures can be viewed as what we should teach the public, or as reflecting what we believe public norms to be. Solutions proposed as reflecting actual behavior (such as the noncooperative theory) require a different treatment and concern for experimental validation. The statement that individuals should select the Pareto-optimal noncooperative equilibrium if there is one is representative of a blend of behavioral and normative considerations. There is a need to clarify the relationship between behavioral and normative assumptions. (5) Little attention appears to have been paid the effects of time. Roth, Murnighan and Schoumaker ( 1988) have evidence concerning the importance of the end effect in bargaining. But both at the level of theory and experimentation clock time seems to have been of little concern, even though in military operational gaming questions concerning how time compression or expansion influences the game are of considerable concern. (6) The explicit recognition that increasing numbers of players change the nature of communication and information 1 4 requires more exploration of numbers between 3 and say 20. How many is many has game-theoretical, psychological and socio-psychological dimensions. One can study how game-theoretic solutions change with numbers, but at the same time psychological and socio-psychological possibilities change. (7) Game-theoretic investigations have cast light on phenomena such as bluff, threat and false revelation of preference. All of these items at one time might have been re garded as being almost completely in the domain of psychology or social psychology, yet they were susceptible to game-theoretic analysis. In human conflict and cooperation words such as faith, hope, charity, envy, rage, revenge, hate, fear, trust, honor and recti tude all appear to play a role. As yet they do not appear to have been susceptible to game theoretic analysis and perhaps they may remain so if we maintain the usual model of rational man. It might be different if a viable theory can be devised to consider context rational man who, although he may calculate and try to optimize, is limited in terms of capacity constraints on perception, memory, calculation and communication. It is pos sible that the instincts and emotions are devices that enable our capacity-constrained rational man to produce an aggregated cognitive map of the context of his environment simple enough that he can act reasonably well with his limited facilities. Furthermore, actions based on emotion such as rage may be powerful aggregated signals. The development of game theory represented an enormous step forward in the realiza tion of the potential for rational calculation. Yet paradoxically it showed how enormous the size of calculation could become. Computation, information and communication
1 4 Around thirty years ago at MIT there was considerable interest in the structure of communication networks. None of this material seems to have made any mark on game-theoretic thought
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capacity considerations suggest that at best in situations of any complexity we have common context rather than common knowledge. There is constant feedback between theory, observation and experimentation. Exper imental game theory is only at its beginning but already some of the messages are clear. Context matters. The game-theoretic model of rational man is not an idealization but an approximation of a far more complex creature which performs under severe constraints which appear in the concerns of the psychologists and social psychologists more than in the game-theoretic literature.
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59:371-404. Mumighan, J.K., A.E. Roth and F. Schoumaker (1988), "Risk aversion in bargaining: An experimental study", Journal of Risk and Uncertainty 1 :95-1 1 8. Nash, J.F., Jr. (195 1), "Noncooperative games", Annals of Mathematics 54:289-295. O'Neill, B. (1986), "International escalation and the dollar auction", Conflict Resolution 30:33-50. O'Neill, B. (1987), "Nonmetric test of the minimax theory of two-person zero-sum games", Proceedings of the National Academy of Sciences, USA 84:2106-2109. O'Neill, B. (199 1), "Comments on Brown and Rosenthal's reexamination", Econometrica 59:503-507. Plott, C.R. ( 1987), "Dimensions of parallelism: Some policy applications of experimental methods", in: A.E. Roth, ed., Laboratory Experimentation in Economics: Six Points of View (Cambridge University Press, Cambridge) 193-219. Powers, I.Y. (1986), "Three essays in game theory", Ph.D. Thesis (Yale University, New Haven, CT). Radner, R., and A. Scholler (1989), "The sealed bid mechanism: An experimental study", Journal of Eco nomic Theory 48: 179-220. Raiffa, H. ( 1983), The Art and Science of Negotiation (Harvard University Press, Cambridge, MA).
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2350 Rapoport, A., M.J. Guyer and D.G. Gordon (1976), The 2 Arbor).
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Roth, B. ( 1 975), "Coalition formation in the triad", Ph.D. Thesis (New School of Social Research, New York, NY). Roth, A. E. (1979), Axiomatic Models of Bargaining, Lecture Notes in Economics and Mathematical Systems, Vol. 170 (Springer-Verlag, Berlin). Roth, A.E. (1983), "Toward a theory of bargaining: An experimental study in economics", Science 220:687691 . Roth, A.E. ( 1986), "Laboratory experiments i n economics", in: T. Bewely, ed., Advances i n Economic Theory (Cambridge University Press, Cambridge), 245-273. Roth, A.E. (1987), Laboratory Experimentation in Economics: Six Points of View (Cambridge University Press, Cambridge). Roth, A.E. ( 1988), "Laboratory experimentation in economics: A methodological overview", The Economic Journal 98:974-l 03 1 . Roth, A.E., and M. Malouf (1979), "Game-theoretic models and the role of information in bargaining", Psy chological Review 86:574-594. Roth, A.E., and J.K. Murnighan ( 1982), "The role of information in bargaining: An experimental study", Econometrica 50: 1 123-1142. Roth, A.E., J.K. Murnighan and F. Schoumaker ( 1988), "The deadline effect in bargaining: Some experimental evidence", American Economic Review 78:806-823. Roth, A.E., and F. Schoumaker (1983), "Expectations and reputations in bargaining: An experimental study", American Economic Review 73:362-372. Sauermann, H. (ed.) (1967-78), Contributions to Experimental Economics, Vols. 1-8 (Mohr, Tiibingen). Sauermann, H., and R. Selten (1959), "Ein Oligopolexperiment", Zeitschrift fiir die Gesarnte Staatswis senschaft 1 15:427-47 1 . Schelling, J.C. (1961), Strategy and Conflict (Harvard University Press, Cambridge, MA). Selten, R. (1982), "Equal division payoff bounds for three-person characteristic function experiments", in: R. Tietz, ed., Aspiration Levels in Bargaining and Economic Decision-Making, Lecture Notes in Eco nomics and Mathematical Systems, Vol. 213 (Springer-Verlag, Berlin), 265-275. Selten, R. ( 1987), "Equity and coalition bargaining in experimental three-person games", in: A.E. Roth, ed., Laboratory Experimentation in Economics: Six Points of View (Harvard University Press, Cambridge, MA), 42-98. Shubik, M. ( 1 962), "Some experimental non-zero-sum games with lack of information about the rules", Man agement Science 8:215-234. Shubik, M. (1970), "A note on a simulated stock market", Decision Sciences 1 : 1 29-141. Shubik, M. (1971a), "The dollar auction game: A paradox in noncooperative behavior and escalation", The Journal of Conflict Resolution 15: 109-1 1 1 . Shubik, M. (197lb), "Games of status", Behavioral Sciences 16:1 17-129. Shubik, M. (1 975a), Games for Society, Business and War (Elsevier, Amsterdam). Shubik, M. (1975b), The Uses and Methods of Gaming (Elsevier, Amsterdam). Shubik, M. (1978), "Opinions on how one should play a three-person nonconstant-sum game", Simulation and Garnes 9:301-308. Shubik, M. (1986), "Cooperative game solutions: Australian, Indian and U.S. opinions", Journal of Conflict Resolution 30:63-76. Shubik, M., and M. Reise (1972), "An experiment with ten duopoly games and beat-the-average behavior", in: H. Sauermann, ed., Contributions to Experimental Economics (Mohr, Tiibingen), 656-689. Shubik, M., and G. Wolf (1977), "Beliefs about coalition formation in multiple-resource three-person situa tions", Behavioral Science 22:99-106. Shubik, M., G. Wolf and H. B. Eisenberg (1972), "Some experiences with an experimental oligopoly business game", General Systems 17:61-75.
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Shubik, M., G. Wolf and S . Lockhart (1971), "An artificial player for a business market game", Simulation and Games 2:27-43. Shubik, M., G. Wolf and B. Poon ( 1974), "Perception of payoff structure and opponent's behavior in repeated matrix games", Journal of Conflict Resolution 1 8:646-656. Siegel, S., and L.E. Fouraker ( 1960), Bargaining and Group Decision-Making: Experiments in Bilateral Monopoly (Macmillan, New York). Simon, R.I. (1967), "The effects of different encodings on complex problem solving", Ph.D. Dissertation (Yale University, New Haven, CT). Smith, V.L. (1965), "Experimental auction markets and the Walrasian hypothesis", Journal of Political Economy 73:387-393. Smith, V.L. (1979), Research in Experimental Economics, Vol. 1 (JAI Press, Greenwich, CT). Smith, V.L. (1986), "Experimental methods in the political economy of exchange", Science 234: 167-172. Stone, J.J. (1958), "An experiment in bargaining games", Econometrica 26:286-296. Tegler, A.I. ( 1980), Too Much Invested to Quit (Pergamon, New York). Zeuthen, F. (1930), Problems of Monopoly and Economic Warfare (Routledge, London).
AUTH OR I NDEX
n indicates citation in footnote. 2175, 2180-2 182, 2189, 2192, 2196, 2200, 2249n, 2259, 2263, 2336, 2348 Austen-Smith, D. 2207n, 221 1 , 2217, 2218, 2222, 2224, 2227 Ausubel, L.M. 1905, 1 9 10, 1913, 1913n, 1915, 1918-1920, 1922, 1928-1930, 1932-1936, 194 1 , 1942 Avenhaus, R. 1952, 1955, 1 956, 1962, 1965, 1969, 1972, 1974, 1980, 1983, 1984 Avery, C. 2245, 2263 Avis, D. 1743, 1756 Axelrod, R. 2016, 2021 Ayres, I. 1908, 1 942, 2239n, 2247n, 2252n, 2263, 2267
Abreu, D. 1881, 1893, 2289, 2290, 2290n, 2291, 2292, 2294, 2296n, 2297-230 1 , 230 l n, 23 12, 23 13, 2320 Adrnati, A. 1922, 1923, 1935, 1938, 1941 Aggarwal, V. 1734, 1756 Aghion, P. 2315n, 2320 Aivazian, V.A. 2247, 2249n, 2263 Aiyagari, R. 1790n, 1797 Akerlof, G.A. 1588, 1589, 1905, 1941 Alger, D. 2344, 2348 Allen, B. 1680, 1685, 1 794n, 1795n, 1798, 2275, 2320 Altrnann, J. 1952, 1984 Amer, R. 2074, 2075 Amir, R. 1 8 13, 1829 Anderson, R.M. 1 787n, 1790n, 1791n, 1798, 2171, 2179, 2 1 82 Anscombe, F.J. 1951, 1984 Arcand, J.-L.L. 2016n, 2021 Areeda, P. 1868, 1893 Arlen, J. 2234n, 2236, 2263 Armbruster, W. 1679, 1685 Arrow, K.J. 1792n, 1 795n, 1798, 21 89, 2200 Artstein, Z. 177 l n, 1785n, 1792n, 1794n, 1798, 2 1 36, 2162 Arya, A. 23 17n, 2320 Ash, R.B. 1767n, 1769n, 1798 Aubin, J. 2162 Audet, C. 1706, 1718, 1745, 1756 Aumann, R.J. 1524, 1530, 153 1 , 1537, 1539, 1544, 1571, 1576, 1577, 1585, 1589, 1590, 1599-160 1 , 1603, 1 606-1609, 1613, 1614, 1636, 1654, 1655, 1657, 167 1 , 1673, 1675, 1676, 1 680-1682, 1685, 1 690, 1 709, 1 7 1 1 , 1713, 1714, 1718, 1763n, 1 764, 1765, 1765n, 1766n, 1770, 177 1 n, 1774n, 1775n, 1 780n, 1 784, 1787, 1789n, 1 793n, 1 797n, 1798, 2028, 2039, 2040, 2044, 2052, 2072, 2075, 2084-2088, 2095, 2 1 17, 2 1 1 8, 21 28-2 1 3 1 , 2135, 2 1 36, 2140, 2141, 2145, 2150, 2 152, 2158, 2159, 2 161-2163, 217 1-2173, 2 173n,
Backhouse, R. 1 992n, 2021 Bag, P.K. 2049, 2052 Bagwell, K. 1 566, 1590, 1861n, 1 867n, 1 872n, 1893 Baiman, S . 1953, 1984 Bain, J. 1 859n, 1893 Baird, D. 223ln, 2232, 2252n, 2263 Balder, E.J. 1783n, 1793n, 1794, 1795, 1795n, 1796-1935 Baliga, S . 1 996n, 2012, 2021, 2313, 23 15, 2319, 2320 Balkenborg, D. 1565, 1568-1572, 1590 Banks, J.S. 1565, 1 590, 1635, 1657, 2207, 2207n, 2210, 2217-2220, 2222, 2222n, 2224, 2227, 2288n, 2293, 2320 Banzhaf ill, J.P. 2052, 2070, 2075, 2253, 2253n, 2257n, 2263 Barany, I. 1 606, 1658 Baron, D. 2218, 2219, 2227 Barton, J. 2263 Barut, Y. 1790n, 1799 Ba�9i, E. 1799 Basmann, R.L. 2340, 2349 Bastian, M. 1734, 1756 Baston, V.J. 1956, 1973, 1984 Basu, K. 1527, 1528, 1544, 1590, 1622, 1658 1-1
1-2 Battaglio, R.C. 233 1 , 2340, 2348, 2349 Battenberg, H.P. 1969, 1980, 1984 Battigalli, P. 1542, 1590, 1627, 1658 Baye, M.R. 1 873n, 1893 Beard, T.R. 2236, 2263 Bebchuk, L.A. 2232n, 2239n, 2247n, 2252n, 2263 Becker, G. 2263 Bellman, R. 1 622, 1658, 2339, 2348 Ben-Porath, E. 1544, 1 566, 1590, 1605, 1622, 1658 Bennett, C.A. 1 952, 1985 Benoit, J.-P. 2260n, 2263 Benson, B.L. 2000, 2021 Berbee, H. 2 1 27, 2136, 2140, 2163 Berge, C. 1773n, 1799 Bergin, J. 1793n, 1 799, 2298n, 23 13, 2320 Bernhardt, D. 1793n, 1799 Bernheim, B.D. 1527, 1585, 1586, 1590, 1601, 1658, 2273, 2320 Besley, T. 2013n, 2021 Bewley, T.F. 178ln, 1799, 1822, 1830 Bierlein, D. 195 1 , 1983, 1984 Bikhchandani, S. 1904n, 1918, 1918n, 1942 Billera, L.J. 2163 Billingsley, P. 1767n, 1777n, 1789n, 1799 Binmore, K. 1 544, 1590, 1623, 1658, 1 899n, 1941n, 1942 Bird, C.G. 1 954, 1985 Birmingham, R.L. 2231, 2231n, 2264 Black, D. 2205, 2227 Blackwell, D. 18 13, 1821, 1830 Bliss, C. 23 18, 2320 Block, H.D. 2062, 2075 Blume, A. 1571, 1590 Blume, L.E. 1550, 1 590, 1629, 1658, 2240n, 2264 Boge, VV. 1679, 1685 Bollobas , B. 177ln, 1799 Bomze, I.M. 1745, 1756 Bonnano, G. 1658 Borch, K. 1 952, 1953, 1985 Borenstein, S. 1 872n, 1893 Borgers, T. 1542, 1590, 1605, 1658 Bonn, P.E.M. 1593, 1 660, 1700, 1719, 1750, 1756, 1757 Bostock, F.A. 1956, 1973, 1984 Bowen, WM. 1952, 1985 Bowley, A.L. 2341, 2348 Boylan, R. 23 18, 2319, 2321
Author Index Brams, S.J. 1799, 1952, 1985, 2259n, 2264, 2268 Brandenburger, A. 1 524, 1544, 1590, 1601, 1608, 1609, 1613, 1654, 1657, 1658, 168 1 , 1685, 1713, 1718 Brander, J.A. 1855, 1893, 2015, 2021 Brewer, G. 2329, 2345, 2348 Brezis, E. 2016n, 2021 Brown, D. 2163, 2 1 8 1 , 2183 Brown, G.W 1707, 1719 Brown, J.N. 2334, 2348 Brown, J.P. 223 1, 223ln, 2233, 2264 Brusco, S. 23 13, 23 1 9-2321 Budd, J.W 1939, 1 942 Bulow, J. 1 858n, 1 894, 2232n, 2264 Bums, M.R. 2010, 2021 Butnariu, D. 2163, 2 1 8 1 , 2183 Cabot, A.V. 1813, 1832 Cabrales, A. 2319, 2321 Calabresi, G. 223 1 , 2242, 2264 Calfee, J.E. 2236, 2264 Callen, J.L. 2247, 2249n, 2263 Calvo, E. 2042, 2052 Camerer, C. 2017n, 2021, 2288n, 2320 Cameron, C. 2249, 2264 Cameron, R. 2009, 2021 Canty, M.J. 1952, 1 974, 1984 Caplow, T. 2337, 2348 Card, D. 1936, 1940, 1942 Carlos, A.M. 1 997, 1 998n, 2021 Carlsson, H. 1579, 1584, 1590 Carreras, F. 2050, 2052, 2074, 2075 Carruthers, B.E. 1999n, 2021 Castaing, C. 1773n, 1 780n, 1782n, 1799 Chakrabarti, S.K. 1769n, 1770n, 1793n, 1794n, 1799 Chakravorti, B . 2297, 23 15, 2316, 2321 Chamberlain, E.H. 2339, 2348 Chamberlin, E. 1799 Champsaur, P. 1795n, 1799, 2163, 2171, 2172, 2 1 8 1 , 2183, 2200 Chander, P. 2276n, 2321 Charnes, A. 1729, 1755, 1756, 1813, 1830 Chattetjee, K. 1905, 1934, 1942 Chen, Y. 23 19, 2321 Cheng, H. 2163, 2181, 2183 Chin, H.H. 1531, 1591, 1697, 1700, 170 1 , 1719 Ching-to Ma, A. 2265 Chipman, J.S. 1796n, 1799 Chiu, Y.S. 2049, 2052
Author Index Cho, 1.-K. 1565, 159 1 , 1635, 1 65 1 , 1658, 1862, 1 894, 1 904n, 1 9 1 8, 19 18n, 1921, 1935, 1942 Chun, Y. 2036, 2037, 2052, 2262n, 2264 Chvatal, v. 1728, 1756 Cirace, J. 2252n, 2264 Clark, C.E. 2339, 2348 Clark, G. 1 999n, 2021 Clarke, F. 2107, 2 1 1 8 Clay, K . 2004, 2021 Coase, R.H. 1908, 19!7, 1930, 1937, 1942, 223 ! , 2241 , 2252n, 2264, 2345, 2348 Coate, S. 2013n, 2021 Cochran, W.G. 1968, 1985 Cohen, L.R. 2249, 2264 Coleman, J.C. 2253n, 2264 Condorcet, M. 2205, 2227 Conklin, J. 2009, 2021 Constantinides, G.M. 1 796n, 1799 Cooter, R.D. 2232, 2236, 2237, 2243n, 2264 Corchon, L. 2297, 23 15, 23 18, 23 1 8n, 2320, 2321 Cordoba, J.M. 1 795n, 1799 Cottle, R.W. 173 1 , 1749, 1756 Coughlan, P. 2225n, 2227 Coughlin, P. 2222, 2227 Coulomb, J.M. 1 8 13, 1830, ! 847-1849 Coumot, A.A. !599, 1658, 1 764n, 1794, 1795n, 1799 Craft, C.J. 2339, 2348 Cramton, P. !90ln, 1 906, 1 907, 1934, 1935, 1938-1940, 1 942, 1 943, 2232n, 2264, 23 !6n, 2321 Craswell, R. 2236, 2264 Crawford, V.P. 1 899n, 1943, 1 998n, 2013, 202 1 , 2226, 2227, 2289n, 2292, 2302, 2321 Cremer, J. 1943 Cunningham, W. 1 992n, 2021 Curran, C. 2236, 2265 Cutland, N. ! 787n, 1799, 1802, 1 804 da Costa Werlang, S.R. 1 663 Da Rin, M. 2012, 2021 Dab!, R.A. 2253n, 2264 Dalkey, N. 1646, 1659 Danilov, V. 2287, 2287n, 232! Dantzig, G.B. 1730, 1757 Dasgupta, A. 2049, 2052 Dasgupta, P. 1530, 1591, 17 16, 17 19, 1 795n, 1799, !905, 1943, 2274n, 2277n, 2280n, 232! d'Aspremont, C. 1 7 17, 1719 Daughety, A. 2232n, 2264
I-3 David, P.A. 1992n, 1997, 2020n, 2021 Davis, E.L. 1997, 2021 Davis, M.D. 1952, 1985, 2259n, 2268 De Frutos, M.A. 2262n, 2264 Debreu, G. !533, !591, 16 12, 1659, 1766n, 1773n, 1787n, 1795n, 1 798-1800 Deke1, E. 1 566, 1 590, 1 60 1 , 1605, 1658, 1659 Demange, G. 2289n, 2321 Demski, J. 2317, 2321 Deneckere, R.J. 19 10, 1913, 1913n, 19 15, 1916, 1 9 1 8-1920, 1922-1924, 1928-1930, 1932-1936, 1941-1943 Denzau, A. 2216, 2227 Derks, J. 2044, 2052 Derman, C. 1974, 1985 DeVries, C. 1873n, 1893 Dewatripont, M. 23 1 5n, 2320 Dhillon, A. 1659 Diamond, D.W. 2010, 2021 Diamond, H. 1 974-1976, 1985 Diamond, P.A. 223 1 , 2231n, 2234n, 2264 Dickhaut, J. 1706, 17 19, 1742, 1757 Dierker, E. 1533, 1 59 1 , 1797n, 1800 Diestel, J. 1 780n, 1781-1784, 1784n, 1785, 1786, 1786n, 1 787-1975 Dincleanu, N. 1782n, 1 800 Dixit, A. 1 860, 1894 Dold, A. 1535, 1591 Doob, J.L. 1 790, 1791n, 1 800 Downs, A. 2220, 2227 Dresher, M. 1528, 1553, 159 1 , 1689, 17 19, 1 95 1 , 1952, 1955, 1956, 1 969, 1 972, 1985 Dreze, J.H. 2028, 2039, 2040, 2052, 2072, 2075, 2 1 92, 2196, 2200 Driessen, T. 2 1 09, 2 1 1 8 Dube� � 1585, 1 59 1 , 1794n, 1795, 1796, 1796n, 1797-1975, 2037, 2038, 2038n, 2052, 2067-207 1 , 2075, 2 1 26, 2 152-2 1 54, 2163, 2 1 8 1 , 2183, 2253n, 2254n, 2264 Duffie, D. 1821, 1830 Duggan, J. 2219, 2220, 2222n, 2227, 2294, 2310, 23 12, 2314, 23 14n, 23 17n, 23 18, 2321 Dutta, B. 2273, 228 1 , 2282, 2287, 2292, 2294, 2297, 2301, 2309, 23 1 0n, 23 19, 232 1 , 2322 Dutta, P. 1 8 13, 1830 Dvoretsky, A. 1765, 1 765n, 1 800, 2144, 2163 Dye, R.A. 1953, 1985 Eaton, J. 1 857, 1894 Eaves, B.C. 1 742, 1 745, 1757 Edgeworth, F.Y. 1800, 2341, 2348
Author Index
1-4
Edlin, A.S. 2236, 2265
Friedman, J. 1 882n, 1 894, 2341 , 2348
Eichengreen, B. 1997, 2021
Fudenberg, D. 1 524, 1542, 1 551 , 1591, 1592,
Ein� E. 207 1, 2075
1605, 1623, 1 634, 1653, 1659, 1680, 1685,
Eisele, T. 1679, 1685
1774n, 1796n, 1 800, 1858n, 186\n, 1 868n,
Eisenberg, H.B. 2341 , 2350
1882, 1 894, 19 10-1912, 1914, 1926, 1943,
Eisenberg, T. 2232n, 2265
1994n, 2022
Eliaz, K. 2319, 2322
Fukuda, K. 1743, 1756
Ellison, G. 1580, 1591, 201 1 , 2022 Elmes, S. 1 646, 1659
Gabel, D. 20!0, 2010n, 2022, 2023
Emons, VV. 2236, 2265
Gabszewicz, J.J. 1717, 1719, 1795n, 1 800, 2182,
Endres, A. 2234n, 2265 Eskridge, WN. 2249, 2265 Evangelista, F. 1712, 1719
2 1 83 Gal, S. 1949, 1985 Gale, David 1796n, 1800
Evans, R. 1923, 1924, 1 943
Gale, Douglas 1956, 1985
Everett, H. 1830, 1836n, 1 840, 1 849
Gamson, WA. 2337, 2348 Garcia, C.B. 1749, 1757
Fagin, R. 1681, 1682, 1684, 1685 Falkowski, B.J. 1 969, 1980, 1984 Falmagne, J.C. 2062, 2075 Fan, K. 1612, 1659, 1766n, 1767, 1800 Farber, H.S. 2013, 2022 Farquharson, R. 2208, 2227, 2289n, 2293, 2322 Farrell, J. 1585, 1591, 1 907, 1943, 2265 Feddersen, T. 2224, 2225, 2227 Federgruen, A. 1830 Fedzhora, L. 2050, 2052 Feichtinger, G. 1 956, 1985 Feinberg, Y. 1 676, 1 676n, 1677, 1685 Feldman, M. l 790n, 1800 Fellner, W 1882n, 1894 Fer�ohn, J. 2212, 2219, 2227, 2249, 2265 Ferguson, T.S. 1 8 13, 1830, 1 83 1 , 1956, 1972, 1985
Garda-Jurado, I. 2050-2052 Gardner, R.J. 2162, 2163, 2182, 2183 Gamaev, A.Y. 1956, 1985 Geanakoplos, J. 1659, 1821, 1830, 1 858n, 1894 Geistfeld, M. 2232n, 2265 Gertne� R. 2232, 2239n, 2247n, 2252n, 2263 Gibbard, A. 2273, 2322 Gibbons, R. 1901n, 1907, 1942, 1943 Gilboa, I. 1745, 1756, 1757, 2062, 2075, 2163 Gilette, D . 1830 Gilles, C. 1790n, 1800 Gilligan, T. 2225, 2227 Glazer, J. 1566, 1 592, 1 899n, 1943, 2245n, 2265, 2283, 2292, 2294, 2300n, 2301, 2301n, 231 ln, 2322 Glicksberg, I.L. 1530, 1592, 1612, 1659, 1717, 1719, 1767, l767n, 1800 Glover, J. 2317, 2317n, 2320, 2322
Fernandez, R. 1 899n, 1943, 2245n, 2265
Golan, E. 2050, 2053
Fershtman, C. 2016, 2022
Goldman, A.J. 1952, 1956, 1985
Filar, J.A. 1 8 1 3, 1 8 14, 1 825, 1830-1832, 1956,
Goldstick, D. 2348
1985
Gordon, D.G. 2332, 2333, 2335, 2336, 2350
Fiorina, M. 2212, 2227
Gorton, G. 2010, 2022
Fischel, WA. 2240n, 2265
Govindan, S. 1534, 1535, 1 565-1567, 1 592,
Fishburn, P.C. 1 609, 1659 Fleming, W 2!05, 2 1 1 8 Flesch, J . 1843, 1 849 Fogelman, F. 2136, 2163
1600, 1640, 1648, 1659 Green, E.J. 1775n, 1793n, 1800, 1877, 1 894, 2008, 2010, 2022 Green, J.R. 2177n, 2 1 83, 2236, 2265
Fohlin, C.M. 2012n, 2022
Green, L. 2340, 2349
Forges, F. 1591, 1 606, 1659, 1680, 1685, 1 8 1 3,
Greenberg, I. 1954, 1985
1820, 1830
Greif, A. 1991n, 1994n, 1995, 1995n, 1 996,
Forsythe, R. 1941n, 1943
1996n, 1999n, 2000, 2001 , 2003n, 2004n,
Fouraker, L.E. 2333, 2341, 2348, 2351
2005, 2007, 2012n, 2014, 2014n, 2016,
Fragnelli, V. 205 1 , 2052
Frenkel, 0. 2334, 2348
2016n, 2017, 2018, 2018n, 2019, 2020n, 2022, 2023
Author Index Gresik, T.A. 1789n, 1795n, 1 800, 1904, 1905, 1905n, 1906n, 1943 Gret1ein, R. 2208, 2227
I-5
1 770n, 177ln, 1775n, 1781n, 1792n, 1793n, 1798, 1801, 1843, 1849, 2033-2035, 2035n, 2039-2042, 2046, 2047, 2047n, 2048, 2052,
Grossman, G. 1 857, 1 894
2064, 2073-2075, 2094--2096, 2098,
Grossman, S.J. 1659, 1 9 1 8n, 1943, 2265
2102-21 1 1 , 2 1 1 3-21 15, 2 1 1 8 , 2 1 3 1 ,
Groves, T. 2285n, 2322
2 1 57-2 160, 2 1 64, 2 1 7 1 , 2172, 2 1 75, 2178,
Gu, W. 1940, 1944
2 1 78n, 2 179n, 2 1 80, 2 1 80n, 2 1 8 1 , 2 1 8 1n,
Guesnerie, R. 1797n, 1 800, 2 1 63, 21 82, 2183
2 1 83
Guinnane, T.W. 2013n, 2021
Hartwell, R.�. 1991n, 2023
Gu1, E 1528, 1534, 1566, 1592, 1659, 1676,
Hauk, E. 1566, 1593
1685, 1796n, 1 800, 1 9 10-1912, 1 9 1 3n, 1914,
Hausner, �- 2338, 2348
19 15, 19 17, 1923, 1 927-1929, 1943, 2046,
Hay, B.L. 2232n, 2265
2047, 2052 Gunderson, �. 1936, 1939, 1942, 1943 Gtith, VV. 1584, 1587-1589, 1591, 1592, 1595, 1955, 1985, 2338, 2348 Guyer, �.J. 2332, 2333, 2335, 2336, 2348, 2350 Haddock, D. 2236, 2265 Hadfield, G. 2239n, 2265 Haimanko, 0. 2 150, 2155, 2156, 2160, 2163, 2 1 64
Haller, H. 1796n, 1797n, 1800, 1 80 1 , 1 899n, 1944, 2245n, 2265 Halmos, P.R. 177 1 , 1 80 1 Halpern, J . Y 1 6 8 1 , 1682, 1684, 1685 Haltiwanger, J. 1 872n, 1894 Hamburger, H. 2333, 2348 Hammerstein, P. 1524, 1592, 1659, 1797n, 1 801 Hammond, P.J. 1 660, 1783n, 1791n, 1795, 1795n, 1796n, 1799, 1 801, 2274n, 2277n, 2280n, 2321 Hansen, P. 1706, 1 7 18, 1745, 1756 Harrington, J. 1 867n, 1 872n, 1894 Harris, C. 1545, 1592 Harris. �. 2232, 2265, 2304, 2322 Harrison, J\. 1936, 1940, 1944 Harrison, G.W. 2265 Harsanyi, J.C. 1523, 1 524, 1530, 1532-1534,
Heath, D.C. 2163 Heifetz, A 1673, 1673n, 1675, 1 676n, 1677, 1682, 1684n, 1685, 1686, 1798 Hellmann, T. 2012, 2021 Hellwig, �- 1545, 1593, 1795n, 1798 Hererro, �- 2293, 2322 Herings, P.J.-J. 1 5 8 1 , 1593, 1660 Hermalin, B. 2231n, 2239, 2239n, 2240n, 2265 Herves-Beloso, C. 1795n, 1801 Heuer, G.A. 1697, 1700, 1 7 19, 1744, 1757 Hiai, E 1785n, 1 801 Hildenbrand, W. 1770n, 1771n, 1775n, 1777n, 1780n, 1781n, 1789n, 1793n, 1 80 1 , 1802 Hillas, J. 1553, 1564, 1565, 1593, 1632, 1634, 1647, 165 1 , 1660, 1719, 1725, 1757 Hinich, �- 222 1 , 2227 Hintikka, J. 1681, 1686 Hoffman, E. 1997, 1998n, 2021, 2243n, 2265, 2345, 2348 Hoffman, P.T. 201 1 , 2020, 2023 Hoggatt, AC. 2339-234 1 , 2348, 2349 Hohfeld, W.N. 2242n, 2265 Holden, S. 1 899n, 1944, 2245n, 2265 Holmstrom, B. 2232, 2265, 2266, 2304, 23 16, 2322 Hong, J.T. 2344, 2349
1537, 1538, 1 547, 1567, 1572, 1574,
Hong, L. 2279n, 2322
1576-1579, 1583-1587, 1589, 1592, 1593,
Hopfinge� E. 1954, 1970, 1972, 1985
1599, 1 606-1608, 1 6 1 8, 1 6 1 9, 1636, 1660,
Housman, D. 1789n, 1793n, 1 802
1678-1681, 1685, 1694-- 1 696, 1704, 1 7 19,
Howson Jr., J.T. 1535, 1594, 1 696, 1701, 1702,
1750, 1757, 1773, 1 80 1 , 1 873, 1 875, 1894, 2034, 2042, 2045, 2052, 2084, 2091 , 2 1 17,
1720, 1725, 173 1 , 1734, 1739, 1740, 1745, 1757, 1758, 1953, 1985
2 1 18, 2 1 7 1 , 2172, 2175, 2 1 80, 2183, 2200,
Hughes, J. 1997, 2024
2277, 2302, 2322
Hurd, AE. 1 787n, 1802
Hart, O.D. 1796n, 1801, 1899, 1 9 17, 1937, 1940, 1944, 2232, 2265 Hart, S. 153 1 , 1541, 1543, 1593, 1 600, 1615, 1660, 1710, 1714, 1 7 1 9, 1725, 1757, 1768n,
Hurkens, S. 1566, 1 57 1 , 1591, 1593 Hurwicz, L. 1795n, 1 802, 2266, 2277, 2279n, 2287n, 2297, 2319, 2322, 2323 Hylton, K.N. 2236, 2266
Author Index
1-6
IAEA 1982, 1985 lchiishi,
T. 2 1 1 6-21 18
Katz, A. 2239n, 2252n, 2266 Katz, 11. 223ln, 2239n, 2265
Imai, H, 2092, 2109, 2 1 18, 2164
Katznelson, Y. 1539, 1590, 1774n, 1789n, 1798
Imrie, R.W. 2255n, 2266
Keiding, H. 1743, 1757
Irwin, D.A. 2015, 2023
Keisler, H.J. 1787n, 1791n, 1802
Isbell, J. 2164
Kelsey, D. 1796n, 1802
Isoda, K. 1 7 16, 1720
Kemperman, J.H.B. 2164 Kennan, J. 1899, 1 899n, 1936, 1937, 1939,
Jackson, 11. 2275, 2277n, 2292, 2294, 2296,
1 940, 1941n, 1943, 1944
2297, 2301, 2302, 2309, 2310, 23 1 3 ,
Kennedy, D. 2242n, 2266
2 3 1 6-23 18, 2323
Kern, R. 2085, 2087, 2088, 2 1 17, 2 1 1 9
Jacobson, �. 1 8 2 1 , 1830
Kervin, J . 1936, 1939, 1943
Jaech, J.L. 1952, 1986
Khan, 11.A. 1767n, 1769n, 1770, 1 77 1 , 1771n,
Jamison, R.E. 1780n, 1802
1772-1777, 1 777n, 1778, 1778n, 1779-1781,
Jansen, 11.J.11. 153 1 , 1565, 1593, 1647, 1 660,
1781n, 1782, 1782n, 1783-1785, 1785n,
1662, 1663, 1696, 1700, 1 7 19, 1721, 1744, 1745, 1750, 1756, 1757, 1759 Jaurnard, B. 1706, 1 7 1 8 , 1745, 1756 Jaynes, G. 1795n, 1802 Jehie1,
P. 1905, 1 944
Jia-He, J. 1663, 1696, 1721
1786, 1786n, 1787-1789, 1 789n, 1790-1793, 1793n, 1 794, 1794n, 1795-1797, 1797n, 1798-2 1 0 1 Kihlstrom, R.E. 1796n, 1 8 0 3 , 1804 Kilgour, D.11. 1952, 1954, 1985, 1986 Kim, S.H. 1 804
Johnston, J.S. 2239n, 2247n, 2266
Kim, T. 1793n, 1794n, 1 804, 23 19, 2323
Jovanovic, B. 1793n, 1796n, 1797n, 1802
Kinghorn, J.R. 20 1 2n, 2023
Judd, K.L. 1802, 2016, 2022
Klages, A. 1953, 1986
Jurg, A.P 1593, 1 660, 1700, 1 7 19, 1750, 1757
Klein, B. 2232n, 2267 Klemm, W.R. 2340, 2349
Kadiyali, V. 1 867n, 1894 Kagel, J.H. 233 1 , 2340, 2348, 2349
Klemperer, P 1858n, 1 894, 1901n, 1907, 1942, 2232n, 2264
Kahan, J.P. 2336, 2349
Knez, 11. 2017n, 2021
Kahan, 11. 2232n, 2235n, 2266
Knuth, D.E. 1745, 1757
Kahneman, D. 2346, 2349
Koh1berg, E. 1 5 3 1 , 1532, 1 5 5 1 , 1553, 1554,
Kakutani , S. 1 6 1 1 , 1 660
1 5 6 1 , 1562, 1593, 1600, 1 6 1 3 , 1627, 163 1 ,
Kalai, E. 1568, 1569, 1 57 1 , 1593, 1660, 1745,
1634, 1635, 1640, 1641, 1 644-1646, 1 649,
1757, 2039, 2052, 2063, 2064, 2066, 2075,
1 660, 1 7 19, 1725, 1745, 1 746, 1757, 1771n,
2082, 2083, 2090, 2099-2 1 02, 2 1 1 8 , 2 1 19,
1775n, 1781n, 1793n, 1 80 1 , 1 8 1 3, 1822, 1830, 1848, 1849, 1 92 1 , 1944, 2 1 6 1 , 2164
2323 Kalish, G. 2349
Kojima, 11. 1756, 1757
Kalkofen, B . 1588, 1589, 1592
Koller, D. 1725, 1750, 1752-1757
Kambhu, J. 2232n, 2266
Kornhauser, L.A. 2231n, 2232, 2234n, 2236,
Kandori, 11. 1580, 1593 Kaneko, 11. 1796n, 1 802 Kannai,
Y. 1 775n, 1777n, 1 802, 2 1 26, 2 1 36,
2164
2237n, 2249, 2252n, 2262, 2264, 2266 Kortanek, K.O. 1954, 1985 Kovenock, D. 1873n, 1893 Kramer, G. 22 1 5-22 1 7, 222 1 , 2227
Kanodia, C.S. 1953, 1986
Krehbiel, K. 2225, 2227
Kantor, E.S. 1993n, 2023
Kreps, D.11. 1540-1542, 1 544, 1549-1 55 1 ,
Kaplan, R.S. 1952, 1986
1560, 1565, 1567, 1 59 1 , 1593, 1 6 1 8, 1623,
Kaplan, T. 1706, 1 7 19, 1742, 1757
1627, 1 629, 1 634, 1635, 1 645, 1 65 1 , 1653,
Kaplansky, I. 1697, 1 7 1 9
1658-166 1 , 1680, 1 686, 1858n, 1862, 1 868n,
Kaplow, L. 2245, 2247n, 2266
1 894, 1 9 1 8 , 1921, 1 942
Karlin, S. 1689, 1 7 19, 1720, 1975, 1986
Kreps, V.L. 1697, 1720
Karni, E. 1797n, 1802
Kripke, S. 1681, 1686
Author Index Krishna,
V.
1-7
1708, 1720, 1900, 1 944
Krohn, I. 1704, 1720, 1740, 1757 Kuhn, H.W. 1541, 1593, 1616, 166 1 , 1699,
Lipnowski, I. 2247, 2249n, 2263 Littlechild, S.C. 205 1 , 2052 Liu, T.C. 2341, 2349
1701, 1720, 1 743, 1751, 1752, 1754, 1758,
Lockhart, S . 234 1 , 235 1
1763, 1763n, 1 804, 195 1 , 1 972, 1 986, 2 1 64
Loeb, P.A. 1780n, 1787, 1787n, 1802, 1 804, 2163, 2 1 8 1 , 2 1 83
Kuhn, P 1 940, 1 944 Kumar, K. 1789n, 1804
Loeve, �. 1767n, 1769n, 1 804
Kurz, �. 2040-2042, 2052, 2073-2075, 2095,
Lucas, R.E., Jr. 1796n, 1 804
2 1 17-21 19, 2 1 62, 2 1 80, 2 1 82, 2 1 89, 21 92,
Lucas, W.F. 1752, 1758, 2075, 2076, 2255n, 2267
2200
Luce, R.D. 1607, 1661, 1678, 1686 Laffond, G. 2220n, 2228
Lyapuno� J\. 1780n, 1 804, 2 1 64
Laffont, J.J. 1 796n, 1803, 1 804 �a, C.-T. 2283, 2292, 2316, 2317, 2317n, 2322,
Lake, �. 2259n, 2264
2323
Landes, VV. 2236, 2266 Landsberger, �- 1954, 1986
�a, T.W. 1766n, 1804
Lane, D. 2236, 2264
�ackay, R. 2216, 2227
Laroque, G. 1795n, 1799
�acLeod, W.B. 200ln, 2023
Lasaga, J. 2042, 2052
�adrigal, V. 1 5 5 1 , 1594
Laslier, J. 2220n, 2228
�ailath, G.J. 1553, 1554, 1580, 1593, 1594,
Laver, �- 2217, 2218, 2228
1632, ! 66 1 , 1 796n, 1 804, 2267
Pl. 1 829-18 3 1 , 1848, 1849
Le Breton, �- 2220n, 2228
�aitra,
Ledyard, J. 2221, 2227, 2318, 2321, 2323
�aj umdar, �- 1785n, 1803
Lefvre,
F. 2164
�akowski, L. 1795n, 1 804, 1900, 1944
Legros, P 23 1 6n, 2323
�alcolm, D.G. 2339, 2348
Lehmann, E.L. 1961, 1986
�alcomson, J.�. 2001n, 2023
Lehrer, E. 1829, 1830, 2070, 2075
�alouf, �- 2332, 2342, 2350
Leininger, W. 1545, 1593, 1 8 1 3 , 1830
�angasarian, O.L. 1 70 1 , 1706, 1720, 1742,
Lemke, C.E. 1535, 1 594, 1696, 1701, 1 702, 1720, 1725, 1 73 1 , 1734, 1739, 1740, 1742,
�arschak, J. 2062, 2075
1749, 1755, 1758 Leong, A. 2236, 2267 Leopo1d(-VVi1dburger),
1 744, 1745, 1758 �ann, I. 2050, 2052 �artin, D.J\. 1829, 1 8 3 1 , 1848, 1849
U. 1587, 1594, 1595
Levenstein, �.C. 1995n, 201 1 , 2023
�arx, L.�. 1661 �as-Cole!!, A. 1763, 1 763n, 1775, 1 775n,
Levhari, D. 1 8 13, 1830
1781n, 1785n, 1788, 1789n, 1793n, 1794n,
Leviatan, S. 2181, 2 1 83
1795, 1 795n, 1796, 1797, 1797n, 1798-1830,
Levin, D. 1 797n, 1802
1 82 1 , 1 830, 2033-2035, 2035n, 2039, 2040,
Levin, R.C. 2010, 2020n, 2024
2047, 2047n, 2048, 2052, 2064, 2075, 2098,
Levine, D.K. 1 524, 1542, 1591, 1592, 1623, 1634, 1653, 1659, 1796n, 1 800, 1 804, 1882, 1 894, 1 9 1 1 , 1 9 12, 1914, 1 926, 1 943 Levine, �.E. 2344, 2349 Levy, J\. 2074, 2075, 2090, 209 1 , 2095 , 2 1 1 7, 2 1 19 Levy, Z. 2046, 2047, 2052 Lewis, D. 1 67 1 , 1681, 1686, 2020, 2023 Liang, �-- Y. 1923, 1924, 1 943
2 1 03-2 1 10, 2 1 14, 2 1 15, 2 1 18, 2 1 3 1 , 2157, 2 1 64, 2172, 2 175, 2 177n, 2179n, 2 1 80, 2 1 8 1 , 2 1 8 1n, 2 1 83, 2297, 2323 �aschler, �- 1537, 1590, 1657, 1680, 1685, 1690, 1718, 1 95 1 , 1954, 1972, 1973, 1977, 1980, 1986, 2 1 10-2 1 12, 21 14, 21 19, 2 1 8 1 , 2 1 83, 2249n, 2259, 2263, 233 1 , 2336, 2337, 2348, 2349 �askin, E. 1530, 1591, 1680, 1685, 17 16, 1 7 19,
Lieberman, B. 2333, 2334, 2349
1882, 1894, 1905, 1 9 1 1 , 1943, 1944, 2264,
Lindbeck, J\. 2222, 2228
2272, 2273, 2274n, 2275, 2277n, 2279n,
Lindstrom, T. 1787n, 1804
2280n, 2282, 2284, 2285, 2285n, 23 1 5-23 17,
Linia1, N. 1756, 1758
2319, 232 1 , 2323
Author Index
I-8
Maskin, M. 1585, 1591
1994n, 1995n, 1999n, 2005, 2012n, 2014,
Mass6, J. 1 793n, 1805
2023
Matsuo,
T. 1904, 1 944
Matsushima, H. 2294, 2296n, 2297, 2299-2301, 230ln, 23 12, 2320, 2323 Maurer, J.H. 1996n, 2016, 2023
Miller, G.A. 2335, 2349 Miller, N. 2210, 2228, 2293, 2323 Millham, C.B. 1697, 1700, 1 7 19, 1720, 1744, 1757, 1758
Maynard Smith, J. 1524, 1594, 1 607, 1661
Mills, H. 1701, 1720, 1745, 1758
McAfee, R.P. 1904, 1907, 1944
Milne,
McCardle, K.F. 1662, 1676n, 1686, 1 7 1 3 , 1720
Milnor, J.W. 1 764n, 1805, 183 1 , 2165, 2349
McCloskey, D.N. 199ln, 2020, 2023
Minelli, E. 1680, 1685
F. 1796n, 1802
McConnell, S. 1936, 1944
Mirman, L.J. 1 8 1 3 , 1830, 2165
McDonald, D. 233 1 , 2340, 2348
Mirrlees, J.A. 1795n, 1805, 1944, 223ln, 2264
McGarvey, D. 2209, 2228 McGee, J. 1868, 1 894 McKee, M. 2265 McKelvey, R.D. 1706, 1720, 1725, 1756, 1758, 2207-2209, 2212, 2227, 2228, 2285, 2287, 2287n, 2293, 2297, 2301n, 2309n, 23 18, 23 19n, 2321, 2323, 2345, 2349 McKenzie, L.W. 1 796n, 1805 McLean, R.P. 1943, 2074-2076, 2090, 209 1 , 2095, 2 1 17, 2 1 19, 2 1 64, 2165, 2 1 7 1 , 2 1 74, 2 1 83 McLennan, A. 1566, 1594, 1659, 166 1 , 1706, 1720, 1725, 1743, 1756, 1758, 1821, 1830, 2066, 2076 McMillan, J. 1796n, 1805 McMullen, P. 1743, 1758 Megiddo, N. 1725, 1750, 1752-1758 Meier, M. 1 674n, 1682n, 1686 Meilijson, I. 1954, 1986 Melamed, A.D. 2242, 2264 Melino, A. 1936, 1939, 1 943
Mitzrotsky, E. 1969, 1986 Miyasawa, K. 1708, 1720 Modigliani,
F. 1 859n, 1894
Mokyr, J. 201 1 , 2023
Moldovanu, B. 1905, 1944 Moltzahn, S . 1704, 1720, 1740, 1757 Monderer, D. 1528, 1529, 1594, 1708, 1709, 1720, 1829, 1 830, 2053, 2060, 2062, 2065, 2066, 2069, 2075, 2076, 2130, 2 1 3 1 , 2 1 5 1 , 2 1 57, 2158, 2164, 2165 Mongin, P. 1682, 1686 Mookherjee, D. 1900, 1 944, 2305, 2305n, 2309, 2324 Moore, J. 2275, 228 1-2283, 2287, 2289, 2291, 2292, 2315-23 17, 2323, 2324 Moran, M. 2268 Morelli, M. 2049, 2053 Moreno-Garcia, E. 1795n, 1 80 1 Morgenstern,
0 . 1523, 1526, 1543, 1 594, 1606,
1 609, 1646, 1663, 1 68 1 , 1 686, 1689, 1 72 1 , 1763, 1790n, 1808, 2036, 2053 Moriguchi, C. 2013, 2023
Melolidak:is, C. 1956, 1972, 1985
Morin, R.E. 2333, 2349
Mertens, J.-F. 1 5 3 1 , 1532, 1548, 1553, 1554,
Morris, S. 1 676, 1686
1559-1562, 1 564, 1566, 1570, 1572, 1593,
Moses, M. 168 1 , 1682, 1685
1594, 1 600, 1629, 163 1 , 1 634, 1635, 1 636n,
Mosteller, F. 2332, 2349
1638-1641, 1644-165 1 , 1654, 1659-166 1 ,
Moulin, H. 1 5 3 1 , 1543, 1594, 1604, 1614, 166 1 ,
1669, 1673, 1675, 1 679, 1 680, 1682, 1686,
1 7 1 1 , 17 14, 1 7 1 5 , 1720, 2209, 2228, 2267,
1745, 1746, 1757, 1758, 1809, 1 8 1 3 , 1 8 16,
2289n, 2292, 2293, 2302, 2323, 2324
1 8 18-1823, 1823n, 1 828, 1 8 3 1 , 1835, 1 840,
Mount, K. 2275, 2277, 2324
1 843, 1847, 1 849, 1921, 1 944, 2 1 3 1 , 2 1 33,
Mukhamediev, B.M. 170 1 , 1720, 1745, 1758
2 1 4 1 , 2 1 46-2148, 2 1 60, 2165, 2 1 80,
Miiller, H. 2 196, 2200
2 1 82-2 1 84, 2201
Mulmuley, K. 1743, 1758
Mezzetti, C. 1900, 1944
Murnighan, J.K. 2342, 2347, 2349, 2350
Michelman, F. 2242n, 2266
Murphy, K. 2012, 2023
Milchtaich, I. 2165
Murty, K.G. 1749, 1758
Milgrom, P.R. 1529, 1539, 1544, 1594, 1661,
Mutuswami, S. 2049, 2053
1680, 1686, 1709, 1720, 1773, 1774n, 1775n, 1781n, 1789n, 1805, 1861, 1 868n, 1 894,
Myerson, R.B. 1 5 3 1 , 1 540, 1547, 1552, 1553, 1594, 1619, 1628, 163 1 , 1 640, 1661, 1680,
Author Index 1686, 1700, 1 720, 1 795n, 1805, 1900, 1903-1905, 1934, 1944, 1 998n, 2023, 2036, 2042, 2044, 2048, 2052, 2053, 2066, 2072,
1-9
Ordeshook, P.C. 222 1 , 2227, 2314, 2324, 2345, 2349 Ordover, J.A. 2236, 2267
2074-2076, 2 1 17, 2 1 19, 2246n, 2267, 2302,
Orshan, G. 2 1 1 2, 2 1 1 9
2302n, 2304, 2316, 2322, 2324
Ortufio-Ortin, I . 23 1 8n, 2321 Osborne, 11.J. 1565, 1595, 1662, 1 899n, 1942,
Nakamura, K. 2206, 2228
1944, 2 1 17, 2 1 19, 2 166, 2245n, 2267
Nalebuff, B. 23 18, 2320
Ostrom, E. 1955, 1987
Nash Jr., J.F. 1523, 1 524, 1528, 1 5 3 1 , 1534,
Ostroy, J. 1 795n, 1804
1586, 1587, 1594, 1599, 1606, 1607, 1 6 1 1 ,
Owen, G. 2028, 2040--2042, 2050-2053, 2062,
1 662, 1690, 1720, 1 727, 1734, 1758, 1764,
2063, 2069, 2073, 2074, 2075n, 2076, 2094,
1764n, 1 766, 1767n, 1771n, 1772, 1783,
2096-2100, 2 1 10--2 1 12, 2 1 14, 2 1 19, 2144,
1785, 1805 , 1 855, 1 894, 2045 , 2053, 2083,
2 1 66, 2 1 8 1 , 2 1 83
2 1 19, 2 1 80, 2 1 84, 2 1 89, 2201, 2338, 2339, 2348, 2349
Pacios, 11.A. 2050, 2052
Nau, R.F. 1 662, 1 676n, 1686, 1 7 1 3, 1720
Page, S. 2322
Neal, L. 1998, 2023
Palfrey, T. 1 805, 2274n, 2275, 2287n, 2292,
Neelin, J. 1941n, 1944
2294, 2295, 2297, 2301, 2301n, 2303n,
Nering, E.D. 2349
2305n, 2307n, 2308-23 12, 2314, 2316,
Neumann, J. von, see von Neumann, J. Neyman, A. 1529, 1594, 1662, 1823, 1830,
23 1 6n, 2317, 23 18, 23 1 9n, 232 1 , 2323, 2324 Pal1aschke, D. 2 1 66
1 8 3 1 , 1 835, 1 849, 2038, 2039, 2053,
Pang, J.-S. 1 73 1 , 1749, 1756
2067-2069, 2075, 2 1 17, 2 1 18, 2 1 26, 2 1 3 1 ,
Papadimitriou, C.H. 1 745, 1756-1758
2 1 33-21 37, 2139, 2148, 2 1 5 1-2153,
Papageorgiou, N.S. 1785n, 1793n, 1 794n, 1803,
2 1 62-2 166, 2 1 7 1 , 2 1 74, 2 1 8 1 , 2 1 83, 2 1 84, 2192, 2200
1805 Park,
1.-U.
1743, 1758
Niemi, R. 2208, 2209, 2228, 2293, 2323
Parthasarathy, K.R. 1767n, 1789n, 1805
Nikaido, H. 1 716, 1720
Parthasarathy, T. 1 5 3 1 , 1 5 9 1 , 1689, 1 697, 1700,
Nisgav, Y. 1955, 1956, 1973, 1986
1701, 1 7 16, 1 7 1 9, 1720, 1725, 1758, 1 8 14,
Nishikori, A. 1954, 1986
1 8 1 8-1820, 1830, 1 8 3 1
Nitzan, S . 2222, 2227 Nix, J. 2010n, 2023 Nogee, P. 2332, 2349
Pascoa, 11.R. 1 785n, 1 789n, 1793n, 1795n, 1 796n, 1801, 1805, 1806 Patrone, F. 2051, 2052
No1deke, G. 1 5 5 1 , 1595, 1662
Pazgal, A. 2 1 66
Noma, T. 1756, 1757
Pearce, D.G. 1527, 1528, 1 534, 1 542, 1566,
Norde, H. 1529, 1589, 1595, 1 694, 1 720, 205 1 , 2052 North, D.C. 1 993n, 1995n, 1997, 1999, 2005, 2012n, 2016n, 2020n, 2021 , 2023 Novshek, VV. 1794n, 1 795n, 1805 Nowak, A.S. 1 8 17, 1 8 2 1 , 1 83 1 Nti, K . 1796n, 1805 Nye,
J.V. 1998n, 201 2n, 2023
1592, 1595, 1601, 1 604, 1662, 1 8 8 1 , 1 893 Pearl, 11.H. 1956, 1985 Peleg, B. 1529, 1530, 1585, 1590, 1595, 1767n, 1768, 1768n, 1 806, 2 1 19, 2297, 23 18, 2324 Perea y 11onsuwe, A. 1662 Perez-Castrillo, D. 2048, 2053 Pedes, 11. 2163 Per1off, J. 2239n, 2267 Perry, 11. 1659, 1 900, 1904n, 1905, 1 9 1 8n,
Ochs, J. 1941n, 1944 Oi, VV. 2232n, 2267 Okada, A. 1 569, 1595, 1662, 1980, 1983, 1 984 Okada, N. 1 954, 1986
1 922, 1923, 1935, 1935n, 1938, 1941, 1943-1945, 2300n, 2322 Pesendorfer, 2227
VV. 1796n, 1 800, 1 804, 2224, 2225,
Okuno, 11. 1795n, 1802
Peters, H. 2044, 2052
O'Neill, B. 1805, 1 949, 1986, 2259, 2260,
Pethig, R. 1955, 1985
2262n, 2267, 2334, 2338, 2348, 2349
Pfeffer, A. 1756, 1757
Author Index
I-10
Picker, R. 223 l n, 2232, 2252n, 2263
Rasmusen, E. 1 994n, 2024, 2239n, 2249, 2267
Pieblmeier, G. 1969, 1984, 1986
Rath, K.P. 177ln, 1774n, 1776n, 1777, 1777n,
Plott, C.R. 2207, 2228, 2344, 2349
1778, 1778n, 1781n, 1785n, 1786, 1786n,
Png, I.P.L. 2236, 2267
1 793n, 1794n, 1803, 1 806
Polak, B. 1662, 1996n, 2012, 2021
Rauh, M.T. 1766n, 1779n, 1796n, 1807
Polinsky, A.M. 2232n, 2236, 2239n, 2267
Raut, L.K. 2166
Ponssard, J.-P. 1565, 1595
Ravindran, G. 1717, 1720
Poon, B . 2332, 235 1
Raviv, A. 2232, 2265
Porter, D. 2288n, 2320
Ray, D. 1585, 1590
Porter, R.H. 1877, 1877n, 1 894, 1 995n, 2010,
Reiche1stein, S . 1900, 1 944, 2297, 2305, 2305n,
2022, 2024 Posner, R. 2236, 2239n, 2266, 2267
2309, 2324 Reichert, J. 2 1 3 1 , 2166
Postel-Vinay, G. 201 1, 2023
Reid, F. 1936, 1939, 1943
Postlewaite, A.W. 1795n, 1796n, 1 800, 1 804,
Reijnierse, H. 1589, 1595
1 806, 2267, 2275, 2279n, 2287n, 2297, 2303,
Reinganum, J.F. 1954, 1986, 2232n, 2264, 2267
2303n, 2307n, 2309, 2323, 2324
Reise, M. 2341, 2350
Potters, J.A.M. 1564, 1589, 1593, 1595, 1647, 1 660, 1663, 1700, 1719, 1750, 1756 Powers, I.Y. 2333, 2349
Reiter, S. 1997, 2024, 2275, 2277, 2297, 2324 Reny, �J. 1544, 1545, 1 55 1 , 1557, 1592, 1593, 1595, 1613, 1 622, 1623, 1627, 1634, 1646,
Prakash, P. 1 806
1654, 1659, 1660, 1662, 1 904, 1905, 1 944,
Prescott, E.C. 1796n, 1804
1 945
Price, G.R. 1 524, 1 594, 1607, 1661 Priest, G.L. 2232n, 2267 Prikry, K. 1793n, 1794n, 1798, 1 804
Repullo, R. 2281-2283, 2285-2287, 2289, 229 1 , 2292, 2324, 2325 Revesz, R.L. 2236, 2252n, 2262, 2266 Re� � 23 15n, 2320
Quemer, I. 2234n, 2265
Ricciardi, F.M. 2339, 2348
Quint, T. 1743, 1758
Riley, J. 1915, 1 945
Quinzii, M. 2136, 2163
Rinderle, K. 1973, 1986
Raanan, J. 2163, 2165
Rob, R. 1580, 1593, 1796n, 1806
Rachlin, H. 2340, 2349
Roberts, J. 1529, 1 544, 1594, 1661, 1680, 1 686,
Ritzberger, K. 1532, 1533, 1595, 1650, 1662
Radner, R. 1539, 1590, 1595, 1773, 1774n, 1789n, 1 792n, 1798, 1 806, 1882n, 1 894, 194ln, 1945, 2344, 2349 Radzik, T. 17 16, 1717, 1720
1709, 1720, 1794n, 1795n, 1 806, 1861, 1867, 1 868n, 1 894, 23 18, 2321 Roberts, K. 1795n, 1806 Robinson, J. 1707, 172 1 , 1 806
Rae, D.W. 2253n, 2267
Robson, A.J. 1545, 1566, 1592, 1593, 1659
Raghavan, T.E.S. 1 5 3 1 , 1591, 1595, 1689, 1697,
Rogerson, VV. 223ln, 2232n, 2237n, 2267
LV. 1725, 1 750, 1755, 1759
1700, 170 1 , 1712, 1716, 1 7 1 9-172 1 , 1725,
Romanovskii,
1730, 1758, 1759, 1 8 14, 1821, 1825, 1 8 3 1 ,
Root, H.L. 1999, 2024
1 832, 1 846, 1850
Rosenberg, D. 1849, 1850, 2252n, 2267
Raiffa, H. 1607, 1661, 1678, 1686, 2342, 2343, 2349 Rajan,
U. 23 17n, 2320
Ramey, G. 155 1 , 1566, 1590, 1593, 1627, 1661, 1861n, 1 867n, 1893 Ramseyer, J.M. 2010n, 2024 Rapoport, Amnon 2050, 2053 Rapoport, Anato1 2332, 2333, 2335, 2336, 2349, 2350 Rashid, S. 1780n, 1787n, 1789n, 1790n, 1 804, 1806
Rosenfield, A.M. 2239n, 2267 Rosenmtiller, J. 1534, 1595, 1696, 1702, 1704, 1720, 1721, 1740, 1757, 2 1 66 Rosenthal, J.-L. 1995n, 1996n, 2008n, 201 1 , 2012, 2020n, 2023, 2024 Rosenthal, R.W. 1539, 1544, 1590, 1595, 1662, 171 1 , 1 7 1 2, 1721, 1745, 1757, 1773, 1774n, 1789n, 1793n, 1796n, 1 797n, 1798, 1799, 1802, 1805, 1 806, 1 873n, 1 894, 2 1 62, 2166, 2294, 2301, 2301n, 2322, 2334, 2348 Rotemberg, J.J. 1870, 1 877, 1895, 201 1 , 2024
Author Index
I-l l
Roth, A.B. 194ln, 1944, 203 1 , 2032, 2053, 2067n, 2070, 2076, 2083, 2095, 2 1 19, 2 1 8 1 ,
1785, 1 785n, 1 795n, 1797n, 1 80 1 , 1802, 1806, 1 807, 2303, 2303n, 2307n, 2309, 2324
2 1 84, 2330-2332, 2342, 2343, 2347, 2349,
Schmittberger, R. 2338, 2348
2350
Schofield, N. 2206, 2207, 2228
Roth, B. 2337, 2350 Rothblum,
U.
2163, 2166
Schotter, A. 1941n, 1 945, 2234n, 2266, 2344, 2349
F. 2343, 2347, 2349, 2350
Rothschild, M. 1 796n, 1805
Schoumaker,
Royden, H.L. 1781n, 1 806
Schrijver, A. 1728, 1759
Rubinfeld, D.L. 2232, 2236, 2240n, 2264, 2267
Schroeder, R. 1 8 1 3, 1830
Rubinstein, A. 1537, 1542, 1579, 1595, 1662,
Schwartz, A. 2231n, 2232n, 2264, 2268
1 882n, 1 895, 1 899n, 1 904n, 1 909, 1 9 10,
Schwartz, B. 2338, 2348
1 9 1 8, 1929, 1933, 1938, 1942, 1944, 1945,
Schwartz, E.P. 2249, 2268
1954, 1986, 2245, 2245n, 2267, 2268, 2300n,
Schwarz, G. 172 1 , 1 807
23 1 1n, 23 1 5n, 2317, 2322, 2325
Seccia, G. 1796n, 1807
Ruckle, W.H. 1952, 1986, 2165, 2 1 66
Sefton, M. 2301, 2325
Rudin, W. 1767n, 178 l n, 1807
Seidmann, D. 2050, 2053
Russell, G.S. 1955, 1986
Sela, A. 1708, 1720
Rustichini, A. 1782n, 1793n, 1 794n, 1799, 1803,
Selten, R. 1523, 1524, 1530, 1 539-1544, 1546,
1 807, 1 906, 1945
1547, 1 549, 1 567, 1572, 1 574, 1 576-1578, 1583-1589, 1592, 1593, 1595, 1600, 1 6 1 8,
Saari, D. 2207, 2228 Saaty, T.L. 195 1 , 1986 Saijo,
T. 2285, 2287, 2297, 2309n, 2325
Sakaguchi, M. 1956, 1 972, 1986 Sakovics,
J. 1909n, 1 945
Saloner, G. 1870, 1877, 1895, 201 1 , 2024 Samet, D. 1568, 1569, 1571, 1593, 1660, 1662, 1673, 1 673n, 1675-1677, 1686, 2039, 2052, 2053, 2063-2066, 2075, 2076, 2082, 2090, 2099, 2 1 00, 2 102, 2 1 03, 2 1 18, 2 1 19, 2165, 2167 Samuelson, L. 1524, 1553, 1554, 1594, 1595, 1632, 166 1 , 1662 Samuelson, P.A. 1 796, 1 796n, 1807 Samuelson, W. 1905, 1908, 1923n, 1934, 1 942, 1945, 2268 Santos, J.C. 2 1 67 Sappington, D. 2317, 2321 Satterthwaite, M.A. 1789n, 1795n, 1 800, 1 804, 1 807, 1 900, 1903, 1904, 1 906, 1 906n, 1 934, 1 943-1945, 2246n, 2267, 2273, 2325
1619, 162 1 , 1622, 1625, 1628, 1636, 1640, 1645, 1659, 1 660, 1662, 1 69 1 , 1 72 1 , 1725, 1 748, 1 750, 1755, 1757, 1759, 1797n, 1801, 1854, 1895, 2 1 17, 2 1 18, 2337, 2341, 2350 Sen, A. 2273, 228 1 , 2282, 2287, 2289, 2290, 2290n, 229 1 , 2292, 2294, 2297, 2298, 2298n, 2299, 2301, 2309, 23 1 0n, 23 13, 23 1 9-2322, 2325 Sercu, P. 2268 Serrano, R. 2312, 2325 Sertel, M.R. 1799, 1806 Shafer, W. 1795n, 1796n, 1807, 2095 , 2 1 19, 2 1 8 1 , 2 1 84 Shaked, A. 194ln, 1 942 Shannon, C. 1529, 1594 Shaprro, C. 2003, 2024 Shaprro, N.Z. 1763, 1 807, 2 1 67 Shaprro, P 2240n, 2264, 2265 Shapley, L.S. 1528, 1529, 1535, 1537, 1 594, 1596, 1704, 1708, 1709, 1720, 1721, 1725, 173 1 , 1734, 1 740, 1759, 1763, 1 764n, 1 800, 1805, 1 807, 1 8 12, 1 8 16, 1 8 3 1 , 2027-2030,
Sauermann, f!. 2341, 2343n, 2350
2039, 2044, 2050, 2052, 2053, 2060, 2062,
Savard, G. 1706, 1 7 1 8, 1745, 1756
2063, 2065, 2066, 2069-207 1 , 2075, 2076,
Schanuel, S.H. 1583, 1595
2082, 2084, 2102, 2 1 16-2 1 19, 2 128-2 1 3 1 ,
Scheinkman, J. 1858n, 1894
2 1 35, 2136, 2140, 2 1 4 1 , 2145, 2150, 2 1 52,
Schelling, J.C. 2343, 2350
2 158, 2159, 2 1 6 1-2163, 2165, 2 1 67, 2 1 7 1 ,
Schelling, T. 2020, 2024
2 172, 2 173n, 2 1 74, 2 1 75, 2 180-2182, 2 1 84,
Schleicher, H. 1954, 1986
2201, 2253, 2253n, 2254n, 2264, 2268, 2338,
Schmeidler, D. 1 5 3 1 , 1593, 1660, 1 7 10, 1 719, 1765, 1766, 1768n, 1770n, 1779n, 1780n,
2348 Sharkey, W.W. 2 1 16, 21 19, 2165
Author Index
I-12
Shavell, S . 2231n, 2232, 2232n, 2235, 2236,
Stem, 11. 1 8 14, 1 8 3 1
2237n, 2239n, 2245 , 2247n, 2252n, 2263,
Stewart, 11. 1936, 1 940, 1944
2266-2268
Stigler, G. 1 869, 1895
Shephard, A. 1 872n, 1893
Stiglitz, J.E. 2003, 2024
Shepsle, K. 2210, 2216-22 18, 2228
Stinchcombe, 11.B. 1548, 1596
Shieh, J. 1793n, 1807 Shilony,
Y. 1 873n, 1895
Stokey, N.L. 1 9 1 5, 1 926, 1 927, 1930, 1945 Stone, H. 1745, 1758
Shipan, C. 2249, 2265
Stone, J.J. 2343, 2351
Shitovitz, B. 2182, 2 1 83
Stone, R.E. 173 1 , 1749, 1756
Shleifer, A. 2012, 2023
Straffin Jr.,
Shubik, 11. 1596, 1743, 1758, 1794n, 1795,
P.D. 2030, 2053, 2070n, 2076, 2253,
2254, 2259n, 2268
1796, 1796n, 1797-2217, 1 8 1 3, 1 8 3 1 , 2030,
Strnad, J. 2206, 2228
2053, 2070, 2076, 2 1 67, 2 1 7 1 , 2 1 84, 2253,
Sudderth, VV. 1829- 1 8 3 1 , 1848, 1 849
2268, 2329, 2332, 2333, 2335n, 2336, 2338,
Sudholter,
2340n, 2341 , 2343n, 2345, 2348, 2350, 2351
Suh, S.-C. 2319, 2325
Siegel, S. 2333, 2341 , 2348, 2351 Simon, L.K. 1530, 1548, 1583, 1595, 1596
P. 1704, 1720, 1740, 1757
Sun, Y.N. 1770n, 177 1-1776, 1776n, 1777, 1 777n, 1778, 1778n, 1779-17 8 1 , 1781n,
Simon, R.I. 2329, 235 1
1782, 1 782n, 1783-1785, 1785n, 1786,
Simon, R.S. 1843, 1850
1786n, 1787-1790, 1790n, 179 1 , 1791n,
Sjostrom, T. 1708, 1720, 2287, 2294, 2297, 23 15, 2316, 23 1 8-2320, 2325
1792-1794, 1794n, 1795-1830 Sundaram, R. 1 8 1 3 , 1830
Smith, V.L. 233 1 , 2340n, 2351
Sutton, J. 194ln, 1 942, 1993n, 2024
Smorodinski, R. 2 1 37, 2166
Swinkels, J.11. 1553, 1554, 1 594, 1 627, 1632,
Sobel, J. 1550, 1 560, 1565, 1 590, 1591, 1593, 1635, 1657, 1658, 1917, 192 1 , 1926, 1 942,
1661, 1662 Sykes, A.O. 2268
1 945, 2226, 2227, 2236, 2265, 2268 Sobel, 11.J. 1 8 1 3 , 1832
Takahashi, I. 1 9 17, 1 926, 1 945
Sobolev, A.I. 2035, 2053
Talagrand, 11. 1782n, 1 807
So1an, E. 1 840, 1843, 1846, 1 846n, 1850
Talley, E. 1908, 1942, 2247n, 2263
Sonnenschein, H. 1 794n, 1795, 1796, 1796n,
Talman, A.J.J. 1704, 1706, 1721, 1725, 1726,
1797-2319, 1 9 1 0-1912, 19 13n, 1914, 1 9 1 5, 1917, 1923, 1 927-1929, 194ln, 1943, 1 944
1745, 1749, 1750, 1755, 1759 Tan, T.C.-C. 1 5 5 1 , 1594, 1663
Sopher, B . 1941n, 1943
Tang, F.-F. 23 19, 2321
Sorin, S . 1 57 1 , 1576, 1 590, 1596, 1657, 1662,
Tarski, A. 1527, 1596
1680, 168 1 , 1685, 1 686, 1721, 1 809, 1 8 16,
Tatarnitani, Y. 2294, 2297, 23 18, 2325
1 823n, 1827, 1829-1832, 1835, 1 846, 1 849,
Tauman, Y. 1 7 14, 1719, 2 1 3 1 , 2 1 33, 2137, 2 1 5 1 ,
1850, 2201 Spence, 11. 1588, 1 596, 1859, 1895
2 1 52, 2 1 61 , 2 165-2167 Tegler, A.I. 2338, 2351
Spencer, B.J. 1855, 1893, 2015, 2021
Telser, L. 1998, 2024
Spiegel, M. 1941n, 1944
Thisse, J. 1717, 1719
Spier, K. 2232n, 2268
Thomas, 11.U. 1955, 1956, 1973, 1 986
Spiez, S. 1843, 1850
Thomas, R.P. 1999, 2023
Spitzer, 11.L. 2243n, 2249, 2264, 2265, 2345,
Thompson, F.B. 1646, 1663
2348 Srivastava, S. 1805, 2274n, 2275, 2292-2295, 2297, 2301, 2303n, 2305n, 2307n, 2308-23 12, 2316, 23 1 8 , 2322-2325 Sroka, J.J. 2 1 67 Stacchetti, E. 1528, 1534, 1592, 1 8 8 1 , 1893 Staiger, R. 1 872n, 1893 Stanford, W. 1528, 1596
Thomson, VV. 2 1 19, 2262n, 2268, 2277n, 2325 Thuijsman, F. 1827, 1832, 1837, 1843, 1 844, 1 846, 1 849, 1850 Tian, G. 1793n, 1808, 2279n, 2297, 2325 Tijs, S .H. 1529, 1530, 1595, 1700, 1717, 1719, 172 1 , 1 750, 1756, 1 825, 1832, 205 1 , 2052 Tuole, J. 1 5 5 1 , 1592, 1659, 1774n, 1 800, 1853n, 1858n, 1861n, 1 868n, 1 878n, 1894, 1895,
Author Index
1-13
1899, 1910-1912, 1914, 1 926, 1943, 1944,
Vincent, D.R. 1899, 1923, 1937, 1 945
1994n, 2022, 2232, 2266, 2316, 2323
Vishny, R. 2012, 2023
Todd, M.J. 1734, 1759 Tomlin, J.A. 1756, 1759 Topkis, D. 1529, 1596 Torunczyk, H. 1843, 1 850 Toussaint, S. 1 783n, 1793n, 1808 Townsend, R. 2276n, 2304, 2322, 2325 Tracy, J.S. 1938-1940, 1 942, 1943, 1945
Vives,
X. 1529, 1596, 1789n, 1795n, 1797n,
1805, 1 808, 2297, 2323 Vohra, R. 1680, 1685, 1793n, 1795n, 1796n, 1 803, 2297, 2 3 1 2, 2322, 2325 von Neumann, J. 1523, 1526, 1543, 1 594, 1606, 1609, 1646, 1663, 1681, 1686, 1689, 172 1 , 1763, 1787n, 1790n, 1 808, 2036, 2053
Treble, J. 1995, 2013, 2024
von Stackelberg, H. 1 977, 1986
Trick, M. 2293, 2325
von Stengel, B. 1725, 1 726, 1 740, 1743, 1745,
Tsitsiklis, J.N. 1745, 1757 Tuchman, B . 2232n, 2266
1749, 1750, 1 752-1757, 1759, 1 972-1974, 1984, 1986
Tucker, A.W. 1763, 1763n, 1 804, 2164
Vorobiev, N.N. 1699, 1701, 1721, 1 743, 1759
Turubull, S. 2316, 23 17, 2323
Vrieze, O.J. 1825-1827, 1831, 1832, 1 837,
Turuer, D. 1868, 1893 Tversky, A. 2346, 2349
Ubi Jr., J.J. 1780n, 178 1-1784, 1784n, 1785, 1786, 1786n, 1787-1939
Ulen,
T. 2236, 2264
Umegaki, H. 1785n, 1801
1843, 1 844, 1 849, 1850 Vroman, S.B. 1936, 1945
VVald, A. 1765, 1765n, 1800, 1808, 2 1 44, 2163 VVallmeier, H.-M. 1704, 1 720, 1 740, 1757 VVang, G.H. 2268 VVeber, M. 1 992n, 2024 VVeber, R.J. 1539, 1594, 1773, 1 774n, 1775n,
Valadier, M. 1 773n, 1780n, 1782n, 1794n, 1799,
1781n, 1789n, 1805, 2045 , 2053, 2060, 2061 , 2066-2069, 2069n, 2070, 2075 , 2076, 208 1 ,
1808 van Damme, E.E.C. 1 524, 1529, 1533, 1548, 1 5 50-1553, 1565, 1566, 1 5 7 1 , 1 576, 1579, 1584, 1585, 1588, 1 590, 1 5 9 1 , 1595, 1600,
21 16, 2 1 20, 2152, 2153, 2163, 2167 VVeibull, J. 1 524, 1527, 1528, 1590, 1596, 2222, 2228
1606, 1608, 1628, 163 1 , 1 640, 1 648, 1663,
VVeiman, D.P. 2010, 2020n, 2024
169 1 , 1693, 1695, 1721, 1725, 1740, 1759
VVeingast, B.R. 1 994n, 1 995n, 1999, 1999n,
van den Elzen, A.H. 1 704, 1721, 1725, 1726, 1739, 1745, 1 749, 1750, 1755, 1759 van Hulle, C. 2268
2005 , 2012n, 2014, 2023, 2024, 2210, 2228, 2268 VVeinrich, G. 1796n, 1808
Vannetelbosch, V.J. 1660
VVeiss, A. 1 566, 1592
Vardi, M.Y. 1681, 1682, 1684, 1685
VVeiss, B . 1539, 1590, 1774n, 1789n, 1798
F. 1955, 1987
Varian, H. 1873, 1 873n, 1895, 2292, 2325
VVeissing,
Varopoulos, N.Th. 1771n, 1799
VVen-Tstin, VV. 1663
Vaughan, H.E. 1771, 1801
VVerlang, S. 1 5 5 1 , 1594
Vega-Redondo, F. 1524, 1596
VVettstein, D. 2048, 2053, 2297, 2324, 2325
Veitch, J.M. 1998, 2024
VVbinston, M.D. 1585, 1586, 1 590, 2177n, 2183,
Veljanovski, C.J. 2243n, 2268 Vermeulen, A.J. 1 564, 1565, 1589, 1593, 1595,
2273, 2320 VVbite, M.J. 2231n, 2239n, 2268
w. 1 8 1 3 , 1 8 3 1
1647, 1660, 1662, 1663, 1721, 1745, 1750,
VVbitt,
1757, 1759
VVieczorek, A . 1797n, 1808
Veronesi, Vial,
J.-P.
P.
1658
1531, 1594, 1614, 1661, 1 7 1 1 , 1714,
1715, 1 720, 1 795n, 1800 Vickrey, W. 1 795n, 1808 Vieille, N . 1 7 17, 1 7 1 8, 1721, 1 827, 1832, 1836, 1839, 1842, 1 846, 1 846n, 1 849, 1 850 Villas-Boas, J.M. 1875n, 1895
VVilde, L.L. 1 954, 1986, 2232n, 2267, 2276n, 2321 VVilkie, S. 2297, 23 15, 23 18, 2321 VVilliams, S.R. 1789n, 1795n, 1807, 1900, 1902, 1 906, 1 906n, 1945 , 2285, 2285n, 2325, 2326 VVilliamson, S.H. 1991n, 2024 VVilson, C. 1588, 1596
Author Index
I-14
Wilson, R.B. 1534, 1535, 1 540-1542, 1544, 1549, 1550, 1 565-1567, 1592, 1593, 1 596, 1 6 1 8, 1 627, 1629, 1645, 1659, 1 66 1 , 1663,
Yang, Z. 1706, 1721 Yannelis, N.C. 1785n, 1793n, 1794, 1795, 1795n, 1796-2042
1680, 1 686, 1 696, 1721, 1725, 1739, 1742,
Yanovskaya, E.B. 1745, 1755, 1759
1745-1748, 1750, 1752, 1755, 1759, 1862,
Yavas, A. 2301, 2325
1 868n, 1 894, 1 899n, 1 906n, 1 9 1 0, 1 9 12,
Ye, T.X. 2075, 2076
1 9 1 3n, 1914, 1915, 1 9 17, 1923, 1927, 1928,
Yoshise, A. 1756, 1757
1936, 1937, 1939, 1 943-1945
Young, H.P 1580, 1596, 2033, 205 1 , 2054,
Winkels, H.-M. 1701, 1706, 1721, 1744, 1759
2 1 1 3 , 2120, 2167, 2269
Winston, W. 1 8 1 3 , 1832 Winter, E. 2040-2043, 2048, 2049, 2052, 2053, 2075, 2076, 2095, 2 1 20, 2 1 7 1 , 2 1 84 Winter, R. 2236, 2268 Wittman, D. 2232n, 2236, 2268, 2269 Woli, G. 2332, 2341, 2350, 2351 Wolfowitz, J. 1765, 1 765n, 1 800, 1808, 2144, 2163 Wolinsky, A. 1 542, 1595, 23 1 5n, 23 17, 2325
Zame, W.R. 1530, 1550, 1583, 1590, 1595, 1596, 1629, 1658, 2167, 2 1 8 1 , 2 1 84 Zamrr,
s. 1 66 1 , 1 669, 1673, 1675, 1679, 1680,
1682, 1686, 1809, 1 8 1 3, 1 8 1 6, 1823n, 1828, 1 830, 1 8 3 1 , 1843, 1850, 1980, 1984 Zang, I. 2 1 67 Zangwill, W.I. 1 749, 1757 Zarzuelo, J.M. 2167
Walling, A. 1961, 1987
Zeckhauser, R. 1915, 1945
Wooders, M.H. 1796n, 1802, 2 1 67, 2 1 8 1 , 2 1 84
Zemel, E. 1745, 1756, 1757
Wu, W.-T. 1696, 1721
Zemsky, P.B . 2245, 2263 Zermelo, E. 1543, 1596
F. 2341, 2351
Yamashige, S. 1776n, 1777, 1777n, 1 806, 1808
Zeuthen,
Yamato, T. 2287n, 2294, 2297, 23 18, 2325, 2326
Ziegler, G.M. 1727, 1736, 1759
SUBJECT I NDEX
y-path extension, 2155
llilocation, 205 1
K-committee equilibrium, 2216
llilocation core, 2218
�-t-measure-preserving automorphism, 2157
llimost complementary, 1701
�-t-symmetric, 2157
llimost completely labeled, 1733
�-t-vlliue, 2125, 2 157, 2180
llimost strictly competitive games, 1 7 1 1
w-potentilli function, 2064
llitemating offer, 1 9 1 8, 1 9 2 1 , 1925, 1926, 1929,
n random order colliitionlli structure vlliue, 2074
1933
n -colliitionlli structure vlliue, 2073
llitemating-offer game, 1909, 1920
n -coalitional symmetry axiom, 2073
amendment procedure, 2212
n -efficient, 2072
amendment voting procedure, 2210
n-symmetric, 2072, 2 1 55, 2156
AN, 2128
n-symmetric vlliue, 2155
animlli behavior, 2340
n -vlliue, 2072
APE, 1920
a--additive, 1682
arms control, 1950
a- -field, 1669
Arms Control and Disarmament Agency
Abreu-Matsushima mechanisms, 2301
artificilli equilibrium, 1733
(ACDA), 195 1 absolute certainty, 1671
assuredly perfect equilibrium, 1 9 1 8
absolutely continuous, 2161
ASYMP, 2 1 24 ASYMP*, 2134 ASYMP*(O, 2154 ASYMP(�), 2154
absorbing state, 1 8 12, 1 8 14, 1843
AC, 2161 additive, 2058 additivity, 2029
asymptotic � -semivlliue, 2154
additivity axiom, 2058
asymptotic vlliue, 2124, 2135, 2152, 2174
adjudication, 2249
atomless vector measure, 1780
Admati, 1922, 1923, 1938
atoms, 1765
Admati-Perry, 1935
attribute sampling, 1 95 1
administrative agencies, 2249
auction mechanisms, 2344
admissibility, 1619-1621, 1653
auctions, 1680, 1908
affinely independent, 1727
auditing, 1952
agency relations, 2002
augmented semantic belief system, 1670
agenda independence, 2213
Aumann, 2039, 2040, 2044, 2172
agenda manipulation, 2344
Aumann and Shapley, 2 1 72
agenda-independent outcome, 22 1 1, 2212, 2214
Aumann-Shapley vlliue, 2 1 5 1
agendas, 2207, 2210
Ausubel, 1 9 1 0 , 1933-1935
agent normlli form, 1541
axiom, 2027, 2029
Agreement Theorem, 1676
axiomatic characterization, 2 1 8 1
agreement/integer mechanism, 2286 airport, 2051
backward induction, 1523, 1621-1634,
Akerlof, 1905, 1906
1650-1653, 1682
Akerlof's lemons problem, 1588
backward induction implementation, 2292
lliarm, 1957
backward induction procedure, 1543
1-15
Subject Index
I-16
balanced contribution, 2036
buyer-offer game, 1909
Banach space, 1781
BV, 2128 bv 'FA, 2130, 2 1 47 bv1 A1, 2 1 24, 2 1 30 bv'NA, 2123, 2130
bankruptcy, 2259-2262 banks, 2012 Banzhaf, 2253 Banzhaf index, 203 1 , 2254-2258 bargaining, 1899, 2046, 2047, 2342 bargaining games, 2338 bargaining mechanisms, 1899, 1934 bargaining problems, 1587 bargaining set, 2249 bargaining with incomplete information, 1680 basic solution, 1737 basic variables, 1737 basis, 1737 battle of the sexes, 1553, 1567, 1689 Bayesian, 1875 Bayesian monotonicity, 2305 Bayesian rational, 1 7 1 3 beat-the-average, 2341 behavior strategy, 1616, 1753, 1 9 1 0 behavior strategy Si , 1541 belief hierarchies, 1 669 belief morphism, 1674 beliefs, 1667 Bertrand, 1857, 1858, 1870, 1878, 1879, 1 882, 1886 best elements, 2308 best reply, 1525 best-reply correspondence, 1525 best-response function, 1777 best-response region, 1 7 3 1 best-response-equivalent, 1 7 1 1 big match, 1 8 1 4 bilateral monopoly, 2341 bilateral reputation mechanism, 2014 bimatrix game, 1689
canonical, 1672 Caratheodory's extension theorem, 1787 Carreras, 2050 carrier, 1 69 1 , 2030 carrier axiom, 2059 Cauchy distributions, 2142 cautiously rationalizable strategy, 1 604 cell and truncation consist, 1574 cell in, 1574 centroid of, 1574 chain store paradox, 1543 Champsaur, 2172 chaos theorems, 2207 characteristic functions, 233 1 Chatterjee, 1905, 1934 cheap talk, 2225 Chin, 2049 Cho, 1935 Chun, 2036, 2037 cliometrics, 1991 closed, 1674 closed rule, 2225, 2226 coalition, 2000, 2041-2043, 2058, 2 1 27 coalition of partners, 2063 coalition structure, 2039, 204 1 , 2072, 2 1 55 coalition-proof Nash equilibrium, 1585 coalitional, 2028 Coase, 1908, 1 917, 2241 Coase Conjecture, 1 9 17, 1921, 1 924, 1927, 1928, 1935, 1937 Coase Strong Efficient Bargaining Claim, 2243, 2245
binary amendment procedures, 2293
Coase Theorem, 2241-2245, 2247-2249
binary voting tree, 2208
Coase Theorem and incomplete information,
binding inequality, 1727, 1736
2245
bluff, 2347
Coase Weak Efficient Bargaining Claim, 2243
Bochner integral, 1782
coherent, 167 4
Borel determinacy, 1829
collectivist equilibrium, 2017
bounded edges, 1702
collusion, 1 869-1872, 1877, 1879-1882, 1884,
bounded mechanisms, 2296 bounded variation, 2 1 28
1885, 20 1 1 combinatorial standard form, 1828
breach of contract, 2237
commitment, 1 908
bromine industry, 201 1
commitment power, 1 977
budget balancing, 1901
common interest games, 1571
business games, 2339
common knowledge, 1576, 167 1 , 1 7 1 3 , 2342
buyer-offer, 1 92 1 , 1926
common knowledge of rationality, 1624, 1656
Subject Index
1-17
common prior, 1675
core, 21 16, 2 1 59, 2179, 2200, 2205, 2248, 2336
communication and signalling, 1680
correlated equilibrium, 1530, 1600, 1612-1615,
community responsibility system, 2005 compact, 1677
168 1 , 1690, 1 7 1 0 correlated strategy, 1709
comparative negligence, 2234, 2236
correlatedly rationalizable strategy, 1601
comparison system, 2 1 1 3
costs, 205 1
complementary pivoting, 1736, 1738
counteroffers, 1908
complementary slackness, 1730
Coumot, 1855, 1858, 1877, 1 878, 188 1-1883,
complete, 1674
1886
complete information, 2342
Coumot-Nash equilibrium distribution, 1775
completely consistent, 2 1 1 2
Cramton, 1907, 1934, 1935
completely labeled, 173 1 , 1743
cultural beliefs, 2017
completely mixed, 1697
culture, 2016
completeness assumption, 1613
curb sets, 1 527
CONs , 2 1 50 conjectural assessment, 1 7 1 3 conjectural variations, 1 872, 1882-1885 connected sets of equilibria, 1644 consistency, 1529, 1589, 2034, 2047, 2109 consistency conditions, 1669 consistent, 1549, 1679, 2035, 2065 consistent beliefs, 23 1 3 consistent NTU-value, 2 1 8 1 consistent probabilities, 1828 consistent solution payoff configuration, 2 1 1 3 consistent system o f beliefs, 1626 consistent value, 2 1 1 1 , 21 1 3 consistent with rr , 2074 constant-sum game, 2189 contestable market, 1882 contested-garment problem, 2260 contested-garment solution, 2261 contested-garment-consistent, 2260 context, 2329 context independence, 1573 context-specific model, 1994, 1996, 2000 continuity, 2128, 2 1 5 1 continuum o f agents, 2172, 2192 contract, 1938, 1939, 2238
Dasgupta, 2049 data verification, 1965 debt repudiation, 1998 deceptions, 2305 delegation game, 2250 demand commitment, 2049 democracy, 2189 Deneckere, 1910, 1933-1935 Derks, 2044 DIAG, 2136, 2147 diagonal, 2152 diagonal formula, 2 1 4 1 , 2 1 5 1 diagonal property, 2125, 2 1 5 1 DIFF, 2147 DIFF(y), 2155 differentiable case, 2173 diffuse types, 2304 diffused characteristics, 1773 dimension, 1 727 direct evidence, 1995, 2010 disagreement region, 2308 disarmament, 1950 discontinuity of knowledge, 1685 discount factor, 1 8 1 1 , 2046 discounted game, 1 8 1 1
contract enforcement, 1999
dispersed characteristics, 1773
contract negotiations, 1940
dissolution of partnerships, 1907
contraction mapping, 1 8 1 6
dissolved, 1907, 1908
contraction principle, 1816
distribution of a correspondence, 1771
controlling player, 1826
distributive politics, 2212
convex, 2049
dividend equilibrium, 2197
cooperation, 2028
dividends, 2034, 2197
cooperation index, 2074
dollar auction game, 2338
cooperation structures, 2039
dominance-solvable, 1529
cooperative game, 2027, 2123
dominated strategy, 1600
cooperative refinements, 1586
domination, 1693
Copeland rule, 2292
double implementation, 23 1 8
I-18 Downsian model, 2220 Dreze, 2039, 2040 duality theorem of linear programming, 1728 Dubey, 2037, 2038 duel, 1975 dumm� 2029, 2040, 2058 dummy player axiom, 2058, 2128 duopolistic competition, 2015 durability, 2316 durable goods monopoly, 1 899, 1930 duration, 1936 dynamic, 1934 dynamic bargaining, 2317 dynamic bargaining game, 1906 dynamic programming, 1 824 East India Company, 2015 economic mechanisms, 2340 economic-political models, 2187 edge, 1701 Edgeworth's 1881 conjecture, 1764 efficiency, 1900-1902, 2123, 2124, 2151 efficiency axiom, 2059 efficient, 1908, 1934, 2059, 2128, 2188 Egalitarian solution, 2100, 2102, 2108 Egalitarian value, 2100, 2101 elaborates, 1673 elbow room assumption, 2197 Electors, 2049 elementary cells, 1574 elimination of strictly dominated strategies, 1604 emotion, 2347 empirical, 1941 empirical evidence, 1936 endogenous expectation, 1572 endogenous institutional change, 2018, 2019 entitlement, 2242, 2244 entry deterrence, 1680, 1 853, 1859-1861, 1 867, 1 868, 1 885 epistemic conditions for equilibrium, 1654--1657 epistemology, 1681 equal treatment, 2197 equicontinuous family, 1778 equilibrium enumeration, 1742 equilibrium payoff, 1835 equilibrium refinement, 1617, 1619 equilibrium selection, 1524, 1572, 1619, 1691, 1996 equilibrium set, 1827 ergodic theorem, 1 815, 1 829 error of the second kind, 1958
Subject Index ESBORA theory, 1588 essential monotonicity, 2287 European state, 2006 event, 1667 evolutionary game theory, 1607 ex post efficient, 1903 ex post efficient trade, 1903 exact potential, 1529 exchangeable, 1700 existence of equilibrium, 161 1 , 1612 expectation, 2238 experimental law, 2345 experimental matrix games, 2329 experiments, 1941 extended solution, 2065 extended unanimity, 2309 extension operator, 2141 extensive form, 2332 extensive form correlated equilibria, 1820 extensive form games, 1523, 1615 extensive games, 1681 external symmetry, 2331
FA, 2128 face, 1727 facet, 1727 fair, 2072 fair allocation rule, 2072 fair ranking, 2037 false alarm, 1957 false alarm probability, 1960 Fatou's lemma, 1772 fear of ruin, 2191 Fedzhora, 2050 fiduciary decisions, 2346 financial systems, 201 1 finite extensive form games with perfect recall, 1540 finitely additive, 2128 finitely additive measure, 1770 First Theorem of Welfare Economics, 2196 fixed points, 1534 fixed prices, 2196 fixed-price economies, 2187 focal points, 2343 Folk Theorem, 1565, 1910, 1925, 1933 formation, 1575 forward induction, 1524, 1554, 1566, 1634, 1635, 1645, 1648-1653 Fourier transform, 2142 Fragnelli, 205 1 Fubini's theorem, 1767
Subject Index fully stable set of equilibria, 1561 fully stable sets, 1646 gains from trade, 1 902 game form, 1541 game in normal form, 1525 game in standard form, 1573 game over the unit square, 1975 game theory, 1523 game with common interests, 1576 game with private information, 1773 games in coalitional form, 2336 games of survival, 1 823 games with perfect information, 1543 games with strategic complementaries, 1529 gang of four, 1680 gap, 1914, 1915, 19 17, 1923, 1925, 1926, 1928, 1933 Garda-Jurado, 2050 Gelfand integral, 178 1 general case, 2173 general claim problem, 2260 general existence theorem, 1827 general minmax theorem, 1829 (general) semantic belief system, 1 670 generalized solution function, 2105 generalized value, 2097, 2098 generic, 1740 generic equivalence of perfect and sequential equilibrium, 1629 generic insights, 1993, 1996 geometry of market games, 2178 Gibbons, 1907 globally consistent, 2 1 1 2 Golan, 2050 good correlated equilibrium, 1 7 1 1 Groves mechanism, 1901 Gul, 2046
H', 2159 H� , 2159 H-stable set of equilibria, 1564 Hadley v. Baxendale, 2239 Halm-Banach theorem, 1683 Harsanyi, 2042, 2045, 2 1 80 Harsanyi and Selten, 1574 Harsanyi doctrine, 1606 Harsanyi NTU, 2092 Harsanyi NTU value, 2175 Harsanyi NTU value payoff, 2091 Harsanyi solution, 2093
I-19 Harsanyi solution correspondence, 2093 Harsanyi--Shapley NTU value, 2193 Hart, 2034, 2035, 2040, 2046-2048, 2172, 2 1 80 hierarchy, 2043 historical context, 1993 history, 1910 holdouts, 1938-1940 holds in, 1670 homogeneous, 2058 homotopy, 1749 Hurwicz, 2273 hyperplane game, 2 1 10, 21 1 1 , 2 1 1 3 hyperplane game form, 2107 hyperstable set, 1646 hyperstable set of equilibria, 1561
i 's expected payoff, 1541 ideal coalitions, 2124 ideal games, 2141 idiosyncratic shocks, 1790 imperfect competition, 2182 imperfect monitoring, 1 846, 1 877, 1878 implementation theory, 2273 impossibility, 2239 incentive compatibility, 1900, 1904, 2277 incentive compatible, 1 900 incentive consistency, 23 12 incomplete information, 1665, 1 843, 1899, 1980, 2 1 17 incomplete information domain, 2302 incredible threats, 1523 independence hypothesis, 1769 independence, statistical, 1602 index of power, 2069 indirect evidence, 1995, 2010 indirect monotonicity, 2290 individual rationality, 1901, 23 1 6 individualist equilibrium, 2017 inefficiency, 1 905 inferior, 1575 infinite conjunctions and disjunctions, 1685 infinite regress, 1679 infinite repetition, 1689 infinitely repeated games, 2020 infinitesimal minor players, 1764 infinitesimal subset, 2187 infinitesimally close, 1788 informal contract enforcement, 2000 informal contract enforcement institution, 2004 information partition, 1 67 1 information set, 1 67 1 , 1750 inspectee, 1949
Subject Index
I-20
inspection, 1680
level structure, 2043
inspection games, 1 949
Levy, 2046
inspector, 1949
lexico-minimum ratio test, 1741, 1747, 1748
institutional foundations of exchange, 2000
lexicographic method, 1741
institutional trajectories, 2017
liability rules, 2242, 2244
insurance, 1953
liability with contributory negligence, 2235
integer games, 2320
limit price, 1859-186 1 , 1865, 1866
interactive analysis, 1994
limit pricing, 1859, 1861, 1867
interactive belief systems, 1654
limited rationality, 2331
interactive implementation, 23 15
limiting values, 2123, 2133
interdependent, 1905
limits of sequences, 2181
interdependent values, 1909, 1923, 1924
UN5 , 2150
interim utility, 2303
linear, 2058
interiority, 2310
linear complementarity problem (LCP), 1731
internal, 2130 International Atomic Energy Agency (IAEA), 1950 international relations, 2016 international trade, 2014 intuitive criterion, 1862 invariant cylinder probabilities, 2148 issue-by-issue procedure, 2213, 2214 iterated dominance, 1 600-1604, 1619-1621 iteratively undominated strategy, 1601 Jackson (1991), 2309 Japan, 2013 joint continuity, 1 8 17 judicial review, 2249 Kakutani's fixed point theorem, 1767 Kalai, 2039 Karl Vind, 2285 King Solomon's Dilemma, 2283 Klemperer, 1907 Knesset, 205 1 knowledge, 1671 Kuhn's theorem, 1616 Kurz, 2040
linear tracing procedure, 1580, 1581 linear-quadratic, 1927 linearity, 2124, 2 1 51 linearity axiom, 2058 Lipschitz condition, 1914 Lipschitz potential, 2105, 2107 local strategy, 1541 Loeb counting measure, 1789 Loeb measure spaces, 1787 logarithmic trace, 1583 logarithmic tracing procedure, 1580, 1583 long-distance trade, 2014 losing coalition, 2069 LP duality, 1730 Lyapunov, 1 820 Lyapunov's theorem, 1780 M, 2128 M-stability, 1564 Maghribi traders, 2001 majoritarian preferences, 23 1 1 majority, 2031 majority rule, 2224 marginal contributions, 2033, 2059, 21 1 1 marginal worth, 2 1 87
label, 173 1
marginality, 2 1 1 3
labor, 1938
market, 2173
labor relations, 2013
market clearance, 2196
large non-anonymous games, 1773
market entry games, 1587
law, 2012
market games, 2125
Law Merchant system, 2005
market structure, 1996, 1997, 2009
leadership, 1977, 1980, 1981
Markov, 1 8 1 1
learning, 2333
Markov decision processes, 1 824
legislation game, 2250
marriage lemma, 1771
Lemke's algorithm, 1749
Mas-Colell, 2034, 2035, 2047, 2048, 2172
Lemke-Howson algorithm, 1535, 1733, 1739, 1748
Maskin, 2273 mass competition, 1763
Subject Index mass markets, 233 1 material accountancy, 1 95 1 , 1962 Material Unaccounted For (MUF), 195 1 maximal Nash subsets, 1744 maxmin, 1690, 1 847 measurability, 2312 measurable selection, 1771 measure-based values, 2157, 2180 measurement error, 1962, 1966 mechanism continuity, 23 1 8 mechanism design, 1899, 1908 memory, 2335 merchant guild, 2015 Mertens value, 2180 Milnor axiom, 2059 Milnor operator, 2059 minimal curb set, 1568, 1570 minimax theory, 2334 minimum ratio test, 1737, 1741 missing label, 1734 MIX, 2124 mixed extension, 1609-16 1 1 mixed strategy, 1525, 1680, 1690, 1726, 2334 mixed-strategy equilibrium, 1766, 1873, 1 875, 1876 mixing value, 2124, 2140, 2152 modal logic, 168 1 model, 1670 modularity, 2038 modulo game, 2309 monopoly, 2010 monotone, 2058 monotonic, 2128 monotonicity, 2033, 2282 monotonicity-no-veto, 2309 moral hazard, 1952 Morelli, 2049 Mount-Reiter diagram, 2275 multi-member districts, 2253, 2257 multilateral reputation mechanism, 2014 multilinear extension, 2096, 2144 multiple equilibria, 1524, 1610 mutual knowledge, 168 1 , 1685 Myerson, 1903, 1904, 1 907, 1934, 2036, 2044, 2048
'NA, 2123 NA, 2128 Nakamura number, 2205 Nash, 2045 Nash Bargaining Solution, 2083, 2099, 2 1 14
I-21 Nash equilibrium, 1523, 1528, 1599, 2279 Nash equilibrium in pure strategies, 1689 Nash implementation, 2278, 2279 Nash product property, 1587 Nash-Harsanyi value, 2341 natural sentences, 1670, 1674 negligence, 2233, 2235 negligence with contributory negligence, 2233, 2234, 2236 negligible, 1763 neutrality, 2032 Neyman, 2039 Neyman-Pearson lemma, 1961, 1963-1965, 1983 no delay, 1929 no free screening, 1928, 1929 no gap, 1925-1927, 1929, 1933 no veto power, 2282, 2285 Non-Proliferation Treaty (NPT), 1951 nonatomic, 1820, 2123 nonbasic variables, 1737 noncooperative games, 1523, 2045 nondegenerate, 1701, 1732, 1739-1741 nondetection probability, 1959 nondifferentiable case, 2178 nonempty lower intersection, 2281 nonexclusive information, 2303 nonexclusive public goods, 2195 nonmarket institutions, 1992 nonstationary equilibria, 1930 nonsymmetric values, 2154 nonzero-sum, 1 8 1 8 nonzero-sum games, 1842 norm-continuity, 1821 normal form, 1524, 1541, 1814 normal form perfect equilibrium, 1628, 1653 normative procedures, 2347 normative theory, 1523 NP-hard, 1756 NTU game, 2079 NTU game form, 2105 NTU value, 2194 nuclear test ban, 1950 nucleolus, 2248 null, 2058 null hypothesis, 1957, 1966 null player axiom, 2058, 2123 offer/counteroffer, 1910, 191 1 offers, 1908 Old Regime France, 201 1 , 2013 one person, one vote, 2253, 2254, 2256
1-22
Subject Index
One-House Veto Game, 2251
pivot,
open rule, 2225, 2226 operator, 2058 optimal threat, 2189
pivotal,
origin of the state, outcome, 1541 Owen,
pivoting,
1636--1 638
2006
2040-2042, 2050, 2051
Owen's multilinear extension, ownership,
2153
1908
2090 2091 p-Shapley value, 208 1 , 2 1 1 6 p-type coalition, 2063 Pacios, 2050 parallel information sets, 17 52 parliamentary government, 2008 partially symmetric values, 2125, 2154 participation constraints, 1900 partition, 2043 partition value, 2151, 2152 partitional game, 2073 partnership, 1907, 1908 partnership axiom, 2063 path, 2 155 path value, 2057 payoff configuration, 2091 payoff dominance, 1577 payoff dominance criterion, 1585 Perez-Castrillo, 2048 perfect, 1748, 1756 perfect Bayesian equilibrium, 1550 perfect equilibrium, 1523, 1547, 1618, 1628-1634, 1692, 2338 perfect infonnation, 1824 perfect recall, 1541, 1616, 1752 perfect sample, 2190 perfectly competitive, 2172 permutation, 2058 Perry, 1922, 1923, 1938 persistent equilibrium, 1524, 1569, 1 57 1 persistent rationality, 1544 persistent retracts, 1568 personal (or non-pecuniary) injury, 2236 perturbation, 1681 Peters, 2044 pFA, 2129 p-Shapley NTU value payoff,
p-Shapley correspondence,
2224 1736, 1737 planning procedures, 2346 player set, changes in, 1639, 1640 players, 2058, 2127 pNA, 2123, 2129 pNA' , 2159 pNA()c), 2156 pNA00, 2123, 2130 pM, 2 l29 podesta, 2007 political games, 2044 political value, 2044 pollution control, 1955 polyhedron, 1727, 1734 polytope, 1727, 1743 pooling equilibrium, 1863, 1 865-1867 populous games, 1763 positive, 2059, 2128 positively weighted value rp"', 2063 positivity, 2123, 2124, 2151 positivity axiom, 2059 potential, 2034, 2 103 potential games, 1528 pre-play communication, 1585, 23 16 pre-play negotiation, 1 606 predation, 1860, 1 861, 1866-1868, 1 882, 1 886 preferences, 2031 , 2332 price formation, 2339 price rigidities, 2196 price wars, 1870, 1 871, 1877, 1878, 188 1 primitive formations, 1575 principal-agent problems, 1 680, 1953 prior expectations p, 1584 prisoner's dilemma, 1690 private values, 1909, 1912 probabilistic values, 2057, 2059 probability space, 1681 Prohorov metric, 1778 projection, 2058 projection axiom, 2058, 2151 prominent point, 2343 proper equilibrium, 1523, 1552, 1619, 1629-1634 proper Nash equilibrium, 1700 property rights, 1908, 1996, 1998, 2008, 2014 property rules, 2242, 2244 proportional solution, 2101 pivotal information,
ordinal potential, 15 28 ordinal vs. cardinal, 2182 ordinality, noncooperative, organizations, 2017
2070 2030
Subject Index public, 205 1 public goods, 2 1 87 public goods economy, 2 1 92 public goods without exclusion, 2 1 92 public signals, 1820 Puiseux-series, 1821 pure bargaining problem, 2080 pure equilibrium, 1689 pure exchange economy, 2 1 73 pure strategies, 1525 pure strategy Nash equilibrium, 1768 purely atomic, 1 820 purification, 1537, 1765, 1875 purification of mixed strategies, 1608 purification of mixed strategy equilibria, 1539 quanta! response equilibrium, 23 19 quasi-concave, 1716 quasi-perfect equilibrium, 1548, 1628, 1630 quasi-strict, 1693 quasi-strict equilibria, 1529 quasivalues, 2057, 2154 quitting games, 1 844 railroad cartel, 2010 random variable, 1770 random-order value, 2057, 2061 , 2081 randomization, 1765 Rapoport, 2050 rational, 1713 rationalizability, 1 600-1605 rationalizable strategy, 1601 rationalizable strategy profiles, 1527 reaction functions, 1856, 1859 reaction to risk, 2346 realization equivalent, 1541, 1754 realization plan, 1753, 1 754 recursive games, 1823, 1828 recursive inspection game, 1969, 1971, 1973 redistribution, 2 1 87, 2189 reduced game, 2035, 2109 reduced strategic form, 175 1 refined Nash implementation, 1523 refinements of Nash equilibrium, 1523, 2288 regular, 1675 regular equilibrium, 1532, 1608, 1695 regular map, 1 695 regular semantic belief system, 1675 regulations, 1996, 1997, 2013 relative risk aversion, 2 1 92 relatively efficient, 2072
1-23 reliance, 2237, 2238 remain silent, 1929 renegotiation, 1586, 2314 renegotiation-proof equilibrium, 2014 repeated game, 1689, 1843, 1869, 1870, 1872, 1873, 1877-1879, 1882, 1884 repeated game with incomplete information, 1680, 1 8 1 3 repeated prisoner's dilemma, 1680 replacement, 1939 replicas, 2 1 8 1 representative democracy, 2220 reproducing, 2 1 30 reputation, 2010 reputational price strategy, 1931 restrictable, 2131 retract R, 1568 revelation principle, 2304 risk, 2032 risk aversion, 2192 risk dominance, 1524, 1576-1578, 1580 robots, 2340 robust best response, 1569 robustness, 23 1 8 Roth, 203 1 , 2032 Rubinstein, 1909, 1933, 1938 S-allocation, 2188
s'NA, 2162
saddlepoint, 2333 Samet, 2039 sample space, 1681 Samuelson, 1905, 1934 satiation, 2196 Satterthwaite, 1903, 1904, 1907, 1934 scalar measure game, 2161 screening, 1936-1938, 1940 searching for facts, 2330 Seidmann, 2050 selective elimination, 2305 self-enforcing, 1523 self-enforcing agreement, 1540 self-enforcing assessments, 1608, 1 609 self-enforcing plans, 1 607, 1 608 self-enforcing theory of rationality, 1526 seller-offer, 19 17, 1925, 1933 seller-offer game, 1909, 19 12, 1914 semantic, 1 667 semantic belief system, 1667 semi-algebraic, 1821 semi-reduced normal form, 1541 semivalues, 2057, 2060, 2125, 2152
I-24 separable, 2215 separable preferences, 2214 separate continuity, 1817 separating equilibrium, 1 863, 1 865-1867 separating intuitive equilibrium, 1 865 sequence form, 1750, 1753, 1755 sequential bargaining, 1909 sequential equilibrium, 1523, 1549, 1618, 1626-1628, 1910, 23 1 2 sequential monotonicity-no-veto, 231 3 sequential rationality, 1626 sequentially rational, 1549 set of best pure replies, 1691 set-valued solutions, 1642-1645 settlement, 1936 Shapley, 173 1 , 2027, 2172, 2 1 80 Shapley correspondence, 2085 Shapley NTU value, 2092, 2174 Shapley NTU value payoff, 2084 Shapley value, 2 1 14, 2123, 2187, 2248, 2253, 2262 Shapley value and nucleolus, 2249 Shapley-Shubik, 2027, 2254 Shapley-Shubik index, 2254, 2255 Shapley-Shubik power index, 2254 sidepayments, 2336 signaling, 1918, 1921, 1936-1938, 1940 signaling games, 1565, 1571, 1588 signals, 1 8 1 3 Silence Theorem, 1929 simple, 2058 simple games, 2030 simple polytope, 1727, 1740 simply stable, 1747 simply stable equilibrium, 1745 sincere voting, 2209 single crossing property, 1 864 smuggler, 1955 so long sucker, 2338 social psychology, 2344 social utility function, 2187 solution T, 1558 solution concept, 1558 solution function, 2105, 2109 sophisticated sincerity, 221 1 sophisticated voting, 2207, 2208 sophisticated voting equilibrium, 2215 Spanish Crown, 2008 speaking to theorists, 2330 stability, 1745 stability concepts, 1524
Subject Index stable equilibrium, 1696 stable set of equilibria, 1561 Stackelberg, 1882 stag hunt, 1576, 1579, 1603 standard form of a game, 1573 state of nature, 1 8 1 1 state of the world, 1667 state space, 1667, 1 8 1 1 state variable, 1912 states, 1667, 1933 static, 1909, 1934 static bargaining game, 1906 stationarity, 191 1, 1925, 1928, 1933 stationary, 1 8 1 1 , 1927, 1929 stationary equilibrium, 1 840, 1910, 1914 stationary sequential equilibrium, 1929 statistical test, 1958 stochastic game, 1 8 1 1 , 1835 strategic complements, 1859 strategic equilibrium, 1599, 1606, 1679 strategic form, 2332 strategic form game, 1561 strategic stability, 1558, 1640-1653 strategic substitutes, 1859 strategically zero-sum games, 1 7 1 1 strategy with finite memory, 1 8 1 5 strategyproofness, 2280 strict equilibria, 1529 strict liability, 2233 strict liability with contributory negligence, 2233 strict liability with dual contributory negligence, 2234 strictly positive, 2063 strictly stable of index 1, 2142 strikes, 1899, 1936, 1937, 1939, 1940 strong efficiency claim, 2249 strong equilibrium, 1585 strong law of large numbers, 1683 strong positivity, 2128, 2151 strong-Markov, 1927 strongly positive, 2 1 5 1 structure-induced equilibrium, 2216 sub-classes, 2037 subgame, 1541, 1572 subgame perfect, 1977, 2047 subgame perfect equilibrium, 1523, 1545, 1625, 1817 subgame perfect implementation, 2289 subsidy, 1739 subsolution, 1531 sunspot equilibria, 1820
Subject Index superadditive, 2058 supermodular, 2 1 1 6 support, 1726, 1742, 2129 swing, 2070 switching control, 1825 symmehic, 2059, 2128 syrnrnehic information, 1813 symmetrized, 1775 symmetry, 2029, 2040, 2124, 2151 symmetry axiom, 2059 syntactic, 1667 syntactic belief system, 1669 syntactically commonly known, 1677 system of beliefs, 1549 tableau, 1737 taking of property, 2239 takings, 2240 Tarski's elimination theorem, 1821 tax inspection, 1954 tax policy, 2191 taxation, 2187, 2 1 89 taxonomy, 2335 test allocation, 2288 test statistic, 1963 theorem of the maximum, 1913 theory of rational behavior, 1523 timeliness games, 1974 tracing procedure, 1524, 1580, 1748 transaction costs, 2241, 2242, 2244, 2247, 2336 transfer, 2057, 2069 transition probability, 1 8 1 1 hiviality, 2037 truncation, 1910, 191 1 truthful implementation, 2274 TU game, 2079 TU game form, 2106 Two-House Veto Game, 2251 two-house weighted majority game, 2136 two-person zero-sum game, 2329 two-sided incomplete information, 1934 type diversity, 2312 type structure, 1679 Uhl 's theorem, 1784 ultimatum, 2338 unanimity rule, 2224 unbounded edge, 1702 underdevelopment trap, 2012 undiscounted game, 1822 undominated Nash equilibrium, 2294
I-25 unemployment, 2196 Uniform Coase Conjecture, 1928, 1929, 1932 uniform equilibria, 1827 uniformly discount optimal, 1824 uniformly perfect equilibria, 1547 uniformly perturbed game, 1574 union, 1899, 1937-1940, 2042 union contract negotiations, 1936 United States, 2049 universal, 1672, 1674 universal beliefs space, 1828 universal type space, 1828 utility function, 203 1 value, xii, 2027, 2028, 2185, 2337 value allocations, 2187 value equivalence, 2171, 2175, 2 1 88 value equivalence principle, 2 1 17 value function, 1926 value inclusion, 2175 value principle, 2172 variable sampling, 1951 variation, 2128 variational potential, 2106, 2107 vector measure games, 2161 verification, 1950, 1982 via strong Nash equilibrium, 23 19 violation, 1957 virtual implementation, 2297 voluntary implementation, 23 16 voting, 2030, 2345 voting game, 2193 voting measure, 2193 voting power, 2253, 2256, 2258 voting system, 2256 w-Egalitarian solution, 2100 w-Egalitarian solution payoff, 2103 w-Egalitarian value payoff, 2100 w-Shapley correspondence, 2089, 2090 w-Shapley value, 2090 w-potential, 2104 wage negotiations, 2013 weak asymptotic �-semivalue, 2153 weak asymptotic value, 2 1 34, 2152 weak correlated equilibrium, 1714 weak efficiency claim, 2246 weak-Markov, 1927 weakly dominated, 1548 Weber, 2045 weight system, 2063 weight vectors, 2063
Subject Index
I-26 weighted majority game, 2136, 2162 weighted Nash Bargaining Solution, 2083 weighted Shapley NTU, 2088 weighted Shapley values, 2057, 2081 weighted value cpw, 2064 weighted values, 2157 weighted voting, 2255, 2257, 2258 Wettstein, 2048 Williams, 2285
winning coalition, 2069 Winter, 2041-2043, 2049 Young, 2033, 205 1 Young measures, 1794 zero transaction costs, 2242 zero-sum game, 1531, 1729, 1960, 1968