STUDIES IN MATHEMATICS AND ITS APPLICATIONS
VOLUME 7
Editors : J. L. LIONS, Paris G. PAPANICOLAOU, New York R. T. ROCKAFELLAR, Seattle
MATHEMATICAL METHODS OF GAME AND ECONOMIC THEORY
JEAN-PIERRE AUBIN Ecole Polytechnique, Universiit Paris IX Dauphine
1979
@ NORTH-HOLLAND PUBLISHING COMPANY - 1979 AII rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted. in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.
North-Holland ISBN 0 444 85184 4
Published by: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK * OXFORD
Sole distrlbutorsfor the U.S.A. and Canada: Elsevier North-Holland, Inc. 52 Vanderbilt Avenue New York, NY 10017
Library or Cangrrscl Cataloging in Publication Data
Aubin. Jean-Piem. Mathematical methoda of g8me and economic theory. (Studies in mathematics and its applications; v. 7) Bibliography: p. 590 Includes index. 1. Game theory. 2. Economics. Mathematical. 3. Functional analysis. 4. Mathematical optimization. 1. Title. 11. Series. QA269.Aal 51S.7 78.18162 ISBN 0-444-85184-4
PRINTED IN HUNGARY
This book is dedicated to N.
PREFACE
It is said that the preface is that part of a book written last, placed first and read least. I do hope, however, that the reader will glance through it to allay any fears that might otherwise be raised by the length of this book. The book has been written with two audiences in mind. (a) Mathematical economists and operations research specialists. Workers in these areas will find that the book provides a solid foundation in non-linear functional analysis at a level just beyond that attained in most mathematical texts intended for economists. (b) Mathematicians. Mathematicians with an interest in non-linear functional analysis will find that the applications offered in optimization, game theory and mathematical economics provide valuable insight into the general structure of the theory. Non-linear functional analysis is the central theme of this book. It not only provides powerful and versatile tools for solving specific problems in economics and the social sciences, but also serves as a unifying theme in the mathematical theory of these subjects as well as in pure mathematics itself. It will be clear that this is a book of mathematics in which formal definitions are given and theorems are proved. However, the interpretation of these theorems in various contexts is given equal weight. The central application treated is the fundamental economic problem of allocating scarce resources among competing agents. This leads us to consider' simultaneously the interrelated applications in game theory and the theory of optimization. No attempt is made in this book at a critical appraisal of the behavioural assumptions implicit in the mathematical models considered nor is any attempt made to justify their importance as objects of study. In particular, no pretence is made that the models studied are immediately applicable and, in many cases, the level of idealization involved in the models is very high indeed. Readers interested in these interpretive questions will have to look elsewhere. Fortunately there are many good texts which study these problems. Applications are introduced as early as possible in order to stimulate the reader and to motivate the theorems. There would have been advantages in vii
viii
PREFACE
gathering the applications together in one place, but I felt that a lively treatment was to be preferred. In the beginning, of course, the applications offered tend to be rather flimsy since they inevitably consist essentially of semantic re-interpretations of mathematical definitions. In spite of this, they can still often be of considerable help to one's intuition. As the book proceeds, the applications given naturally become deeper and of more interest. The reader will find that much of the mathematics in the book will be accessible without the necessity of having a sophisticated mathematical background. However, certain more difficult passages have been marked with an asterisk (*) and may be omitted at a fvst reading. The book is organized into three parts. The first part constitutes a course in optimisation theory and convex analysis. The second covers a number of topics in game theory and mathematical economics. The third part provides and introduction to non-linear analysis and control theory. Purt Z (Chapters 1-5) Linear and convex analysis is developed in the framework of optimization theory. The treatment includes results on the existence and stability of solutions to optimization problems as well as an introduction to duality theory. Numerical results and solution algorithms are not discussed at all.
Purt ZZ (Chapters 6-12) Two-person games are considered first. These prove to be the right framework within which to study the later theorems of non-linear analysis. The treatment continues with Walrasian models for the allocation of scarce resources and with an introduction to n-person non-cooperative games. Most (but not all) important solution concepts for n-person cooperative games are considered in conjunction with their use in Walrasian models. course, many interesting ideas have had to be omitted.
04
Part ZZI (Chapters 13-15) In this part, more advanced issues are considered from a primarily mathematical point of view. A class of functions is introduced which shares the properties of the ordinary convex functions. The treatment continues with an introduction to monotone and pseudo-monotone operators. Duality theory is extended to the case of infinite-dimensional spaces and this allows a short introduction to the calculus of variations and optimal control theory. Impulsive control theory, introduced for solving inventory problems, is also briefly discussed for the deterministic case. In the last chapter, an array of various fixed point and surjectivity theorems are given. These are powerful tools in proving existence theorems. The chapter continues with a short account
PREFACE
ix.
of quasi-variational inequalities and concludes with some further results on correspondences (i.e. “multi-valued functions”).
In developing the material presented in this book, I have benefited from the active collaboration of my friends at the Centre de Recherches de Mathimatiques de la Dkision de I’Universit6 Paris-Dauphine. I am particularly grateful to Ivar Ekeland (who introduced me to game theory), Alain Bensoussan, Frank Clarke, Bernard Cornet, Jean-Michel Lasry and Hem6 Moulin. I also wish to thank the Mathematics Research Center of the University of Wisconsin for the support it has granted me during the last six summers and, in particular, toRichard H. Day (who introduced me to economics). I would also like to point out that the book owes much to the students of the Universit€ Paris-Dauphine and the Ecole Polytechnique whose reactions helped me in improving its pedagogical aspects. Finally it is my pleasure to thank Ken Binmore for correcting the manuscript and thereby drastically improving the text.
SUMMARY OF RESULTS: A GUIDELINE FOR THE READER
We summarize the main concepts and results presented in this book. Of course, this summary is not really meant to be understood all at once, but rather to supply the reader with guidelines and to help him locate the main results in their true perspective. We hope that this summary will provide a rough idea of what this book is about and also that it will indicate the place in the book where a given result is presented. We have tried to compose the chapters in such a way that they can be read independently of each other in so far as this is possible.
Part I. Optimization and Convex Analysis We devote the first part of this book to the study of optimization problems (i.e. one-player games) of the form GC
= inff’(x), XEX
where X represents the strategy set and f the loss function. These together describe the behavior of aplayer. Most of the results presented in this first part are used later on. (1) We begin by studying the existence of a minimal solution. This holds whenever we assume that f is lower semi-continuous (i.e. has closed lower sections) and lower semi-compact (i.e. has relatively compact lower sections) (see Section 2.1). We devote the third chapter to a rather comprehensive study of these required continuity and compactness properties. The important particular case of quadratic loss functions on Hilbert spaces is studied in Sections 2.2 and 2.3. (2) We continue by studying stability properties, i.e. the smoothness of the behavior of the minimal value
xxii
SUMMARY OF RESULTS
when the loss function f ( . , y ) and/or the strategy set X(y) depend upon perturbations y. The most important example is obtained when we perturb a loss functionf defined on a subset X of a topological vector space U by the “simplest” functions, which are the continuous linear functionsp E U*.The function f * defined on U*by
is obviously a convex lower semi-continuous function. It is called the “conjugate¶¶function of f. Convex lower semicontinuous functions are precisely those for which f = f * * (see Section 2) and, consequently, this class of functions plays an itnportant role throughout this book. More generally, we show in Section 4.9 that smoothness requirements on y) guarantee the existence and uniqueness the function y a(y) = inf,,,f(x, of a minimal solution of the minimization problem u(y) for “almost all” perturbations y. This approach allows us to study a whole family of problems rather than a (possibly unstable) problem. When we allow the strategy set to depend upon y, we are led to define and study continuity properties of correX(y). We initiate this studv in Section 2.5 and we ct mtinue it spondencesy in Section 15.3. When the strategy sets X ( y ) are represented by families of inequality constraints, the study of stability properties leads to the “duality theory” for minimization problems. J ( 3 ) We begin to treat duality theory in Chapter 5 (in the case of finite dimensional spaces) and we continue its study in Chapter 14 (in the case of infinite dimensional spaces). Roughly speaking, we can devise dual problems whenever a minimization problem u = infxExf ( x ) can be written in the form
-
-
u = inf sup I(x,p) X € U
PEY’
where I is a function defined on U X V* called the “Lagrangian”. The question arises as to whether it is possible to find jj E V* (called a Lagrange multiplier) such that the initial problem is equivalent to the simpler problem u = = inf,, I(x, p). The dual problem amounts to finding such Lagrange multipliers. n i m a l This will be done ’in (4) It is also useful t o ~ c ~ a r a c t e r ~ ~ ~ m isolutions. Chapter 4, by using several different concepts of dverentiability. For instance, we prove that X is a minimal solution if and only if f ( X ) = f * * ( X ) and iE af*(O), where af*(p) is the “subdifferential o f f * atp” defined by af*(p) = = { x E U such thatT(p)-f*(q) =S ( p - q , x ) for all q}. We devote Section 4.1 to subdifferentiability and show in Section 4.2 and 4.3 how it is related to the usual concepts of differentiability (convexity will play an important role).
SUMMARY OF RESULTS
xxiii
In particular we prove the following “variational principle”. I f f is differentiable on a convex subset X and if T is a minimal solution, then (of (z), 2 - y )
0 for any y E X.
Such problems are called “variational inequalities”. (They will be solved in Section 13.2.) (5) Convexity plays a fundamental role in solving the problems mentioned above. The reason lies primarily in the fact that we implicitly or explicitly perturb minimization problems on topological vector spaces by continuous linear functionals (since the pointwise supremum of linear functions is convex). A secondary reason is that convexity allows the use of the Hahn-Banach separation theorem. We prove the fundamental results of convex analysis (in particular, the fundamental minimax theorem) by using the separation theorem together with the following fundamental property of families of convex functions: iffl, . .,f, are n convex functions defined on a convex set X and ifwe set F(x) = {fi(x), .. , L ( x ) ) E R”,then F ( X ) + R t is convex. In Sections 1.3 and 1.4 we recall elementary properties of convex sets and functions and we devote Section 13.3 to the study of convex cones of functions isomorphic to cones of convex functions. The characterization of convex lower semi-continuous functions is presented in Section 2.4. The minimax theorem for convex-concave functions is proved in Section 7.1.1, Section 7.1.7, Section 7.1.8 and Section 7.1.9 and generalized in Section 13.1.
.
.
Part II. Game Theory and the Walras Model of Allocation of Resources The second part of this book deals with game theory. For simplicity, we present in Chapter 6 the main solution concepts in the case of two-person games and illustrate them with some examples (finite games, Cournot’s duopoly and Edgeworth’s economic game). (1) In Chapter 7 we study two-person zero-sum games. We state and prove two fundamental results, which will be used as basic tools in the rest of the book.
-
Theorem A (minisup theorem). If X is a convex compact subset, Y is a convex R is convex and lower semi-continuous with respect subset and f : X X Y ro x mid concave with resnect to y, then there exists 2 X such that sup/(% y ) = sup inf f ( x , y ) .
Y€Y
Y€Y
X € X
SUMMARY OF RESULTS
XXiV
Note that this implies that suprer inf,,,f(x, y ) = inf,,, s ~ p , ~ ~ f y) ( xand , also the Von Neumann theorem for the existence of saddle points. _-
I-: -
-
.-
4 Theorem B (Ky-Fan’s inequality).
X is a convex compact subset and if f :X X X R is lower semi-continuous with respect to X and concave with respect to y , then there exists I X such that -+
SUP Y € Xf ( x ’ , Y ) YSU €P X f ( Y , Y). Theorem B is actually equivalent to the Brouwer fixed point theorem which asserts that any continuous map from a convex compact subset into itself has a fixed point. But it happens that the Ky-Fan inequality is much more useful and versatile as a tool for proving the main results of game theory and non-convex analysis. We use the Ky-Fan inequality systematically in proving allsubsequent existence theorems. These results are proved in Section 7.1 and are extended and improved in Chapter 13. The rest of Chapter 7 deals with other issues concerning twoperson zero-sum games. (2) We devote Chapter 8 to the description of the Walras model of allocation of scarce commodities among competing consumers and to tlie proof of the Debreu theorem for the existence of a Walras equilibrium (by using the Ky-Fan inequality). This model will be used as the main example of an nperson game. (3) In Chapter 9, we study the non-cooperative concept of equilibrium and prove the Nash theorem on the existence of a non-cooperative equilibrium. More generally, we prove the Arrow-Debreu-Nash theorem in the more general case when the strategy set of each player i depends upon the choice of the multi-strategy implemented by the coalition t of the other players j . Again, these theorems are deduced from the Ky-Fan inequality. As a byproduct, we also prove the Brouwer and Kakutani fixed-point theorems (in Section 9.3). We devote Section 9.2 to the case of quadratic loss functions, in which we need only elementary results. We apply these results to obtain the existence of equilibria in economic models (Section 9.2.5 and Section 9.4). (4) Chapters 10, 11 and 12 deal with the cooperative concepts of n-person games. The ideas are introduced in Chapter 10. Difficult results are proved in Chapter 11 (for games with side-payments) and in Chapter 12 (for games without side-payments). In the case of non-cooperative games, we assume that the players do not “communicate” among themselves. In cooperative games, the players can participate in coalitions. Such participation will, of course, influence their behavior.
I
SUMMARY OF RESULTS
XXV
In other words, cooperative games are described by the behavior of a given family of coalitions instead of by the behavior of the n players regarded as individuals. The main solution concept is the core of the game. To define it, we Will describe how a coalition rejects a strategy. The core of the game is then the subset of strategies which are not rejected by any coalition which is allowed to form. If the only coalition allowed to form is the whole set of players, the core is called the set of Pareto minima. The larger the family of coalitions allowed to form, the smaller is the core. In order to have a core as small as possible, we extend the family of ordinary coalitions of players (described as subsets A of the set N of of players) td the family of “fuzzy editions" (described as fuzzy subsets z of N). In other words, a fuzzy coalition z is a vector z = {zl,..., z, ...,z}, of [0, l]” where xi E [0,1] represents the rate of participation of the it” player in the fuzzy coalition. (For conventional coalitions, the rate of participation zi is either 0 or 1.) By describing the behavior of fuzzy coalitions, we define the concept of fuzzy core of a game. We prove that the fuzzy core is non-empty under reasonable assumptions (again using the Ky-Fan inequality). Furthermore, we prove that, in the case of games with side-payments, the fuzzy core coincides with the “fuzzy value”. This is another “solution concept” defined and studied in Sections 11.3 and 11.4. In the case of games withour side payments, we prove that the fuzzy core coincides with the set of “cooperative equilibria” defined in Chapter 12. In particular, we associate a cooperative game with the Walras model of allocation of scarce resources and define the fuzzy core of an economy as the set of allocations which are not “rejected” by any coalition. We prove that this fuzzy core coincides with the set of Walras equilibria. This result allows an evaluation of the concept of a Walras equilibrium from a normative point of view. (We will not study the alternative approach in which, instead of a finite number of players forming fuzzy coalitions, one considers a continuum of players forming only ordinary coalitions.) Part 111. Non-linear Analysis and Optimal Control Theory
As we have mentioned in the preface, the last three chapters are devoted to improving and completing earlier results. (1) Chapter 13 presents a rather comprehensive study of minimax type inequalities. We use them to define pseudo-monotone and monotone operators and to study associated variational inequalities. In the last section of this chapter, we characterize classes of functions which share the same properties as. convex functions.
xxvi
SUMMARY OF RESULTS
(2) Chapter 14 is devoted to duality theory in infinite dimensional spaces. In the first section, we extend the duality theory devised in Chapter 5. Section 14.2 devoted to minimization problems for (nonconvex) integral criteria. Section 14.3 is a short introduction to the calculus of variations in the framework of convex analysis and to the Pontriagiii principle for the optimal control problem. In Section 14.4 we present a short introduction to the dynamic programming approach for optimal control problems as well as to stopping time and impulsive control problems. We devote the first section of Chapter 15 to the proof of a collection of fixed point and surjectivity theorems using the Ky-Fan inequality. We begin by proving the existence of critical points x , i.e. solutions x of multivalued equations 0 E S(x), where S maps a space X into a vector space U.In section 15.2 we study quasi-variationalinequalities. We continue this chapter by giving examples of lower semi-continuous correspondences. Finally, we prove the Michael continuous selection theorem and characterize semi-continuous correspondences with convex images.
CONTENTS OF OTHER POSSIBLE COURSES
A) Convex analysis (a) Optimization and convexity: Chapter 1. (b) Continuity of convex functions : Sections 2.1,2.4,3.3and 3.4. (c) Differentiability of convex functions: Sections 4.1, 4.2 and 4.3. (d) Minimax Theorem: Sections 13.1.1 and 13.1.2: Existence of a conservative solution; Sections 7.1.7, 7.1.8 and 7.1.9: The minisup theorem; Section 6.5.3 : perturbation by linear functions; Section 13.1.3:Existence of a minisup under weaker compactness assumptions. (e) Duality for optimization problems : Sections 5.1, 5.2 and 14.1. (f) y-convex functions: Section 13.3. (g) Applications: Sections 14.2 and 14.3. ( B ) Correspondences
(a) Upper semi-continuous correspondences: Sections 2.5.1, 4.1.4, 2.5.3 and 2.5.4. (b) Lower semi-continuous correspondences: Sectious 2.4.2, 15.3.1, 15.3.2, and 15.3.3. (c) Semi- and hemi-continuity: Sections 2.5.1, 15.1.4 and 15.1.5. (d) Monotone correspondences : Sections 13.2.6 and 13.2.7. (C) Non-convex analysis (prerequisite: course B : correspondences) (a) The Ky-Fan inequality: Sections 13.1.1, 13.1.2, 7.1.2, 7.1.3, 7.1.5, 7.1.6, 13.2.1 and 13.2.3. (b) Existence of critical points: Sections 15.1.1, 15.1.3 and 15.1.4. (c) Fixed point and surjectivity theorems: Section 15.1.2. (d) Variational inequalities: Sections 13.2.4 and 13.2.5. (e) Quasi-variational inequalities : Sections 9.3.1 and 15.2.
xxvii
xxviii
CONTENTS OF OTHER POSSIBLE COURSES
Table of economic illustrations and applications Production sets, profit, cost and production functions: Sections 1.4.5, 1.4.6 and 2.4.5. Marginal profit : Sections 4.2.5 and 5.1.4. Analysis of the duopoly and oligopoly: Sections 6.2,9.2.4 and 9.2.5. The Edgeworth box: Section 6.4. The Walras model: Chapter 8, Sections 9.4, 10.3.5, 10.4.7, 11.4.1 and 12.3.
NOTATIONS
(1) The reader who prefers to avoid the intricacies of infinite dimensional
spaces can assume in the two first parts that the vector spaces are finite dimensional. Most of the.examples used in economic theory or game theory are, of course, finite dimensional spaces. Otherwise, Appendix A provides a short summary of what one needs to know in linear functional analysis. R the cone of non-negative vectors a = {al, . .., a,} (i.e. We denote by : R satisfying ai z- 0 b' i), by R:+ the cone ofpositive vectors a (i.e. satisfying a E : and a # 0) and by 8" the cone of strongly postive vectors a (i.e. satisfying a, =- 0 for all i). We set a z- b if a- b E : R (a is not smaller than b), a w b if a- b E g, (a is greater than b) and a = b if a - b E &+(ais strongly greater than b). If J is a subset of R",we set A , = A+R$
and
A,
= A+RZ.
We denote by Rn*= B(R", R) the dual of R"and by RY and p : the cone of non-negative and strongly positive linear forms. The duality pairing on R"*X R" is denoted by (p, x ) = ~ ~ o l p i x i .
(2) A correspondence (or multivalued map) from X into Y associates with any Y,called an image or value of the correspondence S.
x E X a subset S(x) of
(3) If a topological space is not metric (or not metrizable), we have to replace ~ elements x, of X by the use of generalized sequenthe use of sequences { x , } , ~of ces (or nets) { x ~of elements } ~ ~ xP~of X. These are defined as maps p E M t-F-- x,, E X, where A? is a set of indices with a preorder == such that we can associate with any pair {pl, p2} another p such that p max (PI,p2). of the If 0 2 is another set of indices, a generalized subsequence {xpv€.,} generalized sequence is the map Y I-+ p , I--+ xPvwhere the map Y I--+ pv satisfies the property: V p , YO such that Y 3 YO implies p, a p. Recall that compact spaces are those with the property that we can extract from any generalized sequence a convergent generalized subsequence. xxix
XXX
NOTATIONS
(4) We list below other symbols we shall use.
coalitions (or finite subsets) number of elements of the coalition A decision rules vector spaces duality operator linear operators set of players subsets correspondences strategy or commodity sets multilosses loss functions multiloss operator Lagrangian linear functionals strategies or commodities duality pairing semi-distance semi-norm family of coalitions families of decision rules space of continuous functions space of maps from X into Y space of continuous linear operators from U into V subset of discrete probabilities on X family of finite subsets, space (cone) of all (positive) functions on X family of fuzzy coalitions space of bounded functions Lebesgue space strong, Mackey and weak topologies barycentric operator Dirac operator constraints fuzzy coalitions measures or linear functionals on functions spaces
NOTATIONS
4w
fuzzy value (Shapley value) gauge of X upper and lower support functions of X indicator of X subdifferential topological dual of U transpose of L conjugate function off (f- sharp)
4 x , *1 a # ( X ; .), ab(X; .) '
Y(x; .)
a
U* L*
f* f Y X ) = SUPy,Yf (x, Y ) v# = inf,,, sup,,,f (x, Y ) h ( Y ) = inf,,,f (x, Y ) vb = SUPYEYinf,,xf (x, Y ) vU = SUP,,, inf,,, sup,,,f(x,
f IS f"
xxxi
( f- flat) Y)
loss function of the dual problem linear extension off lower section of,f S(f 4 Df (4 gradient off minimal (maximal) set off Mbcf, ( M # ( f ) ) Lf,gL product off by L inf convolution off and g f o g co ( X ) , G ( X ) , cos ( X ) (convex, closed convex, symmetric closed convex hull) P(X> cone spanned by X PAX) recession cone of X X#(Xb) upper (lower) polar set of X X+(X-) positive (negative) polar cone of X XI orthogonal (annihilator) of X A^ = [ A complement of A , adverse coalition, i = N--i d = closure of A , A interior o f A A + = A+R"+,+ = A+@+ if A c R" XA = xi,X N = x'= XA x X" = x'xXf strategy set of a coalition A X(A) multiloss operator of a coalition A FA = {SF, . . .,f;;'> game described in strategic form F) cooperative game described in strate{X(A), F A )A ot gic form cooperative game (with side-payments) described in characteristic form 9
nieA
{-v%
,
xxxii
NOTATlONS
fuzzy cooperative game (without side payment) core of a game (of a fuzzy game) (exchange) economy economy with producers
CHAPTER 1
MINIMIZATION PROBLEMS AND CONVEXITY
This chapter is largely concerned with generalities about optimization problems and includesan introduction to convexity. In Section 1.1 we recall the general form of an optimization problem inf,,, f( x ) and give some examples of strategy sets X and loss functionsf. In Section 1.2 we introduce the “product from the left” of a function. Wc use the notation Lf for this concept and define. (Lf)(Y) = inf f ( x ) . LX=y
We also discuss the related concept of the inf-convolution of functions. The next topic is the identity
which yields some useful general decomposition principles for optimization problems. It will come as no surprise that convexity plays a key role throughout this book. In Section 1.3 we motivate the introduction of convexity ideas by drawing attention to the usefulness of replacing an initial minimization problem a = inff(x) X € X
by an extended minimization problem rl.
=
inf fA(ni),
m€JK/ll(X)
in which X is embedded in the convex subset M ( X ) of probability measures (called “mixed strategies”) and where f A ( m )is the corresponding expected loss. In defining convex functions by the usual barycentric inequalities, we observe that the initial and the extended minimization problems are equivalent for convex functions. In the remainder of Section 1.3 we state some elementary properties of convex sets and functions. In Section 1.4, we introduce the indicator function yx, the support function:c and thegauge function zxfor a closed 3’
3
4
MINIMIZATION PROBLEMS AND CONVEXITY
[Ch. 1,s 1.1
convex subset X c U which contains 0. The indicator function is defined by yx(x) = 0 if x E X and motherwise. The support function u$ is defined on the dual U* by u s ( p ) = sup,,, (p, x). The gauge function a, is defined by nx(x) = inf {A =- 0 such that 1 - l ~€ X}. These three functions are all convex. The indicator and the gauge chmacterize X in the sense that
+
X = {x E U such that yx(x) = 0 )
and X = {x E U such that sX(x)6 1).
In Section 1.4.2, we reformulate the fundamental Hahn-Banach theorem as the assertion that X = { x f U such that (p, x) 4 uz(p) for all p c U*).This
representation of a closed convex set will play a fundamental role at a later stage. Finally, we mention an economic interpretation of these concepts in the framework of producfion sets.
1.1. Strategy sets and loss functions
We shall definea minimization problem a = infXE,f(x) in which Xis regarded as a strategy set and f as a loss fimction. In economic models, X often describes a set of available commodities. The minimization problem can then be viewed as a method for selecting a commodity from those. Usually such minimization problems are given more explicitly. We shall comment on the following special case in which X = {x E U such that Lx E Y} (where U and Y are vector spaces, L E Q(U,V)and Y c V)and f = gM where ME 2(U,W) and g is defined W. For the sake of simplicity, we shall allow loss functions to take the value +-.Iff:U+]a,+-]andX=Domf={x€Usuchthatf(x)i+m}, the restriction to Xis an ordinary function from X into R. In other words, a loss function from U into ]- m, + -1 involves the description of both the strategy set X and the actual loss function f. We end this section by recalling the definition of sections and epigraphs of functions. 1.1.1. Optimization problem
We begin by considering the case of a single player game (or a single decision-maker decision problem). In fact a player can represent a team of players. In this case, the “normative rule” for selectinga strategy is to solve an optimization problem defined by the following items:
Ch. 1, 0 1.11
STRATEGY SETS AND LOSS FUNCTIONS
5
(a) a set U,representing a set of unconstrained strategies (or unconstrairzed decisions). (b) a subset X of V,representing a set of (feasible) strategies (or decisions). In many cases, X is defined by a set of constraints. (c) a function f : X R, associating With a strategy (or decision) x a loss f (x) (or a cost or disutility). We call -f (x) a payoff (or a proJit or utility). Thus, the optimization problem amounts to selecting a strategy 2 E X which minimizes the loss function f (x) as x ranges over the strategy set X . In other words, we have to find Z such that
-.
(i) 2 E X, (ii) f ( 2 ) = infXcxf (x). We shall say that Z is an optimal strategy (or decision). It is clear that such a general optimization problem can “model” many instances. We mentioned the vocabulary used in (parlor) games or in optimal decision problems, but the implications range far beyond, to mathematics, economics, politics, etc. 1.1.2. Allocation of available commodities
In particular, we consider the basic problem of economics by which we mean the problem of allocating available commodities among competing ends. This may be modelled in various ways of which we give some examples. In these examples - U describes the set of “commodities”, - Y c U describes the set of “available commodities”. An optimization problem can be viewed in this context as a selection procedure for choosing a commodity from those available which minimizes a given loss function f (provided such minimizing commodities exist). Example. Commodity space R’. In most examples, we describe the commodity space U by a finite dimensional space R’. If the I goods are 1abelled.by i = 1, . . ., I, a commodity vector x = { X I , . . ., xl} R‘ represents XI units of the first good , . . .,x, units of the I& good. The elements ei = (0, . . ., 0, 1, 0, . . ., 0 ) of the canonical basis of R are the “unitary commodities”, since e‘ represents one unit of the ith good. Note that we can write the commodity vector x = c:=lxzd as the sum of xi units of the first unitary commodity , . . .,of xI units of the I” unitary commodity. In most examples, the real line represents a set of values, where the basis { 1 represents the ‘‘unit of account” of the model (for instance, dollar, franc,. . .
6
MINIMIZATION PROBLEMS AND CONVEXITY
[Ch. 1, Q 1.1
Thus the dual R” of R‘ can be regarded as the space of prices p associating (linearily) with any commodity x E R’its value ( p , x ) = p(x). The dual canonical basis of R‘*is spanned by the forms e; defined by (2)
e;(ek) =
0 if k # i, 1 if k = i.
For any x E R , e,*(x) = xi. Any p E R’* can be written
(3)
p =
I
C pie;
i=l
where p i = ( p , e’).
In other words, the it” component pi denotes the value of one unit of the i‘h good, i.e. what is usually called “the price of the it“ good”, or “unitary price the good”. The duality pairing can be written
c I
(4)
( P . x> = P ( X ) =
P‘Xi.
i= 1
This formula speaks for itself. 1.1.3. Resource and service operators
A more explicit general form for an optimization problem can be described in the following way. We introduce the following sets : (a) the unconst&ined strategy set U; (b) a set Y of resources; (c) a set W of services (or “outcomes”, “attributes”, “acts”, LL criteria”, “characteristics”, “observations”,etc.) ; (d) the resource operator L : U V which associates with any strategy x E U the resource L(x) E V needed to implement x ; (e) the service operator M : U --r W which associates with any strategy x E U a “service” M ( x ) E W produced by x ; (f) a subset Y c V of “availableresources”(for instance, Yis the set of resources smaller than or equal to y for a convenient ordering); (g) a function d mapping W into R which associates with any service v E W a “distance” d(w, u) to an “objective” u E W. We then define the strategy set X by
-
X = { x E U such that L(x) E Y)
Ch. I , § 1.11
STRATEGY SETS A N D LOSS FUNCTIONS
7
(i.e. X is the subset of strategies whose required resources are feasible) and the loss function by
f ( x ) = d(Wx), .) where the loss f(x) associated with the strategy x is the “distance” to u of the service M ( x ) performed by x. In particular, we shall restrict our attention to the case where - U,V, W are finite dimensional vector spaces (or, more generally, topological vector spaces) -L, A4 are linear operators (or continuous linear operators.) A special treatment will be offered in the case when
I
(i) W is a Hilbert space, (ii) d(v,u) = $ l l v - u ] 1 2 = +((w-u, w-u)),
where ((w, w ) ) denotes the scalar product on W (see Section 2.3.1). Example. Take U = R’to be a commodity space as in the above example. We shall regard U as the “output” space. Take V = Rk and interpret this as a space of “resources” (or an “input space”, space of “primary goods”, etc.). The map L E L!(U, V) describes a “linear production operator”, which associates with any output x E U the input Lx E V which is used to produce x. If x = l&‘ x,d, the input Lx = CrElx,kJ is the linear combination of inputs &’used to produce the unitary commodities ej. Then the entries (5)
a{ = (e;, Le>
of the matrix ofL represent the amounts of units of the it’ input used to produce one unit of the j”‘ output. They are called the “technical coeficients” of the production process. Usually, the “production operator” Lis assumed to besurjective,i.e.any input y E Rk can produce at least one output x E Rk. If p E V * = Rk*is an input price and if L’ is the transpose of L, then L*p is the output price such that the value (L*p, x ) of any output x is equal to the value ( p , Lx) of the input Lx needed to produce x. Note that (6)
v j = l ,..., 1,
k
$=Caw. i=1
The unitary price q’ of the jth output is the sum of the values ./pi of the a: units of input i used to produce j.
8
MINIMIZATION PROBLEMS AND CONVEXITY
[Ch. 1,s 1.1
Since we assumed L to be surjective, L* is injective. If the input price p E Rk’ is non-zero, the output price L*p E R“ is also non-zero.
Example. We shall regard
W
= Rm as a space
of “services”
and M f d ( U , W ) as a linear map associating with any commodity x E R the service Mx E R” produced by x. The entries b: = (e;, M d ) of the matrix of M represent the numbers of units of the z* service produced by a unit of the J*“ commodity. Usually, it is assumed that a service operator M is injective, i.e. that any non-zero commodity x E R’produces a non-zero service Mx. If q E Rm*is a service price, then the price M*q E R’*is the price such that the value (M’q, x) of x equals the value (q, M x ) of the service Mx produced by x. Since we assumed a service operator M to be injective, the operator M* is surjective. If p E R’* is a commodity price, there exists at least one service pricep E Rm’ such that q = M*p. The problem of choosing a service price p satisfying the equation 9 = M’p is called a “pricing problem”. 1.1.4. Extension of lossfunctions
It will be useful to extendfto a functionfx mapping the set Uof unconstrained strategies into 1- -, + -1 by setting
0
+-
i f x E X, ifx 6 X.
The optimization problem (1) is then equivalent to finding 2 E U such that
In other words, the extendedfunction fx “involves” both the strategy subset X c U and the original real-valued lossfunction f.
Dewtion 1. Let U be an unconstrained strategy set andf(resp. g) a function mapping U into 1- a, -1 = R U {+ -} (resp. [+ -I = R U {- a}). We define the “domainy’Dom f of f (resp. Dom g of g ) by
+
(9)
03,
Dom f = {x E U suchthatf(x) < + -}. (resp. Dom g = {x E U such that g(x) z -
10.
1)
Ch. 1, § 1.11
9
STRATEGY SETS AND LOSS FUNCTIONS
We exclude once and for all extended functions f or g with Dom f = 0 or Domg = 0. Occasionally we emphasize this exclusion by referring to our extended functions as “proper” extended functions. I f f is a loss function defined on the strategy subset X of U,fx is called its “extension” to U. It is clear that
The use of a finite loss function defined on a strategy subset X i s equivalent to the use of its extension mapping U into 1- m, -1, where the domain Dom f of f represents the strategy subset X .
+
It will be useful to use the second approach when, for example, a given loss function is constructed as a pointwise supremum of functions. More generally, whenever a function f is constructed according to certain formulas, these specify the domain off implicitly, by specifying whether f(x) is or not. In the first approach, one would always need to describe the domain o f f explicitly before the values off on that domain could be given.
+
Another instance where it is useful to consider functions mapping U into ] - a, -1 occurs when the strategy set X ( y ) depends upon a parameter y and we have to study the function
+
a(y) = inf f(x). X€XCv)
We shall adopt the following rule for defining a ( y ) when X(y) is empty. We write (11)
inf f ( x ) = + m,
supf(x) = -
00.
X€0
XE!d
(The loss becomes infinite when it is impossible to select a strategy.) We extend the usual 3rithmetic operations to + -1 by setting 00,
=-+a
a+
a-
m
for - m < a < +
= 00
= - m+a = - w
a m = m a = m ’ , a(-
a
m= w
(I= -
m;
O m = -0 = 0 = O ( -
-
;or - m G a -= w,
m)
a(-
m,
= (- m)a = -
w)
m)=(-
inf 0 = + a; sup 0 = - w.
= (-
m)
m)O;
for 0 -= a 4 m,
a = w for - m G a -(-
m)
=+
m,
< 0,
provided the sums a+b are not the forbidden
00
-
00
or
-
- + -.
1.1.5. Sections and epigraphs
Definition 2. Let f : U k- ] - 00, (12)
S(f,A)
+ -1
be a fundion. The subsets
= {x E U such thatf(x)
4
A}
are called the ‘‘(lower) sectionr” off. The subsets &(f , A) = {X E U : f ( x ) < A}, S”(f, A) = {x E U : f ( x ) a A} and A) = {X E U : f ( x ) =- A} are called respectively the (lower) open sections, the upper sections and the upper open sections ofJ The subset
s”(f,
,(13)
L p ( j ) = { {x, A} E U x R such thatf(x)
-S
A}
is said to be the “epigraph” off. Proposition 1. Let f = sup,,,fi be the pointwise supremum of u family of functionsfi : U k-- 1- 0 0 ) -1. Then
+
u.}icI
Proof. This is left as an exercise.
Remark. If we are only concerned with minimization problems, loss functions are only used via their lower sections. Indeed, only the subsets S ( f , f ( X ) ) consisting of those y E Xsuch thatf(y) = = f ( x )are relevant. These sections are nothing other than the fower sections of the total preordering “y is preferred to x” if and only f ( y ) =sf(x), i.e. y E S(f , f ( x ) ) . It is clear that the sections S( f,f (x)) are invariant when f is replaced by p f where p is a (strictly) increasing function from R into itself.
-
Ch. 1, 0 1.21
11
DECOMPOSITION PRINCIPLE
* 1.2. Decomposition principle Let U and V be two vector spaces and let L belong to J ( U , V). We can associate with any function g defined on V the comFosition product gL defined on U. Symmetrically, we will associate with any function f defined on U the product (from the left) Lf defined on V by Lf ( y ) = infLXeyf (x). Such products yield ways of constructing new loss functions. For instance, the inf-convolution of functions can be constructed in this way. The identity inf
LX€Y
( f (4+g(Lx)) = inf (Lf(Y) + d Y ) ) Y€Y
is obvious but yields some useful general decomposition principles. We deduce for instance that (ML)f = M ( L f ) and that inf
Ax+By=s
[ f ( x )+g(y)l = inf [ A f(z- BY) +S(Y)l. Y
1.2.1. Product of a loss function by a linear operator
Let g : V F- 1 - OD, + -1 be a function defined on V. If L E &(U, V ) is a map from U into V, we shall denote by gL the function from U into 1- m, -1 defined by x +-g[L(x)]. It is clear that
+
Dom [gL] = L-l[Dom g ]
(1)
It will be convenient to refer to gL as the “product from the right” of g by the map L . We now define the “product from the left” L f of f by a map L as follows. Definition 1. Let L be a linear operator mapping U into V. Let f : U F+ ] - -, + -1 be a function defined on U . We shall associate with f and L the function L f : V -c 1- -, -1 defined by
+
(2)
VY E V ; Lf(Y) = inf
Lx=y
f(4.
This is called the “productfrom the left” off by L . Since the infimum over the empty set is + m, we shall set
(3)
Lf ( y ) = +-
if y
4 L Dom (A.
12
MINIMIZATION PROBLEMS A N D CONVEXITY
[Ch. 1, 0 1.2
Then Dom Lf = L Dom f:
(4)
Remark. We can of course define Lf even for non-linear operators but all the examples we shall study involve only linear operators and so we consider only this case. 1.2.2. Example: Inf-convolution of functions
+
Cdnsider n functionsA mapping U into 1- w, -1. We associate with these functions a function f : U"k-+1- -, -1 defined byf(x) =fl(x1)+fi(x2)+ +fnoand we let the operator L be the operator I: from U"into U defined by 2%= c(;E~x'. Then
---
+
* Definition 2. We shall say that the function Xf defined by (5) is the inf-convolution of the functionsf,. We shall set (6)
g=flOf20
... 0 f n
=
l3.L
I=1
Notice that we can write the inf-convolution of two functionsf and g in the, following way
( f 0g) (4 = y +inf ( f (y)+g(z)) z=x = inf (f(Y)+g(x-Y)) YEU
= inf [ f ( x - z)+g(z)]. ZEV
Examples. Consider the case when
U = U1XU2, L = LI+L2 where Li E 2 ( U i , V) (i = 1,2). Then (7)
( ( & + L d f ) (Y) =
inf
LlXl +Ltxz=Y
f(x1, X Z ) .
In particular, consider the case U1 = Uz = V , L1 = 1 and La =
-
1.
Ch. 1, 9 1.21
DECOMPOSITION PRINCIPLE
13
Then ( G + L d f ) ( y ) = inff(x,y+x).
(8)
X€
v
Taking
v
= Ul, L1 = 1, Lz = 0,
we say that
(9) is the “projection off onto U1.” Interpretation. We have already met the product from the right f M when M is a service operator.
Consider the case when L E &(U, V) is a resource operator from U into a resource space V . Iff is a loss function associating with any strategy x E U its loss f ( x ) , the loss function Lfassociates with any resource y E V the smallest possible loss which can result from the implementation of a strategy x available for the resource y (i.e. Lx = y). The inf-convolution can be regarded in the following way. We suppose that n players i share a resource y, i.e. choose strategies xi such that E=lx’ = y. The inf-convolution Uy=lh associateswith any y the smallest total loss occurred during the sharing.
I .2.3. Decomposition principle Any minimization problem of the form (PI:
inf (f(x)+g(Lx))
LX€Y
can be decomposed into two minimization problems, i.e. and
(Q): inf (LS(y)+gW) Y€Y (RCV)) : inf f ( x ) Lx-y
for a convenient y.
14
MINIMIZATION PROBLEMS AND CONVEXITY
[Ch. 1, 5 1.2
holds and the two following statements are equivalent: (1 1)
1
(a) 2 minimizes f ( x ) + g ( L x ) on the set X = L-I(Y) (b) j = LX minimizes Lf ( y ) + g ( y ) on Y and X minimizes f ( x ) on the set L-l(jj).
Proof. Since the set X = { x E U such that Lx E Y } is the union of the sets L-'(y) = { x E U such that Lx = y } as y ranges over Y,we deduce that Lx=y
It is clear that, if2 E X minimizes f(x)+g(Lx) on X , then p = LX E Y minimizes Lf ( y ) + g ( y ) over Y and f (?) = ( L f ) (7). Conversely, suppose that j j minimizes L f ( y ) + g ( y ) over Y and that X minimizes f ( x ) over L-l(jj). Then 2 minimizes f(x)+g(Lx) on X . To see this, observe that f(Z)+g(LX) = L.(?)+g(J) 6L f ( L x ) + g ( W e f ( x ) + g ( L x )
when x E X . 0 From this general principle we deduce the following consequence. Proposition 2. (12)
( M - L )(f)= M * [ L f l
Furthermore, the two following statements are equivalent: (1 3; a)
X minimizesf under the constraint MLx = z
and
(13; b)
(i) j minimizes Lf under the constraint M y = z, (ii) X minimizesf under the constraint rX = J .
Proof. The result follows from Proposition 1 with g = 0 and Y = M - 4 . 0 In particular, consider the problem
(PI:
inf
Z'- 1 L,(x')=Y
n
1f;:(xi).
i= 1
Ch. 1, 5 1.21
15
DECOMPOSITION PRINCIPLE
We shall decompose this problem into a “centralized problem” for the allocation of y ”
and n “decentralized problems” (Ri(y’)):
inf .fi(xi).
L,Xt=y(
Proposition 3.
The two following statements are equivalent (15; a)
and (16; b)
X = (9, . . .,2”)minimizes Ef=lJ ( x i ) under the constraints
ELlL’X’ = y
y’ = Liz’ minimizes E7=l (Lcfi)(y’)
ELl
under the constraints yi = y and, for any i = 1, 2‘ minimizesf,(x) under the constraints Lixi = 7.
. . ., n,
Interpretation. The above proposition can be interpreted in the following way. under the We seek strategies x’ which minimize the loss function z=lf;(x‘) constraint that the sum of the resources L,(X’)needed to implement x’ is equal to the scarce resourc :y. This problem is equivalent to first, allocating the resource y = J’ among the n players by mini(L&) (y‘) mizing the loss function and then, each player implementing the strategy X iwhich minimizes his loss function A over the set of strategies xi which can be implemented with the resource J’.
-
x;=l
ELl
1.2.4. Another decomposition principle
Let U,V and W be three vector spaces. Let A E 2(U,W ) , B E J ( V , W), and let f and g be two functions defined on U and .V respectively. ’Consider the problem (P) :
inf
AX+ B y = z
(f’(x>+g(y)).
16
MINIMIZATION PROBLEMS AND CONVEXITY
[Ch. 1, 0 1.2
This can be decomposed into the two following problems (Q): inf ( A D - BY)+g(Y)) Y
and
(R(y)):
inf
AXPZ-BY
f(4
Proposition 4. (17)
inf
Ax+By=r
( f ( x ) + g ( y ) )= inf Y
(Mz---BY)+~Y)).
The twofollowing statements are equivalent (18; a)
and (18; b)
{Z, y } minimizes (P)
(i) 7 minimizes (Q) (ii) X minimizes R(y3
Proof.
inf
Ax+By=r
( f ( x ) + g ( y ) )= inf
(Af (zd+Bg(zz>)
zl+~:==z
= inf ( A f ( z - 22)+Bg(zd) 2:
= inf (Af(z-By)+g(y)). Y
(BY (10))
It is clear that, if X andJ are solutions of the first problem, then Jis a solution of the second problem and 3 minimizes f under the constraint Ax = z-BJ. The converse is also obvious.13 Interpretation. We can regard U and V as the consumption space and the production space of a consumer and of a producer respectively. The consumer. and the producer have to share a (scarce) resource z by minimizing the sum of their losses f (x) and s(y)- This problem can be solved in two steps. In the first place, the producer chooses his production y to be optimal for a new loss function. This new loss function is the sum of his first loss function g(y) and of the loss Af @--By). The latter loss represents the smallest of the losses possible for the consumer when his available resource is z-By, i.e. that which remains once the producer has implemented y . Secondly, given that 7 is the actual production implemented, the consumer chooses x to minimize his loss under the constraint Ax = z-BY. In following this program, the producer and consumer achieve the optimal consumptionproduction pair.
Ch. 1, Q 1.31
MIXED STRATEGIES AND CONVEXITY
17
Remark. We can use several of the above decomposition principles successively. For instance, consider the following problem involving n consumers and m
We then have the problem:
n
This can be decomposed into the following problems :
n
and Si(zi):
inf fl(x').
A& =rC
In other words, the optimal solution {%Iy . . ., ZnYp1, . . .,of problem if'} (P) can be obtained by finding the optimal production {jP, . . .,J m } as a solution of problem (Q), then by allocating the rest of available resources in resources 51, . ., k as a solution of problem R ( 3 , . ..,7) and finally, by letting each consumer i choose (in a decentralized way) his optimal consumption by solving the problem Si(Zi).
.
1.3. Mixed strategies and convexity
ixed strategies via the following extension problem. Can we replace the minimization problem a = inf,,, f ( x ) by an "extended minimization f A ( m )in which a = di and X is embedded in M(X) problem" E = infmcM(m by an injective map 6 1 Since any optimal solution X of the initial minimization problem is mapped by 6 into an optimal solution 6% of the extended minimization problem, it is advantageous to extend a given minimization problem to a minimization problem which always has an optimal solution, especially if we can take M(X)to be convex and f " to be linear. This desirable result can be achieved by taking M(X)to be the set of discrete probability measures. In terms of game theory, these are called mixed strategies. Note that there are other extension possibilities which will be examined in more detail in the framework of two-person games (see Section 7.2). We shall only check in this chapter that, for finite strategy sets, the subset of mixed strategies is equal to the subset of infinite sequences of pure strategies. 4
18
MINIMIZATION PROBLEMS AND CONVEXITY
[Ch. 1,s 1.3
It is natural to devise sufficient conditions for the initial and the extended minimization problems to be equivalent. This can be done by associating with any map b :M(X)-, X , such that 6s = 1, the subset of functions f such that f(/3m)e f A ( m )for a11 m E ,/n(X). The main example is provided by the barycentric operator /3 when Xis a convex subset. The functions satisfying the above inequalities are the conveF functions. Other classes of functions satisfying analogous properties are introduced in Section 13.3. We end this section by recalling some elementary properties of convex subsets and functions. We mention here only the following property which will play a fundamental role throughout this book. Let f i , . ..,f, be n convex functions defined on a convex subset X and let F be the associared operatorfrom X into R“&@zed by F(x) = {f&), ...,h(x)}Then F(x)+R: is a convex subset of R“. 1.3.1. Motivation: extension of strategy sets and lossfunctions
It is often the case that an optimization problem with no solution can be converted into one with a solution by enlarging the strategy set. In formal terms, we propose to “enlarge” the strategy set X (by embedding it in a larger space M(X))and to “extend” the loss function f defined on X to a new loss function f ” defined on X(X)whenever there is no solution of the initial problem. A minimum requirement for such an extension is that the minimal and maximal values be invariant, i.e.
Indeed, this property obviously implies the following result.
-
&oposition 1. If 8 X M(X)is the injective map embedding X into M ( X ) and if property ( 1 ) is satisfid, then any minimal (resp.maximal) solution X E X o f f is mapped onto a minimal (resp. maximal) solution 8(5) o f f A .
The converse holds whenever the following property is satisfied by the loss function. (2)
There exists a map from M ( X )into X such that f(/3m)*fd(m) for any m E M(X).
We therefore obtain the following result.
Ch. 1, 0 1.31
19
MIXED STRATEGIES AND CONVEXITY
Proposition 2. If properties ( I ) and (2) are satisfid, then any minimal (resp. maximal) solution m 5 M(X)of f A is mapped onto a minimal (resp. maximal) solution Bm off
”.
Among all possible extensions satisfying properties (1) and (2), we single out the most important example in which (a) M(X)is the convex subset of discrete probability measures; (b) 6 is the Dirac operator; (c) f” is the linear extension off to JZ(X); (d) B is the barycentric operator; (e) functions satisfying property (2)are convex functions. 1.3.2. Mixed strategies and linearized lossfunctions Definition 1. Let X be a set and 8 ( X ) = RX be the vector space of real-valued functionsf defined on X (supplied with the topology of pointwise convergence). We shall denote by 6(x)the “Dirac measure”
(3)
w :f I--+ f(4
which is a (continuous) linear functional on &X). We shall call any linear combination (4)
m=
n
C ;p\l’S(xi):f I--+
i=l
(m,f ) =
C A’f(xi) I
i=1
a “discretemeasure” on X . We shall say that (5)
m = C AQ(xi)is “positive” if, for any i, ‘A I
0
i=1
and that (6)
m is a “discreteprobabilitymeasure” if m is positive and EEI 1’ = I .
We shall often use the word “probability” as a synonym for “probability measure”. We denote by
(7)
4.
II
S*(X)the vector space of discrete measures, S : ( X ) the convex cone of positive discrete measures, M(X) = &(X) the convex set of discrete probabilities, 6 :X M(X)the injective map associating with x E X the Dirac measure 6(x) E M(X).
-.
20
[Ch. 1,
MINIMIZATION PROBLEMS A N D CONVEXITY
5
1.3
In other words, the Dirac operator 6 is an embedding from a set X into a convex subset M(X). Following the terminology of game theory, we make the following definition. Definition 2. We shall say that theconvex subset M(X)of discrete probabilities is the set of “mixed strategies” on X . The map m C - (f,m) is a linear functional defined on S ( X ) and, in particular, on d(X>.
-.
Definition 3. Let f : X R be a loss function defined on a strategy set X. We shall define the “linearized loss” function f’ :8*(X)* R by (8)
vm
=
i=1
xi) E &*(XI,
fA(m)
= (m,f) =
c nif(xi). n
i=l
The minimum requirement (property (1)) is clearly fulfilled.
Proposition 3. Let f be a loss function defined on a strategy set X and f’ be the linearized loss function defined on the mixed strategy set A(x>. Then (1)
inf f ( x ) = inf f”(m); supf(x) = sup f A ( x ) .
X€X
mE./n(X)
XEX
)
’
W m EJ
Proof. It suffices to prove the first statement. Since 6 embeds X in M(X)and
f ( x ) = f”( a(-V)),
inf f d(m) =s inf f ( x ) . ~JI€Jn(rn X€X On the other hand, for any
we have
and thus,
inf f ( x ) =S inf f A ( m ) .
X€X
mE J n W )
Remark. In fact when Xis compact, we shall embed d ( X ) in the convex subset 2 ( X ) of Radon probabilities (which is compact for a suitable topology) and extendf(when continuous on X) to a functionf‘ which iscontinuous on -@(X). Since this construction involves several results which will be proved later, the construction is postponed until Section 3.1.6.
Gh. 1,
1.31
MIXED STRATEGIES AND CONVEXITY
21
1.3.3 Interpretation of mixed strategies
The usual interpretation of a mixed strategy is that of a “random strategy” or
“my’ m=
n
C Aid(xi)
i=l
where A’ is the probability of selecting the ith strategy. In this case, (m, f) = ?$(xi) is the “expected loss”. With the above interpretation, a “mixed strategy” would be implemented by a probability mechanism (for instance, toss of coin, roll of a die; “wheel of fortune”, table of random numbers). Instead, we shall use a more operational interpretation, by assuming that the game can be “repeated”. In a large number of repetitions of the game, the coefficients A’ represent the proportion of times that xi is expected to be played. Assume in the first place that the coefficients Ai = pi/p are rational. In other words, we assume that the game is repeated p times. In this case we can interpret the mixed strategy m= ( p i / p )d(xJ as the strategy X I played p l times, . . ., the strategy xi played pi times, . . ., the strategy ‘‘4, played ’’ pn times in a game repeated p times. Since any real number Ai can be approximated by rational numbers, we shall Ai8(xi) as the strategy X I played ‘‘A1 times” regard a mixed strategy m = ,...,the strategy xn played “A” times” in an infinitely repeated game. Now, iff is a loss function defined on X, and if m = Aid(xi)is a mixed Atf(x,)= (m,f)equal to “A1 times” strategy, we associate with m the loss the lossf(x1) plus . . . plus “Antimes” the lossf(x,). When A‘ = pi/q is a rational number, the loss A‘f(xi)= ( l / p ) x ; = l p i f ( x i )is nothing other than the average loss. = I
cy=l
c;=l
c;=,
cy=,
1.3.4. Case of finite strategy sets
Suppose that X = {I,
. . ., n )
-.
is a finite set of n elements. Since any function f E 8 ( X ) is defined by iE X f ( i ) E R, then 8 ( X ) = RX is isomoprhic to R” and the cone S + ( X ) to the positive orthant R;.
22
MINIMIZATION PROBLEMS AND CONVEXITY
[Ch. 1, 0 1.3
The space S'(X) of discrete measures m = EE1 A'6(xi) is isomorphic to the dual R"' of R".In this case
c n
f"(4 = (w-) = Mi) I-1
is the familiar duality pairing on R"*XR". Furthermore, the cone S:(X) is isomorphic to the positive orthant RT of R"' and the set M(X)is isomorphic to the probability simplex M nof R" defined by (9)
M n= {A
=
{A1, . . ., An} such that A'== 0 and CYel A[ = 1)
which is compact. In summary, when X = (1, . ., n}, we shall make the following identications (10) S ( X ) = R", S*(X) = Rn*, S:(X) = RY, A(X)= A?"
.
andf"(m) = ( m , f ) =
xx1 Afr, whenever
m =
A&i) andf = {A}.
* 1.3.5. Representation by infinite sequences of pure strategies We shall prove that any mixed strategy m E &(X) = M" of a finite set X
of n elements can be written
C a,6(xt) OD
(1 1)
m =
where xt 5 X and a -
t=l
In other words, any mixed strategy can be implemented by playing an infinite sequence {x,} of pure strategies and the expectation ( m , f ) = C E l a t f ( x t ) is a "generalized discounted" summation of the lossesf (xt) of pure strategies x,. Notice first of all that a, a 0 and that
We denote by XNthe set of all sequences of elements of X. Proposition 4. Let X be a f i i t e set of n elements and n: the map from XN into &(X) deJind by
c 00
(13)
ac{.t}t>
=
t=l
atqxt) E
Then the map a,' is surjective.
MW,.
Ch. 1, 9 1.31
23
MIXED STRATEGIES AND CONVEXITY
xzSl
Proof. Let m = ak6(k)be a probability on the strategy set X = (1, . . ., k, . . ., n}. We define'recursively a sequence of pure strategies x, and nonnegative measures m, =
zz1 ai6(x,) as follows. For t = 1, n
114)
Note that x1 is well-defined since ak 3 0,z & k = 1 and at least one of the is not smaller than l/n. Therefore, ml is a non-negative discrete measure such that n
k=l
n- 1 n
Similarly, if a positive measure m, satisfies n
(n- 1)'
k-1
nr
=pi=-
'
then at least one of the acsisnot smaller than (l/n) ((n- l)/n)' = a,+l and so x , + ~is well-defined. Thus m,,, is a non-negative measure satisfying n
C ui+l k=l
= nat+l-a,+l =
(n - l)t+l
t+l
*
Since we can write m1+l= m-(a16(xl)+az6(xz)+
. . . +afd(xI))
,
we deduce that
c;=l
Therefore as6(xs)=sm. Since both measures, they are equal. 0
zzl as6(xs) and m are probability
24
MINIMIZATION PROBLEMS AND CONVEXITY
[Ch. 1, Q 1.3
1.3.6.Linearized extension of maps and the barycentric operator
We can also extend a map F from X into a vector space U to a linear operator F mapping 8*(X)into U.
Definition 4. Let F be a map from X into a vector space U . We shall say that the linear operator FA mapping S*(X) into U defined by (16)
vm =
2 Aid(xi),
is1
FA(m)=
i=1
AiF(xi)
is the “linearized extension of F”. It is clear that this linear operator FA is the unique operator FA mapping S*(X)into U such that (17)
F = FA8
and that the map F -, FA is a one to one correspondence between the space &(X, U)of maps from X into U and the space 2 ( S ’ ( X ) , V )of linear operators mapping &‘(XI into U. In particular, we shall study the linearized extension of the canonical injection from a subset X of a vector space U into U. We assume now that (18)
X is a subset of a vector space U.
Definition 5. We shall say that the map @ from the space &*(X) of discrete measures on X into U defined by (19)
ym =
A‘d(xi) E k l
s*(x), bm
Aixi E
= i= I
u
is the “barycentric operator”. In other words, /3 is the linearized extension id of the canonical injection i from X into U. The following proposition, together with propositions 1 and 2, explain the importance of convex subsets and convex functions in optimization theory. Proposition 5 . Let X be a subset of a vector space U. Then the “convex huil” co ( X ) o j X is equal to the image under /Iof the set of mixed strategies, i.e. (20)
B(J4X)) =
(X).
Ch. 1, 5 1.31
25
MIXED STRATEGIES AND CONVEXITY
Thus
(21)
Xis “convex” if and only i f B maps M ( X )into X .
A function f deJined on a convex subset X is “convex” i f and only i f
(22)
V m E M(X),
f (Bm)
and f is ajine if and only i f
(23)
-
V m E M ( x ) , .f(Bm) = f ”(4.
A function f : U
(24)
(m,f) = f’(m)
- 1-
00,
+ -1
is “convex” i f and only if
V m = ZAiS(xi) E M(U), f (pm) =sf ”(m)=
A function f : U
--c
[-
00,
+ -[
-
n
C A’f(xi).
i=1
is “concave” i f -f is convex.
It is clear that the domain Dom f of a convex function is convex. Also, if X c U is convex and f :X R is convex, its extension fx : U -,1- -, + -1 is convex. 1.3.7. Interpretation of convex functions in terms of risk aversion
We regard (25)
I
(i) X as a strategy set, (ii) M(X)as the set of mixed strategies m = CZl I’ &xi) (iii) t!?m = Aixi as the expected value of the mixed strategy m.
xy=l
We consider the following equivalence relation. Two mixed strategies ml and m2 are equivalent if and only if they yield the same expected value pml = Bm2. W e call Brn the “certainty equivalent” of the mixed strategy m. In this framework, a player with a convex loss function f is said to be “risk averse”, in the sense that he prefers the certainty equivalent Pm of a mixed stategy m to m itself becausef (pm)=s (m, f) 1.3.8. Elementary properties of convex subsets and functions
Proposition 6. The image and the pre-image of any convex subset X by a linear operator L are convex. The intersection of any family of convex subsets is convex. Any product of convex subsets is convex.
26
MINIMIZATION PROBLEMS AND CONVEXITY
[Ch. 1,
1.3
If X1 and Xz are convex subsets, the set alX1+wJ2 = (
~ I X ~ + ~ ~ ~ X ~ } ~ ~ E X , , ~ ~ E X ,
is also convex. Any linear subspace of a vector space is convex.
Proof. This is left as an exercise. 0 Definition 6. A subset X of a vector space U is called a cone if AX c X for any 1 a 0. A convex cone is a cone which is convex. Proposition 7. A subset X is a convex cone if and only if ,(26)
BJXX>= x
i.e. if and only ifs m =
zf=lAixi
c:=la's(x,)E s : ~ . '
=
E X for all positive discrete measures m =
Proof. This is left as an exercise. 0
Next we recall some of the main properties of convex functions.
Propition 8. A function f : U -, ]- a, + -1 is convex if and only i f its epigraph Cp(f ) is convex. The pointwise supremum of convex functions is convex. The lower sections and the lower open. sections of a convex function are convex. Proof. This is left as an exercise. 0
Proposition 9. The products from the right g L and from the left Lf of convex functions f and g by a linear map L are convex. Proof. The first statement is obvious. We prove that Lf is convex by estimat(1 - 4 ~ 2 ) . ing Lf (h+ We assume that both L , ( y l ) and L f ( y z ) are finite (if not, the barycentric inequality is trivial). This implies that we can associate with any E =- 0 elements x1 and x2 such that
Lxk = y k and f ( x k ) 4 L f ( y k ) f & (k = 1,2). Therefore x = Axl+ (1 - 1)x2satisfies the constraint Lx = 1y1+ ( 1 - A)y2.
Ch. 1,s 1.41 Hence
27
INDICATORS, SUPPORT FUNCTIONS AND GAUGES
+
+ +
Lf (AYl (1 - 4 y 2 ) =s. f ( x ) = f (i1x (1 - f ) x z ) (x1) (1 - 4 f ( x 2 )=sam) (1 - I)Lf(Y2)
Y +
+
8.
Letting E tend to 0, we obtain the inequality
Lf( h + (1 -4Y2) -c ALf (Yl)
+ 0,-1)Lf
(Y2).
0
Delinition 7. Let X be a convex subset of a vector space U and let
F :x E X t+ F(x) = { f i ( ~ ). ,.., f ( x ) } E R" be a map from X into R". We say that F is a "convex operator" if and only if its componentsf, :X -, R are convex. Proposition 10. Let X be a convex subset of U and let F :X -,R" be a convex operator. Then
(27)
F(X)+RI is a convex subset of R".
cr=l
Proof. Let c = ak(F(Xk)+hk)E co ( F ( X ) + R t ) be a convex combination of elements of F(X)+R:. Consider x = E,$ E X (which is convex) and write
c = F(x)+d
cr=l
whered =
c akbk+ m
m
k=l
k=l
akF(Xk)-F(X).
The components d/ of d are non-negative since
and the bk's belong to R: while the functions4 are convex. 0 1.4. Indicators, support functions and gauges The aim of this section is to associate with any closed convex subset X containing 0 some convex lower semi-continuous functions which characterize it. We first introduce the indicator yx of X defined by y x ( X ) = 0 if x E X and yx(x) =+ m if x 8: X . The domain of yx is X . The second function is the support function a$ defined on the dual U Lof U by
28
MINIMIZATION PROBLEMS A N D CONVEXITY
[Ch. 1,s 1.4
We associate with it the polar subset X* of X defined by X+ = ( P E U*such that &(p)
=sl}.
We shall see in the second part of this section that the Hahn-Banach theorem (see Appendix A) amounts to saying that X = (x E U such that (p, x)
4
u$(p) for all p E V } .
Wecontinue by proving the bipolar theorem, which states that X*# = = co (XU(0)). In particular, we obtain the following generalization of the Farkas lemma: If X is a closed convex subset containing 0 and i f L ( X )is closed, then L ( X ) = [L*-l(X")]*. We pursue this point in the fourth part. We associate with Xits recession cone P J X ) = U a n d its barrier cone P(X*)= = AX* and prove that P,(X) = P(X")". We also show that P ( X x ) is closed when P,(X) has a non-empty interior. In the fifth part, we interpret the above results when X is regarded as a production set. We state the main assumptions usually made and note that the indicator is the net cost function and that the support function is the profit function. We end this section by introducing the gauge szx(x) of X defined by
nAao
UnBo
m ( x ) = inf (1 > 0 such that 2-l x
E X}.
This is a third function characterizing X: We have that
X
= {x
E U such that nx(x) =S l}.
1.4.1. Indicators and support functions
Definition 1. Let X be a subset of U. We shall say that the function y x = = y(X; from U into 10, + -1 defined by 0 )
is the "indicator" of the sdbset X . The following proposition is obvious. Proposition 1. The indicator of X is convex (resp. lower semicontinubus) whenever X is convex (resp. closed). It satisjes (2)
y ( x ; x)+Y(Y;
= y(xny ; XI.
Ch. 1,
8
1.41
INDICATORS, SUPPORT FUNCTIONS AND GAUGES
29
Definition 2. Let U and U’ be two paired spaces and let X be a non-empty subset of U.With any p E U*, we shall associate (3)
a$(p) = a#(X; p ) = sup ( p , x) E ] - -, X€X
+ -1
and (4)
a&(p)= a b ( X ; p )= inf ( p , x) E [*EX
00,
+ -[.
-
We shall say that the functions p !-+ a#(X; p ) and p ab(X;p ) are respectively the “upper and lower support functions” of X . The upper support function 0 : is clearly lower semi-continuous, convex and positively homogeneous, since it is the pointwise supremum of continuous linear functioh. It is non-negative i f 0 E X . It is related to the indicator of X by the fundamental relation (5)
a#(X; P ) = SUP [(P,x)-Y)(X; X € U
41.
Proposition 2. The upper and lower support functions are related by (6)
o#(X;p ) =-aO”(X; - p )
The upper support function satisfis the following properties. If1 and p are positive, then (7)
o#(AX+pY;p) = Aa’(X; p)+pa#(Y;p).
Also, (8)
u q x - Y ;p ) = aqx;p)-ab(Y; p).
I f X c Y , then
(9)
a#(X, PI
u#(Y,PI.
If { X I } I Eis, a family of subsets X,, then
Proof. The first statements are obvious. We prove (17). Since Xic X = X,, a#(Xi;p ) =sa*(X; p ) and thus,
UiEr
sup oyxi;p ) e a#(X; p ) i€I
Conversely, let x E X =
uicr Xi.
Then x E Xi for some i E I.
30
MINIMIZATION PROBLEMS AND CONVEXITY
[Ch. l , § 1.4
Defdtion 3. We shall say that the closed convex subsets of U defined (11)
{ (ii)( i )
X’ = { p E U*such that d ( X ; p ) =S l } Xb = {p E U*such that d ( X ; p ) a- l }
are the ‘bpper and lower polar subsets” of X respectively and that (12)
P(X#) =
E U*such that a#(X;p) <
+ -}
= Dom a;
is the “barrier cone” of X . Definition 4. Let X be a subset of U.We shall say that (13)
{ (ii)(i) X-
X+ = Cp E U*such that ( p , x ) 0 for all x E X}, = { p E U*such that (p, x ) =s0 for all x E X }
are the “positive and negative polar cones of X and that the closed subspace (14)
X’- = { p E U’ such that (p, x ) = 0 for all x E X )
is the “orthogonal complement of x” or Lrannihilatorof X”. The following relations are trivial:
It is also clear that, if X c Y, then (16)
Y * c X # , Y + c X + and Y 1 c X . L .
Proposition 3. Suppose that both X and Y contain 0. Then (17)
(X+Y)+ = x + ~ Y + .
Proof. Let p E X + nY + and z = x+y E X + Y , where x E X and y E Y . Then (p, x+y) = (p, x)+(p, y ) 0 i-e., P E (X+Y)+. 0 whenever x E X and Y E Y. Conversely, if p E (X+Y)+, then @, x+y) By taking y = 0, we deduce that p E X+ and by taking x = 0, that p E Y+. Thusp E X + n Y + . 0 Remark. If X is a cone, then clearly X- = X” and if X is a vector space, then X I = X - = X’
Ch. 1, 0 1.41
INDICATORS, SUPPORT FUNCTIONS AND GAUGES
31
Proposition 4. Suppose that X is a convex coHe in U. Then
-
Proof. In the first place, .#(A', p ) = supxcx(p, x) 0 since 0 belongs to X . If p E X - , then ( p , x) 0 for any x E X and thus, o#(X,p) =s 0. Therefore, a#(X, p ) = 0 whenever p E X-. If p $ X-, then there exists x E X such that ( p , x) = 8 > 0. Since X is a cone, Ax E X for any il w 0. Hence
a#(X, p ) z=sup ( p , Ax) = sup 18 = + m. 1-0
1-0
Proposition 5. Suppose that X is a convex cone and that Y is any subset. Then
and
I .4.2. Reformulation of the Hahn-Banach theorem Since p is a linear form, supxEx( p , x) = S U ~ , ~ , , ( ~ () p , x). Since p is continuous, supxEx( p , x) = supxcp( p , x), where denotes the closure of X . Therefore, (21)
ayx;p ) = a#(co ( X ) ;p )
where CO (X) denotes the closed convex hull o f X . The Hahn-Banach separation theorem amounts to characterizing closed convex subsets by their support functions.
Theorem 1. Let X be a subset of a Hausdog locally convex space U..Then (22)
G ( X )=
Proof. Denote by M =
(23)
{x E U such that ( p , x) {x E U such that ( p , x)
ope
for all p E U*}, ab(X;p ) for all p E U*}.
=s a#(X; p )
Kp the intersection of the subsets
Kp = {x E U such that ( p , x)
=so+(X,$)}.
32
MINIMIZATION PROBLEMS AND CONVEXITY
[Ch. 1, 0 1.4
We notice that the subsets Kp are closed and convex. Therefore, M is also a closed convex subset. Since X is obviously contained in K, for each p , we deduce that X ' c M . Thus, CO (X) c M. Assume that CO (X) f M, i.e. that there exists x E M which does not belong to CO (X). The Hahn-Banach separation theorem states that there exist a non-zero p E U' and E w 0 such that VY E 65 ( X ) , (P,Y>-s ( P , x)- c: This implies that aX(X;p) =s ( p , x)- E =s a#(X; p ) tion. (24)
E
which is a contradic-
Remark. Closed convex subsets in infnite dimensional spaces. Let U be a Hausdorff locally convex space and let U' = &U, R) be its dual. We know that U' is a1 o isomorphic to the space 2(U,,R) when U is supplied with the weakened topology a(U, U') (see Appendix A, Theorem 6). In other words, the space of continuous linear forms on U coincides with the space of weakly-continuous linear forms on U. Theorem 1 implies the following proposition. Proposition6. Let U be a Huusdorfl locally convex space. A convex subset X of U is closed if and only if it is weakly closed.
Now consider the case when U and U ' are paired spaces. The Mackey theorem states that U' is isomorphic to the space 2(U,R) of continuous linear forms whenever U is supplied with the Mackey topology z(U, U') (see Appendix A, Theorem 9). Theorem 1 implies the following result.
Proposition 7. Let U and U ' be two paired vector spaces. A convex subset X of U is closed in the Mackey topology ifand only if it is weakly closed.
' 1.4.3. The bipolar theorem We now prove another characterization of closed convex subsets. Theorem 2. Let X be a subset of a H a u s h f l locally convex space U. The bipoIar X # # of X is the closed convex hull of X u (0). The bipolar cone X + + of X is the closed convex cone spanned by X . The biorthogonal X I of X is the closed vector space spanned by X . Proof. Since X1 c X - c X", (see (15)), we deduce that X " c (X-)# = = X - - c (XI)" = XIL.On the other hand, X U (0) is clearly contained in X"#.
Ch. 1 , s 1.41
INDICATORS, SUPPORT FUNCTIONS AND GAUGES
33
(a) We have that CO (XU (0)) is contained in the closed convex subset X"+. We shall assume that there exists x E X#*which does not belong to CO (XU(0)) and obtain a contradiction. By the Hahn-Banach separation theorem, there exists p E U*,p # 0, and E > 0 such that (25)
(p, r> (p, 4 - 2 E
vy E
=(XU (0)).
On taking y = 0, this implies that (p, x ) 2~ > E. Put q' = p / ( ( p , x)- E). We deduce from (25) that
Therefore, taking the supremum over X,we deduce that sup.@, j ) -G 1, i.e. q E X+.
Y€X
since x f X", (q, x ) as 1. But (q, x ) = ( p , x ) ( p , x)- E ) > 1 by the very debition of q. This is the required contradiction. (b) The closed convex cone P spanned by X i s contained in X--.We assume that there exists x E X-- which does not belong to P and obtain a contradiction. By the Hahn-Banach theorem, there exists p E U*,p # 0 and E z 0 such
(27)
( p , y ) -& (p, 4
- 8
VY
E p.
Taking y = 0, we deduce that 0 < E 6 (p, x). On the other hand, supycp(p, y ) = = 0 since it is bounded above (see Proposition 4). Thus p E X- and (p, x ) 6 0 because x E X--.This is the required contradiction. (c) We prove the last statement of the theorem as in (b) above except that P is taken to be the closed subspace spanned by X and X-- is replaced by by XLL.0
As a consequence, we have the following result. Proposition 8. Let L belong to 2(U,V) and let X be a convex subset of U containing 0. Then (28)
L(X)* = L*-1(X#).
Furthermore, ifL ( X) is a closed convex subset containing 0, we obtain that
(29) 5
L(X)= [(L'-'(X+)]*.
34
MINIMIZATION PROBLEMS AND CONVEXITY
[Ch. 1,s 1.4
Proof. To say thatp belongs toL(X)* amounts to saying that V X E X, (p, Lx) = (L'p, x ) G 1,
i.e. that L'p belongs to X#, or that p belongs to L'-l(X*). This implies that [L*-'(X*)]" = L(X)"* = L(x).0
Remark. We can restate eq. (29) by saying that there exists x E X such that Lx = y if and oniy if (30) (p, y ) =s 1 whenever L'p E X*. t
1!44 Recexxbn coaes mrdbmrkr conex
Definition 5. LetX be a subset of U containing 0. We shall say that
1
(31)
'
(i) P ( X ) =
UIBo AX is the ''cone spanned by X",
nlPO
(ii) P&) = X is the "recession cone" of X , (iii) P(X*) = UAB0 AX# is the "barrier cone" of X.
Proposition 9. Suppose that X is a closed convex subset such that 0 E X. Thmt (32) P,(X) = P(x*)-.
-
Proof. Let x E P,(X) and p f P(X#). Since Ax E X for all I Z- 0, we deduce that A(p, x) -s o'(X, p ) -= for any A =- 0. This implies that (p, x ) =s0. Conversely, let x E P(X")-. We prove that Ax E X for any il =- 0. Since P ( X x ) is a cdne, then Ax E P(X*)- for any A > 0. Thus Ax E X*# since
+
P(X#)- c (X")- c X # # . Since 0 E X and X is closed, X*'
= X. Thus, ilx
E X for any A =- 0.
The converse is true under additional assumptions. We need that U is either a finite dimensional space or a topological v s t o r space supplied with the
Mackey topology z(V, U*). For this purpose, we use the bipolar theorem:
Theorem 3. Suppose that X is a closed convex set containing 0 and that the interior of its recession cone is non-empty (in the Mackey topology). Then its barrier cone P(X") is closed and (33) P(X*) = P,(X)-. Finally, P ( X x ) is spanned by the compact subsets P = (p E P(X+)such that (p, 2 ) =- 1) (34) where Z
E
Int P.-(X)-.
Ch. 1, $ 1.41
INDICATORS, SUPPORT FUNCTIONS A&D GAUGES
35
Proof. Suppose that I belongs to the interior of the recession cone P J X ) . Then there exists a neighborhood K* of a such that Z+KX c P&). Such a neighborhood is the polar subset of a ball of positive radius K(which is compact) when U is a finite dimensional space or the polar subset of a weak compact convex subset containing 0 when U is supplied with the Mackey topology. Firstly, we prove that
(35)
( p , 2 ) < 0 whenever p E P(X*), p # 0.
If not, there exists that
PO E
P(X*), PO f 0 such that @o, I) = 0. This implies
sup (PO,X)<
xEK+
sup ( P 0 , X ) ~ O
X€P,(X)
since P J X ) = P(X*)-. But this is impossible because supXcKx(PO,x) > 0. Secondly, we check that P defined by (34) spans P(X*).I f p E P(X*),( p , I) =
= -I where I w 0 and thus, p = Aji where ji = p / ( p , 2 ) belongs to P.
Thirdly, we prove that P is weakly compact. For this purpose, we notice that the inclusion 2+ K* c P,(X) implies that, ifp E P,then
- 1 + ~ * ( K * ; p ) = ~ * ( z + K # ; pS) U # ( P , ( X ) ; ~ )G O since ( p , 2) = - 1 and p E P J X ) = P(X,)-. Thus o*(K*; p ) =s 1, i.e. p E K** = K by the bipolar Theorem 2 since K is a (weakly) compact convex subset. Finally, we prove that P ( X * ) is closed, Let {p,,} be a generalized sequence of elements p,, E P ( X * ) converging top. We prove that p belongs to R(X*). Either I p = 0 E P(X*) or p f 0. In this case, we can suppose that p,, # 0 for all p. Then q,, = -p,,/(p,,, 2 ) belongs to the compact subset P. Hence a subsequence q, converges to an element q E P. On the other hand, the subsequence (pv,3) converges to ( p , I)=s 0. Therefore the subsequence p y = (p,,, 2)q, converges to p =- (p, 2) q which belongs to P(X#). 0 1.4.5. Interpretation :production sets and projt functions
Let U be a commodity space. We shall represent each particular realization of the pcoduction of a firm by a pair {x, y } consisting of an output vector x E U and an input vector y E U. Such a pair {x, y } is called a production process. Thus the “production technology” set T of a given firm is the subset of UX U consisting o f all possible production processes. If T is a given production 5.
36
MNIMIZATION PROBLFMS AND CONVEXITY
[Ch. 1, 0 1.4
set, we define the “productioncorrespondence” which associates with any output
x € W the subset
(36)
L(x) =
E U such that {x, y} E T}
of inputs which can be used to produce x. Conversely, we can describe a firm by a production correspondence L mapping U into itself and define T to be the set of pairs {x, y } where y € L(x) and x ranges over U (Tis said to be the graph of the correspondence). (We have already introduced linear operators mapping U into itself to describe a firm (see Section 1.1.3)). Actually, we will use only the image Z of T under the map (x, y ) I-- x- y to describe a ikm. We shall say that Z is the production set of the firm.When ’v = R’, we agree that, if the h* component z h of z is positive, good h is an output, if z h = 0, good h is not used in the production process and i f z h < 0, good h is used as an input. The @st assumption we usuaIIy make is that (37)
O E z,
ie. it is always possible for the firm to engage in no production. The second assumption which is usually made is that
(38)
Z is closed.
This amounts to assuming that, if a given production z is arbitrarily close to possible productions, then the production Z is also possible. The third assumption we will often need is that (39)
2 is convex.
This is a much more restrictive assumption. It is related to the more elementary assumption of additivity and decreasing returns to scale. Additivity means that, whenever z1 and z2 belong to Z (i.e. are possible), P+zZalso belongs to Z. We say that Z exhibits descreasing (resp. constant) returns to scale if Az E Z whenever z E 2 and 1 f 10, I[ (resp. 1 > 0). The assumption of additivity is
inevitable. Decreasing return to scale is not always realistic, since activities are not necessarily divisible even if we assume that the commodities are divisible. Note that additivity and decremkg return to scale imp& that Z is a convex cone (and thus, that 2 exhibits constant returns to scale). Thef w t h assumption which is often required is that
(40)
-R: is the recession cone of 2.
Ch 1, 0 1.41
INDICATORS, SUPPORT FUNCTIONS AND GAUGES
37
This implies in particular that Z-R: = Z i.e. Z “&its free disposal”. This means that, if the production z is possible, then the net production of smaller quantities z ’ e z is also possible. This can be done by disposing of (or throwing away) some commodities. This assumption implies that (41)
Int z = i = Z -
Pi
and that the barrier cone of Z is the cone RT of non-negative prices p (see Theorem 3). Two other assumptions are sometimes made. The first of these is that (42)
z n R ! + = (0)
which means that it is impossible to produce positive amounts of all goods without inputs (the “land of Cockaigne” does not exist). Finally, the property (43)
z n -2
= (0)
represents the irreversibility of the production possibilities,i.e. the impossibility of producing the original input as output by consuming the original output as input. (Note that one important source of irreversibilityis the existence of labor or other indispensable factor of production). Net cost function. - We shall interpret the indicator yz of Z as the “net cost function” of the production set X. The cost yz(z) of z is either 0 if z is possible or infinite if z is impossible to produce. Projt function. -Since we interpreted the indicator yz as a loss function, we shall say that its support function uz is the projt function, associating with any price p E R’* the maximum profit
(44)
U R P ) = SUP (P, 2) ZEZ
which can be obtained when the price p prevails. The profit function is lower semi-continuous, positively homogeneous and convex. If 0 E 2, it is non-negative. If we assume that 2 is convex and closed, then the production set Z can be described by budgetary constraints, i.e. (45)
Z = (z E R’ such that ( p , z) -= uz(p) for all p E R‘*}
i.e. a net production z is possible if and only if, for each price p, its value is not larger than the maximum profit.
38
[Ch. 1,
MINIMZATION PROBLEMS AND CONVEXlTY
6
1.4
Remark. Suppose that Z is the image of the graph T of a production correspondence L. If T is closed andconvex$ ischaracterized by its support function a$ defined by
(46)
4qPY
v)l
q) = SUP [(P,x)+(q, ~ Y ) E T
where P, q E
u*.
The profit function is equal to
We shall say that
is the cost function of x, i.e. the smallest value of the inputs y which can be used to produce x.
Therefore, we can write
’1.4.6. Gauges Definition 6. Let X be a convex sui>set of a vector space U.We assume that 0E
x.write
(50)
and (51)
nx(x) = n ( X ; x ) = inf
{ (ii)(i) Bx(A) bx(A)
{A
c=
0 such that I-lx EX} E [0, -1
U such that n ( X , x) =sA}, = {X E U such that n ( X ; x) -= A}. = {x E
We shall say that nx = z ( X , .) : U
-
+ -1
[0,
is the “gauge of X”.
Proposition 10. Let X be a convex subset such that 0 E X. The gauge z(X, .) satisfies the following properties. (52) rf
(X) { (ii)(i) PPm(X)
U such that z ( X ; x ) < + -}, = { x E U such that x ( X ; x) = O}.
= {x E
U is a topological vector space, then
(53)
Int (x> c &I) c x c B ~ ( I c ) X
Ch. 1, p 1.41
INDICATORS, SUPPORT FUNCTIONS AM) GAUGES
39
and (54)
Ax) { (ii)(i)nn((XX;;x+y)
= I n ( X ;x) , =sn ( X ; x)+n(X; y).
Moreover, the subsets Bx(1) and BX(1) are convex. Therefore, if X is a closed convex subset containing 0, we can represent it in the form (55)
X = { x E U such that z(X;x ) =s 1).
Proof. Consider the subset
Z(x) = {I =- 0 such that A-1 x E X )
+
+
I f x 6 P ( X ) , then Z(X) = { -} and z ( X , x) = -. If x E P ( X ) , then Z(x) is a half-line because, if I E Z(x) and p p E Z(x) since p-fX
Thus
=- 1, then
= (p-lI)I-% = (1 - p-lI)O+(p-1A)1-1x EX.
n ( X ; x) = inf A. aEI ( X )
(a) To say that n ( X ; x ) + 00 amounts to saying that there exists A z.- 0 such that A-lx E X , i.e. x E AX c P ( X ) . (b) To say that ~ ( p x ) ,= 0 amounts to saying that for all I =- 0, I-lx E X , i.e. x E AX for all A =- 0. (c) Suppose that Int (X) = k f 0 and choose x E 2. There exists q 3, 0 such that x+qx = (l/(l+q)-l)x belongs to X . Thus 3t(X; x ) =s l/(l+q) -= 1, i.e. x E Bx(l). (d) Let x belong to kx(l). Then a(% x) -C 1 and thus, 1 E Z(x). Therefore x E x. (e) If x E X , then 1 E Z(x) and thus, $(X; x ) =s 1, i.e. x E Bx(l). ( f ) If x E Bx(l), then a ( X ; x) 6 1 and thus, A-lx E X for all A =- 1. Since x = limA+l A-lx, we deduce that x belongs to the closure X of 1. (g) Since Z(8x) = 8Z(x),we deduce that n ( X ; Ox) = Bn(x;X ) for all 8 > 0.
(h) We prove that z ( X ; x+y) e z ( X ;x)+a(X; y). If either x or y does not belong to P ( X ) , then the right-hand side is infinite and the inequality holds. Suppose that both x and y belong to P ( X ) . For all E > 0, there exist A 0 and p =- Osuch that x E AXandy E pXand I = s a ( X ;x)+d E, p ==z(X; y)++ E .
40
MINIMIZATION PROBLEMS AND COMrEXITY
[Ch. 1,s 1.4
Therefore, x+y E (A+p)Xand thus, st(x;x+y) = ~ A + p - s a ( X ; x ) + l t ( X ; y ) + +2+ E for all E z 0. (i) Finally, the gauge a(X, *) is convex and thus the subsets B d l ) = = S(a(X; 1) and fix(l) = S(a(X; *), 1) are convex. 0 a),
Interpretation. Productiofifunctions. Suppose that the production set Z satisfies
Z is closed, convex and contains 0.
(56)
Then its gauge az defines a “production function”f z = a, - 1 in the sense that Z = {Z E R’Such that f&)
(57)
= ~ Z ( Z ) -1 4 0).
Productions z such that fz(z) = 0 (or zz(z) = 1) are called “efficient”. The formula
a,(z) = inf {A
(58)
Z-
0 such that z E AZ}.
has an obvious interpretation. The number 3tz(z) is the smallest scalar A by which the production set 2 must be “increased” in order to produce z. 0
We also show that certain properties of convex subsets induce properties on their gauges. Proposition 11. Let X be a convex subset such that 0 E X. If X is symmetric, then a(X,x ) = a(X, - x). If a(X,x) = a(X; - x ) for all x, then BA1)is symmetric. If X is closed, then a(X; is lower semi-continuous. If s ( X , is lower semi-continum9then Bx(l) is closed. If0 E 2,then a(% is uniformly continuous and there exists a semi-norm P and a constant M such that a)
0 )
0 )
(59)
In(% x)-a(X; y)l
6
MP(x--y) for all x, y .
Proof. If X is symmetric, then Z(x) = I(-x) since, if A-lx E X,then A-l( -x ) E E -X=X. Thus a(X,- x)= a(X, x). If X is closed, then X = Bx(l) and thus the sections BAA) = S(a(X; A ) are closed. This amounts to saying that a),
71(x; *) is lower semi-continuous. If 0 belongs to the interior of X,there exists a semi-norm P for the topology of U and a constant M > O such that P(z)/M E X for all z such that P(z) > 0. Thus
Ch. 1, Q 1.41
INDICATORS, SUPPORT FUNCTIONS AND GAUGES
If x and y are given and if&--y)
Z-
41
0, then
Proposition 12. Let X be a closed convex subset such that 0 E X . Then
(60)
vx
E u, v p E U', ( p , x ) 4 4 X ; X ) B Y X , p ) .
Proof. This inequality holds trivially if either x 4 P(X)or p 4 P ( X * ) , since the right-hand side is infinite. Assume therefore that x E P ( X ) and p E P(X"). Ifn(X; x) Z- 0, then x/n(P,x ) E X . Thus ( p , x / z ( X ; x)) -G o # ( X ; p ) and (60) holds. Ifn(X;x) = 0, then ( p , x ) =s0 for all p because, if there exists p E U* such that ( p , x ) = 6 =- 0, then Ax E X for all A =- 0 (since z ( X ; Ax) = Az(X, x) = 0). Thus w#(X, p ) 3 SUP^,^ [ ( p , Ax)] = sup,,, A0 = + This contradicts the assumption that a"(X; p ) -=
+
00.
00.
CHAPTER 2
EXISTENCE, UNIQUENESS AND STABILITY OF OPTIMAL SOLUTIONS
This chapter deals with general results concerning the existence and uniqueness of an optimal solution (and the structure of the optimal set). The stability properties of optimal solutions are also considered. By this is meant their regularity with respect to perturbations of the loss function and/or the strategy sets. We begin by noticing that the optimal set of the minimization problem a = infxcxf(x) is equal to A) where S ( f , A) = {x: f ( x ) A} is a lower section off. Hence any property of the lower sections which is preserved under intersection holds for the optimal set. For instance, iff is convex, its sections S(f, I.) are convex and so the optimal set is convex. Since a decreasing sequence of non-empty compact subsets has a non-empty intersection, an optimal solution exists whenever all the sections S(f,A) are compact, i.e. closed and relatively compact. A function with closed sections is said to be lower semi-continuous. We shall say that a function with relatively compact sections is lower semi-compact (Section 2.1). Another instance of a decreasing sequence of sets with non-empty intersection occurs when the sets are closed, non-empty subsets of a complete metric space whose diameters converge to 0. This property holds when f is a quadratic function defined by f ( x ) =$llx--~11~,where 1 1 - 1 1 isthenormofaHilbertspace U a n d X i s a closed, convex subset of U. We deduce the existence of projectors of best approximation (Section 2.2). In Section 2.3, we consider the case when X = {x E U such that Lx = w}, where w belongs to a Hilbert space Vand L E d(U,V). The optimal solutions x’are given by a nice formula: p = u- J-l L*j,where p = (U-lL*)-l (Lu- w) and J E &(U, U*) is the duality operator from U onto U*. In Section 2.4, we study the perturbation of the loss function by the “simp l ~ t ’ 7loss functions (i.e. the continuous linear forms p :x ( p , x)) as p ranges over u*.Iff is a proper loss function, we set inf,,, [ f ( x ) - ( p , x)] = =- f * ( p ) SO that f* :p f * ( p ) is a lower semi-continuous convex function from U* into 1- -, -1, called the conjugatefunction off. If we consider the conjugate f ** = (f*)* off *, we obtain a lower semi-continuous convex
nJCf,
+
-
-
42
Ch. 2,
0 2.11
EXISTENCE AND UNIQUENESS OF AN OPTIMAL SOLUTION
43
-
function smaller than f. We prove that f =f ** if and only iff is lower semicontinuous. Thus f f * defines a symmetric, one to one correspondence between lower semi-continuous convex functions defined on U and U’ respectively. This conjugacy operation will play a useful role throughout this book. In many situations it proves to be the key which allows us to take advantage of Duality Theory. Finally in Section 2.5, we study more general perturbations. We examine the continuity properties of the function a defined by a(y) = infx,s(yl f ( x , y) in terms of the continuity properties of the perturbed loss function { x , y} I-+ I-+ f ( x , y) and of the correspondence y S(y). For that purpose, we give a short introduction to correspondences. Further properties of correspondences are described in Sections 8.2, 8.3, 9.3, 13.2 and in Chapter 15.
-
2.1. Existence and uniqueness of an optimal solution
nAza
From the fact that S( f, A) is the minimal set of inf,,. f (x), we deduce the existence of an optimal solution when the proper function f : U -1 - + -1 is lower semi-continuous (i.e. S(f, A) is closed) and lower semi-compact (i.e., S(f,A) is relatively compact). We notice that these two requirements strain in opposite directions in the sense that the number of lower semi-continuous functions increases as the topology is strengthened while the number of lower semi-compact functions decreases. We also mention that, if n functions f;: : X R are lower semi-continuous and a t least one is lower semi-compact, then F(X)+R: is closed, where F(x) = {fl(x), . . .,f,(x)} E Rn.We have already pointed out that the optimal set is convex when f is a convex function. This implies that there exists at most one solution when the function f is assumed to be strictly convex. A function f has the non-satiationproperty if, for any x E Dom f, there always exists y E Dom f satisfying f(y) -= f (x). Iff is also a convex function, we show that the minimal set is contained in the boundary of the stategy set X on which f is minimized. 00,
-
-
2.1.1. Structure of the optimal set
Let f : U 1- a,+ -1 be a function with domain X. We shall denote by a = inf f ( x ) = inf f ( x ) (1) X€
u
X€X
the ,,minimal value off” and by M b ( f ) = {x E U such that f (x) = a} ‘(2)
EXISTENCE, UNIQUENESSAND STABILITY OF OPTIMAL SOLUTIONS [Ch.
44
2 , s 2.1
the “minimalset” off, i.e. the subset of minimal solutions of the minimization problem (1). In the same way, iff :UI- [- -, -[, we denote by
+
the “maximal value off‘ and by MYCf) = { X E U such thatf(x) = B} (4) the “maximalset off”, i.e. the subset of “maximal solutions” of the maximization problem (3). It is clear that the minimal set off is equal to
mf)n~ ( fA).,
(9
=
rlwa
Therefore, any property of subsets which is preserved under intersection satisfied by the sections off is also satisjied by the minimal subset M b ( n .
+
Definition 1. Let f be a function mapping U into ]- -, -1 (resp[- a, -1). If U is a topological space, we shall say that (a) f is “lower semi-continuous” (resp. “upper semi-continuous”) if its sections Scf, A) (resp. S y ( f , A)) are closed. (b) f is “lowersemi-compact” (resp. “upper semi-compact”) if its sections Scf, A) (resp. S&, A)) are relatively compact. If U is a vector space, we shall say that (c) f is “quasi-convex” (resp. “quasi concave”) if its sections S ( f , A) (resp. P c f , A)) are convex.
+
The following are some obvious results concerning the extensions f x of real valued functions defined on X.
Proposition 1. Let X be a subset of a topologicdspaceU,f be a real valued function on X and f x its extension to U. If X is relatively compact, then f is lower semi-compact. If X is closed and f is lower semi-continuous on X,then fx is lower semi-continuouson U. Proof. The sections S(f,A) are contained in X = Dom fxand thus are relatively compact when X is relatively compact. When f is a lower semi-continuousfunction on the topological subspace X of U,then the sections S(f,A) = SY-,, A) are closed in X,and, since X is closed, are closed in U.0 Of course, there are examples where fx is lower semi-continuous on U when
Ch. 2,s 2.11
EXISTENCE AND UNIQUENESS
OF AN OPTIMAL SOLUTION
45
X is not closed in U and examples where f x is lower semi-compact when X is not relatively compact. 2.1.2. Existence of an optimal solution From ( 5 ) and Definition I we deduce the following theorem. Theorem 1. (Weierstrass). Let U be a topological space and let f be a function mapping U into 1- m, -1. I f f is lower semi-continuous and lower semicompact, then the minimal subset M b o f f is non-empty and compact.
+
Proof. Since f is both lower semi-continuous and lower semi-compact, the sections S ( f , A) are compact subsets for A E ]a, A,]. Since S(f,Al) = = S(f,min,=,,. ..,” At), any finite intersection of sections S(f,A) is non-empty. S ( f , A) is non-empty and comTherefore the intersection M b ( f ) = pact. 0
In view of the above theorem, we shall study su5ciency conditions for f to be lower semi-continuous and/or lower semi-compact. A sufficiency condition for the former will be called a “continuity condition” and a sufficiency condition for the latter, a “compactness condition”. 2.1.3. Continuity versus compactness The two requirements of continuity and compactness pull in opposite directions. The stronger the topology defined on U,the more lower semi-continuous functions there are and the fewer lower semi-compact functi0ns.l When U is a subset of a finite dimensional vector space, there is no problem, since all vector space topologies are equivalent. Otherwise, if we are to use the above Theorem 1, the problem of the existence of an optimal solution amounts to finding a topology on the unconstrained stategy set U strong enough to ensure the lower semi-continuity of the loss function f and weak enough to ensure the lower semi-compactness ofJ
* 2.1.4. Lower semi-continuity of convex functions in infinite dimensional spaces Let U be a Hausdorff locally convex space and let U* = &(U,R) be its dual. A weakly lower semi-continuous function is one whose sections are weakly In other words, let U,(i = 1,2) be a set U supplied with two topologies i = 1.2, where the first is stronger than the second.This amountsto saying that theidentity map is continuous from U,onto U,.Therefore any relatively compact subset of U, is relatively compact in U, and any closed subset of U, is closed in U,.
46
EXISTENCE, UN~QUENESSAND STABILITY OF OPTIMAL SOLUTIONS[Ch.
2,§ 2.1
closed. Any weakly lower semi-continuousproper function on U is lower semicontinuous. Proposition 1.2.1 implies that the converse is true for convex and quasiconvex functions.
Proposition 2. A quasi-convex function f : U b- ]- a, + -1 is lower semicontinuous if and only if it is weakly lower semi-continuous. Proof. The sections S(j,iZ)are convex subsets and so they are closed if and only if they are weakly closed by Proposition 1.2.1. 0 The following result for paired spaces is proved similarly but using Proposition 1.2.2.
Proposition 3. Let U and U' be two paired vector spaces and f be a quasi-convex function. It is lower semi-continuousfor the weak topology if and only if it is lower semi-continuousfor the Machey topology. Remark. We will devote Section 3.1 to the study of lower semi-compactness of functions. 2.1.5. Fundamental property of lower semi-continuous and compact functions
The following property of a family of lower semi-continuous and lower semi-compact functions is worth mentioning.
Proposition 4. Consider n funcfionsfi : U (6)
-+
]-
-,
+ -1
satisfying
semi-continuous, { (ii)(i) Qi,fi is lower that & is lower semi-compact. 3 io such
nys1
Let X = Domfi andlet be the operatorfrom X into R" defined by F(x) = = { f d x ) , - .., f,(x)): Then
(7)
F ( X )+R; is.closed in R".
Proof. Let (dkXcN be a sequence of elements of F(x)+R: which converges to d E R". Write dk = F(*+Ck. For k 3 ko, (8)
A&")
4
4,
die+ 1
and since A,, is lower semi-compact, we deduce that 2 stays in a relatively compact subset of X. Hence a subsequence x' converges to an element x of U.
Ch. 2, tj 2.11
EXISTENCE AND UNIQUENESS OF AN OPTIMAL SOLUTION
47
Since fi is lower semi-continuous, we deduce that (9)
J ( x ) =S lim infrfi(xl) =S lim dfl = di for all i.
Hence d E F(X)+R”. CI 2.1.6. Uniqueness of an optimal solution We now consider how convexity assumptions can imply uniqueness of the minimal solution. First, we deduce from (5) and Definition 1 the following result. Proposition 5.
Iff
is quasi-convex, then the minimal set Mb(f) is convex.
Convex functions are, of course, quasi-convex. Definition 2. We shall say that a real valued function f defined on a convex subset X of a vector space is “strictlj convex” if
(10)
VX, y E
x, f($ ( x + y ) )
+
( (x)+f (y)) unless x
= y.
Proposition 6. I f f is a strictly convex function mapping a convex subset X into R, then there exists at most one minimal solution. Proof. I f x and y are minimal solutions, then f ( x ) =f (y) = a. Since f is strictly convex, then f($ (xi-y)) -=.+(f( x ) + f ( y ) ) = a. But Mb(f ) is convex. Thus 1 2 ( x + y ) E M b ( f ) . Therefore, f (+ (x+y)) = a and we obtain a contradiction unless x = y. 0 Proposition 7. Let g and h be two convexfunctions and let f be de$ned by f (x) = = [ g ( ~ ) ] ~ + h (Then x ) . g and h are constant on the minimal set Mbcf) off. Proof. Let x1 and x2 be two elements of M b ( f ) . Since the function a+ h is strictly convex, either g(x1) = g(x2) or else
and this is a contradiction. 0
48
EXISTENCE, UNIQUENESS AND STABILITY OF OPTIMAL SOLUTIONS [Ch.
2,s 2.2
2.I.7. Non-satiation property We shall poinf out a property off which implies that the minimal set is contained in the boundary of the strategy set X. Dellnition 3. We shall say that f has the “on-satiation” property if (1 1)
Vx
E Dom f , there exists 5 f Dom fsuch thatf(t)
c:f
(x).
,,Proposition 8. Suppose that (12)
f is convex
and that
(13)
f has the non-satiationproperty.
If X minimizesf on X , then X belongs to the boundary of X . Proof. Suppose that 2 belongs to the interior of X. We shall obtain a wntradiction. For that purpose, let 5 E Dom f satisfy f (4) -cf(3. Since 2 E Int X, there exists 8 =- 0 small enough to ensure that y = X+ O(5-Z) = (1 - O)Z+OE belongs to X. The contradiction follows from the fact that
f ( r )= $ ( ( I - e)z+ e5) (1 - e)f(z)+ef(E) -= ( i - e ) f ( g + w - ( z ) =f(x3Remark. We could replace the convexity assumption (12) by the weaker assumption (14)
Wheneverf (4) -= f (x) and 8 E 10, 11, then f((i-e)x+e~) -=f(x).
Such a function is said to be semi-strictly quasi-convex. 2.2. Minimization of quadratic fmctionals on convex sets We shall prove that a quadratic minimization problem has a unique solution which we shall characterize. Then we define the projectors of best approximation and study them.
Ch. 2,§ 2.21
MINIMIZATION OF QUADRATIC FUNCTIONALS ON CONVEX SETS
49
2.2.1. Hilbert spaces Let U be a Hilbert space. This is a vector space supplied with a bilinear form {x, y} E U X U t-. ((x, y)) E R called the scalar procu& which satisfies (1)
{
vx, Y
u, ((x, Y ) ) = ((Y, .I)
E
v x = 0,
(symmetry),
(<x,x)) s- 0
and whose topology is defined by the norm and which is complete for this topology. The fact that I I 11 is a norm follows from the Cauchy-Schwurz inequality
-
(3)
VX,Y E
u,
((.,Y)) ==
llxll .Ilvll.
We shall prove the existence of a unique optimd solution of the “quadratic minimization problem” (or quadratic program) a = inf f ( x > (4) XEX
where (5)
(i) Xis afo{ (ii)f(x) +IIx-ull2.
U
=
The problem amounts to minimizing the distance between the set X and an objective u E U.
2.2.2. Existence and uniqueness of the minimal solution For this purpose we use the fact that, in complete metric spaces, the intersection of a decreasing sequence of closed, non-empty subsets whose diameters converge to 0 is equal to one point. Proposition 1. Let U be a Hilbert space supplied with the norm 11 xII, let X be a closed convex subset of U and let f be the quadraticfunctional defied by (6)
f(x) =
l l ~ - u 1 1 ~where u isgiven in U .
Then there exists a unique minimal solution 1 E X. Ilx-yll. The sections SY; A) Proof. Let An = a + l / n and let s, = are closed subsets of the complete metric space X and so to prove that
(7) 6
M b ( f >=
n W,
An)
n
=
{?I,
50
EXISTENCE, UNIQUENESSAND STABILITY OF OPTIMAL SOLUTIONS [Ch.
2, $ 2.2
(i.e. that Z is the unique minimal solution) it is enough to show that s, converges to zero. Let x, y belong to S(f,A). Adding the equations Ilx*Y1l2 = 11x112+llYIl2&2((*,y))
we obtain that
Since X is convex, $ (x+ y) belongs to X and thus f(+ ( x + y ) ) a. Therefore, if x and y belong to S ( f , i) then , [Ix-y1I2 =s8(A-a). Thus the diameter 6, of S(f, I.,) is smaller than or equal to
and converges to 0. 0
2.2.3. Characterizationof the minimal solution The following characterization of the optimal solution plays a fundamental role. Proposition 2. An element Z E X is an optimal solution if and only if it is a solution of the variational inequalities
(8)
((Z-u, Z - y ) )
G
0 for ally
E X.
Proof. Lety belong to Xand 0 E [0, 11. Then (1-0)XfBy = x+B(y-2) longs to X because X is convex. Thus I 11 2-u 2
~-
112-
11 X+ e(Y6
1)- u) I 12
=s0
be-
for any 0 =- 0.
+
Since I 12- u I I p - I I x"+ 0(y-2)- u I l2 = 2B((x'- u, 1- y)) B2 I ly- XI 12,we obtain (8) by letting 6 converge to 0. Conversely, if X is a solution of (9, it satisfies 11 1-ul12- ((X-u, y - u ) ) = = (((X-u, 2 - u + ( u - y ) ) ) = ((Z-u, 1 - y ) ) =s0 forally E X . Therefore, applying the Cauchy-Schwarz inequality, we obtain IIX- u Il2 e ((Z- u, y -u)) =IS =.s IlX-ull Ily-ull and thus, f ( Z ) = = f ( y ) . 0
Ch. 2, 0 2.21
MINIMIZATION OF QUADRATIC FUNCTIONALS ON
CONVEX SETS 51
Proposition 3. Let X be a closed convex cone. Then 1 E X is an optimal solution if and only if
(9)
{ (ii)(i)
( ( 2 - u , x ) ) = 0,
((X- u, y ) ) 2 0 for ally E X .
Let X be a closed subspace. Then X C X is an optimal solution
(10)
if and only if
((X-u, y ) ) = 0 for ally E X .
Proof. Let X be a closed convex cone. If X satisfies (9 (i) and (ii)), it satisfies (8) (by subtracting (9 (ii)) from (9 (i)). Conversely,let X be a solution of (8). Since Xis a cone, 0 and 2Z belong to X: we obtain ((X-u, 2)) e 0 and -((X-u, X)) s 0, and thus ((X-u, 2)) = 0. ‘ Hence ((X-u, y ) ) 2 0. If X is a closed subspace, equations (10) are clearly equivalent to inequalities (9(ii)).
-
2.2.4. Projectors of best approximation Definition 1. The map u imation on X.”
2 = tx(u) E Xis called the “projectorof best approx-
It is a continuous map. In fact, it satisfies the following properties Proposition 4. The projector of best approximation t, satisfies (1 1)
IItx(x)--x(~)tI=s Ilx-Yll.
If X is a closed convex cone, t, (12)
also satisfies
(i) t x ( l x ) = l t x ( x )for any x E X , ;Iz=0, (ii) llx1I2 = I I t ~ ( x ) 1 1 ~ + 1 1 ( ~(x>1l2, -t~) (iii) I I tx(x)I I =sI Ix I I *
If X is a closed subspace, then tx is linear. Proof. Since tx(x) and tx(y) belong to X , we deduce from the inequalities (8) characterizing tx(x) and t,(y) that ((t,(x)-x, t,(x)-t,(y))) =s 0 and ((t,(y)-y, tx(y)- tx(x)))=s0. Adding, we obtain that ((x-y- (tx(x)- tx(y), tx(x)-tx(y))) *O. This implies that JIt,(x)-tx(y)JJ2e ((x-y,t,(x)-tx(y))) =s I I x-y I I I I tx(x)- tx(y)I[ by the Cauchy-Schwarz inequality. 6’
52
EXISTENCE, UNIQUENESS AND STABILITY OF OPTIMAL SOLUTIONS [Ch.
2 , $ 2.2
If X is a cone and A =- 0 we deduce from inequalities (9(i) and (ii)) characterizing tx(Ax) that
((.-
1
MW,
1
tx(Ax))) =
nz1 ((Lrx(Lx), t x ( h ) ) )= o
and.that, for any y E X ,
Finally, if X is 8 subspace, equations
Ei((X,-tx(xi), y ) ) = 0 (where y E X ) imply that
(( C mixiThus tX(Caixi) =
C aitx(xj), y ) ) = o
for all y E X.
1aitx(x,).
Remark. When X is a cone, we can show that (1 - t x ) is the projector of best approximation onto the closed convex cone @'A
= y E
{
U such that sup ((x, y ) ) =sO } . X€X
When Xis subspace, 1- tx is the projector onto the closed subspace Xo = {y E U such that ((x,y ) ) = 0 for all x E X}. 2.2.5. The duality map from an Hilberi space onto its dual
We shall denote by U* either the space &(U, R) of continuous linear forms or a Banach space isomorphic to B(U,R).
Ch. 2, g 2.21
MINIMIZATION OF QUADRATIC FUNCTIONALS ON CONVEX SETS
53
Recall that a Banach space U* is isometric to J ( U , R) i f cnd only if there exists a bilinearform { p , x } E U* X U I-+ ( p , x ) E R satisfving (i) i f ( p , x ) = 0 for any x E U,thenp = 0, (ii) if ( p , y ) = 0 Ibr an y p E U*, then x = 0, (iii) V p E U*, IIpllu* = S U P ~ E UI(P7 x)l/llxll.
(13)
The bilinear form -) on U*X U is called the duality pairing. The isometry j from U* into a ( U , R)is defined by p E U* k - j ( p ) E 2(U,R)where (a,
= (P,x).
A P ) :x + j P W
Note that the duality pairing on 2(U,R)X U is defined by {p, x} k- p(x). It is often useful to choose for the dual a space U* isometric to a ( U , R) instead of J ( U , R) (when U* “looks like” U for instance). Once such a space U* is chosen (explicitly or implicitly), we shall say that it is the dual of U . If U is a Hilbert space, there exists an isomorphism J E 2 ( U , U*) from U onto U*defined by (14)
vx,y E
u,
( ( x ,Y ) ) = (Jx, Y )
called the duality map. This means that for any x E X , Jx is the continuous linear form y t- (Jx, y) = ((X’Y)). It satisfies 1
(15)
(Jx, y ) = lim 211x+~Y112-+11x112
e
e-o
We can supply the dual U* with the “dual scalar product” (16)
KPY 4))* = ((J-%
J-14)) =
(P, J-14).
The associated norm coincides with the dual norm
thanks to the Cauchy-Schwarz inequality. Example. Let U = R’ be finite dimensional. We shall denote its dual by
u* = R‘*.
Let k = (0, basis.
. . ., 0,
1,0,
.. ., 0}
denote the i& element of the canonical
54 EXISTENCE, UNIQUENESS AND STABILITY OF OPTIMAL SOLUTIONS [Ch. 2,
2.3
Since any bilinear form ( ( x ,y)) can be written (18)
( kY ) ) =
I
C
i,j=l
xiyi((ei,4 )=
it is characterized by a matrix J = {((e', e'))}'*'='* from R' into R'*. The bilinear form ((x, y)) is a scalar product if and only if (i) ((ei,4)= (W,e')), (ii) 'dx f ; 0, j=l ((ei,el))xixj =- 0.
Cf,
The matrix J is said to be symmetric and positive definite. Therefore, we can write 120)
( ( x ,y ) ) = (Jx, y ) where J = {((el, e9)}i.j=1,..., I .
A scalar product on R' is thus characterized by a symmetric positive definite matrix from R' into R'*. 23.Minimization of quadratic fmctionals on subspaces We shall prove that the optimal solution of a = infLx=, I [ x - u [ [ is defined by Z = u- J-lL*F and 3 = (LJ-lL*)-l (Lu- w). This implies that L f = = J-lL*(LJ-lL*)-l is a linear continuous right-inverse of the surjective operator L E d(U,V ) and that M- = (M*JM)-lM*J is a linear continuous left-inverse of an isomorphism M from U into W.We deduce that a problem a = inf,,,, IIMx-ull is equivalent to the problem a = infLM-==,IIz-uII. 2.3.1. Thefundamental formula Let
L E 2(U,V) be a surjective continuous linear operator from a Hilbert space U onto a Hilbert space V. and let
(2)
u
E U,w E V be fixed.
Theorem 1. Given (1) and (2) the unique solution 3 of the minimization problem (3)
min +llx-u112
Lx= W
Ch. 2 , s 2.31
MINIMIZATION OF QUADRATIC FUNCTIONALS ON SUBSPACES
55
is defined by (4)
{ (ii)(i) pX
L*p, = (LJ-'L*)-' (Lu= u- J - l
W) E
V*.
As a corollary, we mention :
Proposition 1. The unique solution X of the minimization problem (5)
min
(P. x)=r
+
llx-u112
i s defined by
Proof. We leave this as an exercise. CI
-
Proof of Theorem 1. If Z minimizes I]x - u on the subset A' = {x E U such that Lx = w}, we deduce from (1-8) that f E X satisfies
(7)
((2-u, X-x))
0 for any x E X .
But x E X if and only if y = X - x E Ker L. Thus X is the optimal solution if and only if (8)
(JZ- J u , ~ = ) ((X-u, y)) = 0 for any y E Ker L,
i.e., if and only if (9)
JX- Ju E (Ker L)-L = Im L*.
[The closed range theorem implies that Im L* is closed and equal to (Ker L ) I ] . Therefore, there exists (a unique) p E V* such that J Z - Ju = -L*p, i.e. such that 2 = u- J-lL*(10) P. By appIying L to both sides of this equation and recalling that LZ = w, we deduce that j is solution of the equation (1 1)
(LJ-'L*)j = Lu-L? = Lu- W.
It remains to prove that LJ-'L* is invertible. This follows from Lemma 1 below. Lemma 1. I f ( 1 ) holds, then LJ-lL' is an isomorphism.
56
EXISTENCE, UNIQUENESSAND STABILITY OF OPTIMAL SOLUTIONS [Ch.
2, Q 2.3
Proof. We use the Lax-Milgram Proposition 9.2.1 below which asserts that LJ-lL* is an isomorphism when it is V*-elliptic, i.e. when there exists a positive constant c such that (12)
vpE
v*, ((LrlL*)P,p )
I$*-
c IIP I
But (W-lL*p, p ) = (J-'L*p, L*p) = llL*pll& On the other hand, L* is a bijective continuous map from V*onto its closed range ImL* which is a Hilbert space. The Banach open mapping principle implies that there exists a positive constant c such that I I L*p IIU+ 2 fz I Ip I I The V*-ellipticityis therefore proved. 0 *2.3.2. Orthogonal right inverse Proposition 2. Suppose that (1) holds. Then the map L+ = J-lL*(W-lL*)-l € &(V, U) is a continuous right inverse of L. The product from the Ieft L 11.- I[ of the norm of U by L is equal to the product from the right of the norm by L+, i.e. (13)
VY €
v, GI1 .Il)(Y)
= IIL+YlI = inf
L*=y
llxll.
Furthermore,
Proof. We have that LL+ = ( L P I L * (LJ-IL*)-l ) = 1. Thus L+ is a continuous right inverse of L. We can write the optimal solution X of (3) in the form (15)
2 = u-L+Lu+L+w.
In particular, by taking u = 0, we deduce that L+w minimizes x -I the constraint Lx = w. Therefore
(LII
ID (w) = IIL+wll.
Finally,
since we can easily check that
IIx 1I under
Ch. 2, 5 2.31
MINIMIZATION OF QUADRATIC FUNCTIONALS ON SUBSPACES
57’
* 2.3.3. Orthogonal left inverse Let (17) where (18)
M E 2(U,W) be an injective continuous linear operator from a Hilbert space U onto its closed range
W is a Hilbert space for the scalar product (( ., and K E B(W, W*) denotes the duality map.
a))
We consider the case of a loss function f defined by f ( x ) = IJMx-u /I2 The optimization problem may be interpreted as that of minimizing the distance from a “service” Mx produced by x to an objective u in the “service space” W . Proposition 3. Suppose that (17) holds. Then the unique solution X of the minimization problem
is given by (19)
X = (M*KM)-lM*Ku.
Proof. We have that MX minimizes
aII
z - u 1 l2 on the range M(U). Therefore,,
((MZ- u, Mx)) = (KMF- Ku, Mx) = (M*KMX-M*Ku, x ) = 0 for any x E U.
Hence Z is the solution of equation
(M*KM)x’= M*Ku. Lemma 1 with L = M* and K = J-l shows that M*KM = L*J-lL is invertible. Hence (19) follows.
Proposition 4. Suppose that (17) holds. Then the map M - = (M*KM)-lM*K E f 2 ( W , U)is a continuous left inverse of M . Furthermore (i) (M-)* = (M*)+,
(ii) M = (M-)+. Proof. We have that M - M = 1. Also (M-)* = KM(M*KM)-l= [K-l’J-l(M*)*[(M*) [K-’]-l ( M * Y ] - l = = (M*)+.
EXISTENCE, UNIQUENESS AND STABILITY OF OPTIMAL SOLUTIONS [Ch.
58
2,§ 2.3
This implies in particular that
M-K-yM-)' Therefore
= (M*KM)-'.
( M - ) + = K-l(M-)* [M-K-1(M-)"1-1 = = M(M*KM)-'[(M*KM)-l]-l = M .
*2.3.4. Another decomposition property Consider the minimization p r O b h n
(21)
min
LX=w
+IIMx-u\J~,
where U , V , W are Hilbert spaces and (22)
V )is surjective f (ii)(i) LM EC:L(U, L(U, W ) is injective and has a closed range.
Let M - = (M*KM)-IM*K E a ( W , U ) be the left inverse of M . The position is illustrated below.
We have the following decomposition property. Proposition 5 . Suppose that (22) holds. Let X E U be the unique solution of (21). It is deJned by
(i) X = M-u- (M*KM)-~L*~T, (ii) p = [L(M*KM)-1L8]-l(LM-u- w). It is also defined by (24)
2 = M - 5 where Z minimizes f 1 z - u 1 Iz under the constraints LM-z = w,
i.e. by the formula
(25)
i= M-U- M - ( L M - ) + L M - u f M - ( L M - ) + w .
Ch. 2,p 2.31
MINIMIZATION OF QUADRATIC FUNCTIONALS ON SUBSPACES
59
Proof. Let X be a solution of (21). Since Lx = w if and only if X - x Ker L , we deduce that ((MX- u, Mx)) = (KMZ- Ku, M x ) = (M*KM?-M*Ku, x ) = = 0 whenever x E Ker L. Hence M*KMZ-M*Ku belongs to (KerL)I = = Im L* and so there exists jj such that M*KM%-M*Ku = L*jj, i.e., such that 2 = (M*KM)-lM*Ku- (M*KM)-lL*jj = M-u- (M*KM)-IL*p. Writing L% = w, we obtain the relation
[L(M*KM)-lL*JP= LM-U-
W.
Applying Lemma 1 to J = M’KM, we deduce that L(M*KM)-lL* is invertible. Thus, formula (23) is proved. Notice that L M - = L(M*KM)-lM*K and (LM-)* = KM(M*KM)-lL*. It follows that (LM-)K-l(LM-I)* = L(M*KM)-lL*. Therefore, we can write (26)
jj = [(LM-)K-l(LM-)*]-l (LM-u- w).
Also,
M - K - ~ ( L M - ) * = (M*KM)-lL* Therefore, we obtain
(27)
3 = M-(u-K-l(LM-)*jj) where jj = [(LM-)K-l (LM-)’]-l (LM-U- w).
By Theorem 1, i = u- K-l(LM-)*jj minimizes LM-2 = w. 0
iII z-
u 112 under the constraints
*2.3.5.Interpretation We can, of course, describe many economic problems by the minimization problem minLxSw ) ) X - U ) ) ~ . For instance, we can regard U as the output space, u E U as a “demand” in output, V as the input space, L as the production operator, w E V as the “supply” of input, Lu E V as the input necessary to produce the “demand” u, i.e., the “demand in input”. Hence jj = (LJ-lL*) (Lu- w) can be regarded as an input-price, which depends linearily upon the difference between the demand and the supply of input (in other words, this relation describes the law of supply and demand). Then 4 = L*jj is the associated output price. Therefore the production X = u- J-lS is the closest to the demand u either under the production constraint Lx = w or the budgetary constraint (4, x) = (q, X).
60
EXISTENCE, UNIQUENESS AND STABIUTY OF OPTIMAL SOLUTIONS [Ch.
2,s 2.4
2.4. Perturbation by hear forms: conjugate functions
Let U be a topological vector space. The class of “simplest” functions is the dual U* = Pe(U, R), i.e. the space of continuous linear forms in U. Let f be a proper loss function mapping U into 1- 00, -1. It is natural to associate with the minimization problem
+
the perturbed problems -fL(P) = inf [f(x)- (p, 41 E [ - O0 9
(2)
X€
u
+
OD
t
for the continuous linear forms p E U*. We shall study the relations which exist between the problem and its perturbations. We begin by studying the properties of the function p I---f ‘(p) defined on the dual U’ of U.A proof is given that a function f is convex and lower semi-continuous ifand only iff is equal to (f*)*. Explicit forms for the conjugates of support functions and functions 11) of norms are obtained. Finally, some other elementary properties are discussed. 2.4.1. Conjugatefunctions
First we notice that, f:U+ p E
u*,
-
OD
-= (p, x0)-f
1-
+ -1
0,
(xo) Rs SUPI(P, X€ u
and xo E Dom f , then, for all
4-f(-41.
Definition 1. We shall say that the function f* : U*I--+ 1associates with any p E U’
(3)
f+(P) = SUP[(p, X)-S(X)l XE u
E 1-
- +Y
-
1
+
-,
+-1
which
- -+
is the “conjugatefunction” of the function f:U 1- =, -1. In the same way, we define the “conjugatefunction’’ g* of a function g : U* ]- , -1 bY The “biconjugatef **” of a function f is therefore defined on U byf ** = ( f *)*. First, notice that (5)
-f*(O)
= inf f ( x ) = a X€ u
Ch. 2,
0 2.41
PERTURBATION BY LINEAR FORMS
61
and that the biconjugate f ** satisfies (6)
f**Wss PEU* SUP [(P, 4- [(P, 4-f(x)l] -=f(4
by (2) and (3). Also, we have the so-called FencheI inequality
(7)
( P , x) = = f ( X ) + S ' ( P )
vx E u, V P E u*.
Sincef is the pointwise supremum of a family of continuous affine functions p { p , x)- f (x), we have that -t
(8)
the fmctions f * and f ** are convex and lower semi-continuous.
2.4.2. Characterization of lower semi-continuous convex functions
It follows from (8) that a necessary condition for f =f ** is that f be a convex lower semi-continuous function. This is actually also sufficient. "heorem 1. A function f dejined on a topological vector space U is convex and lower semi-continuous if and only i f f = f **.
-
Remark. In other words, the conjugacy operation f f * induces a symmetric, one to one correspondencebetween the cones of convex lower semi-continuous functions defined on U and U* respectively.
Proof. (a) We assume that R -= f (x). Since the pair { x , R} does not belong to the epigraph dp(f)off, there exists a continuous linear form (p, a) E U*x R which strictly separates {x, R } from the closed convex set & p ( f ) (see Theorem 2.3.1): there exists E 1 0 such that
(b) We note that aa0;if not, by taking y = xo E Dom f (which is nonempty by assumption) and I = f(xo)+ n, we would have - an =S( p , x ) - aR- ( p , xo) + af (xo) - E, which is impossible if 01 -= 0 and n is large enough. (c) We consider the case where xEDomf. By taking y = x and I =f (x), we deduce from (9) that a ( f ( x ) - R ) =- E. Hence a > 0; by dividingby a Z- 0 and settingp = Flu, we obtain
62 EXISTENCE, UNIQUENESSAND
STABILITY OF OPTIMAL SOLUTIONS [Ch.
2,s 2.4
By taking the supremum with respect toy, we deduce that f*(jj)e@-,x)-R-a/a.
(11)
This implies that E Dom f * and also, that R 4 f **(x) for any R -==J(x.) Hencef(x) =f**(x). (d) We consider thecase where x 6 Dom $. If a =- 0, assertions (10) and (11) still hold true and we deduce that R 4 f **(x)for ail R. Hence f **(x) = It remains to consider the case where a = 0. Inequalities (9) imply that
+
(12)
try EDomf,
01.
(p,y-x)+~~o.
Let us take 3 E Dom(f*) (which we proved to be non-empty). By multiplying inequality (12) by n == 0 and adding it to inequality (j?, y) - f " ( p ) - f ( y ) =s 01 we obtain
VY E Domf,
(13)
( p + np, v)-n(p, x) + ne-.P(jj) -fb)-= 0.
It implies, by taking the supremum with respect to y E Domf, (14)
Y@+ np)
E s
n ( p , 4- nE +f*(P).
Therefore,
By letting n
+
-, we deduce that f**(x) = + m .
2.4.3. Examples of conjugatefunctions
We begin by pointing out that support functions and indicators are related by conjugacy relations. Proposition 1. Let X be a subset. Then its support function o$ is the conjugate of its indicator, i.e. o$ = y;. Conversely, if X is closed and convex, yx = if$*. Proof. The fact that 02 = y$ follows immediately from their definitions (Section 1.4). Since yx is convex and lower semi-continuous if and only if X is convex and closed, the converse result follows from Theorem 1. 0 We next show that the conjugate function of (6(11-11) is ~$*(11.11*). We begin with a function (15)
-
y :R, R, which is continuous, strictly increasing and satisfies. ~(0) = 0 and 1imA+- ~ ( 2 = ) 00.
Ch. 2, fi 2.41
PERTURBATION BY LINEAR FORMS
63
We associate with 2 the convex function @ defined by (16)
@(t)=
if t == 0,
~ ( 1d)l
if t -= 0. We shall consider the convex function f defined on the Banach space U by (17)
f ( x ) =@(llxll).
Proposition 2. The conjugate function o f f is deJned by (1 8)
f * ( x ) = @*(llxll*).
where @* is the conjugate function of 0 and I IpI I+ = sup I(p, x)l/ll x 1 1 is the dual norm of u*.
Proof. We have that
Example. Consider the case when ~ ( 2 = ) Aq-I where q z 1. Then
and -tq*
+-
if t a 0, ifte0,
where l / q + I / q * = 1. This follows immediately from the well-known fact that, for any a, b 1 ab =s- aq 4
+ 41
bQ*
> 0,
64
EXISTENCE, UNIQUENESS AND STABILlTY OF OPTIMAL SOLUTIONS [Ch.
2,$2.4
and
1 1 1 ab = -@++bq* if b = 4 9 Therefore, the conjugate function of
1 4
= - Ilxll'
is the function
(20)
1 fq*(p)= 4' IIpII:'
where
1 1 -+T 4 4
= 1.
For q = 2, we find that (21)
. a x ) = +llx211; f,'(P> = +llpll:.
2.4.4. Elmentary properties of conjugatefunctions
Proposition 3. I f f e g, then g* =sfa. u g ( x ) =f (x-xo)+(po, x)+a, then xo)+a). g*(p) = ~ * ( P - P o ) + ( Pxo>-((po, ,
Proof. The first statement is. obvious. The second is obtained as follows. sup C(p,x ) -g(x)l= u
X€
SUP xE
[(P -PO, x) -f (x -XO)] - a
= SUP [(P --par x -xo) - f (xxE u
XO)]
-a
= f * ( p - p o ) + ( p , x ~ ) - a - ( p O . xo).
+( p-PO,
XO)
o
We can also prove the following property.
Proposition 4. Let L C 2(U,V)and let f be a functiimfrom U into ]-
Then
(22)
( L . * =f*L*.
Proof. If p E U*,we have that
0 ,
+ -1.
Ch. 2, $ 2.41
PERTURBATION BY LINEAR FORMS
65
Remark. Under suitable assumptions, we shall prove that (gL)* = L*g*. (See Proposition 5.2.9.) In particular, we can compute the conjugate function of an inf-convolution. Proposition 5. The conjugate function of the inf-convolution is the sum of the conjugatefunctions:
(23)
(fm* (PI = f *(p)+g*(p)
Proof. I f p f U*,we have that
Remark. Under suitable conditions, we shall prove that (f+g)* (p) = (f*Og*) (p). (See Proposition 5.2.10.)
-
-
Proposition 6. Let U = UIX U2 be the product of two spaces, f : U1X U2 1- 00, -1 and L = L1+L2 E ~ ( U I X U ZV), where LI E 2,(U, V). Then
+
In particular, if V = U,LI= 1, Lz = 0, then
-
2.4.5. Interpretation:cost and profit functions
+
We may interpret function g : U 1- -, -1 as a “cost function” of a producer. The domain Z = Dom g can then be viewed as the “production set”. In Example 1.5.5, we considered the particular case when g = yz is the indicator of 2. If p E U*is a price system, ( p , 2)-g(z) is the profit associated with z. Then the conjugate function g* associates with any price system p the maximum profit g*(p) = sup,,, [(p, z)-g(z)] obtained when price p prevails. Now, consider m producers j = 1, . . ., m, described by their cost fvnctions 2’(where 2’ = Dom g-‘is the prog,. If they have to produce z E 2 = duction set of thePh firm), their minimal loss is equal to
7
,
66
EXISTENCE, UNIQUENESS AND STABILITYOF OPTIMAL SOLUTIONS [Ch.
Hence the profit function
2 , § 2.5
m
is the sum of the profit functions of each producer by Proposition 5.
2.5. Stability properties: an introduction to correspondences
-
We investigate in this section the problem of stability of minimization problems, that is the regularity of the function cc defined by
and of the correspondence M b defined by W ( y ) = {x E
X such that a(y) =f(x, y)}
when the loss function f( y) and the strategy set S(y) depend upon a parameter y. We are led to introduce “continuity” properties of correspondences S mapping elements y € Y into subsets S(y) of a topological space R. These are the so-called “upper semi-continuity” and “lower semi-continuity” properties. Both generalize the concept of a continuous map. For instance, we shall prove that iff is lower semi-continuous and S upper semi-continuous with non-empty compact images, then a is lower semi-continuous. Also, we shall prove that if f is upper semi-continuous and S lower semi-continuous with non-empty images, then a is upper semi-continuous. Finally, in order to prove that the correspondence M b is upper semi-continuous, we study the relations between closed correspondences (correspondences with closed graphs) and upper semicontinuous correspondences. We end this section by studying the construction of upper semi-continuous correspondences. We shall prove the upper semicontinuity of the subdifferential of a continuous convex functions in Section 4.1. A further study of correspondences is undertaken in Section 15.3. a,
2.5.1. Upper semi-continuous correspondences JDefinition 1. We shall say that a correspondence S :Y continuous” at yo € Y if (1)
-R
is “upper semi-
for any neighbourhood N(S(y0)) of S(yo), there exists a neighbourhood N(y0) of y o such that for all Y E WO), S(y) c N(S(y0)).
Ch. 2, 4 2.51
67
STABILITY PROPERTIES
A correspondence is upper semi-continuous on Y if it is upper semi-continuous at any point of Y. It is clear that such a property generalizes the usual notion of the continuity of a map.
Theorem 1. Suppose that
(2)
-
(i) the map f is lower semi-continuous on R X Y (ii) the correspondence S :X R is upper semi-continuous at yo E y J(iii) S(y0) is compact and the images S(y) are non-empty when y E Y.
Then the function a :y t- a(y) = infxES(,.) f(x, y) is lower semi-continuous at YO.
Proof. We have to prove that for any N(y0) of yo such that (3)
.(yo)-
E 4
a(y)
E
w
0, there exists a neighborhood
for all y E N(y0).
Sincef is lower semi-continuous at any pair {x, yo}, we can associate with any x (open) neighborhoods N(x) of x and Nx(yo)of yo such that (4)
f (x, YO)- E - c f ( z , y ) for all z E N(x), Y E N,(YO).
Since S(y0) is a non-empty compact subset of U,we can cover it by a finite number n of neighborhoods N(x,). Therefore, (5)
N(S(y0)) =
U;=lN(xi)is a neighborhood of S(y0).
Since the correspondence S is upper semi-continuous at yo, there exists a neighborhood N&o) such that n
Consider (7)
the neighborhood N(y0) = No(y0)r;
0N,,(yo) of yo. I,
i=l
When y E No(y0) and x E S(y), x belongs to a neighborhood N(x,) (by (6)) and thus, by (4), we obtain f(xi, YO)-
f ( x ,V )
E =z
since x E N(xJ and y E NX,(yo). 7.
68 RXISTENCE,
UNIQUENESS AND STAJNLITY OF OPTIMAL SOLUTIONS
[Ch. 2,§ 2.5
Thus
when y E NCyo) and x €
Sb).We may therefore deduce (3). 0
Definition 2. Let U and U*be two paired spaces and let S be a correspondence from Y into U.We shall say that S is “upper hemi-continuous” if (8)
V p E U, y -I
u+(SGy), p ) is upper semicontinuous
The following proposition will play an important role later (Chapters 8, 9 and 15).
Proposition 1. Let S be a correspondence with non-emptyvaluesfrom a topological space Yinto a Haaw’orf locally convex space Usupplied with the weak topology. If S is upper semi-continuous, then it is upper hemi-continuous.
Roof. Let p E U* and B = s { - p , p}*, which is a neighbourhood of 0 in U for the weak topology. Since S is upper semicontinuous, there exists a neighB as y ranges over N@o). There bourhood N(y0) of yo such that s@)c s@~)+ fore (9)
U#(SCy),
This shows that y
I-+
p)
Q
qsDo), P)+U*(B, P) 6 a*(S(yo), P)+&.
u*(S(y), p ) is upper semi-continuous. 0
2.5.2. Lower semi-continuous correspondmces
The following concept of lower semicontinuous correspondences is also
a generalization of the idea of a continuous map.
Definftion 3. We shallsay that a correspondence S : Y contimcous” at yo € Y if (10)
-
R is “lower semi-
for any x E Solo) and for any neighborhood N(x), there exists a neighborhoodN(y0)of yo such that, for all y E N ~ O ) , nsw z 0.
w)
A correspondenceis lower semi-continuous on Y if it is lower semi-continuous at any point of Y.
Ch. 2, 0 2.51
69
STABILITY PROPERTIES
Definition 4. We shall say that a correspondenceS from Y into R is “continuow” at y o (resp. on Y) if it is both upper and lower semi-continuous at yo (rap. on Y). Theorem 2. Suppose that
(11)
(i) the map f is upper semi-continuous on RX Y, (ii) the correspondence S :Y R is lower semi-continuous at yo E y, (iii) the images S(y) of S are non-empty whenever y E Y . +
Then thefunction y I-+ a(y) = infxEsOf(x, y) is upper semi-continuous at yo.
Proof. We have to prove that for any N(yo) of yo such that (12)
a(y) =S a(yo)+s
E
z- 0, there
exists a neighborhood
for all y E N@o).
Choose an element xo E S(y0) such that
Since f is upper semi-continuous, there exist open neighborhoods No(y0) and N(xo) such that
Since the correspondence S is lower semi-continuous, we can associate With the neighborhood N(XO) a neighborhood NIQo)such that (15)
s(y)nN(xo)# 0 for any y E Ndyo).
Consider (16)
the neighborhood N(y0) = No@o)nNl(yo).
If y E N(yo), there exists x E S(y) which belongs toN(xo) (by (15)) and which therefore satisfies (14). We deduce that a@) =
inf f ( u , y ) e f ( x , Y )
uE W)
ef(x0,
yo)+: 2
2.5 aQo)+T = a@O)+e.
70
EXISTENCE, UNIQUENESS AND STABILITY OF OPTIMAL SOLUTIONS[Ch.
2,
2.5
2.5.3.Closed correspondences These two results give sufficient conditions for the continuity of a. We shall prove further that the minimal sets Mb(y) = {x E S(y) such that f ( x , y )
=~
(y)}
&fine a correspondence which is upper semi-continuous. Theorem 3. Suppose that
(i) the function f is continuous on RX Y , (ii) the correspondenceS is continuous on Y. (iii) the subsets S(y) are non empty and compact.
(17)
-
Thrm the function y t-.a(y) = infxcs(,,)f(x, y)is continuous and thecorrespondM6(y)= { x E S(y) such that f (x, y ) = a(y)} is upper semi-continuous.
ence y
Proof. The first statement follows immediately from Theorems 1 and 2. To prove the upper semi-continuity of the correspondence Mb, we write (18)
M ~ ( Y= ) XY)
n T(Y),
where 119)
T(y) = {.x E U such thatf(x, y)-a(y)
=s 0).
Since the functionsf and a are continuous on R.X Y and Y respectively, the subset (20)
G(T) = {{y, x } E Y X R such that x E T ( y ) }
is a closed subset of Y X R beacause, by (19), it can be written as (21)
G(T) = { b,x ) E Y x U such that f ( x , y ) - aQ
SG
0).
Therefore, Theorem 3 is a direct consequenceof the following Definition 5 and Proposition 2. Definition 5. Let T be a correspondence from a topological space Y into a topological space R. The subset
(22)
G(T)= { { y , x} E Y X R such that x E T(y)}
is called the graph of T . A correspondence T is said to be “closed” if its graph G(T)is a closed subset of YX R.
Ch. 2, § 2.51
71
STABILITY PROPERTIES
Proposition 2. Let S and T be two correspondencesmappinga topologicalspace Y into a topological space R. Suppose that
(23)
(i) the correspondenceS is upper semi-continuous at yo, (ii) the correspondence T is closed, (iii) S(yo) is compact and the subsets S ( y ) n T ( y )are non-empty as y ranges over Y.
Then the correspondence S nT :y
-
S ( y )fl T ( y ) is upper semi-continuousat yo.
Proof. We have to associate with any open neighborhood N[S(yo)nT(yo)] of S(y0)nT(y0)a neighborhood N(y0) of yo such that (24)
S(Y)n
w) c “s(yo) n
when Y E ~
~ ( ~ 0 ) i
( ~ 0 1 .
If NIS(yo)nT(yo)]contains S(yo), then (24) follows from the upper semicontinuity of S at yo. I f not, we introduce the non-empty subset (25)
K = s ( y 0 )n c N P S ( Y ~n)T ( Y ~ ) I
which is compact (since S(y0) is compact by assumption). Now, since K n T ( y 0 ) is empty by the very definition of K , then the pairs {yo,x } does not belong to G(T)for any x in K . Therefore, since the graph G(T) o f T is closed, we can associate with any x E K open neighborhoods N,(yo) and N ( x ) of yo and x respectively such that G(T)n (N,(yo)XN(x)) = 0. Hence, these neighborhoods satisfy (26)
N ( x ) nT ( y ) = 0 for any y E N,(Yo).
Since K is compact, there exists a finite sequence {XI, K such that (27)
K c N(K) =
. . .,xn} of elements xi of
UN(xi). n
i=1
Therefore (28)
T(Y) nN(K) = 0 when Y E
n
N.&o).
i=l
On the other hand, N ( K ) UN[S(yo)nT(yo)]is an open subset containing S(y0). Since S is upper semi-continuous at yo, there exists a neighborhood No(y0)
72
EXISTENCE, UNIQUENESSAND STABILITY OF OPTIMAL SOLUTIONS [Ch.
2, 0 2.5
such that (29)
S(Y) c N O UW"y0) nW 0 ) l
when Y E No(Yo).
Define the neighborhood N(yo) by (30)
1-1
Then for any y E N(yo), (31)
= NO im[Sbo)nT ( Y ~ ) I , { (ii)(i) S(Y) T(y)n N ( K ) 0 =
by (29) and (28) respectively. Thus, (31) implies (24) as required. The latter proposition is very often used in the following particular case.
Proposition 3. Any closed correspondence T mapping Y into a compact space R is upper semi-continuous. Proof. Take S to be the constant correspondence defined by S(y) = R for all
Y E Y. 0 The converse statement is also true.
Proposition 4. Suppose that a correspondence S is upper semi-continuous with non-empty compact images. Then S is closed.
Proof. We shall prove that the complement of the graph G(S) is open. Let {XO, yo} fJ G(S). Then xo does not belong to the non-empty compact S(y0). Thus there exist disjoint neighborhoods N(xo)and N[S(yo)]of xo and Solo). Now, since S is upper semi-continuous, there exists a neighborhood N(y0) of yo such that S(y) c N[S(yo)]as y ranges over N(y0). This implies that the intersection of the neighborhood N(x0)XN(yo) of {XO, yo} and of the graph G(S)is empty. We also mention the following property.
Proposition 5. Suppose that S is upper semi-continuous with non-empty compact images. I f Y is compact, then S(Y)= u y c y S ( yis) also compact.
Proof. We shall prove that any open covering {U,,},,,, of S(Y)contains a finite covering.
Ch. 2, Q 2.51
73
STABILITY PROPERTIES
First, since S(y) is compact, it can be covered by a finite number n(y) of such U,. We write
(32)
S(Y)
=
UY
=
u
lsisn(y)
ua,
Now, since S is upper semi-continuous, there exist open neighborhoods N(y) of y such that S(N(y)) c U,,. Since Y is compact, it is contained in a finite union N ( y j ) of such open subsets. Hence, S(Y)can be covered by the finite number p of such S(N(y,)) c U,,,, 1.e.
uls,sp
1('
u
=
uYJ
1sjsp
u u
ui,'
lsjsp l s i s n b J )
*2.5.4. Construction of upper semi-continuous correspondences
We begin by showing that the product of upper semi-continuous correspondences is upper semi-continuous.
Proposition 6. Let S : Y line their product T S Y (33)
-
R and T : R W by
-
W be two correspondences. We de-
u T(x).
TS(Y) =
XES(Y)
If S and T are upper semi-continuous, their product is also upper semi-continuous. Proof. This is trivial. If N is an open neighborhood of TS(yo), the subset M = {x E R such that T ( x ) c N} is open since T is upper semi-continuous. Therefore, since S is also upper semi-continuous, there exists a neighborhood N(yo) such that S(N(y0)) c M. Then, for any y E N(yo),S ( y ) c M and TS(y)C c N. Hence TS is upper semi-continuous. 0
-
We now consider n correspondences S, : Y R' with non-empty images. X S,,to be the correspondence associating with any y E Y Define S = SIX the product S ( y ) = Sl(y)X XS,,(y) c R = nZlR'.
--
---
--
Proposition 7. The product S = S I X XS,, of n closed correspondences S,, is closed. If the correspondencesSi satisfy (34)
{ (ii)( i ) Si
is upper semi-continuous, the images Si(y) are compact,
then the product S = SIX
-
XS,, is upper semi-continuous.
74
EXISTENCE, UNIQUENESS AND STABILITY OF O ~ M A SOLUTIONS L [Ch.
2,s 2.5
Proof. The first statement is trivial. Ify, converges t o y and x, E S(y,) converges to x E R, then x: E Si(yp) converges to x' E R' for any i. Since S, is closed, x' E Si(y). Hence x = belongs to S(y) = nblSi(y). We now prove the second statement. Let N be a neighborhood of the product S(y0) of compact subsets Si(yo). We can find n neighborhoods N' of S,(yo) such that
(35)
S(y0) c
IjN' c N.
i=1
Since the correspondences Si are upper semi-continuous, there exist neighborhoods Ni(yo) of y o such that Si(Ni(yo))c N'. Thus the neighborhood No(yo)= Ni(yo)of y o is mapped into N' c N . 0
nlsisn
n;=l
Proposition 8. Consider n correspondences Simapping Y into a topological vector space U.If,for all i, Si is upper semi-continuous with non-empty compact images, then the sum S y F+ S(y) = SlW+ ..*
+S,Q
.ofthe correspondences Si is upper semi-continuous.
Proof. This is left as an exercise. 0
Finally, we mention the following extension property. Proposition 9. Let
(i)D be a dense subspace of a topological space Y , (ii) X be a compact subset, (iiQ S :D -c X be a closed correspondence. Then there exists a closed correspondence (37) V X E D, S(X) = S(X).
s mapping Y into X satkfying
Proof. We have to construct S such that its graph G ( S ) is cbsed. The graph G ( S ) c D X X is closed in D X X , but not necessarily closed in Y X X . Let G(S) be its closure in Y X X . We define S by G(S) = G(S), i.e., by V x E Y, S(y) = {x E Xsuch that {x, y} E Go}. (38) Since G(S) is closed in DX X , we have
G(S) =G(S)n(D X Y) = G(S))n(DXY). This implies that S ( y ) = S ( y ) for any y E D. Furthermore, S is closed since its graph G(S) its closed by construction. 0
CHAPTER 3
COMPACTNESS AND CONTINUITY PROPERTIES
We proved in the preceding chapter that functions which are lower semicontinuous and lower semi-compact achieve their minimum. We devote this chapter to a more extensive study of these compactness and continuity requirements. We begin in Section3.1 by constructing examples of lower semi-compact functions. We prove that semi-coercive functions f (satisfying liml,xl,4J'(x)/~I x I I > 0) and functions f such that f* is continuous a t 0 are lower semi-compact.
We also study results of less immediate importance.
In Section 3.2 we study the compactness of strategy sets of the form X =
= {x E R such that Lx E Y}. Such a set can be written as X = zRG, where3tR
is the projection of R X Y onto R and G = { { x , y } E R X Y such that Lx-y = 0) is the kernel of L- 1. Thus X is compact when G is compact. We are therefore led to consider maps M from A into B for which the pre-image of any point is compact. A continuous map from a topological space A into a topological space B is said to be proper if the pre-images of points are compact and if it maps closed subsets.
After characterizing proper maps, we obtain several sufficient conditions for a map (x,y ) c- Lx- y from RXY into Y to be proper.
We devote Section 3.3 and Section 3.4 to the continuity of convex functions (Section 3.4 can be passed over in a first reading). We begin by proving that a convex function f is lower semi-continuous at xo only if we can associate with any E =- 0 a semi-norm p and a constant M such that
fixo) = ~ f ( x ) + M p ( x - x ~ ) + ~for all x and that we can take E = 0 when f is co2tinuous. In this case, we can even prove that I J'(X)-~(XO)I =s Mp(x- XO) when (x- xo) =s for a convenient rj i0. We go on to prove that a convex function is continuous on the interior of its domain when it is bounded above on some non-empty 75
76
COMPACTNESS AND CONTINUITYPROPERTIES
[Ch. 3,§ 3.1
open subset. This implies that any convex function on R”is continuous on the interior of its domain and that any convex lower semi-continuous function on a Frdchet space is continuous on the interior of its domain. We continue in Section 3.4 by showing that any lower semi-continuous convex function is continuous on the strong interior of its domain. Using estimates which can be interesting in themselves, we prove that a convex function f is lower semi-compact (for the weak topology) if and on& if its conjugate f * is continuous at 0 for the Mackey topology. We end this section by proving that any convex continuous function defined on a compact subset achieves its maximum at an “ektremal point” at least.
3.1. Lower semi-compact functions We devote thfs section to the study of lower semi-compact functions. For the sake of simplicity, we begin with the case of finite dimensional spaces and prove that semi-coercivefunctionsf (i.e. those which satisfy limllxll+w f (x)/l I x I I =- 0) and functionsf whose conjugate functionf is continuous at 0 are lower semi-compact. Also, we introduce the so-called “constraint qualificationhypothesis” 0 E Int (L(R)-Y), where R c U,Y c V and L E B(U,V). This hypothesis will be met throughout the book. It implies that, if F is a function defined on R X Y,then function p +-- F*(-L*p, p) is lower semi-compact. We extend these results to the case of infinite dimensional spaces. Finally, we embed X in a convex compact subset z ( X ) of mixed strategies and we extend a bounded functionf on X into a continuous linear functionf’ such that GC = inff(x) = inf fA(m)=fd(iii). MXiiCY)
X € X
3.1.1. Coercive and semi-coercivefunctions
We give examples of lower semi-compactfunctions. We begin with the simple case of finite dimensional spaces.
Definition 1. Let X be a subset of a normed space U and let f be a function defined on X. We shall say that f is “coercive” if (1)
lim
IIx11-t-
f (4 = + -
__ IIXII
and “semi-coercive”if (2)
Jim
llxll--
llxll
=- 0.
Ch. 3,g 3.11
77
LOWER SEMI-COMPACT FUNCTIONS
Propition 1. Let X be a subset of R".Then any semi-coercive]unction ir Zower semi-compact.
Proof. We shall prove that the sections of a semi-coercivefunction are bounded and use the fact that bounded subsets are relatively compact in R". Suppose that
Then, there exists R such that, for any
(1 XI(
3
R,
We obtain the inequality
If x belongs to S ( f , A), then
and so S( f,A) is bounded in R". 0
Example. The main examples of coercive functions are those derived from norms. Proposition 2. The functions x I-+ coercive when p = 1.
I I x-
u II p are coercive when p
Z-
1 and semi-
Proof. We can write
Hence, ~ ~ X ~ ~ ~ - no smaller than 1 i f p = 1.
~
converges ( ~ - ~ ~to U+ ~ ~when / ~ p ~=-X1 and ~ ~ to ) a~ scalar
3.1.2. Functions such that f * is continuous at 0 We now use continuity properties of the conjugate functionf * off to deduce the lower semi-compactness of$
78
COMPACTNESS AND CONTINUITYPROPERTIES
[Ch. 3, § 3.1
+
Proposition 3. Let f : R"I-+ ] - DJ, -1 be a proper function. If its conjugate functionf is continuous at 0 E R"*,then f is lower semi-compact.
Proof. I f f * is continuous at 0, there exists a ball of radius bounded by 1 :f * ( p )=s 1 when 11 p 11 * e s 7. Therefore, (6)
on which f* is
V x E S(f,A), V p such that llpll* ~ 7 ,
( P , x> Q f ( X ) + f + ( P )
=s A+
1.
Thus, V p E R"*,
Hence, Vx E S( f,A),
The sections off are bounded, and thus, relatively compact.
Remark. Proposition 3.4.5 below shows that the converse is true when f is convex and lower semi-continuous. 3.1.3. Lower semi-compactnessof linear forms Let p E R"*be a linear function on R"and let X be a nonempty subset of R"Let p x :R" F+ ] + -1 be the function defined by 00,
x)
(9)
ifx E X,
Proposition 4. I f p belongs to the interior of the positive polar subset X b of X , then p x is lower semi-compact. Proof. Indeed, since p E Int(Xb), there exists a ball B(9) of radius 9 such that p+B(q) c X b = - X", i.e. such that
(10)
vq E B(rl), q-P E X".
This amounts to saying that (11)
vq E
w, 0 a q - P )
1
and thus, that q ++ a$(q-p) is continuous at 0.
Ch. 3, 9 3.11
LOWER SEMI-COMPACT FUNCTIONS
79
But, a,#(q-p) = p:(q) because
(12)
q q - P ) = SUP ((49 X>-(P, X€X
4)= xERm sup ((4, x)-px(x>).
Therefore, p x is lower semi-compact by Proposition 3. 0 Proposition 5. Let p belong to the interior of the positive polar subset X b of X. Then Xis the union of the relatively compact subsets H(X, p , A) defined by
(13)
H ( X , p , A) = { x E Xsuch that ( p , x)
=s A}.
Proof. The subset H(X,p, A) is nothing other than the section S(px, A). 0 3.1.4. Constraint qualijication hypothesis Consider also the following example which we shall study thoroughly in Chapter 5. Let
(14)
U and V be two topological vector spaces, L E B(U, V ) .
and
-.
(i) R c U and Y c V be two subsets of U and V , (ii) F : R X Y R be a function defined on R x Y . Proposition 6. Suppose that ( 16)
0 E Int (L(R)- Y ) (constraint qualification hypothesis)
and that (17)
V is a finite dimensional space.
Then thefunction p E V*I--- F*( -L*p, p ) is lower semi-compact, where F* is the conjugate function of F dejined by
Proof. We have to prove that the subsets S(A) = { p E V * such that F*(-L*p, p ) -s A } are relatively compact, i.e. bounded, since V* is a finite dimensional space. For any z E V assumption (16) implies that there exist E =- 0, x E R and y E Y
80
COMPACTNESS AND CONTINUITY PROPERTIES
[Ch. 3,
3.1
such that z = ~ ( y - L x ) .Therefore
and thus, that S(2) is bounded. 0
We now give some examples for which theconstraint qualification assumption is satisfied. Proposition 7 . Suppose that
(21)
L(R) f ?Int Y
z
0.
Then (16)
0 E Int (L(R)-Y).
Proof. Assumption (21) implies that 0 E L(R)- Int’ Y c Int (L(R)- Y ) .
Remark. When Y = v-R!!+ c V = R”
assumption (21) is known as the “Slater condition”. It can then be written in the form (22)
32 E R such that, Q j = 1, . . ., n, Lj(2) -= y,.
Proposition 8. Suppose that R and Y are convex cones. Then 0 E Int (L(R)- Y ) i f and only if (23)
V = L(R)- Y .
In particular, this assumption is satisfied when (24)
X = U, Y = {0} and L iswjective.
Ch. 3,§ 3.11
81
LOWER SEMI-COMPACT FUNCTIONS
Proof. This is left as an exercise. 0 The constraint qualification assumption will play a crucial role for proving the existence of a solution to dual problems (see Chapter 5 and Section 14.1). It implies also that the map L' 1 from Tx X R" into V' is proper (see Proposition 3.2.10 below). Finally, it implies that the correspondence S defined by S ( y ) = { x E R such that Lx E Y- y } is lower semi-continuous (see Theorem 15.3.1).
+
*3.1.5. Case of infinite dimensional spaces
(a) Case of dual of Banach space Suppose that (25)
U = E' = 2 ( E , R) is the dual of a Banach space.
This is the case when, for instance, U isa Hilbert space. Werecall that any bounded subset M c E' is relatively compact for the weak topology of the dual B' of E (see Corollary 4 of Appendix A). The proofs of Section 3.1 can therefore be applied.
-
-
Proposition 9. Suppose that U = E* is the dual of a Banach space and that the ] is either semi-coercive or else has a conjugate f ' function f : U 3 continuous at 0. Then f is lower semi-compact when U = E' is supplied with the weak topology. 03,
+
*(b) Case of locally convex spaces Consider two paired spaces U and U'. We can extend proposition 9 when
U is supplied with the weak topology a(U, U') and 17'is supplied with the Mackey topology z(U*, U ) (the strongest topology on U' such that U is the dual of U'). Recall that the neighborhoods of U' for the Mackey topology are the polars of the symmetric compact convex subsets of U (for the weak topology).
-.
Proposition 10. Let U and U*be two paired spaces and let f : U 1- -, + -1 be a properfunction. If its conjugatefunction f ' is continuous at Oon U* supplied with the Mackey topology, then f is lower semi-compact on U supplied with the weak topology. In particular, if U = E' is the dual of a Frkchet space or, more generally, a barreled ' = E, space E, paired with U 8
82
[Ch. 3 , s 3.1
COMPACTNESS AND CONTINUITY PROPERTIES
then the Mackey topology t(V*, U) is the initial topology on E and the weak topology a(V, U’)on U is the weak topology of the dual E* of E.
*3.1.6.Extension to compact subsets of mixed strategies If X is a compact space, the subset &(X) of mixed strategies we introduced in Section 1.3.2 is not necessarily compact. We shall embed it in a larger subset JN’(X)which is compact (and remains convex). The construction of x(X)is analogous to the construction of M(X)made in Section 1.3.2. The space 8 ( x ) of all functions on X is replaced by (26)
@(X), the vector space of continuous functions defined on X
which is a Banach space when supplied with the norm
We shall assume that (28)
) is the cone of continuous positive functions, { (ii)(i) 8@E, ( Xe(X) is the constant function equal to 1;
and introduce (i) the dual @*(X) of the Banach space @ ( X ) supplied with the weak topology, (ii) the polar cone @:(X) of @ + ( X ) : @:(X) = {m € @*(A’) such that (m, f ) Z- 0 whenever f €@+(X)}. The elements of @*(X) are called “Radon measures”.
Definition 2. W e denote by M(X) the subset of @:(X) satisfying (m, 0) = 1. The set 2 ( X ) is the set of all (Radon) probability measures. Proposition 11. Let @*(X) be supplied with the weak topology. Then 2 ( X ) is a compact convex subset of @(X).The set A”(CX)is a dense subset of -@(X), and inf (m, f ) = inf f ( x ) ; sup (m,f)=sup f ( x ) . (30) m E ZW)
X € X
rnC3Gm
*€x
Proof. The function 8 clearly belongs to the interior of the convex cone @+(X). Therefore, @ + ( X )being the positive polar cone of @:(X), Proposition 5 and Proposition 9 imply that the subset (m E @:(X) such that (m, 8) I } is compact. Therefore,J?(X) is also compact, since it is a closed subset. It is obvious that B ( X ) is ’ convex.
Ch. 3, 0 3.21
PROPER MAPS A N D PREIMAGES OF SUBSQTS
83
Now any Dirac measure 6(x) defines a continuous linear form in x(X), namely f E @ ( X ) E - (d(x),f ) =f ( x ) E R. Therefore M ( X ) is contained h J(X). Moreover, M ( x ) is dense in &(X). If not, there exists rno E x(X)outside the closure of M(X)in @*(X).By the Hahn-Banach extension theorem, there then exists a non-zero f E @ ( X ) [regarded as the dual of @ * ( X )supplied with the weak topology] such that
We shall exhibit a contradiction by proving that (32)
inf f ( x ) = inf ( r n , f ) . m EJ n i ~ )
X€X
The inequality u = infx6, f ( x ) == infmEX(*)( m , f ) is obvious. It remains to prove that the inequality can be reversed. But f--uO E @ + ( X )and m E 2 ( X ) is positive. Thus (m, f - d ) = (in, f)-a(rn, 6 ) = (my f ) - u 2 0 for all m E X(X). 3.2. Proper maps and preimages of compact subsets This section is devoted to the compactness of strategy sets or subsets of allocations defined by
X = {x E R such that L x E Y } where R c U,Y c V are subsets of topological vector spaces and L E a(U,V). If G denotes the graph of L defined by G = { { x ,y } } E R x Y such that Lx- y = 0 } ,
we prove that X is the projection onto R of G and so it is compact whenever G is compact. Since G is the preimage of (0) by the map M = L- 1 from R X Y into V , we are led to introduce “proper maps” M which are closed continuous maps for which the preimages of points are compact. We begin with the following characrerimtion : A map M is proper if and only if, whenever a generalized sequence y, = Mx, converges to y E B, we can extract a generalized subsequence x , converging to x E A such that Mx, converges to y = Mx. We then show that, if R c R’is bounded below and Y c R’ is bounded above, then the map { x , y } E R X Y I-+ x - y E R‘ is proper. 8.
84
COMPACTNESS
AND CONTINUITY PROPERTIES
[Ch. 3,§ 3.2
The above result can be generalized to the case where Y is a closed convex subset containing an element w such that (Y-w) n : R = (0). This result can be improved when Y = Ziis the sum of closed convex subsets.These results will be used throughout our study of the economic models of Chapters 8 and 9. We end this section by showing that the constraint qudifcation hypothesis 0 E Int (L(R)- Y ) implies that the map (p, q } E Y* X R" L*p+q is proper.
-
3.2.1. Proper maps
Proposition 1. Let X and Y be two topological spaces and let L be a continuous map from X into Y. The two following statements are equivalent. (1)
L is closed (ie., maps closed subsets onto closed subsets) and, V y E Y , L-l ( y ) is compact.
(2)
From any generalized sequence x, of X such that Lx, converges to y, we can extract a convergent generalized subsequence.
Proof. It is easy to prove that (2) implies (1). Let B be a closed subset of X.
Let {L(x,)} be a generalized sequence of elements of L(B) converging to y. We can extract a subsequence xP,E B converging to x , which belongs to B since B is closed. Then y = L(x) E L(B) since L is co,ntinuous. Hence L(B) is closed. Next; let x, be any generalized sequence from a preimage L-lQ. Since a x , ) = y, we can extract a subsequence x,, which converges to an element x, satisfyingL(x) = y because L is continuous. Hence L-l(y) is compact. Conversely, suppose that (1) holds. Let y = lim L(x,). Then y E L(V& where V, = (x,},~,. Since L is closed and continuous, L(V,) = L(V,). Since y E L(VJ, the subsets L-l(y) n Vpare non-empty compact subsets, which have the finite intersection property. Therefore L-l(y) n V, is not empty. Any element x belonging to this subset satisfies Lx = y and is a cluster point of the sequence {x,}. 0
n,
np
n,
Definition 1. Any continuous map L from X into Y satisfying one of the two above equivalent statements is said to be &proper".
Proposition 2. Let L be a proper mapfrom X info Y.Then the pre-hage L-'(K) of any cotnpazt subset K of Y k compact.
Proof. Let {x,} be a generalized sequence of L-l(K). We can extract a convergent generalized subsequence of the sequence L(xJ which lies in the compact
Ch. 3,$ 3.21
85
PROPER MAPS AND PREIMAGES OF SUBSETS
subset K. Since L is proper, we can again extract a subsequence of {x,} which converges to an element x. This limit belopgs to L-l(K), since L-l ( K ) is closed because L is continuous.
Proposition 3. If X is compact, any continuous map &@zedon X is proper. If X is compact, the projectionfrom X X Y onto Y is proper. Proof. This is left as an exercise. 0 We also mention the following obvious properties.
Proposition 4. Let L be a continuous map from X into Y and M a continuous map from Y into Z. Then (3)
1
(i) i f L and M are proper, M L is proper,
(ii) i f M L is proper, then L is proper, (iii) if M L is proper and L is surjective, then M is proper.
€'roof. This is left as an exercise. 0 We now study several examples of proper maps.
3.2.2. Compactness of some strategy sets We shall study examples for which the map {x, y } E R X S t-.x+y is proper, where R and S are subsets of a vector space V. We use these results to prove the compactness of various subsets of allocations.
Proposition 5. Cmsider n subsets R' c R' satisfving (4)
V i = 1,
.. .,n, R'
is closed and bounded from below.
Then the map x E
nIlsian R' I---&~,~,, x' E R' is proper. In particular,
(5)
R' is closed.
lsisn
x=l
Proof. Let {X'm}mENbe n sequences of elements 2, E R' such that ym = 2; converges to an element y of R'. Since R' c E'+R!+, we obtain that, for all i, (6)
t i e x ; =ym-c jzi
XA
eym-c
v.
j#f
Since ym converges, it is bounded and thus, the sequence {2m} is bounded. We can therefore extract subsequences (also denoted by) gmwhich converge
86 to
[Ch. 3 , § 3.2
COMPACTNESS AND CONTINUITYPROPERTIES
elements
x' of R' (since R' is closed). Thus y
=
E=lx'
and the map
xi is proper.
{Xi}ls,snk-+Clsisn
Proposition 6. L e l
(7)
R be a closed subset of R' which is bounded below
and let Y be a subset of V = R' satisfving (i) Y is convex and closed, (ii) 3o E Y such that ( Y - o ) n R :
Then the map ( x , y } E R X Y
-
= (0).
x- y E R: is proper.
Proof. Let uk = xk-yk be a sequence converging to u in R', where xk E R and Y.We begin by proving that a subsequence of yk converges to y . If not, there would exist a subsequence (also denoted by) yk such that llykll
-
yk E
Letz,=-
yk
llykll
03.
belong to the unit sphere S(1) c R'. Since S(1) is compact,
there exists a subsequence (again denoted by) zk which converges to an element 5 E S(1). Since u, converges to u, it is bounded and there exists m E R' such that #k s m. Also, because R is bounded below, R c E+R:. Therefore, yk = xk-#k E-m. Hence, the inequalities z i z= ( E - m ) / ~ ~ yimply k ~ ~ that
z E.:R
On the other hand, since llykll a 1 for k large enough, and since o E Y , we have that (l/llVkll)yk+(l-l/llvkJI) o E Y (because Y is convex). Letting k tend to 00, we deduce that Z+w E
Y (because Y is closed).
We have therefore proved that Z E ( Y - o ) n R : . B y assumption (8(ii)), we deduce that Z = 0 and this is a contradiction. Thus a subsequence {yk} converges to an element y of Y (which is closed). Therefore the subsequence Xkr = uk'+yk' converges to x = u+y which belongs to X since X is closed. Propositions 5 and 6 imply the following result.
Proposition 7. Suppose that (9)
V i = 1, .. ., n, R'
c R'
and that Y satisfy properties (8).
is closed and bounded below
Ch. 3,
5 3.21
87
PROPER MAPS AND PREIMAGES OF SUBSETS
Then the map (xl, . . ., 2,y } E fl:=l R' X Y I-+ c"i = l x i - y E R' is proper. In particular, the subset Ri- Y is closed, the subset X = ( x E R' such x' E Y } is compact anditsprojectionsa' = R ' n ( Y - G # , Rj) are comthat
c;=l
c;=l
flel
pact.
Proposition 8. Consider m subsets d of R' satisfving (10)
1
i) Q j = 1, . . .,mj, Zj is closed and 0 E Zj, Zj is closed and convex, ii) Z = iii) zn -2 = (0) and ZnR: = (0).
Then the map z = {zl, . . .,z"} E
Zj I--
zj
E R' is proper.
Proof. Let uk = eel z i be a sequence of Xy=l Zj converging to u E R'. Then uk is bounded and there exists a vector d E R' such that (11)
z i = ~k
d.
We have to prove that for all j , there exists a convergent subsequence of the contrary. There exists jo such that the sequence zf does not converge. Hence
zi. Let us assume (12)
Uk
= max 11zlkll goes to
00
lejern
while the sequence y i = Zi/uk stay in the unit ball of R'. Therefore, there m, yiconexists a subsequence (again denoted by) y i such that, V j = 1, verges to y'.Then, we deduce that
-..,
(cy=l +
Since 0 belongs to the convex subset Z , then 1 /ak z i ) (1 - 1/ak)0 E Z . By taking the limit, we deduce that ' & y j E Z since Z is closed. The assumption that Z n R\ = (0) implies that Cy=l1" = 0. This can be written as
Since 0 belongs to Z j for all j , the left-hand side belongs to Z and the righthand side belongs to - Z . Hence the assumption that zn -2 = (0) implies thaty' = - ( C j f i y j ) E Z n - 2 = (0). We have now proved that V j , y i = zi/akconverges to 0.This implies that the equations ak = I I zi I I can hold for at most finitely many k for fixed j . But this
88
COMPACTNESS AND CONTINUITY PROPERTIES
[Ch. 3, 0 3.2
is a contradiction, since the very definition ofakimpliesthat such equations must hold for at least onei for each k, and hence, for infinitely msny k for somej . 0 Propositions 5 and 8 imply the following consequence.
Proposition 9. Suppose that the subsets R’ c R’ satisfv V i = 1, .. .,n, R’ is closed and bounded below (15) and that the subsets Zj c R‘ satisfv (i) V j , ZJis closed and C/mEIZJ= Z is closed and convex, (ii) ZnR: = {0}, (iii) Z rl - Z = {0}, (iv) V j = 1, . . ., m, 0 E Zj
n
(17)
f
1=1
Ri-
2Zj+ R: is closed. rn
j=1
Also, the subset
{x, z ) such that
n
rn
i=l
j=1
C xi - C zj =sw
and
*3.2.3.Examples where the map L’+ I is proper
Proposition 10. Suppose that (i) R is a subset of the dual U = F* of a Frkchet space F (i.e. metrisable and complete) (ii) Y is a subset of a barreled space V and that (21)
0 E Int (L(R)- Y ) (constraint qualification hypothesis).
Ch. 3, Q 3.21
PROPER MAPS AND PREIMAGES OF SUBSETS
89
Then the map (p, q } E Y" X R" F+ L*p+q E F is proper (when Y" X R" is supplied with the weak topology and F with the initial topology). In particular, this implies that the subsets L*Y" R* and the subsetsL*Y- +R- are closed and that the subsets { p E Y+ such that L*p-q E - Ri} are compact.
+
Proof. Since F is a Frkchet space (i.e. metrisable and complete), we have to check that for any denumerable sequence r,, = L*pn+q,, (where p, E Y x and q,, E R") converging to r in F, we can extract subsequences of p , and qn converging t o p and q. Sincep, belongs to the dual V* of a barreled space, we have to check that the sequencep,,is bounded and thus, relatively compact (see Corollary 4 of Appendix A). By assumption (21), we can associate with any z E V an E =- 0, x E R and y E Y such that z = ~ ( yLx). - Hence
/'
p\
n,
-v z\
= ( p n , y - ~ x )= (-L*pn, X ) + ( p n , Y>
= ( q n , x)+ ( P n , Y)- ( r n ,
=s 2-(rn,
X>
x) e 2+c
since (q,,,x ) -G 1, (p,,, y ) 1 and l(r,,, x)I 4 c because any convergent denumerable sequence is bounded. Therefore, we can extract from p,, a generalized subsequence which converges to an element p E Y". Hence, we can extract from q,, = r,,-L*p, a generalized subsequence which converges to an element q of R'. 0 When U is no longer the dual of a Frtchet space, we have to strengthen the constraint qualification hypothesis (21).
Proposition 11. Let {U, U*} and {V, V * }be two paired spaces,L E 4 U ,V)and R C U , Y C V.Ifweassumethat
n Int ( Y ) f 0, then themap Cp, q } E Y" x R" iL*p+q E U*isproper (when U*,V* aresup-
(22)
L(R)
plied with the weak topologies and V with the Mackey topology).
Proof. Let re = L*pP+qP(wherep p C Y* and q, E R") be a generalized sequence converging to r in u*. Let x E R have the property that L x = y belongs to the interior of Y. We can write (r4, x ) = (L*PP+9P, 4 = ( P P , LX)+(9P, x ) = ( P P , r)+( 9 P , 4.
'90
[Ch. 3, Q 3.3
COMPACTNESS AND CONTINUITY PROPERTIES
On the other hand, there exists po such that, for any p z==PO, [(rp,x>- ( r , x ) I =s E . Since (q,, x ) =s 1 (for q, E R* and x E R), we deduce that (23)
(pp, - y ) =s 1-(r,
X)+E
whenever p
PO.
Since - y belongs to the interior of - Y c (Y")b, we infer from Proposition 1.5 that the subset of p, satisfying (23) is compact. We can therefore extract a subsequence (again denoted by) p , which converges to p E Y*. Thus qg = = r,- L*pp converges to q E R". *3.3. Continuous convex functions Consider a topological vector space U whose topology is defined by a family
r of semi-norms. A function f is lower semi-continuous at xo only if we can associate with any z 0 a semi-norm p E r and a constant q w 0 such that E
f
(XO)
4
f (x)+
E
whenever p(x-
XO) e r).
Iff is also convex, we deduce that &
f (x0) e f ( x ) + -p(x- xo) whenever P ( X r)
XO)
rl
When, in addition, f is continuous, we obtain that &
f(xo) sf(x)+-p(x-xo) r)
for all x E U
and ] f (x0)- f ( x ) I
==
-5 p(x-
r
XO)
whenever p(x-
XO)
r).
In fact, we prove that a convex function is continuous on the interior of its domain if and only if it is bounded from above over some non-empty open subset. This implies that a convex function f is continuous on the interior of its domain when U = R' or when U is a Fr6chet space and f is lower semi-continuous. We end this section by proving that, if V is supplied with the seminorms Lp (where L E a ( U , V ) ) ,a function g is continuous on V if and only if gL is continuous on U.Also, iff is a continuous convex function, we prove that Lf is continuous and convex. 3.3.1. A characterization of lower semi-continuous convex functions Proposition 1. Suppose that a convex function f : U .+ 1- 00, -i--1 is lower semicontinuous at xo E Dom f. Then, for any E w 0, there exists a semi-norm p
Ch. 3, 0 3.31
CONTINUOUS CONVEX FUNCTIONS
91
Proof. We have that f i s lower semi-continuous at xoif and only if (l(i)) is satisfied. In order to prove inequality (1(ii)), we assume that p(x- XO) =- 7 and consider y = Ox+(l-O)xo where 8 = q/p(x-xo) < 1. Then, p(y-xo) = dp(x-xO) = 7. We deduce from (I(i)) and from the convexity off that
f ( x o )e f ( y ) + E
ef(x)+(l-w(xo)+E.
Hence E
f(x0)
s f ( x ) + i =f(x)+-p(x-xo). 11
0
3.3.2. A characterization of continuous convex functions In the case where f is continuous, we obtain the following inequalities. Proposition 2. A convex function f is continuous at a point xo E Dom (f)if and only i f f is bounded above on a neighborhood of XO. In this case, there exists a semi-norm p and a constant c =- 0 such that
(i)f(xo) =sf (x)+cp(x- xo) for all x E U, (ii) f (x) =S f (xo) -i-cp(x- xo) when p(x- XO) =S 7. Furthermore, f is continuous on the interior of its domain.
Proof. It is obvious that a functionfcontinuous at xo is bounded above on an open neighborhood of XO. In particular, this implies that xo belongs to the interior of the domain off. Conversely, let us assume that there exists a semi-norm p and scalars q > 0 and a > 0 such that (3)
f ( y )e a
whenever p(y- xo)
=S
v.
(a) In order to prove inequality (2(i)), we associate with any x E U the element
92
COMPACT"
AND
CONTINUITY PROPER^
-
[Ch. 3 , s 3.3
n u s p ~ - x o ) = q since y -x o = ((1 -e)/e) (x-x0) = ( q / p + - ~ ) ) ( x - x o ) , We deduce from (3) thatfb) 6 cl and from the convexity offthat (5)
HenCe
f(x0)
= f (@+(1 - W)
ea+ (1 -
e)f(x).
(1 -e)f(x0)
6
Of
o+(1 - e)s(x)
e (a-f(xo)) +(1 - e)f(x).
Dividing by (1 -e), we obtain that
(6)
f ( x 0 ) aG
e 1-8(a- f O )+ f ( x )
= a-f(xo) p(x - xo) +f(x).
11
We have therefore proved (2(i)) with c = (a-f(xo))/r]. (b) In order to prove inequality (qii)),we associate with any x satisfying p(x- XO) 4 r] the element
(7)
Y=
where 6 = p(x0-x)
x-(l-e)xO
e
1.
7
Thus p(y--x0) = 7 sincey-x0 = (x-xo)/O = (q/p(x-xo))(x-xo). Wededuce from (3) thatfb) 6 a. The convexity off implies that
f ( 4=f(eY+(l-Qxo) 6 e(a-f(xo))
4
ef(v)+(~-~)f(xo)
+f(xo) = a-f (xo) p(x- xo)+.f(xo).
7
We have therefore (2(ii)) with c = (a-f(xo))/r]. Inequalities (2(i) and (ii)) imply that (8)
If(x)-f(xo)l
4
a-f rl(xo) p(x-
XO)
whenever p (x-
XU) -G
r]
and thus, that f is continuous at XO. (c) We, now prove that f is continuous on the interior X = Int Dom of the domain off: Let XI belong to Int Dom (f).To prove that f is continuous at 'xl it is enough to show that
(n
(9)
fQ-G b whenever p(y-
xl) s A7 and I Z- 0.
Since x1 E Int Domf, there exists 1 -= 1 such that
Ch. 3,# 3.31
CONTINUOUS CONVEX FUNCTIONS
93
belongs to Domfsince (1 1)
A p(x2- x1) = -p(x1- xo). l-A
Let y satisfy p(y- x l ) c Aq. We associate with y the element 1 1 2 = ++;IXo-x$ =T(y-(1-Ab2). (12) Notice that
(13)
1 p(z-xo) = ; I P ( Y - X d
6 11.
By (3), we deduce that f(z)r; a. Hence the convexity off implies that
f 0 =f &+(I - W 4
2 )
Q
Y(z)+ (1 - A ) f (xz)
Aa+(l-A) f ( x 2 ) = b.
Thus property (9) is satisfied. 0 0
3.3.3. Examples of continuous convex functions The last result has the following important consequences.
Proposition 3. Any convex function f defined on an open subset X of R" is continuous on X. Proof. Let xobelong to X and let N(x0) c X be a neighborhood of xo contained in X. We can find n elements x, E N(x0) such that the vectors X,-XO are linearily independent. The function f is bounded above on the convex hull K = co (XO, XI, . . ., xn) because
Since the vectors x,-xo are lineady independent, the interior of K is nonempty and thus contains a neighborhood of XO. By Proposition 2, the function f is continuous on X. 0 In the case of Frkhet spaces. we obtain the following result. Proptsition 4. Suppose that
(14)
U is a Frkchet (complete metrizable) space.
Any convex lower semi-continuousfunction is continuous on the interior of its domain.
94
COMPACTNESS AND CONTRJUITY PROPERTIES
[Ch. 3 , $ 3.3
Proof. Suppose that the interior of the domain X = Dom f is non-empty. By Proposition 2, we have to prove that f is bounded on an open subset of X. For this purpose, we consider the closed subsets F, = { x E X such that f ( x ) r~ =sn}. Then X = u;=lFn. IfXhas a non-empty interior, the Bake theorem1 iinplies that at least one of the closed subsets F, has a non-empty interior fin. Cl
The study of,continuous convex functions is continued in the next section. We conclude this section with a study of the continuity of the functions gL and LJ 3.3.4. Continuity of gL and Lf
Consider the case when L E' B(U,V ) is surjective. If 4 is a semi-norm on U it is clear that Lp is a semi-norm on V. If U is a locally convex space whose topology is defined by a family P of semi-norms p, we shall supply the space V with. the topology defined by the family LT of semi-norms Lp. This topology is called the image by L of the topology of u. The operator L is clearly continuous for such a topology. Proposition 5 . A map g from V into a topological space Z. is conlinuous if and only i f the map gL from U into 2 is continuous. Proof. It is clear that, ifg is continuous, then gL is also continuous. Converse&, assume that gL is continuous from U into 2.We prove that g is continuous at ya E V, i.e. that we can associate with any neighborhood N of g(yo)in 2 a semi-norm p of P and r] =- 0 such that (15)
Lpdy-yo) s 7 implies that g(y) E N.
But we know that gL is continuous at a point xo satisfying LXO= yo. Thus there exists p E r and r] such that (16)
p(x-
XO)
=S
2-q implies that g(Lx) E N.
Now, we can associate with any y satisfying Lpb-yo) =s4 an element x satisfying (17) Lx = y and p(x- X O ) =G Lp(y-yo)+r] == 2.7 Therefore inequality (15) follows from (16) and (17). 0 The Baire theorem states that in a complete metrizable space (or a locally compact space), if the union of a denumerable sequence of closed subsets F, has a non-empty interior, then the interior of at least one of the subsets F, is non-empty.
Ch. 3, § 3.41
95
CONTINUOUS CONVEX FUNCTIONS
Next we prove that L f is convex and continuous whenever f is convex and continuous. Proposition 6. Suppose that f is convex and continuous at a point xo satisfying LXO= y o . Then L f is convex and continuous at yo. Proof. By assumption, we can associate with any and 7 =- 0 such that
(1 8)
E
z0
a semi-norm p on U
f (x) = s f ( x o ) + & whenever p ( x - X O ) =s 7.
On the other hand, we can associate with any j j an element 2 satisfying
(19)
+
L2 = y ; p ( 2 ) =s Lp(jj)+ q.
Now, by taking ji = y-yo where p(y-yo) =s +q, we can find an associated x = 2 + x o which satisfies L x = y and p ( x - X O ) =s2;. r]. = r]. Therefore, Lf(y) sf ( x ) - - ~ ( X O ) + E
whenever Lp(y-yo) ~ $ 7 .
Since Lf is convex, this implies that Lf is continuous at
XO.
0
*3.4. Continuous convex functions (continuation)
In this section we continue the study of continuous convex functions. (The section can be passed over in a first reading.) Using the characterization of continuous convex functions of the preceding section, we deduce that any convex lower semi-continuous function is continuous on the interior of its domain for the strong topology p(U, U*)of U . We shall prove the following important characterization : a convex function f is lower semi-compact if and only if its conjugate functionf* is continuous at 0. This implies, for instance, that a lower support function 0; is continuous at PO if (and only if) the subsets { x E K such that (PO,x ) =s A} are compact. To prove the above result, we obtain the following estimates for a lower semicontinuous convex function :
f ( x )= = ~ ( X O ) + C Z ( X - - O )
iff(x) s f ( x o ) + c ,
f ( x ) a f (xo)+ c z ( x -
iff(x) a f (xo) c,
+ where z is the gauge of the subset {x E U such that f (xo+x ) =sf (xo)+c}. xo)
We continue this section by defining the extremal points of convex subsets and by proving that a continuous convex function defined on a convex compact
96
COMPACTNESS AND
comm~y PROPERTIES
[Ch. 3,g 3.4
set achieves its maximum a t an extremal point (at least). Finally, we prove that any convex compact subset is the closed convex hull of the set of its extremal points. 3.4.1. Strong continuity of lower semi-continuous convex functions
' be two pairedspaces. Thenany lower semi-continuous Propo&on 1. Let U and U function (for any topology compatible with the pairing) is strongly continuom on the strong interior of its domain Dom ( f )provided this is non-empty. Proof. Recall (see Section 4 of Appendix A) that strong neighborhoods of U are convex symmetric (weakly) closed and absorbing subsets of U.Therefore, if xo belongs to the strong interior of Dom (f), then, for any x E U,there exists 8, > 0 such that xo+8x E Dom(f) whenever 8 =sax,i.e. such that (1)
f (xo+Ox)
+
whenever 8 4 Ox.
We set
where c =- 0. Note that M is a closed convex subset which contains 0, sincef is convex and lower semi-continuous. If we prove that M is absorbing, then M n -M is a strong neighborhood (being closed, convex symmetric and absorbing) on which f is bounded above. This implies that f is continuous at xo by Proposition 3.2. To prove that M is absorbing, we have to associate with any x E U a scalar qx > 0 such that (3)
f(xo+ ex) Gf (xo)+c whenever 8 4 qx.
By (l), the function 8 -t f (xo+ Ox)- f (XO) is defined on the subset ] - t9-x, Ox[ of the real line and thus is continuous at 0 by Proposition 3.3. Thus there exists qx z 0 such that I f (xo+Ox)- f (xo)I 4 c whenever I 8 1 e qx, i.e. (3) holds. 0
Proposition 2. Let U be a barreled space. Any lower semi-continuous convex function is continuous on the interior of its domain. Proof. If U is barreled, the strong topology p(U, U') coincides with the initial topology (see Corollary 3 of Appendix A). Therefore, Proposition 2 is a consequence of Proposition 1. 0
Ch. 3,
0
3.41
CONTINUOUS CONVEX FUNCTIONS
97
3.4.2. Estimates of lower semi-continuous convexfunctions We shall need the following result for characterizing continuous convex functions. Proposition 3. Let f be a lower semi-continuous convexfunction and xo € Domf. Let M be convex subset deJined by (4)
where c hold.
M = { x E Usuch thatf(xo+x)ef(xo)+c} > 0. Let
a
= zM be the gauge of
M. Then the following inequalities
(i)f ( x )e f(xo)+cn(x--co) whenever f ( x ) =G f (xo)+c, (ii) f ( x ) f (xo)+cn(x- X O ) wheneverf(x) a.f (xo)+ c. Proof. By setting 4(x) = f(x+xo)- f(xo), we can write M = {x E U such that 4(x)e c}. If d(x) =sc, then a(x)6 1. Thus
since t$(x/z(x))= c. If 4(x) =- c, then z(x) w 1. Thus
Therefore, &(x) ca(x) whenever +(x)=- c. We obtain inequalities (5) by taking x = y-xo since 4(x) =f Q-f(x0). Proposition 4. Let f be a lower semi-continuous convexfunction which is bounded below. Let a = ( I f(x)and xo belong to Domf.Let (8)
M = {xE U~~chthatf(xo+x)~f(~o)+~}.
Then, ifn = zM denotes the gauge of M , we have that (9) 9
for any x E U, ] ( x ) z- cz(x-xo)-c+a.
98
COMPACTNESS AND
comm~y PROPERTIES
[Ch. 3, $ 3.4
Proof. We set 4(x) =f ( x o + x ) - f ( x o ) . Then M = {x E U such that 4(x) =s G c}. If 4(x) =s c, then c(z(x)- 1) = ca(x)-c == 0. Since a-f(xo) = = inf,,, 4(x), we deduce that cz(x)- c + a - f (xo) =s 4(x) if $(x)
If 4(x)
G
c.
=- c, it follows from (7) that cz(x) =s4(x). Since a- f
- c =z 0, we conclude that cn.(x)-
c+a- f (xo) =s4 ( x )
(x0)-
c =s 4(x)-
if 4(x) e c.
Thus, setting x = y-XO, we obtain (9). 0 3.4.3. Characterization of continuous convexfunctions Proposition 5. Let U and U* be two paired spaces and let f be a lower semicontinuous convex function from U into ]- -, -]. The two following statements are equivalent.
+
(1 0)
(i)f is continuous at xofor the Mackey topology t ( U , U*) (ii) the sections S(f *, XO, A) = (p E U ' such that f *(p)- ( p , X O ) 6 + A } are weakly compact.
Proof. We begin by proving that (lO(i)) implies (lO(ii)). Let N be a neighborhood of 0 E U such that f ( x o + x ) s 1 whenever x € N. Let K = N# be its polar subset, which is weakly compact by the very definition of the Mackey topology. If p belongs to S(f*,XO, A) we deduce from the relation (1 1)
(P,xo+x) a f W + f ( x o + x )
that (12)
(p, x) s A+ 1 whenever x € N.
+
Therefore, (12) implies that p belongs to (A 1)K. Conversely, suppose that the non-empty subset S = S(J*,XO, A) is contained
in a compact subset of U*. We first show that
is finite. Since S is bounded and
f *(PI- ( p , XO) a ( p . x-
x0)-
f(x) if x E Dom f
Ch. 3,
5 3.41
99
CONTINUOUS CONVEX FUNCTIONS
it follows that inf ( f * ( p ) - ( p , XO))
P€S
-
inf [(p, XO)-
P€S
f(x)] =- -
00.
Therefore, xo E Dom (f).Now let POE S and let M be the closed convex subset defined by (14)
M = {p E U* such thatf(p0fp)-(p, XO) =Z f*(po)+c} = S(f *, XO, +PO where c = A+(po, xo)-f*(po).
By assumption, M is contained in a compact subset K. From Proposition 4 with x t - f ( x ) is replaced by p F+ f *(p)- ( p , X O ) , we obtain that (15)
ca#(K" ;p-po)- c - ~ ( x o e ) ca"(M# ;p-po)- c-~(xo) f *(PI- ( p , XO) whenever p E U*,
since the gauge n(K;p) = aX(K"; p ) of K is smaller than or equal to the gauge a ( M ; p ) = a#(M"; p ) of M and -f (XO) = inf,,, [f *(p)- ( p , XO)]. Therefore, for any x E U and p E U*, we have that (16)
(P,xo+x)-f*(p)
(p, x ) - c ~ # ( K " ;p-po)+c+f(xo).
Taking the supremum over U*, we deduce from (16) that
Therefore, the function x I--.f ( xo + x) - (PO, x ) is bounded above on a neighborhood K # of 0 and is therefore continuous at 0. This implies that .fis continuous at xo. 0 3.4.4. Continuity of support functions
We apply the preceding result to the support function and indicator of a closed convex subset K.
Proposition 6. Let 17and U* be two paired spaces and let K c U be a closed convex subset. The lower supportfunction 0; is continuousatpo E U*(suppliedwith the Mackey topology z(U*, U ) ) i f and only i f the subsets {x E K such that (PO,x ) e A} are compact (when U is supplied with the weak topology a(U, U')). An element xo E K belongs to the interior of K (for the Mackey topology z(U, U')) if and only if(K-xoJ* iscompact (for the weak topology b(U*, U)). 9-
100
COMPACTNESS AND C O ~ N U I T YPROPERTIES
[Ch. 3 , s 3.4
Pmf. To prove the first statement,we apply Proposition 5 to the function p F .+ a g ( - p ) =-c&p). This function is continuous if and only if the sections { x E U such that y ~ ~ ( x ) - ( - px,,) , e A } = { x E K such that (PO, x ) 6 A } are compact. To prove the second statement, we apply Proposition 5 to the function fpar. This function is continuous at xo(i.e. xo E Int if and only if the sections (p € U’ such that a*(& p)- ( p , x,,) = a+(K- XO, p ) -G A } are compact, i.e. if and only if the subsets ~(K-xo)*are compact. 0
(a)
3.4.5. Maximum of a convex function: exlremal points
Definition 1. Let X c U be a convex subset of a vector space U.A point x E X is said to be an extremal point of X if, whenever elements y and z of X satisfy x = Ay+ (1- A)z, where A€ [0, 11, then either A = 0 or 1 = 1. We prove next that a convex continuous function achieves its maximum at an extremal point. Proposition7 . Let U be an Harrsclfl locally convex space, X a non-emp@ compact convex subset of U and f an upper semi-continuous convex function d e m d on X. Thenf achieves its maximum at an extremal point.
Proof. (a) To prove this theorem, we shall use Zorn’s lemma applied to the family F of subsets K of X satisfying (1 8)
K is non-empty, closed and x E K,y , z E X and x = Ay+(l-A)z implies y and z belong to K.
This family F is non-empty because the set X belongs to F. Of course, an element x is extremal if and only if the subset { x } belongs to F. The family F ordered by 3 is inductive: Let C = { K f } f Ebef a chain. Then K = Kf is a lower bound of C. The subset K is non-empty and closed because { K f } is a decreasing sequence of non-empty compact subsets. If x E K can be written x = Ay+(l-A)z, then y and z belong to K. Hence K belongs to P and Zorn’s lemma implies that
nfEf
(19)
any non-empty subset of F contains a minimal element.
(b) Recall that
M + M , f ) = { x € K such thatf(x) = sup fb)} YEK
Ch. 3,g 3.41
CONTINUOUS CONVEX FUNCTIONS
101
We prove that (20)
If K E F, then M + ( K , f ) belongs to F.
Now, M y ( K ,f)# 0 because f achieves its maximum on K, being upper semicontinuous on the non-empty compact set K. Clearly, M'(K, f)is closed. It remains to check that, if x E M+(K, f)is written x = Ay+ (1 - A)z, then both y and z belong to M * ( K , f ) . Sincef is convex, we have that
This implies that f ( y ) = f ( z ) = suprEKf(y). Since K belongs to F, we know that both y and z belong to K. Thus y and z belong to M + ( K , f ) . (c) Now, we use properties (19)and (20)to prove that (21)
K E F is minimal if and only if K
= {x} contains a single element
x (and is therefore an e x t r e d point).
It is obvious that, if {x}isextremal, then it is a minimal element of F. Conversely, suppose that a minimal K in F contakp two distinct points x # y. Then there exists a continuous linear form p E U* such that (p, x ) < ( p , y). But then M*(K,p) is a proper subset of K which lies in F. This is a contradiction. (d) Finally, properties (19), (20) and (21) imply the theorem. Given any upper semi-continuous convex functionf, the set M*(X, f) belongs to F by (20)and contains an extremal point x by (21). Such an extremal point x satisf i e s m = WPyexf(Y). 0 Proposition 8. Let X be a compact convex subset of an Hausdorf locally convex space U . Then,for any p E U*,there exists an extremal point x E X satisfying
(22)
u#(X;P ) = ( p , x).
Finally, we obtain the fundamental result of Krein and .Milman. Proposition 9 (Krein-Milman). Let X be a convex compact subset of an Hawdo@ localkj convex space U. Then X is the closed convex hull of the set of its extremal points. Proof. Let Y be the closed convex hull of the set of the extremal points of X. Then Y c X. If Y # X, there exists x E X such that x $ Y . By the Hahn-
102
COMPACTNESS AND CONTINUITYPROPERTIES
[Ch. 3 , s 3.4
Banach separation theorem, there then exists a non-zero continuous linear formp E U*such that
(23)
SUP ( P , Y€Y
v) -= ( P , ).
=sSUP (P,x ) X€X
= a+(K P)
This is impossible since o"(X, p ) = (p, y ) for at least one y E Y by Proposition 8. 0
CHAPTER 4
DIFFERENTIABIJJTY AND SUBDIFFERENTIABILITY: CHARACTERIZATION OF OPTIMAL SOLUTIONS The aim of this short chapter is to introduce the various concepts of differentiability in the framework of optimization theory. We will characterize in Section 4.1 the minimal set of a convex function f as the subdgerential af *(O) of its conjugate function f at 0. The subdgerenlial of a functionf at xo is defined as the (possibly empty) subset af (xo) = (p E U*such that f (xo)- f ( x ) 6 ( p , x 0 - x ) for all x E U}.
It is not a stringent concept: we prove that any convex faction f continuous at xo has a non-empty compact subdgerential Elf (xo). We show that when f is convex and lower semi-continuous the correspondaf ( x ) and af * :p af *(p) are inverse. Furthermore, we ences af :x af ( x ) is an uppersemi-continuouscorrespondenceon the interior prove that x of the domain ofJ In the case of convex functions, the subdifferential af (xo) is a generalization of the concept of gradient Of(xo). When the gradient exists, it is a continuous, linear functional satisfying
-
-
-
for all x. We prove, in fact, that iff is convex and differentiable at XO, then af(x0) = = {~f(xo)l. More generally, if a function f is both dx$Terentiable and subdgerentiable at xo, then af(x0) = (Df(x0)). In this case, we prove that xo E X minimizes f on X if and only if
(Df(xo), X O )
-
(Df(xo), x) for all x E X .
This latter minimization problem can be considered as a “linearized minimization problem”, although, as opposed to the linearized extension to the mixed 103
[Ch. 4
DIFFERENTIABILITY AND SUBDIFFERENTIABILITY
104
strategy subsets constructed in Section 1.3.2 and Section 3.1.6, the linearized functional Of(xo)depends upon the (unknown) minimal solution xo. We also observe that, when f * is differentiable at p , 3 = Df ‘ ( p ) is the unique solution minimizing x t-- f ( x ) - ( p , x). We may therefore interpret E as a “marginal profit” wheneverf is regarded as a loss function and f ’ as a profit function. In the next section, we introduce the “intermediate” concept of a function dflerentiablefrom the right. The derivative from the right off at xoin the direction x is defined by
The main esult states that any convex function continuous at xo is differentiable from the right and that D~(xo)(x)is nothing other than the upper support function of the subdifferential, i.e. Df(xo)(x) = u y a f (xo);x ) . The main justification of this concept lies in the following result. Under appropriate assumptions, the pointwise supremum A x ) = suppdpf (x; p ) of a set of differentiable functionsf (-;p ) is differentiable from the right and
where P ( x ) = E P such that g(x) =f(x, p ) } . Finally, in the last section, we introduce a very weak concept of differentiability: we shall say that f is locally &differentiableat x if there exist q z 0 and p E U* such that ,f(E)
< . I ( x ) + ( p , ~ - - x ) + ~ ~ ~ ~whenever - x I l lIZ-xlI(
=S
7.
Tzlis concept is of interest because a lower semicontinuous function on a Hilbert space U is 1oGally E-subdgerentiableon a dense subset of U. This is quite a powerful result, which we will use for studying families of perturbed problems
as p ranges over a Hilbert space U’.We shall prove that, if the perturbations are “smooth” and the functions x ++f (x, p ) are lower semi-continuous, then a unique minimizing x exists for “almost all” p (i.e. for every p belonging to a dense denumerable intersection of open subsets). This is obviously of interest, since if a minimization problem a(p) has no unique optimal solution, we may simply choose a small enough perturbation, insignificant for practical purposes, such that the new problem has a unique solution.
Ch. 4, Q 4.11
SUBDIFFERENTIABILITY
105
4.1. Subdifferentiability 4.1.1. Definitions In order to characterize the minimal set of a proper functionf defined on a topological vector space U, we need the definition of the subgradient of a function f at a point xo.
Definition 1. Let f be a proper function defined on y topological vector space U and let xo E DomJ The “wbd~erential”off at xo is the (possibly empty) subset af(xo) of the dual U* defined by af(xo) = { p E U+ such thatf(x0)- f(x) S. (p, x0-x) for all x E U } .
(1)
We call the elements p E af(xo) the “subgradients” off at XO. We say thatfis subdflerntiable at xo if af(xo) is non-empty. We define also the “superdflerentiul” af(x0) by (2)
8f(x0) = {p E U* such that f(x0)- f(x) a (p, XO-x) for all x E U}. = - a(-.f) (x0).
The following is obviously an equivalent definition for af(x0).
(3)
p E x
ifand only iyxo minimizes f(x)- (p, x) on U
af(x0)
+
The first justification of the concept of subdifferential is its relation with the minimal set of a function.
Proposition 1. Let f be a proper function defined on a topological vector space U. Then xo minimizes x F+ f (x)- (po, x) on U (4) i f and only if (5)
xo E af*(Po), { (ii)(0 f(x0) f**(xo). =
Proof. Suppose that (4) holds. Then the inequalities .f(Xo)-
(Po, XO) ==f( 4 - (Po, x)
106
DIFFERENTIABILITY AND SUBDIFFERENTIABILITY
[Ch. 4, g 4.1
imply that
(6)
f '(Po)
= (Po7 xo)-f(xo).
But, for all p E U',
?(PI
3
(P, Xo)-f(Xo).
We deduce that
(7)
f *(Po)-f+(P)
==
(Po-P, xo>
(i.e. that xo E af*(po)).Furthermore, (7) implies that
f '*(xo) = (P7 XO) --f+(Po)
(8)
and thus, by (6),f (xo) = ~ " ( x o ) . Conversely, suppose that ( 5 ) holds. Since xo E af*(po), ?(Po)-
0709
x d ef*(d(P7
and we deduce that
f **(xo)-
(PO, XO) = -f*(po) = inf [ f ( x ) - ( p , XE
u
41.
Since f (xo) =f**(xo), we have proved that xo minimizes xt--. f (x)-(p, x ) on U.0 We can reformulate Proposition 1 as follows. Proposition 2. The minimal subset of: x -t (9)
{X
f (x)- (p, x ) is equal to
E a f ' ( p ) such thatf(x) =f "(x)}.
We deduce from Theorem 2.4.1 the following consequence. Proposition 3. Suppose that f is a proper convex lower semi-continuousfunction. Then the minimal subset of the perturbed function x :I - 4 f (x)- ( p , x ) is equal to af *(PI. This result has the alternative formulation (1 0 )
p o E af(xo) if and only i f x 0 E af*(po)
i.e. the correspondence af * is the inverse of a$ 4.1.2. Examples of subdiflerentials
Let X be a non-empty convex subset of a vector space U and let yx be its indicator. Then p belongs to ayx(x) if and only if y x ( x ) =S yx(y)+(p7 x - y )
Ch. 4,g 4.11
107
SUBDIFFERENTIABILITY
for all y E U,i.e. if and only if (i) x E X , (ii) (p, x - y )
Z-
for ally E X .
Definition 2. We shall say that the subdifferential ayx(x) (when x E X) is the normal cone to X at x.
Proposition 4. Let X be a non-empty dosed convex subset of U . For each p E .U*, au,#(p) consists of the elements x (if any) where the linear function x I--- (p, x ) achieves its maximum on X . Proof. This follows from Proposition 3 withf = yx since f *
= uz. 0
Next we characterize the subdifferential of a function x ++-@(I on a normed space.
I x 1 I) defined
Proposition 5. Consider the convex function f ( x ) = @(I I x I I), where @(t) = = j:,(P(;l)dA (and satisfies (2.4.15)). Then its subdgerential is equal to
(12)
a f ( x ) = { p E U*such that ( p , x ) = IIpII; IlxlI and IlPllL = 4
Proof. Suppose t h a t p E U* satisfies ( p , x ) = llpll* llxll and Then, since +(t) = (DD)(t), we have that
ll~ll~=4~llxll).
(P, x ) = IIpll*llxll = llxll r n ( I l X l l ) =
@~ll~ll~+@*(~~ll~ll~)
=@
~ l l ~ l l ~ + @ * ~ l l=p fl(lx*)~+ f ’ ( p ) .
This implies that p E af(x). Conversely, suppose that p E af(x). Then (13)
@(llxlI)-@(Ilull)=s(P,X - Y )
Ifwe takey = llxl\(z/\\zll),then llyll = (
for all Y E
u.
( ~ ( 1 . We deduce from (13) that
108
DIPFEREN~AB~LIT~ AND SUBDIFFERBNTUBILITY
[Ch. 4, # 4.1
To prove that IIpII+ = $(llxll), we takey = (l+e)x where t9 € 1-1, +l[. Then (13) implies that (16)
O(PY x ) -@W
+fa1 I) -*I XI
I x II).
Therefore, for any 8 E 10, I[, we deduce that
Proof. By Proposition 5,
4.1 -3. Subdjgkrentiability of continuous convex functions
Proposition 7. Suppose that a convex proper function f is continuous at a point x of the interior of its domain. Then af (x)is non-empty and weakly compact. Proof. Since f is continuous at x , GpJhas a non-empty interior .(ifN(x) is an open neighborhood contained in Dom f , then N ( x ) X [ c , c+ 1[ is an open subset contained in dpcf) (providedthat c is an upper bound off on N(x)). Since {x, f ( x ) } does not belong to the interior of the convex set Ep (f),the Hahn-Banach separation theorem implies that there exists a non-zero continuous linear form (p, a } € U*XR such that
(49)
VY E
u, V c aff(~), (p, y)-ac
6
(P,x ) - ~ ( x ) .
Ch. 4,§ 4.11
SUBDIFFERENTIABILITY
109
Taking y = x and c =f (x)+1, we deduce that a 0. Actually, a =- 0 since a = 0 implies that ( p , y - x ) e 0 for all y E DomJ This implies the contradiction p = 0 because x E Int DomJ Taking c = f (y) and dividing both sides of the above inequality by a =- 0, we deduce that f '(p/a) @/a, x)-f(x), i.e. that p/a E af (x). We next prove that the closed convex subset af (x)is compact in U*(for the weak topology o(U*, U)). For this purpose, we note that
af (x) = { p E U*such thatf *(p)- (p, X )-s -f (xo)} (20) is a lower section off *. Since f = (f')' is continuous at x , Propositions 3.1.3 and 3.1.10 show that p t- f ( p ) - ( p , x ) is lower semi-compact. Thus af(x) is relatively compact. We shall prove (Proposition 13.2.10) that, in Hilbert spaces, the set X of elements x E Dom ( f ) such that a f ( x ) z 0 is not empty and that its closure is equal to the closure of Dom 0. 4.1.4. Upper semi-continuity of the subdirerentid
-
Suppose that a convex function f is continuous on an open subset X. Then the correspondence x i3f ( x ) is an important example of an upper semicontinuous correspondence.
Proposition 8. Suppose that a convex function f is continuous on an open subset X of a vector space U (for the Mackey topology z(U, U)'). Then the correspondence x af (x)from Uinto U*(supplied with the weak topology o(U*,U)) is upper semi-continuous. -+
Proof. We shall prove that the correspondence af maps a neighborhood xo+N of xo into a compact subset K of U* and that the restriction of af to xo+N has a closed graph. Hence Proposition 2.5.2 implies that the restriction of af to xo+N is upper semi-continuous. Since X is open, this implies that af iS upper semi-continuous on X. Sincef is continuous at x,, there exists a neighborhood x0+2K" of xo such that (21)
If(xo)-f(y)I==+ forallyExo+2K#
where K is a symmetric convex compact subset of U*.(Recall that the neighborhoods of U are the polar subsets of the weakly compact subsets of U'.) Now,,if x E xo+K*, p E a f ( x ) and z E KX,then y = z+x E x0+2K* and
110
DIFFERENTIABILITYAND SUBDIFFERENTIABILITY
[Ch. 4,s 4.1
we obtain (22)
- 1~f(x)-f’f(Xo)+f’(Xo)-f(X+Z)
(P,
-4
by applying (21) twice. Therefore, for any x E X O + K” and p f af (x), (p, z) e 1 , i.e. p belongs to K. Thus af maps x o f K’ into the weakly compact subset K . Furthermore, the restriction of af to XO+ K” has a closed graph. To prove this, suppose that xfl converges to C xo+K* and p , < af(x,) converges to j in U* (for b(U*, U)). We deduce from the inequalities
f(x,O-f(r)
6 (PP,
x,-r>
that f@)-f(Y)
(I, Z-Y>*
Here we have used the fact that f (x,) converges to f (a), that (p,,, y ) converges to @, y ) and that ( P ~x,,) , converges to @, Z) (becausep, ranges over the weakly compact subset K and xfl converges to x for the Mackey topology). 0 *4.1.5. Characterization of subdgerentiable convex functions
Prdposition 9 . Suppose that f is subdigerntiable at
(23)
XO.
Then
f(x)+a+(af(xo); xo-x) for all x E U.
f’(x0) =s
Proof. When p ranges over 8f (xo), we have that
f(xo)-I’(x)
( P , xo-x) =s sup ( p , xo-x) = u*(af(xo); xo-X). 0 PE
a m
We now prove a converse result for which we need “the min-sup theorem’’ (Theorem 7.1.6) proved in Chapter 7.
Proposition 10. Let f be a convex proper function and let xo E Dom‘f. exists a weakly compact convex subset K c U*such that (24)
f (xo) =zf (x)+a*(K; X O - x ) for any x E U,
then af(x0) is not empty
Proof. Consider the function (25)
4 ( x , p ) =f ( x ) + ( p , X O - x ) defined on X X K
If there
Ch. 4, Q 4.21
DIFFERENTIABILITY AND VARIATIONAL INEQUALITIES
iir
where X = Dom f i s the dumain ofj: Inequality (24) implies that
Now Xis convex and is convex with respect to x . Also K is convex and compact and q!J is continuous and concave with respect to p . We deduce the existence of a max-inf po € K, i.e. f ( x o ) =G inf sup 4 ( x , p ) = inf q!J(x,P O ) x € X PEK
(27)
X€X
= inf ( f ( x ) + ( p o , X O - x ) ) X€X
Therefore, P O belongs to af (xo). 0 4.2. Differentiability and variational inequalities 4.2.1. Definitions
Subdifferentiability is related to differentiability as follows. Definition 1. Let f be a proper function mapping a topological vector space U into 1- m , -1. We shall say that f is Gdteaux-diyerentiable (or, in short, differentiable) at xo E Int Dom f i f
+
(i) Vx € U,lim,,o, (f(xo+ex)-f(xo))/8 = Df(x0) ( x ) exists, (ii) the functional x ++ Df(x0)( x ) is linear and continuous. We shall set in this case (2)
Df(x0) ( x ) = (Df(xo), x ) where Qf(x0) E U*
and call Df (xo) the “gradient” or “derivative” off at XO. We shall say thatfis differentiable on a subset x if it is differentiable at any point xo of x . The main justification of the concept of differentiability (in the framework of optimization theory) is the variational principle which allows us to replace a minimization problem by an equivalent problem in which the loss function k linear.
112
DIFFERENTIABILITYAND SUBDIFFERENTIABILITY [Ch. 4,s
4.2
Propition 1. Let us assume that f is dgerentiable on a convex subset X of a topological vector space U.I f f € X minimizesf on X,then (3)
(Of (Z), Z) = min (Of (Z), x). XEX
In particular, if1 belongs to theinteriorof X,thiscondition implies that Of (1)= 0.
Proof. Since Xis convex, y = 1 + 8 ( x - z ) = (l-O)x'+Bx belongs to X whenever x € X and 8 € 10, 11. Therefore, since Z minimizes f over X , we obtain (4)
Taking the limit when 8 T that ,(5)
-
0+, we deduce from the differentiability off at
(Df(X), x-1)
0 for any x E X.
In particular, if Z belongs to the strong interior of X, then for any y E U, there exists E Z- 0 such that x = 1 + e y belongs to X . Thus ( 5 ) implies that for any y E U,(Of(T), y) a 0. Therefore, Of (2)= 0. 4.2.2. Dgerentiability and subdiflerentiability Conversely, any solution Z of (3) minimizesf over X provided that f is subdifferentiable at 1. Proposition 2. Suppose that f is both diyerentiable and subdgerentiable at a point 1 of its domain X = Dornf. Then (6)
afw
= {Df(3}
In this case, any solution of
(7)
{ (i)
.
1 E X, (ii) v x E (Of(Z),Z-x)
x,
<0
minimizesf.
Proof. Let p E t?f(Z) be any subgradient off at 2. Then, for any x E X and any 6 E lo, I[,
Ch. 4, $ 4.21
DIFFEREN~ABILITY AND VARIATIONAL INEQUALITIES
113
Taking the limit when 0 converges to 0, we deduce that for any x E U, (9)
(P, 4
6
(Of(Z),.)
Therefore, p = Of (Z). If 2 is a solution of (7), then the fact that Of(5)is a subgradient implies that f (2)- f( x ) =z (Of(Z), Z- x) =s 0 for any x E U. For convex functions, the subdifferentiability hypothesis is redundant.
Proposition 3. I f a convexproperfunction f is direrentiable at apoint Z, then it is subdverentiable and af(Z) = {Of(Z)} Proof. From the convexity off, we deduce that, for any x E Dom f and 8 E 30, 11, (10)
e
e =f ( x ) - f (3
Therefore, taking the limit when 0 f a t Z that (11)
ww-m
f(i+ e(x- z))-~(z) d (1 - e)f(z)+
(Of(,), x- Z)
-
0, we deduce from the differentiability of
==f(x)-f @I?>,
i.e. that Of (2)belongs to the subdifferential off at Z. 0 4.2.3. Legendre transform
Proposition 4. Suppose that the conjugate function f’ is duerentiable at p. Then Z = Of ‘ ( p ) is the unique solution minimizing x k--- f (x)-(p, x ) if and only if (12)
.f**[9f*(P)l =f [Of‘ ( P I
Remark. Note that the function p k-+( p , Df*(p))-f[Df*@)] is called the “Legendre transform” off.
In particular, when f is a convex function, we obtain the following consequence. Proposition 5. Suppose that f is a proper convex lower semi-continuousfunctiop. If-f * is dverentiable at p E U’,then x = Of* ( p ) is the unique solution which minimizes x F-+f (x)- ( p , x). 10
DIFFERENTIABILITYAND SUBDIFFERENTIABILITY [Ch. 4,
114
Q 4.2
These results motivate the further study of differentiabilityand/or subdifferentiability of functions and their conjugate functions. 4.2.4. Interpretation:marginal projit
Proposition 5 admits an economic interpretation of the unique minimal solution 2 = Df ‘(p)o f x I-+f (x)- ( p , x ) as the so-called “margindprojit” atp. Iff (x)represents the loss or the cost of x, we saw that f ‘ ( p ) can be regarded as the maximum profit obtained when price p E U’ prevails. Hence Of *(p)is called the “marginal profit”. For any q E U’,
measures the marginal increase of profit when p moves towards p+q. More generally, we will interpret the fact that a minimal solution 3 of f( .)-(p, is a subgradient of the “profit function” f at p (Proposition 1.1) by saying that 3 is a “marginal profit”. a)
4.2.5. Variational inequalities
We consider now the case when f = g+h and (i) g is convex and differentiable on X, (ii) h is convex.
Proposition 6. Let X be a convex subset pf a vector space U.Suppose that (13) holds. Then f minimizes f = g+ h on X if and only iff is a solution of (i) f E X , (ii) V y E X , (Dg(x’),S-y)+h(Z)-h(y) s 0.
Proof. Let 3
f
X be an optimal solution and y f X . Then, if 8 E 10, 11, x =
= f + 8(y- 2) E X and thus,
g(q
+
==
g(~+eb-~))+(i-e)h(~)+ehb).
Dividing this inequality by 8 h(Z)-h@)
+- + +
=f(z) +(x) =f ( n e(y 3) = g(f+eb- 3)) h(x e(y- z))
-s
w
0, we obtain
g(f+e(Y-f))-m
e
We let 6 converge to 0 and obtain (14).
Ch. 4,
4.31
DIFFERENTIABILITY FROM THE RIGHT
115
Conversely, since {Dg(X)} = ag(2). we deduce from (14) that, fcr any y E X ,
f ( x 3 -f (v) = dx’)-g
(15)
=S
+
m h(x’)- h(y) (Dg(X),Z-Y)+h(Z)-h(y)
=G
0.
These results motivate the following definitions. Definition 2. Let X be a convex subset of a topological vector space U and let g and h satisfy (13). A solution X of
is called a stationary point off = g + h.
More generally, if S is a map from X into U*, inequalities of the form
[ (ii)
(9 Z E X , V x E X , (S(x’),2-x)
(17) or
{ (ii)(i) V X E X ,
=S
0
2 E X,
(18)
(S(T), Z - ~ ) + h ( X ) - h ( x ) s 0
will be called “variational inequalities”. The existence of solutions to such inequalities will be studied in Section 13.2. 4.3. Differentiability from the right We shall extend slightly the concept of differentiability and show how it is related to the concept of subdserentiability. 4.3.1. Definition and main inequalities
Definition 1. Let f be a proper function mapping a topological vector space U into]-1. We shall say that f is “dgerentiable from the right” at xo E Dom f if 00,
exists. 10‘
+
116
DIPFERENTIABILITY AND SUBDIFFERENTIABILITY
[Ch. 4,§ 4.3
The positively homogeneous functional x I--+ Df(x0) ( x ) is called the “derivative from the right” off. We shall say that f is differentiable from the right on a subset M if it is differentiablefrom the right at any point x E M. We begin by proving that any subdifferentiable convex function at xo is differentiablefrom the right.
Proposition 1. Let f be a convex p r o p function subdi@2rentiable at XO. Thenf is di@$erentia5le from the right at XO. The function x t--- Df (xo) ( x ) is convex‘ and positively homogeneous and
Proof. For all p E 8f(x0),x E U and 1 =- 0, we have that 1 ( P , x> = I ( P , xo+-Ax-xo)
(3)
4
+
f(x0 2x1-f(xo)
1
On the other hand, the function I k-+ 8(A) =f (xo+Ix)- f (xo) is a convex function vanishing at 1 = 0. Therefore, A I---
e(A) is increasing. n u s , I
Since this inequality is true for all p E a f ( x ~ )we , obtain inequality (2). The convexity of x k-4 Of (xo)( x ) is obvious. 0
Proposition 2. Let f be a convex proper function dflerentiable from the right at XO.
(5)
Then,
Of(X0)
(x--xo) e f ( x ) - f ( x o )
proof. The first inequality follows from ( f (XO+ e(x- xo)) - f ( X o ) ) / e 6 &.f(x)-f(xo) by letting ego to O+. To prove the second inequality. we Write
Ch. 4,s 4.31
DIFFERENTIABILITY FROM THE RIGHT
117
The convexity off implies that
We deduce that
from which we obtain the second inequality by letting 8 go to O+ 0 . 4.3.2. Derivatives from the right and the support function of the subdi&Terential
-.
Proposition 3. Let f : U 1- -, + -1 be convex and lower semi-continuous. Suppose that af(x0) is non-empty. Then
Proof. Since af(xo) is a closed convex subset, (7) is equivalent to E U'.
f (xo+Ax)-f(xo) I
Of (xo)(x) = inf
a=-0
[
Df(x0)' ( p ) = sup ( p , x>- inf X€
u
9
f (xo+
w-f (xo)
a>o
1
= sup sup 5 [ ( p , Ix)-f(xo+;Ix)+f(xo)l xcu
1-0
(9)
I f p E af(xo), thenf(p)+f(xo)-((p, xo) = O and thus, Df(xo)* ( p ) = 0. I f P Q af(x01, then f '(PI+ J(xo)-(p, X O ) = 8 > 0 and Df (xo)' ( p ) = = Supa+o 012 = a*. Therefore, Df(x0) ( p ) is.the indicator y(af(x0); -) o f af(x0). c)
+
118
DIFFERENTIABILITY AND SUBDIFFERENTIABILITY
[Ch. 4,s 4.3
Proposition 4. Let f be a convex proper function continuous at xo E Int Domf. Then the map (10)
x I-
Df (xo) ( x ) = u"@f (xo); x ) is continuous on U.
Proof. Since
and 4 is continuous at 0, Of (xo) is bounded above on a neighborhood of 0. "%&refore,x I-- Df (xo) (x) being a convex function bounded from above on a neighborhood, it is continuous (by Proposition (3.3.2). This implies in particular that
(see Proposition 3). fl
Proposition 5. Let f be a convex proper function defined on U. If (13)
f is continuous at xo and af (xo) contains a unique element,
thenf is dgerentiable at xo and af (xo) = {Of(xo)}.
Proof. I f f is continuous at xo and if af(x0) = {PO} contains the unique element po, then Proposition 4 implies that Of (xo)( x ) = ~ ' " ( { p o x) } ; = (PO,x } for any x. Thus p o = Of (xo) is the derivative off at XO. 0
' (14)
(i) P is compact, (ii) there exists a neighborhood U of x such that for any y E p c- f( y ; p ) is upper semi-continuous, (iii) Qp E P, y I-+ f ( y ; p ) L convex and dgerentiable from the
v,
Ch. 4,
0 4.31
DIFFERENTIABILITY FROM THE RIGHT
119
and the subset
(16)
Po = { p E P Such that g(x) = S(X; p)}.
Then g is dgerentiable from the right at x and
Letting 8 tend to 0, we deduce that
We now prove the converse inequality. Put a = Dg(x) (y)- E where E =- 0. We prove that there exists p PO such that a =s D f ( x ; p ) ( y ) . The function g is convex and so
Therefore, for any 8 =- 0,
Let U be the neighborhood of x appearing in assumption 14 ($1. Then there exists O0> 0 such that x+Oy E U for any (3 E [0, eo]. Since p k - - f ( x + e y ; p ) is upper semi-continuous, Be is a closed subset of P. Furthermore, 0 F+ Be increases with 8. To prove this let 81 =s(32 and p E Bel. The convexity of x f ( x ; p ) implies that
-.
+
since x+ y = (1 - (31lO2)x ((31/(32)(x+ (3u)and f ( x ;p)-g(x) =s 0 for allp by definition of g. Because P is compact, there exists 3 belonging to the non-empty intersection B,. Therefore, for any 0 E E 10, O,], we have 8a = s f ( x + ey; p)-g(x). Letting 0 tend to 0, we deduce that
noces80
DIFFERENTIABILITYAND SUBDIFFERENTIABILITY [Ch. 4,$
120 g(x) e f ( x ; write
1)and thus, that j j E PO.Therefore,
(22)
a==
f(x+@; P)-f(x;
e
This implies thata
=sDf (x; j j )
g(x) =f(x;
4.4
p ) and we can
1)
(y). 0
*4.4. Local E-subdifferentiability and perturbed minimization problems
We deal in this section with minimization problems for which the loss function f is lower semi-continuous and bounded below, but not necessarily lower semi-compact. For such problems f does not necessarily achieve its minimum. However, when U is a Banach space, we shall prove the existence of approximate solutions: We show that, for any E Z- 0, there exists P such that
(*I
1 (9
f (2)=S inf.,xf(x)
+E,
fm= 5 f ( Y ) + f i I IP-Y
(ii) VY E x,
I I.
If X is convex andf is differentiable this &plies that x’ is a solution f o the variational inequalities. (**)
V Y € X ( ~ f C P ) > , X ’ - Y==fi ) 113- YII.
In particular, iff is differentiable on its (open) domain X,and bounded from below, we exhibit x, such that
(**I
(0 iIlfx,x f (4+ { (ii)-iiDf(xJII* ii. f ( X J =s
8,
=S
In other words, the equation Df (x) = 0, although it need have no solution, always has an “approximate solution” x,. We also use (*) to prove that any lower semicontinuous function on a Hilbert space is “locally &-subdifferentiable” in a dense subset of its domain. (We say that f i s locally E-subdifferentiableat P if there exist 7 > 0 andp E U’ such that f ( Z ) - f ( x ) -= @, P-x)fe IlZ-xll whenever IlZ-xll =S q.) This fact is used to study the family of perturbed problems
as p ranges over a Hilbert space U*. We require only a weak assumption on x F-f ( x , p), namely that x I-- f(x, p ) is lower semi-continuous and bounded from below for anyp E U*
Ch. 4,$ 4.41
121
LOCAL E-SUBDIFFERENTIABILITY
In addition, however, we need a set of strong assumptions pertaining to the smoothness of the perturbations (for the precise assumptions, see Theorem 3 below). We then prove that for "almost aN" minimization problems ~ ( p there ) exists a unique optimal solution, in the sense that the property holds when p ranges over a dense denumerable intersection of open subsets. As an example, we apply this result to linear perturbations, and prove that any continuous convexfunction on a Hilbert space is d1@7erentiable on a dense denumerable intersection of open subsets. Finally we note that a best approximant to x E U by an element of a closed bounded (non convex) subset U exists and is unique for"'a1most all" x.
4.4.1. Approximate optimal solutions in Banach spaces
+
Consider a proper function f : U I-+ ] - m , Then, for any E =- 0, there exists x, E U such that a = inff(x) e f ( x , ) G a+ E. (1)
m
] which is bounded below.
X € U
Proposition 1. Let f be a lower semi-continuous proper function mapping a Banach space U into 1- -, -t-1. Then, for every x, satisfying ( I ) and every I =- 0, there exists some X E U such that (2)
I
0) f(@ f ( X e L (ii) ~ l x ~ - ~i' l1,l (iii) b'z # X , f ( Z )
The proof of this result is based on a device due to Bishop, Phelps and Browder. Let p =- 0 be given. We define an ordering on U X R by
(3)
{yl, al} .S {YZ,az} if and only if(az-a1)+eIlyz-ylII
0.
This relation is easily seen to be reflexive, antisymmetric and transitive. Also, its upper sections { ( y , b} =={x, a}} are obviously closed. We proceed to show that every closed subset of U X R has a maximal element, provided it is "bounded below".
Lemma 1. (Bishop-Phelps-Browder). Let S be a closed subset of U x R such that @ = inf {a such that {x, a} E S} =- (4)
-
Then,for any by b}.
b,b} E S, there exists a maximal element { X , ci}
E S larger than
122
DIFFERENTIABILITY AND SUBDIFFERENTIABILITY
[Ch.4,$ 4.4
Proof. We define by induction a sequence {x,,, a,,} E S with {xl, al} = {y, b}. Suppose {xn,a,} is known. We set f5)
Clearly, 18, (6)
[
(9 Sn = {{x, a } such that {x, a } {xn, Y n } } , (ii) pn = inf {a such that {x, a} E S,,}.
p. We now define {x,+~,a,,+l}to be any point of S,, such that
an-an+l*
+(i-(an-Bn).
All the subsets S,, are closed and non-empty and S,+l c S,, for every n. More over, we deduce from (6) that (7)
~(1/2~)Ib-pI.
“+Ian-BnI
Ian+l-Bn+lI
Hence, for every {x, a} E S,,, we obtain from (5(i)) that (8)
[ ((09
Ian+l-al
lb-81,
=s(W)
I l ~ n + 1 - ~ 1 1==
-
(Ve.29lb-Bi.
This proves that the diameter of S,, converges to 0 as n 0 0 . Since U is a complete metric space, the sets S,, have a unique element {%, a} in common, i.e. (9)
fi s, = {Z, a}.
n=l
By definition of S,,, we have {F, ii} a {x,,, u,,} for all n. In particular, for n = 1, this implies that {X, CS} {y, b}. Suppose that {X, h} is not maximal. Then there exists {x*, a*}E S strictly larger than {Z, a}. By transitivity, {x*, a * } is larger than {x,,, an}for anyn. Hence, {x*, a*}E S,,, i.e. {x*, a*} = {Z, a}. This proves that {Z, ii} is indeed maximal. 0
n;=l
Proof of Proposition 1. We take S = & p ( f ) to be the epigraph of J It is closed since f is lower semi-continuous. Take e = &/A and { y , b ) = {xs, f(x8)}. Lemma 1 implies the existence of a maximal element {Z, a} in S satisfying {X, a} a {x~,f(x,)}. Since {Z, a} E S, we also have {Z, f (X)} z= {Z, H}. But, {x, 6)being maximal, this implies that ii = f ( Z ) . The maximality of {%,f(%)} means that (10)
(a-f(Z))
+nIIx-Xll &
unless x = 1 and a = f ( Z ) .
>0
whenever {x, a } E S
Ch. 4,$, 4.41
123
LOCAL 8-SUBDIFFERENTIABILITY
Taking a = f ( x )yields (2(iii)). Now, since (55) = {%f ( 2 ) )Z- {%f we have that
(X,)),
Hence, of course, f ( 5 ) ef(x,) (i.e., (2(i))). It follows from (1) that f ( 2 ) 2 a f (xJ- E . Substituting this result in (1 I), we obtain that (&/A) I I X - x,J1 =s E . But this is (2(ii)) and so the proofk complete. 0
4.4.2. The approximate variational principle We have seen that, if a differentiable function f achieves its minimum at some i?in the interior of its domain, then Of (2)= 0. We shall prove that, if a differentiable function is bounded below (although it need not attain its bound) then, for any E > 0, there exits x, E X such that II Of(x,) I I =s E (i.e. its derivative can be made arbitrarily small).
-.
Theorem 1. (Ekeland). Let X be a convex subset of U and let f : X R be a dgerentiable function which is bounded below. Then,for any E > 0, there exists x, such that (12)
f(xJ
a+
&
where a = inf f ( x ) . X€x
Further, for any ;I=- 0, there exists 1 such that
(i)f (2)--f(x,) =s a+ (ii) 11 2- xeII =s A
E,
and
If we also assume that 2 belongs to the interior of X , inequalities (14) become
Proof. By Proposition 1, we know that there exists 2 such that, for any z E ,'A
124
DIFFERENTIABILITY AND SUBDIFFERENTIABILITY
[Ch. 4,
4.4
Hence,taking z = Z+O(y-?), where y E X and 8 E ]0,1[, we obtain that
+ - ( x +eCv- 3)+: el Iy - 2 II. Therefore, dividing by 8 and letting 8 converge to 0, (17)
0 es (Of(?), v-q+;
Ily-fll.
If f belongs to the interior of X,we take y = I+ Buywith u in the unit ball. Then (18)
(ma4
+
IIuIL
which implies that I IOf(Z) I It =se/1. 0
4.4.3. Local E-subd@erentiabiIity
We shall need to introduce both a stronger concept than differentiability and a weaker concept than subdifferentiability of studying the regularity of perturbed problem. We begin by recalling the concept of a Fr6chetderivative.
Dehition 1. Let U be a Banach space. We shall say that a differentiablefunction f is ‘%r~chet-d~erenti~le” at f if
Remark. One can give s a c i e n t conditions fo,r a differentiable function to be Frkhet-differentiable. For instance, one can prove that, i f f is a djgerenriubIe function with open ahmain X = Domf and if x I--- Of(x) is continuousfrom X into the Brmach space U*,thenf is Fr&chet-d@erentiable on X . We now define our weakest concept of differentiability.
De5ition 2. We shall say that a function f is “locally e-subdi~erentiable” at f if there exist a continuous linear form p E v‘ and a positive number q such that (21) f ( Z ) - f ( x ) =s@-, i-x)+eIlZ-xll whenever ~ ~ f * - q. x ~ ~
Ch. 4, Q 4.41
125
LOCAL E-SUBDIFFERENTIABILITY
Of course, p E U* and =- 0 depend upon E. Note that iff is locally Esubdifferentiable at 2, it is locally d-subdifferentiable for every E’ Z= E. This shows that we can associate with any 2 E U a number EO E [O, -1 such that f is locally &-differentiableat x for E =- EO but not for E i80. The more regular functions have smaller values of EO. = 0, we shall say that f is “locally subdiferentiable” at Z. With E = 0 and q a, (21) reduces to our familiar concept of subdifferentiability (Definition 4.1.1). The definition can be stated in another way: f is locally E-subdifferentiable at 2 if there existsp E U*suchthatthefunctionx+f(x)-(j?, x)+EJJZ-XJJ achieves a local minimum at 2. The concept of local E-subdifferentiability is so weak that any lower semicontinuous function satisfies it “almost everywhere”(in the senseof the following theorem).
+
ve =+
Theorem 2. (Ekeland). Let f be a proper function mapping a Hilbert space U into
1- -, i- -1. Iff is lowersemi-continuous,the set of points where f is locally Esubdiferentiable is dense in Dom f.
Proof. Let xo E Dom f, 6’ =- 0 and E =- 0 be given. We shall find a point X such that 1 I 2- xoII Q q, where f is locally 2 fi-subdifferentiable. Since f is lower semi-continuous, there exists 8 e 6’ such that f (x0)- a =sf(x)
whenever II x- xo11
6.
Consider the function Q, defined on U by
The domain of Q, is an open ball of radius 8 which we denote by B(S). Furthermore, (23)
Q, is
Frdchet-differentiable on Dom Q, = h(6).
(Its derivative is equal to (2/(d2- IlxJlz))~,(x)Jx,where J is the duality operator from U into U’.) We introduce a new function g d e h e d by (24)
+
g ( 4 = f ( x > Q,(x- xo)
whose non-empty domain is Dom fn(xo+h(6)).It is bounded below by From Proposition 1 (with I = @), we deduce that there exists
f(xo)-a.
126
DIFFERENTIABILITY AND SUBDIFFERENTIABILITY
[Ch. 4,s 4.4
2 E Domg c xo+B(G) such that
(25)
{ (ii)(i) g(Z)
4
f ( x 0 )- a
g(2) s g(x)
+
E,
+u‘i11 X- x I I for any x E Dom g.
Substituting (24) in (25(ii)), we obtain that
(26)
f(x3-.f(x)
eG
q4x-
x0)-
v(2- xo)+
fi I I x- x I I.
But, since is Fdchet-differentiable, there exists q such that, With p = = -Dq(2- Xo),
(27)
I ~ ( 2 xo)&-XO)+@, 2-
fi IIixII
x ) I =s
whenever
III-xll sq. Hence (26)and (27) imply that f is locally 2 fi differentiable at Z E xofB(6). This completes the proof. 0 Remark. Since we used the assumption that U is a Hilbert space only via (23), Theorem 2 remains true whenever there exists a function q satisfying (23). If a Banach space U has a norm equivalent to a Frdchet-dBerentiable norm, we can still use the function Q defined by (22). Remark. When f is convex, we can prove a global version of this theorem. 4.4.4. Perturbation of minimization problents
We consider now a family of minimization problems
I281
a(p) = inf f (x, PI. X€X
We shall prove that, for‘“almost all” p E V*,(i.e. for everyp of a dense denumerable intersection of open subsets), the minkation problem has a unique solution ?(p) to which any minimizing sequence {xn} of a ( p ) converges. Of course, the necessary assumptions are quite strong, but do not unduly restrict the class of perturbations to which the result applies. Theorem 3 (Ekeland- Lebourg). Suppose that (29)
V is a Hilbert space, X is a closed subset of a Banach space
and that
(i) Vp E V*, x I--+ f ( x , p ) is lower semi-continuous, (ii) p I--- a(p) is continuous
Ch. 4,0 4.41
127
LOCAL E-SUBDIFFERENTIABILITY
Suppose also that
(31)
Q xE X , p
+f (x,p ) is Frkhet-duerentiable on V*
and that Q p E V * , V E =- 0, 37 =- Osuch that IIx1-x2)I provided that f (xi,p ) e u ( p ) + q (i = 1, 2) and I I Of (x1, PI- Of ( X I , PI I I =s17.
G E
Finally, suppose that (33)
Then there exists a dense denumerable intersection T of open subsets of V such that, V p E T ,
(35)
the minimization problem has a unique solution X(p), { (ii)(i) any minimizing sequence x,, converges to x(p). u(p)
Before proving this theorem, we give an application to the case of perturbation by linear functionals. Proposition 2. (Asplund). Let X be a closed bounded subset of a Hilbert space U. Let f :X ++ R be a lower semi-continuousfunction which is bounded below and let a be defined on U by
(36)
.(P> = inf
X€X
[f(X)-(P,
41.
Then there exists a dense denumerable intersection T of open subsets such that (35)holds. PrQof.We check the assumptions of Theorem 3. Note that, in this particular case, the functions p I- f (x)- ( p , x) are k-Lipschitzian on U*, where k = = supxEx[ 1 x 11. This is finite because Xis bounded. The “equicontinuity assumption” (34) follows. It follows that the function u, which is finite because f is bounded below and X is bounded, is k-Lipschitzian, and hence continuous.
128
DIFFERENTTABILITYAND SUBDIFFERENTIABILITY [Ch. 4,§ 4.4
Therefore assumptions (30) are satisfied. If we set f ( x , p ) =f(x)-(p, x), we see that Df (x, p ) = -x. Thus
(37)
llDf(x,p)ll* = 11x11is bounded Vx E A’, V p E U*.
We deduce that assumption (32) is satisfied, with rl = E, since IlDf(x,l p)-Df(xz,p)ll = Ilxl-xzll. Finally, assumption (33) is obviously satisfied because we always have that (38)
f(x,4)-f
(x, p)-(Df(x, PI, 4-P) = 0-0
Remark. Suppose moreover that (39)
f’is convex.
Since a ( p ) =-f;(p), condition (35(i)) amounts to saying that aJ;(p) has a unique element, namely the derivative of f i at p. In fact, by refining the arement a little, we obtain the following result. This sheds new light on the regularity of convex functions (compare with Proposition 4.3.4).
Proposition 3. Let U be a Hilbert space. Any continuous convexfunction defined on U is diflerentiable on a dense denumerable intersectionT of open subsets.
Proof. We shall prove this result for the conjugatef, of a convex function f, wheref is assumed to be continuous. We apply Theorem 3 to the problems
where X = Dom f is no longer assumed to be bounded. Again, all the assumptions of Theorem 3 are trivially satisfied except for the “equicontinuity assumption” (34) which we check directly. Let 1 E B and {pn,xn} be a sequence such that
Ch. 4, Q 4.41
129
LOCAL E-SUBDIFFERENTIABILITY
Since pn converges to p a r d f * is continuous at p , there exists q such that (42)
f * ( r )e f * ( p ) + when I I r-p I I { (ii)( i ) IIp-pnII =sq w h e n n a N . 1
> 0 and
N
2q,
=S
Hence, for any q E B(q) and for any n z= N , r,, = pn+q belongs to B(27). We deduce from (41) that for any n z= N, (q, xn? e f * (p )+ A + 1 when q E B($. Hence the sequence x,, is bounded. Thus I f(p,,, xJ- f ( p , x,,) I -=I (p-p,,, x,,) I e llP-Pnll* Ilxnll 4 kllp-Pnll* convergestoO* 0 We also mention another application of Theorem 3. This concerns the existence and uniqueness of a best approximant to given points by elements of a closed bounded (non-convex) subset of a Hilbert space. Proposition 4 (Edelstein). Let X be a closed bounded subset of a Hilbert space U. There exists a dense denumerable intersection T of open subsets such that, for any p E T , there exists a unique point nearest to p and a unique point farthest ,from p in X (where a nearest point minimizes the distance to X and a farthest point maximizes the distance to X). Proof. We apply Theorem 3 to the function
(43)
f(X,P)
=f+llx-PI12
for which a minimum x’ is either the nearest point to p or the farthest point to p . The functions given by (43) are Frkchet-differentiable and (44)
Df (x, P> = fJ(x-P)
where J is the duality map from U onto U*. Hence we have that
Df(x,P)-Df(Y, P ) = f 4 x - y )
((
f ( 4 , X ) - f ’ ( P , x ) = f 4-p, 2P + q x)) f ( 9 , 4-f (P, 4- (Of@,
$9
4 - P) = fI 14-P
I l2
Everything now follows easily from the fact that X is bounded. For instance, the derivative Df(x, p) is locally uniformly bounded, hence the functions p I--+ f (x, p ) are locally uniformly Lipschitzian. Thus f is locally Lipschitzian and therefore continuous. 0 11
130
~~FFERE~ABUXTX
SUl3XXFEEXSNTlABlLlTY
Ich. 4, 9 4.4
4.4.5. Proof of Ekeland-Lebourg's theorem
The proof of Theorem 3 is quite involved. We divide it into four steps. First step. We prove the following result, which is interesting in itself.
Proposition 5 . Suppose that (31) and (33) hold. If the function a is locally subdflerentiable at p E V', then (45)
E-
IIof(xi, p )- of(xz, p ) 1I 4 68 provided that f ( x i , p ) s a(p)+8 (i = 1,2)for some 0.
Proof. Since u is locally &-subdifferentiableat p E V",there exists u E V = V** and q1 =- 0 such that (46)
=(PI ~ 4 I ) + ( P - 4 ,u)+&llP-!7ll*
whenever IIp-411, s 71.Choose q2 = min (7,q1) where 7 appears in (33). Let x E X be any point such that (47)
f (x, P ) =s=(PI+
Ella.
We write part of (33) as an inequality (48)
f(x9
4)- - f ( x , p ) + @ f ( x , PI. 4-P)+~114-PII*.
Taking into account (47) and the inequality a(q) a;f (x, q),we deduce from (48) that (49)
=sa(p)+(Df(x, PI, 4--P)+2&72.
Now, (46) and (49) imply that (50)
(U-
of(X, P). 4-P)
=s38%.
Taking the supremum when q ranges over p+B(qz), we deduce from (50) that (51)
lIu--~?(x, P)ll
=G 3 E .
If y E X is another point such that f ( y , p ) 6 a(p)+ E (52)
IlU-Df(V,
PI11
we also have that
~ Z
63E.
Hence (45) follows from (51) and (52) with 8 = 8qa 0 Second step. We use Proposition 5 and Theorem 2 to prove a further result
(Proposition 6). We require the following definition.
(53)
T. = { p E V* such that there exists 0 w 0 such that Ilxl-xz!l = s ~ w h e n e v e r f ( x ~ , p ) = s a ( p ) += 6 (1,2)}. i
Ch. 4, Q 4.41
131
LQCAL E-SUBDIFFERENTIABILITY
Proposition 6. Suppose that assumptions (30), (34, (32) and (33) hold. Then, for any E > 0, the set T. is dense in V*. Proof. By assumption (32), there exists
r]
such that
I I XI-
x2
II =S E whenever
f (xi, p ) == a(p)+q and IIDf(x1, p)--Df(x2, p ) ] =S q. Proposition 5 and Theorem 2 (with E = q/6)imply that the latter property is true asp ranges over a
dense subset T,of V. 0 Third step. Proposition 7. Assumption (34) implies that the subsets T,are open. Proof. Let p E T,. Then there exists 8 =- 0 such that 11.7c1-x2ll =s E whenever f ( x i , p ) =Sa(p)+O(i = 1,2). Taking I = a(p)+8, wededucefrom (34)theexistence of 7 Z- 0 such that
(54)
If(x, q ) - f ( x , p ) I 4 8 provided that I I q-p 11 =sq and f (x, q) =S il
This clearly implies that (55)
when IIp-q/I s q .
a(q)-ca(p)+$O
Take any q such that I I q-p I I =s q. Proposition 7 will be proved if we can show that q belongs to T,. For this purpose, let xi(i = 1,2) be any two points in X such that (56)
f(xi, 4)
a(q)+ $6.
Taking (55) into account, we deduce that (57)
f ( x i , 4) =s a(p)+g
8 -= 1..
We can therefore apply (54) and obtain that (58)
f ( x i , p ) - - ' f ( x i , q ) + + e = = a ( p ) + 8 (i = L2).
Since p belongs to T,, 1 I X I - x2 I I =sP . Hence (56) implies that q E T,. Fourth step : Proof of Theorem 3. We define T c V as (59) 11.
T=
nT m
n=l
n
~=, ~ T.. E>O
132
DIFFERENTIABILITY AND SUBDIFFERENTIABILITY
[Ch. 4, 0 4.4
Since V* is a complete metric space and the sets TI,,,are dense and open (Propositions 6 and 7), the Baire theorem1 implies that T is dense. Let p E T.Consider a minimking sequence {xn} for a(p), i.e. V n,f(x,, p ) -s =sa(p)+ I/n. Since p E Tefor any E 7 0, we deduce from (53) that
(60)
V E Z- 0, 3Nsuch that ~ ~ x n - Q x pE when ~ ~ n, p a N.
Hence (x,,} is a Cauchy sequence. This converges to X E X, because X is a closed subset of a Banach space. Sincefis lower semi-continuous With respect to x, we have that
(61)
f ( 5 , p) 4 lim inff(x,,, p) = a ( p ) . n
It follows that x is an optimalsolution of problem a@). This optimal solution is unique. If there were another solution 7, we would havef(X,p) = f ( y , p ) = = a(p). Since p E T. for all e =- 0, this would imply that IlX-jjll =S E for every E =- 0, and thus that X = 7. 0
See note in Section 3.3.3.
CHAPTER 5
INTRODUCTION TO DUALITY THEORY
This chapter is devoted to an introduction to duality theory. We will continue this study in Chapter 14. In Section 5.1, we state the framework in which duality theory arises. We assume that the strategy set
X = {x E R such that, V p E P,y(x, p ) =s0) is a subset of R defined by a family of constraint inequalities y(x, p) =G0. We assume that
[
(i) P is a convex cone, (ii) Vx, p -, y(x, p ) is positively homogeneous.
Since minimization problems on R are easier to solve than minimization problems on X,the question arises whether it is possible to replace
w = inf f ( x ) (consrained minimization problem) X€X
by a minimization problem on R: w = inf [ f ( x ) + y ( x ,p)] (unconstrained minimization problem) xER
for a convenient p E P (called a Lugrange multiplier), by adding to the initial loss function f a (well chosen) constraint y ( - , p ) which measures a "cost of violation of the constraint x E T".The problem of finding a Lagrange multi. plier is called the dual problem of the initial problem. Once the dual problem is solved and a Lagrange multiplier p is obtained, it remains to solve the unconstrained minimization problem. We shall check that 1 E X minimizes f on X if and only if
{
(i) 1 minimizes x t---f(x)+y(x, (ii) y(2, p ) = 0.
1)on R,
Lagrange multipliers have another fundamental property which emphasizes their importance. Let V* (the dual of a topological vector space V )be the vector 133
134
INTRODUCTION TO DUALITY THEORY
[Ch. 5
space containing P. We associate with any y E T the perturbed subset
X ( y ) = { x E R such that y(x, p ) + ( p , y ) d 0 V p E P}.
Note that X = X(0). Let v be the. function defined by v(y) = inf f ( x ) X€XCv>
which associates with any perturbation y E V the minimal loss v(y) on
Then we shall prove that the Lagrange multiplier j j satisfies
I
E
Xb).
a m
i.e. p measures the marginal increase of loss when the strategy set is perturbed by Y.
We devote Section 5.2 to the case where the constraints are defined by y(x,p)= ( p , Lx)- af(p), where a; denotes the upper support function of a closed convex subset Y c Y and where L E d(V,V) is a linear operator. In other words, X = {x E R such that LX E Y}. If the loss function f is lower semicontinuous and convex, we shall prove that
I E X is an optimal solution and jj E P (which is the barrier cone of Y) is a Lagrange multiplier if and only if
5 E af *(-L*p) and jj E i3yY(L3 where ay(LZ) denotes the normal cone to Y at LZ. These relations are called the “extremality relations”. They lead to the followingf h n t a l formula. This is the multi-valued equation
o E aa;(p)-Laj-*
(-L*I).
of which the Lagrange multipliers are solutions. Oncejj E P is obtained, minimal solution can be found from
In the rest of this section we investigate several examples and applications of this formula. For instance, if we assume that f is differentiable on -L* P
Ch. 5,
5.11
135
DUAL PROBLEM AND LAGRANGE MULTIPLIERS
then the minimal solution Z is equal to Df*(--L*p), where the Lagrange multipliers p E P are solutions of the variational inequalities
(--LDf+(--L*j), p - p ) +u$@)--Op(p)
e i 0,
v p E P.
These variational inequalities become (-LDf+(--L*p3+0, p - p )
4
0, Qp E P
when Y = w-P+ and thus reduce to the equation
rnf*(-L*) = w when Y = {w}. We prove in Section 5.3 two existence theorems for Lagrange multipliers (the Fenchel and Uzawa theorems) in the case where V is a finite dimensional space. Lagrange multipliers exist iff, R and Y are convex and if the constraint qualification assumption 0 E Int(L(R)-
Y)
holds. 5.1. Dual problem and Lagrange multipliers
We introduce the Lagrangian of a m h h h t i o n problem where
'u
= inf,,,f(x)
X = {x E R such that y(x, p) =S0 V p E P}. The Lagrangian is defined by
I(x, p ) = f ( x ) + y ( x , p ) when x E X,p E P. We check that f x ( x ) = SUP 4x9 p ) PEP
and we define the Lagrange multipliers 1 by inf I(x, p ) = inf sup I(x, p).
x€R
x€R
PU'
Then we show that 2 E Xis an optimal solution if and only if
+
(i) x' minimizesf ( x ) y(x, p ) on R, (ii) y(X, 1)= 0.
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INTRODUCTION TO DUALITY THEORY
[Ch. 5 , $ 5.1
This being done, we check that
E
jj
av(o)
where
v Q = inf f ( x ) X€X(Y)
and X ( y ) = {x E R such that y(x, p ) + ( p , y)
-= 0 for all p E P } .
We end this section by interpreting the case where X = { x E R such that L ( x ) E Y } is the set of x satisfying the constraints ( p , L ( x ) ) - o $ ( p ) ~We . say that Xis represented by the family of constraints. 5.1.I . Lagrangian
In many instances, the strategy set Xis a subset of an “unconstrained” set R defined or “represented” by a family of inequality constraints, i.e. (1)
X = {x E R such that y(x, p ) =s0 for all p E P)
where we shall assume once and for all that
I
(i) P is a closed convex cone of a dual topological vector space
v+,
(ii) y is defined on R X P and is positively homogeneous with respect to p (i.e., V l a 0, y(x, Ap) = Ay(x, p ) ) .
The aim of the following approach is to replace an optimization problem on X by an equivalent problem on R (by modifying the loss function) whenever an optimization problem on R is simpler than an optimization problem on X . The idea is to “measure” the fact that x belongs to X or not by the values of the constraints y(x, p ) . In other words, by adding to a loss function f defined on R a constraint y(., p ) , we “penalize” the fact that a strategy x does not belong to X. Definition 1. Consider the subset X of R &@ed by (3)
X = { x E R such that y(x, p) == 0 for all p E P}.
Let f be a loss function defined on R. We shall say that the function l defined on R X P by (4)
4 x , P) =f (4+V(XY PI
Ch. 5 , § 5.11
DUAL PROBLEM AND LAGRANGE MULTIPLIERS
137
is the “Lagrangian” associated with the minimization problem a = infxExf(x) and the representation of X by the constraints y( -,p ) . Of course, as we shall see later on, there are many ways of defining a subset X by means of inequality constraints (see Section 14.1). The following proposition gives a precise meaning to the penalization procedure we described.above. Proposition 1. Suppose that (2) holds. Let Z(x, p ) be the Lagrangian of an optimization problem. Then (5)
fXW
= SUP I(x, PI. PEP
Proof. If x E X , then y(x, p ) -s 0 for any p E P and y(x, 0) = 0 since 0 E P Therefore,
Now, if x 6 X , there exists a t least one elementp E P such that y(x,p) = 8 w 0. Since P is a cone, the elements ;Iji belong to P whenever 3, =- 0. Because y is positively homogeneous with respect to p , y(x, IF) = 28. Thus suppEpl(x,p) suplpo I ( ~3 ,, ~ =f(x)+supA,o ) 28 = =fx(x).
+
5.1.2. Lagrange multipliers and dual problem
We deduce from (5) that (7)
inff(x) = inf sup I(x, p ) .
XEX
x E R PEP
The problem now arises as to whether there exists an element that
(8)
p E P such
inf f (x) = inf sup I(x, p ) = inf l(x, p).
*EX
x € R PEP
xER
If so, the initial minimization problem on X is equivalent to a minimization problem on the unconstrained set R. Detinition 2. We shall say that the elements F E P (if any) satisfying (8) are the “Lagrange multipliers” of the problem or the solutions of the “dual problem”. If such a multiplier p exists, we obtain the following characterization of a minimal solution iE X off.
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INTRODUCTION TO DUALITY THEORY
[Ch. 5 , s 5.1
Proposition 2 . Suppose that (2) holds. Then j j E P is a Lagrange multiplier and 3 E X minimizesf over X if and only if
Proof. Obviously, (9) implies that x' E X (by (9(ii))) and that I minimizesf on X because, if x E X , y ( x , p ) -s 0 and 110)
f ( 3= f ( 3 + 0 e f ( x ) + y ( x , p ) e f ( x ) + O
(by (9(iii) and (i))). Inequalities (9) imply also that (8)
inf f ( x ) = inf l(x,jj). x€R
X€X
Conversely, suppose that jj is a Lagrange multiplier and that I minimizes Jon X . Since Z E X,y(Z, p ) e 0 for all p f P and, in particular, ?(I,p ) =s 0. On the other hand, by (8),f(x3 = infXERI(x,p)=s l(15p)=f (Z)+y(I,jj). Thus, ~ ( I , P ) 0. Therefore y ( I , p ) = 0. This implies that
f (I)+~(I, p) =f ( ~=) inf I (x, j j ) = inf (f(x)+y(x, p)). x€R
*€R
Definition 3. We shall say that (1 1)
fl*(p)=
-inf I(x, p ) x€R
is the lossfunction of the dual problem. We always have 112)
inf f (x)+ inf f '*(p) 0. PEP
X€X
An element jj E P is a Lagrange multiplier if and only if
(13)
{
(i) inf f(x)+inf f'*(p) = 0 x'x
P€P
(ii)f l * ( p ) = inf f J * ( p ) PEP
Remark. If the function f is strictly convex and the constraints y ( . , p ) are convex for all p E P , then the function f+.y(i, p ) is also strictly convex. Hence if p is a Lagrange multiplier, there exists at most one solution x' E X which minimizesf on X . If it exists, it is necessarily the unique solution which
Ch. 5 , $ 5.11
DUAL PROBLEM AND LAGRANGE MULTIRPLIERS
139
minimizes x ++ f( x )+y(x, p ) on R. Hence it necessarily satisfies the condition y(5,jj) = 0.
Remark. More generally, the question arises as to whether any solution X
minimizing x -.-I ( x )+y(x, jj) is a minimal solution of the initial problem. For this purpose, we replace the minimization problem v = inf,,, f( x ) by the equivalent minimization problem v2 = inf,,, w(x)l2.Its Lagrangian is defined by 4x7 PI = [ f ( X ) 1 2 + Y ( X , PI.
Proposition 3. Let us assume that there exist a minimal solution of the problem v2 = inf,,, [f(x)I2and a Lagrange multiplier j. Then any 2 which minimizes x V(x)l2+y(x, jj) on R is a minimal solution.
-
Proof. By Proposition 2.1.6, the functions f and y( -,jj) are constant on the minimal set M b = {x E R such that f ( ~ ) ~ + y jj) ( x ,= v}. Thus f(2) = v and y(5,jj) = 0 for any 5 E Mb. 0 The equivalence property (8), which we used to introduce Lagrange multipliers, is not the only fundamental property of Lagrange multipliers. 5.1.3. Marginal interpretation of Lagrange multipliers
We have already studied the perturbed problem
obtained by adding linear functionals q to the original loss function (see Section 2.4). When X is represented by inequality constraints y(x,p) =s 0, where P is a subset of the dual space V', we perturb X by associating With any y E V the subsets X ( y ) defined by (14)
X ( y ) = {x E P such that y(x, p)+(p, y ) =S0 V p E P}.
We shall study the behavior of the function v(y) = inf f ( x ) (15) XEX(Y)
and its relations with the Lagrange multipliers.
Proposition 4. Suppose that there exists a Lagrange multiplier PO E P of the problem
140
INTRODUCTION TO DUALITY THEORY
+
[Ch. 5, 0 5.1
+
for the Lagrangian f (x) y(x, p ) ( p , YO).Then (1 7)
P O belongs
to the subdiFerentia1 av(yo).
Remark. We shall prove the converse statement when p I-+ y(x, p ) is concave and upper semi-continuous (see Section 14.1.6). In this case,we shall prove that po is a Lagrange multiplier of the problem v(y0) if and only if PO E av(y0) and v(y0) = w**(yo). This result is analogous to Proposition 4.1.1. More generally, we shall associate with families of optimization problems
(where A is a proper function defined on the product UX V of two vector spaces) a “generalized” Lagrangian defined by 4x9 P ) = inf (19) Y>-(P, Y)l. YE V
An analogous duality theory can be devised in this case. 5.1.4. Example
Consider the framework described in 1.1.1, where the set of feasible strategies is constructed from a subset of available resources. We assume that (20)
The space V of resources is a topological vector space
“resourceprices’’ p (associating with any resource y E V its value ( p , y) E R). We define a closed convex set Y of
and we regard its dual V* as the space of available resources by
(21)
Y = {y E Y such that ( p ,y ) 4 r(p) V p E P}
Ch. 5 , § 5.11
DUAL PROBLEM AND LAGRANGE MULTIPLIERS
141
where (i) P is a (closed convex) cone of V*, regarded as the cone of feasible prices (ii) r :P R is a positively homogeneous function associating with any feasible price p a maximum profit r(p).
-
In other words, the set Y of available resources is described by “budgetary constraints”, i.e. as the set of resources J’ E Y whose values do not exceed the maximum profit for any feasible price. Let R be the unconstrained strategy set and L : R V be the resource operator. Then the strategy set X defined by -+
(23)
X = { x E R such that L(x) E Y}
is represented by the constraints y(x, p ) = ( p , L(x))- r(p) since statement (23) is equivalent to (24)
X = { x E R such that V p E P, ( p , L ( x ) ) - r ( p )
=s0).
We regard the value (p, L(x)) of the resource L(x) needed to implement the strategy x , as the “cost” of the strategy x when the price p prevails. Hence, we interpret (p, L(x))-r(p)as the value of the loss occured by implementing the strategy x under the price p. This loss is positive whenever the cost of x is larger than the maximum profit allowed. It is quite natural to add this loss penalizing the fact that x requires a non-available resources L(x) to the initial loss function of an optimization problem, i.e. to introduce the functions (25)
+
x F+ 4 x , P) =f ( 4 (P,L(x))- r(P)
as new possible loss functions, which depend upon the prevailing price. Proposition 2 implies the following result. Proposition 5 . Suppose that (22) and (24) hold. Then p is a Lagrange multiplier of the Lugrangian l ( x , p ) = f ( x ) + ( p , L(x))-r(p) and? minimizesf on X i f and only i f
I
+
(i) x’ minimizes x k+f ( x ) ( p , L(x)) on X , (ii) VP E P,(P, W ) =s ) (iii) (p,L(X)) = r(p).
m,
In other words, if the price p is a Lagrange multiplier (when such a multiplier exists), then an element X E X minimizes the loss function on X if and only if it
142
INTRODUCTION TO DUALITY THEORY
minimizes on R the sum of the loss functionfand the cost ( p , L( price j j and the cost ( p , L(.f))is equal to the maximum cost r(p). The subsets (27)
[Ch. 5, $ 5.2 a))
under the
X(y) = { x E R such that L(x)+y E Y }
are represented by the constraints
(28)
Y(X, P)+(P,
Y ) = (P,L(x)+y)-r(p).
This is because L(x)+y belongs to Y if and only if ( p , L(x)+y)-r(p) e 0 for all p E P.
Regard Y - y as the set of available resources obtained by removing a new resource y E V. Then (29)
v(y) =
inf
W)€ y-Y
f(x)
describes the behavior of the minimal loss with respect to perturbations of the set of available resources Y obtained by removing resources. Proposition 4 implies that the price j measures the marginal increase of loss when a new resource y is taken from the set Y of available resources (see Section 4.2.1). 5.2, Case of linear constraints: extremdity relations
We devote this section to the case where
X
= { x f R such that Lx E Y}
and L is a linear operator. Let yr denote the indicator of Y . Since we can write v = inf f ( x ) = inf If(x)+yr(Lx)] XER
X€X
we are led to a more general study of minimization problems of the form ZJ
= inf F(x, Ax). XEX
where F maps R X Y into R. We define a Lagrange multiplier p as the solution of the problem
{
(i) v* = inf F*(-L'p, p ) = F*( -L*p,p), PEP
(ii) v* -I-v = 0.
Ch. 5, 0 5.21
143
CASE OF LINEAR CONSTRAINTS
We prove that when F is convex and lower semi-continuous, ji is a Lagrange multiplier and I is an optimal solution if and only if the “extremality relation” {-L*p, p} E W(Z, LZ) holds. In the case where F ( x , y ) = f ( x ) + g ( y ) , the extremality condition can be written (i) z E af(-L*p)nL-l ag*(@), (ii) o E ag*(p)-L ar(-L*p). This formula leads, first of all, to a solution of the dual problem (0 E ag*(p)-L aj’*(-L*jj))and secondly, to an optimal solution X. Iff* is differentiable on -L*P,the Lagrange multipliers are solutions of the variational inequalities
{
(-LDf*(-L*p), p - p ) + g * @ ) - g * ( p ) =S 0 for allp E P. Hence X = Df*(-L*jj). We specify the particular form of the above variational inequalities in the following special cases : g*(p) = a m ) , g * m = O,#_P+(P) and g*(p> = O,#(P) = (PY 4Since the decomposition principle implies that
we check that these two equivalent minimization problems have the same dual problem. We also emphasize the so-called “decentralization” principle. When CJ .= = U‘, R = R‘ and f ( x ) = zy=lJ(d),Lx = Lid, the extremality relations of the minimization problem
ny=l
ny=l
c;IIcl
can be written (i) i = 1, . . ., n, E afi*L;(-j?), (ii) C;=l Liz’ E ag*(p), i.e. the relations “decentralize”. The rest of this section deals with examples of conjugate functions. 5.2.I . Generalized minimization problem Let (1)
(i) U and V be topological vector spaces, (ii) L E 2(U,V ) be a linear continuous operator, (iii) Y c V be a closed convex subset of V , (iv) R c U be a convex subset of U .
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INTRODUCTION TO DUALITY THEORY
'
[Ch. 5, 9 5.2
We assume that the strategy set Xis defined by (2)
X = { x E R such that Lx E Y } .
Since Y = { y E V such that ( p , y ) =s $ ( p ) for all p E V * } ,we deduce that X is represented by the constraints y(x, p ) defined on RX V* by
(3)
74x7 P ) = (P, W - a , # ( p ) .
-.
Recall that ay" = y~; is the conjugate function of the indicator of Y . Hence, -1, we can if R = Dom f is. the domain of a proper function f : U 1write 00,
+
(4)
The loss function of the dual problem can be written f'*(p) = - inf [l(x,p ) ] X € U
= SUP [(-L*P, x)-f(x)+Y,*Y(P)l X€ u
f
+
= *(- L*P) Y&P).
With a view to other applications, it is worthwhile to study more general problems of the type (5)
v = inf F(x, Lx) X€X
-
where F is a function detined on RXY. We shall say that p F*(--L*p,p) the loss function of the dual problem, where F* is the conjugate function of F defined by
Its minimal value is denoted by
(7)
v* = inf F*(-L*p, p). P€V*
We notice that
since, for all x E R and p E V*, we always have that (9)
0 = (-L*p, x ) + ( P , Lx)-S F(x, Lx)+F*(-L*p, p).
Ch. 5, $ 5.21
145
CASE OF LINEAR CONSTRAINTS
Definition 1. We shall say that p is a "Lagrange multiplier" of the minimization problem (14) or a "solution of its dualproblem" if
(i) v+v* = 0, (ii) v* = F*(-L*p,
p).
This definition is consistent with the previous one because, if we take F(x,y ) = =f ( x ) y y w , then
+
F*(9, P) = . f * ( d + 4 ( P ) .
Hence v* = inf F*(-L*p,p) = inf [f+(-L*p)+y;(p)] (11)
P
P
=
inff'*(p). P
An intermediate case occurs when (12)
F(x, y ) =f(x)+g(y), R = Dom f, Y = Dom g.
Since F*(q,p) =f *(q)+g*(p), we obtain that (13)
v = inf [f ( x ) + g ( ~ x ) ] , o* = t n f $f*(-~*p)+g*(p)]. XEX PE v*
In this case, we can say that p is a Lagrange multiplier if and onlj if (14)
+
v = inf [f($ (p,L x ) ]-g*@) X
since we can write (15)
v = - 2)' =-f *(-L*p)-g*(p) =-sup [( --L*p,x ) -f(x)] -g*(p). X
5.2.2. Extremality relations
-
Proposition 1. Suppose that F is a lower semi-continuous convex function whose domain is R X Y. Then jj is a Lagrange multiplier and 2 minimizes x F(x, Lx) on X if and only IY (16)
{ -~ * pp ,) E aqx', LZ).
If F(x, y ) =f (x)+g(y) where f this means that either
andg are lowersemi-continuous convex functions,
146
[Ch. 5 , § 5.2
INTRODUCTION TO DUALITY THEORY
or (18)
z E af*(-L*p) and Lz E
If Fcx, y ) =f ( x )+y&) (1 9)
ag*(p).
these relations become
Z 6 af *(-Lop)and 1 belongs to the normal cone to Y at fi.
Proof. The pair {-Lop, ji} belongs to W(Z, L?) if and only if (20)
( p , LZ) = FfZ, LZ)+ F*(- Lop,ji).
0 = (- L*p, Z)+
Recalling that inequalities (9) hold for any x C R and p C V*,we deduce that (20) implies that (21)
F(2, &?) = min F(x, Lx) and F*(-L*p,p) = min Po(-L*p, p) P
X
Conversely, if F(2,L.z) = w, F*(-L*p, p ) = w* and v+w* = 0, then (20) holds. Relations (17) or (18) are clearly true when F(x, y) = f ( x ) + g ( j ) and relation (19) clearly holds when g = yr. 0
Definition 2. Relations (16) (or (17), (18), (19)) are called “extremality relations”.
5.2.3. Thefundamental formula We give an equivalent formulation of the extremality relations (17).
Proposition 2. Let f and g be two lower semi-continuous convex properfunctions deJined on U and V respectively and let L E l(U,V). Then iminimizes x f(x)+g(Lx) on X and p is a Lugrange multiplier v a n d only i f
-
(i) x E af*(-L*p)nL-lag*@) (ii) 0 E ag*(p)-L af’(-L*ji)
Laf*L i lag;*
Remark. The following scheme illustrates eq. (22). XEU--
1-03, +-1--v*
af*l
f*
I
.
L
L*
v
VI=-I--,
P
+-1
Ch. 5, § 5.21
147
CASE OF LINEAR CONSTRAINTS
Remark. We can interpret the above result as follows. To solve the minimization problem, we begin by looking for a solution p of the multivalued equation (23)
0 E ag*(p)-L af *(-L*p).
Then, we look for a solution X of (24)
2E
ap~*(-p)nL-1 ag*(p).
Proof of Proposition 2. Let f and p satisfy-extremality relations (18). Then LX E ag*(p) and LX E L af *L*(-p). Thus 0 E a g * ( p ) - L af *L*(-P). Conversely, let p be a solution of 0 E ag*(P)-L af *L*(-P).There exists J E a g * ( p ) n L af*L*(-p). Therefore, there exists X E af*L*(-p) such that J = LX E ag*(p). Thus the extremality relations (18) are satisfied. 0 Proposition 3. Suppose that
(25)
f is direreentiable on
-L*P where P = Dom g'.
Then the correspondence af * is a map Df * (see Proposition 4.2.3). Thus relations (22) can be written
(i) X = Df *L*(-p), (ii)p E P = Domg*, (iii) V p E P, ((-L)Df+(-L*P), p-p)+g*(P)-g*(p)
(26)
0.
Proof. Statements (26(i) and (iii)) amount to saying that (27)
L D ~ z L * ( - E~ )ag*(p),
i.e. that 0 E ag*(p)-LDf *L*(-jj). Remark. The following lemma provides -a sufficient condition for assumption (25)Lemma 1. Suppose that (28)
{
(i) f is strictly convex, (ii) tlp E P , the function x F--f ( x ) + ( p , Lx) is lower semi-cornpact and semi-continuous.
Them the property
(25) is-satisfied. 12'
f * is dverentiable on -L*P
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Proof. Assumption (28(ii)) implies that for any q =-L*p E -L*P and for any il E R the subsets { x E U such thatf (x)- (q, x ) s A} are (weakly) compact. Proposition 3.4.5 implies that f* is continuous on -L*P and thus, subdifferentiable at any point of -L'P. Sincef is strictly convex, the subdifferentials af*(-L*p) contain a unique element. Proposition 4.3.5 then implies that this unique element is nothing other than the gradient Df*(--L*p). 0 ik5.2.4. Minimization problem under linear constraints
We consider the case when (29)
X = {x E R such that Lx- w E - P + >
where (i) R is the domain of a lowersemi-continuous convex proper functionf , (3) P is a closed convex cone of V*,'P is its polar cone and w belongs to V, (iii) L E B(U, V ) .
(30)
Proposition 4. Suppose that (28) and (29) hold and that (25)
f
.
is direrentiable on -L*P+
Then 2 E X minimizesf on.X and p is a Lagrange multiplier if and only if (31)
[
(i) 2 = Df*L*(-p), (G)Y E P , (iii) b'p E P , ((-L)Df*(-L*p)+w,p-p) =so.
f f P = V* (i.e. P+ = {0} or X = { x E R such that Lx = w}), these relations become (i) Z = Df*L*(-F). (3)LDfLL*(-P) = W . *5.2.5. Minimization of a quadraticfunctional under linear constraints
Suppose that (33)
U is a Hilbert space and that f (x) =
11 x-
u I12.
Let J E B(U,U*)be the associated duality map from U onto U*(see Section 2.2.5).
Ch. 5,
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CASE OF LINEAR CONSTRAINTS
Proposition 5. Let g be a lower semi-continuous convex proper function and L E (U,V).Then 2 minimizes x t-c$ Ilx-ul12+g(Lx) and p is a Lagrange multiplier if and only if (i) X = u- J-lL*p, (ii)p E P = Dom g*, (34) (iii) V p E P , ((LJ-lL*)p-Lu,p-p)+g'(p)-g(p) 0.
I
Proof. From the fact that f ( x ) = 1) x- u 112 we deduce that D f ( x ) = J ( x - u), Df*(p) = u+J-'p and that LDf*L*(-p) = Lu-LJ-lL*p. Thus eq. (34) follow from eqs. (26). 0 We shall now generalize Theorem 2.3.1 and obtain an explicit formula for the minimal solution on subspaces. Suppose now that X = { x E U such that L x - W E -P+}
Proposition 6. An element 2 E X minimizes x t(i) x = u-J-IL*p, (ii)p 6 P , (35) (iii) ((LJ-lL*)p- (Lu- w), p-p)
1
II x- u I l2 on X if and only if =S
0, p E P.
When L is surjective, ji is the orthogonal projection on P (for the scalar product ( ( p ,4))". = ((L*p,L*q)),,) of theLagrange multiplier Po = (LFL*)-~(LU- W ) of the minimization probleiq with equality constraints (case where P+ = (0)).
Proof. If g(y) = y(w-P+ ;y ) , then
Therefore, (35(iii)) follows from (34(iii)). If P+ = {0}, the Lagrange multiplier
PO satisfies the equation (LJ-lL*)po = Lu- w. When L is surjective, we deduce that PO = (LJ-lL*)-l (Lu- w). We can write (36)
((LJ-l.L*)(P-Po),P--p)
= ((p-po,p -p))y* -s 0, v p E P . 0
*5.2.6. Minimization problem under linear equality constraints
By taking g = y{,,,) to be the indicator of the point w, we obtain .v = (Lf>(w) and w* = -(f*L*)* (w) (37)
since g*(p) = ( p , w).
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Proposition 7. Consider the case where g = y(,,,).Then jj is a Lagrange multiplier of the minimization problem v = ( L f )(w) if and only if (i) ( L f )(4= ( f *L*)* (4= (Lf)**(w), (ii) - p E a ( L f )(w).
If we assume furthermore that f is convex and lower semi-continuous, then X minimizes f under the constraint L x = w, if and onIy if (39)
I E af*L*[Laj-*L*]-1 (w).
Proof. We have that p is a Lagrange multiplier if and only if v = ( L f ) ( w )= - v* = (f*L*)*(w) = sup [(-p, w)- f *L*(-p)] = ( - F , w)- f 'L*(-p). P
This amounts to saying that p E - a(Lf) ( w ) because f *L*= (Lfl*(see Proposition 2.4.4) and we can write (40)
(Lf) ( w ) = (-P, w)- (Lf+)* (-PI.
Since g*(p) = (p, w), the extremality relations can be written (41)
2 E af*L*(-p) and Laf+L*(-p) = w.
Remark. This formula is an extension of a formula we proved in the case of the quadratic functionalf(x)
=
f I I x- u I l2 (see Theorem 2.3.1).
5.2.7. Duality and the decomposition principle Consider the case when F(x, y) = f (x)+g(y). R e d l that
It is clear that these two equivalent problems have the same dual problem. In the first case, we take F(x, y ) =f(x)+g(y). Then the loss function of its dual problem is
Ch. 5,
5 5.21
151
CASE OF LINEAR CONSTRAINTS
In the second case, we take U = V, L = 1 and G(x,y ) = (Ln(x)+g(y). Then the loss function of its dual problem is P I-+
(Lf)f(-p)+g*(p) =f+L*(-P)+g*(P)
since (Lf)*=f*L*. The extremality relations for the second problem are given by
(44)
p E ag(7) and - p E a(Lf) (7).
5.2.8. The decentralizationprinciple
Consider the case when (45)
n;=l
u= U', f(x) = C;=lf;(~i) where5 : Ui I- m , Lx = C;=l Lixi where Li E ,P(Ui, V). +
+ -I,
The domain R off is the product R = fl;=lR' of the domains R' of the functions A. Proposition 8. Let
E V * be a Lagrange multiplier of the minimization problem
This amounts to writing
v = --v* = -
(47)
n
=
C
cfi*(-GP)n
i=1
g*(P)
inf [fi(x')+(L;p, xi)]-g*(p).
i=l d € R {
In this case, X = {Zl,. . .,X} is a solution of the minimizatior problem (46) if and only if (i) V i = 1, . .., n, X' E afrL;(-p),
Proof. This is left as an exercise. Interpretation. We interpret the functions A as loss functions of n players i = 1, . . ., n. Extremality relations (48(i)) show that knowledge of the value of a Lagrange multiplier p (if one exists) allows one to deduce that the component
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Ti of an optimal solution 2 = (3,. . .,P} lies in the subdifferential Elf ;L;(-p) o j the lossfunction of player i.
In particular, wheneverf f are differentiable on -L:P (where P = Dom g*), optimal solutions 2‘ and the Lagrange multiplier p are solutions of the equations
(49)
(i) V i = 1, . . ., n, Zi = Df;c(-Lrp) where p is a solution of
We interpret relations (48(i)) by saying that 2 “decentralizing” problem (46)by p .
=
{Zl,. . ., P}is obtained by
5.2.9. Conjugatefunction of gL
Proposition 9. Consider (50)
L E a(U, V ) and g : V I-+
1-
00,
+ -1.
Suppose that the following in&um
isjnite. Then an element p E V* is a Lagrange multiplier if and only if‘ (52)
-v =
v*
= (L*g*)(4)= g*(p) where L*fi = q.
ikforeover, ifr E a(L*g*)(q) and g is convex and lower semi-continuous, we obtain that
(53)
(Y) = ( L ’ W ) (Y).
Proof. The minimization problem we are considering is the particular case when f (x) = - (4,x). Since f * ( p ) = ~ ( - , ~ ( pis) the indicator of -4, we deduce that the loss function of the dual problem is defined by F*(-L*p,p) = = y(-,) (-L*p)+g*(p) = y+,)(L*p)+g*(p). Therefore v* = inf,.,,,g*(p) = (L*g*)(4). Thus, p is a Lagrange multiplier if and only if =7
(54)
(gL)* (4)= g*(p) = inf g*(p) where L*fi = q. L.p=q
It is clear that we always have (55)
L*W(Y)
= akL1 (Y).
Ch. 5 , s 5.21
153
CASE OF LINEAR CONSTRAINTS
To prove this write q = L*p wherep E ag(Ly). Then g(Ly)-gW
(P, J3-W = (4, y - 2 ) .
= (L*P,y - 4
* **
Conversely, suppose that y E a(L*g*)(q), i.e. that q E a(L g ) ( y ) = ** ** = a[g L ] ( y ) = a(gL) (y). We deduce from (54) that (56)
=( g o
(47
(v)+ (gL)*(q) = ALYl + g * m
where L*P = q. This implies that (57)
(P7
LY) = (L*& u) = (4, Y ) = g[Lyl+g*(P)
and thus that p E ag[Ly].Hence (58)
q = L*p E L*ag[Ly].0
5.2.10. Conjugatefunction of f i
+
f2
Proposition 10. Consider (59)
two proper functions jland f 2 from U into ] -
00,
+
]
and assume that the following inJtnum (60)
v = - (f1+f2)*
(4) = inf [ f i ( x ) + f ~ ( x )(4,x)I XE
u
isfinite. Then an element p E V* is a Lugrange multiplier if and only if (61)
- 21 = v* = (fi* 0fi*) (4) =f;(q-p)+fi*(Ji).
Moreover, iffl andf 2are convex and lower semi-continuous and y E then (62)
w 1 +f2)
a[f;O f,’](q),
(v) = afl(Y)+af2(u).
Proof. The minimization problem we are considering is the particular case where V = U,L = 1, f is defined by f (x) = fi(x)-(q, x) and g =fz. Since , f * ( p ) =f:(q+p), the loss function of the dual problem is defined by F*(-L*p, p ) = f:(q-p)+f2(p). Therefore (63)
?J* =
inf[fi(q-~)+fi(p)l = [I? fiI(q). P
Thus p is a Lagrange multiplier if and only if ( 64)
(fi+f2Y
(4) =fi(q-P)+X(P)
= [fi”
fi’I(q).
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[Ch. 5, $ 5.2
INTRODUCTION TO DUALITY THEORY
It is clear that we always have
+w 2 m = w-1+ f 2 ) (v). Conversely, suppose that y E a[f: 0 f:] (q), i.e. that q E a[f: 0 f;l*(y) = = [ah**+ f,”] ( y ) = a(fi+ f 2 ) (y). We deduce from (64) that q = (q-p)+p
(65)
satisfies
0 = (4,Y>’- (fi” 0fi”) ( 4 ) - [(fl(v)+fi(~)l = [(q-P 7
v)--fi,(q-P) -fdv)l
+ [(P, U)-fi”(P)--f20.’)I.
Since each of these two terms is non-positive, we deduce that‘both are equal to 0. This implies that 4-p E afi(y) and p E af2(y), i.e. that q = q-p+ F E afl(Y>+v2(r). cib *5.2.11. Minimization of the projection of a function
-..
Proposition11. Let U1 and Uabe two vector spaces andf : Ulx U2 ] - m ,+ -1 be a proper function. An element p1 E (Ui)* is a Lagrange multiplier of the problem
if and only if *@7)
-v
= v* =
inf f *(ply0 ) =f
P’E(U’)*
Proof. We take U = U 1 x U 2 , V F(x, y ) = Ax1, x2) Y{O)(Y).Then
+
(68)
=
*(pl,0).
U1. We define Lx =-x1 and write
w = inf f (xl, x2) = inf [f (xl, x2)+y(o)(-xi)]. -x1=0 xe€ up
+
The loss function of the dual problem is defined by F*( -L*p, p ) = f *( - p , 0) f O . Thus (69)
w* = inf f * ( p l , 0). CI P‘E (U9*
*5.2.12. Minimization on the diagonal of a product
Proposition 12. Let V be a vector space and f : V X V function.
+
]- m ,
+
a
] be a proper
,Ch. 5, 0 5.31
EXISTENCE OF LAGRANGE MULTIPLIERS
155
An element ji is a Lagrange multiplier of the problem
(70)
w = inf f ( x , x ) XE
v
if and only i f (71)
-
w = w* = inf f * ( p , - p ) = f *(p, -p). PE
v*
Proof. We take
u = vxv,
Lx =-x1+x2
whenever x = {xl, x z } E U. We set F(x, Y ) = f W , X ~ ) + Y { ~ ) ( Y ) . Since Lx = 0 if and only if x1 = x2, we can write v = inf,, f (x, x). The loss function of the dual problem is F*(-L*p,p) = f * ( p , -p)+O. 0 5.3. Existence of Lagrange multipliers in the case of a finite number of constraints
In this section we prove two existence theorems. The first deals with minimization problems of the type w
=
inf F(x, Lx),
x€R
where F is a convex function defined on R x Y , R is a convex subset of a vector space U,Y a convex subset of V = R". The Fenchel theorem states that the constraint qualification assumption 0 E Int ( L ( R ) - Y )
implies the existence of a Lagrange multiplerp. We also check that the constraint qualification assumption implies the stability of Lagrange multipliers in the sense that, if F depends upon A, then the Lagrange multipliers Is, stay in a compact subset under reasonable assumptions. We apply this result by proving the subdifferentiability of Lf on Int [ L Dom g ) ] and of gL whenever 0 E Int [L(U)- Dom g ] .Finally, we prove the Uzawa theorem which establishes the existence of Lagrange multipliers of
w = inf f ( x ) AxsO Bx=O
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when f is a convex function, A is a copvex operator and Bis an affine operator. The required constraint qualification assumption is that (i) 32 such that A(1) cc 0 and B(2) = 0 (Slater condition), (ii) 0 E Int B(R) (surjectivity condition). 5.3.1. The Fenchel existence theorem
We shall prove an existence theorem for Lagrange multipliers, in the case when V is a finite dimensional space.
.
Theorem 1. Suppose that (1)
V = R” is a finite dimensional subspace
and that (2)
F is a convex function from RX Y into R where R c U and Y c V are convex subsets.
If (3)
0 E Int (L(R)-Y),
then there exists a Lugrange multiplier p E V*. Remark. Recall that the “constraint qualification” assumption ( 3 ) implies that p I--+ F*(-L*p, p ) is lower semi-compact on V* (see Section 3.1.4 and Proposition 3.1.6). Remark. We do not ;lave to assume that F is lower semi-continuous when we ask only for the existence of a Lagrange multiplier. Proof. We introduce the following operator C#J from R X Y into R X V defined by
(4)
9 ( x , v) = {F(x, v);
-
v}
and the following items (5)
{w, { ((i)i i )theth ecovectorn eQ=]O,
0} E R X V , -[X{O}c
RXV.
It is clear that the convexity of F and the linearity of L imply that (6)
C#J(R X Y)+Q is a convex subset of V
Ch. 5, 0 5.37
EXISTENCE OF LAGRANGE MULTIPLIERS
157
(see the proof of Prop6sition 1.3.10 for instance). Also,
(7)
4 #@xY)+Q
{v, 0)
[If {v, 0)E b(RXY)+Q, there would exist x E Randy E Y such thatLx-y = 0 and v r F(x, y ) = F(x,Lx). This contradicts the definition of v.] We may therefore use the separation theorem for a point and a convex subset in finite dimensional spaces. There exists a non-zero linear form {a,p } E RX V* such that (8)
av
=S
({a,PI, { v , O ) )
=S
0*(4(RX Y)+Q; {a,P I )
-
= cb(b(RXQ;{a,pI)+ab(Q, {a,
PI).
Since Q is a cone and d(Q; {a,p}) is bounded from below, we deduce from Proposition 1.4.7 that {a,p} E Q+, (i.e. that a 0) and that d(Q;{a,p}) = 0. Thus, inequality (8) becomes
We must have a 0s
f 0. Otherwise,
inf
xER, Y € Y
(p,Lx-y) =
inf
u€UR)-Y
(p,u).
Since 0 f Int (L(R)- Y ) , this implies that p = 0 and thus, that {a,p } = (0,0) which is a contradiction. Therefore, a > 0. Dividing both sides of inequality (9) by a > 0 and setting p = pla, we obtain that v = inf inf [ F ( x , y ) - ( - L * , x € R ICY
x)-(p,y)]
=-F*(-L*jj,F).
Since F*(-L*p, j?)
v* a
- v,
we deduce that p is a Lagrange multiplier. 0
Remark. We shall extend this result in the case where V is an infinite dimensional space (see Section 14.1).
*5.3.2.Stability properties The constraint qualification hypothesis implies with appropriate assumptions that the Lagrange multipliers stay in a fixed compact subset when the loss function F depends upon a parameter.
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-
Proposition 1. Suppose that (1)and (3)hold. Let A and F: R X Y X A R satisfy (10)
1
(i)A is a compact subset of a topological vector space W, (ii) A I-+ F(x,y;A) is upper semi-continuousfor all {x, y } E R X Y , (iii) {x, y } ++ F(x, y ; A) is convexfor all I E A, (iv) 3 d E W* such that, VA E A, (A, d)=s inf,,, F(x, L x ; A),
Then the Lagrange multipliers pAof the problems (1 1)
v(A) = inf F(x, Lx; A) X€X
stay in a compact subset POof P.
Proof. Let p AE V* be a Lagrange multiplier of v(I) where (12)
v(A) = - F*( -L”p2,PA).
Such a pA exists by Theorem 1. We shall prove that pa remains in a bounded (and thus, relatively compact) subset of P, i.e. that, for any z E V , SuPAcn (PA7 4 =s 4 4 -= + O0 Since 0 E Int ( L ( R ) - Y ) , there exist E z 0, x E R and y E Y such that z = & ( ~ - - L x ) . Thus (PA, -4 = E[(-L*PA7 X ) + ( P A , Y)1 =s E(F*(--L*Pa, pa; A>+F(x, Y ; A)) E[F(X,y ; 2)- v(4l == 4% y ; 1)- ( I , d)) d sup [ F ( x ; y ; A)-(& d)]=sEM. 0 A€A
*5.3.3.Applications to subdirerentiability Application : Subdverentiability of Lf.
+
Let f be a convex proper functioxt from U into ] - a, 00 ] and L E &(U, V ) . When V is a finite dimensional space, we know that Lf is continuous (and thus, subdifferentiable) on the interior of its domain Dom (Lf>= L Dornf. This result can also be obtained by applying the above theorem to the case when F(x, y ) =f ( x ) + y ( { w } , y ) , since the “constraint qualification” assumption (3) then reduces to 0 E Int ( L Domf- w) = Int Dom Lf- w. Application:Subd@erentiability of gL.
Proposition 2. Suppose that (1)holds and 0 E Int (L(U)-Dom g). (13)
Ch. 5, Q 5.31
EXISTENCE OF LAGRANGE MULTIPLIERS
I f g is convex and q
159
E Dom (gL)*,then there exists jj such that
(gL)+(q) = (L*g*)(q) = g*(p) where L*p = q.
(14)
Furthermore, if g is lower semi-continuous, then = L+W L y )
(15)
Proof. We apply Theorem 1 to the case where F(x,y) =f ( x ) + g ( y ) with f ( x ) =-(q, x ) and R = Domf = U. Then there exists a Lagrange multiplier jj whenever (gL)*(4)-= This satisfies the above properties by Proposition 2.9.
+
Qo.
Application :Subdifferentiability of f1+ fi Proposition3. Let f1 and f2 be two convex proper functions from V into 1+ -1. Suppose that (1) holds and that 00,
0 E Int (Dom f1- Domfz).
(16)
Ifq E Dom (f1+f2)*,then there exists p E U* such that (17)
(f1+f2)*
(4) = ( f i * q * ) (d =f:(q--P)+.&*(p).
Furthermore, if f1 and f2 are lower semi-contimow, then (18)
awl
+fz)(r) = afl(rl+afi(r).
Proof. We apply Theorem 1 to the case when L = 1 and F(x, y ) =f (x)+g(y) with f ( x ) = f l ( x ) - (4,x) and g(y) =f 2 ( y ) .Assumption (3) amounts to writing that 0 E Int (Domfi-Dom f 2 )since R = Domf = Domfl and y = Dom g = = Dom f 2 . Thus there exists a Lagrange multiplier p whenever (f1+f2)* (4)< . This satisfies the above properties by Proposition 2.10. 0 <
+
~o
5.3.4. Case of nonlinear constraints; The Uzawa existence theorem
Consider (19)
i
(i) a convex subset R of a vector space U, (ii) k convex functions a, defined on R, (iii) m- k afine functions bi defined on R, (iv) a convex functionf defined on R.
We associate with these items the subset
(20)
X
= {x C R such that a@)
k+l e j e m }
=s 0
for 1 e i s k and bj(x) = 0 for
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Consider the problem
Proposition 4. Given the convexity assumptions (19), the "Slater condition" 3 1 such that ai(2) -c 0 V i = 1, . . ., k and bj(2) = 0, V j = k + l , ..., m
(24)
and the surjectivity condition:
0 belongs to the interior of B(R) in R"'+k),
(25)
it follows that there exists a Lagrange multiplier
{p,q} E Rk+'X Rc"'-k)*.
Proof. This is analogous to the proof of Theorem 1. Consider the operator 4 mapping X into RkXR"-k defined by +(x) = {f(XI, 49,B(x)).
(26)
Write
{ (ii)(i) c
8 = {v, 0, 0} E R x R ~ x R ~ - ~ , = R, x R: x (01.
(27)
Then (28)
~
e 6 +(R)+c.
If 8 belongs to +(R)+C, there would exist 1 E R such that
v =- f (3, A(P), 0 = B(1).
0
This is a contradiction of the definition of v. Convexity assumptions (19) imply that +(R)+C is convex (see Proposition 1.3.10). Since +(R)+C is contained in u finite dimensional space, we can use the separation theorem. Thus there
Ch. 5, 0 5.31
EXISTENCE OF LAGRANGE MULTIPLIERS
161
exists a non-zero triple I = {a,p, q } E RXRk*XR("'-')* such that
The fact that C is a cone implies that
can then be written
then we can divide both sides of inequality (30) by a and setp= p / u and 4 = gla. Then (32)
-s
inf [ f ( x ) + (P, 44)+ (4, B(x))l
X€X
as required. It remains to prove (31). For this purpose, we assume that u = 0. The Slater condition (24) then implies that p = 0 and condition (25) implies that q = 0. But this contradicts the fact that 1 = {a,p, q } is non-zero. [Tosee this, let u = 0 and let 2 E Xbe the element of Xappearing in the Slater condilion. Then we deduce from (30) that
(33)
0 =s(P, 4 2 ) ) =
cP W 3 k
i=1
Since q(2) -= 0 and pia=0, we deduce that p' = 0 for all i, i.e. that p = 0. Because p = 0, (30) reduces to the inequality (34)
0 =S(q, B W ) Vx
E R.
But 0 belongs to the interior ofB(R) and so this implies that q = 0.1 0
13
CHAPTER 6
TWO-PERSON GAMES: AN INTRODUCTION
This chapter is devoted to an account of most of the important concepts of game theory for two-person games. In Section 6.1, we describe two-person games in “normal” or “strategic” form in terms of the following items :strategy sets X and Y of the two players, a subset U c X X Y of feasible pairs of strategies and a biloss operator F : U R2 associating with any pair {x, y } E U the biloss F(x, y) = { f ( x ,y), g(x, y)}, where f (x, y ) is the loss of the first player and g(x, y ) the loss of the second. The influence of a player on the other appears (a) in the fact that the loss functions f and g depend upon the strategies played by both players ; (b) and in the fact that U c X X Y is not necessarily the product of two strategy sets: if x is chosen, then the choice of y is required to belong to the set S(x) = { y E Y such that {x, y } E U}. We begin by introducing two‘kinds of “values” of the game. The fist is the “shadow minimum” a = {a,, aY} E R2defined by
-
ax =
inf f ( x , y ) ,
{“;Y)EU
ay
= inf g(x, y). {X.Y)EU
The definition implies that F(U) c a + R t when both 01, and oly are finite. Usually, a does not belong to F(U). The “Conservative values” :v and v g are defined as follows in the case when u =XXY. If the first player (named Xavier) has no information whatsoever about the strategy choice of the second player (named Yvette), he calculates the worst possible loss associated with a strategy x E X by the formula f ” ( x ) = = sup,,,f(x, y). Then v: = infXcxf*(x) = inf,,, ~up,,~f(x,y ) is called the conservative value for Xavier, and v; = inf,,, g(x, y ) is called the conservative value for Yvette. If we set v x = {v$ vy“} E R2,the subset V of strategies defined by
V = { {x, y } E U such that F(x, y ) =sv+} 165
166
TWO-PERSON GAMES: AN INTRODUCTION
[Ch. 6
is the “useful” part of V, in the sense that Xavier, for example, will “reject” x iff(x, y ) > v x (since he can do better in the worst situation). Notice that F(V)is contained in the square ( a + R t ) n(w* -. ) : R We next define the following concepts of “stability” and “equilibrium”. A pair {Z, J} E X x Y is individually stable if neither Xavier nor Yvette have any incentive to modify their own choice of strategy unless the other player modifies his choice. This means that
We say that {z, j j } is a “non-cooperativeequilibrium”. It is collectively unstable, if both Xavier and Yvette can find a pair of strategies {x, y } E U such that f ( x , y ) -= f(R, 7) and g(x, y) < g(Z, j?). The collectively stable pairs of strategies are called the Pareto minima. The ideal situation occurs when there exists a unique pair of strategies {Z, p} which is individually and collectively stable, i.e. a non-cooperative equilibrium which is Pareto optimal. Unfortunately, this situation is quite exceptional. We are therefore led to let aside the concept of individual stability while retaining that of collective stability (Pareto minima exist under quite weak assumptions). We relax the requirement of individual stability then have to devise selection procedures for Pareto minima. For instance, we require that Pareto minima belong to V, i.e. are rejected neither by Xavier nor by Yvette: the subset of such pairs of strategies is called the core of a (two person) game. In Section 6.2 we examine several examples of games for which some of the above concepts apply and other do not. These examples show that there is no hope of defining a unique “good” concept of solution. In Section 6.3 and Section 6.4, we examine the two basic economic models. The first illustrates the behavior of a duopoly. In this case, X and Y are production sets, U = X X Y (i.e. the productions can be chosen indepedently) but the production costs f(x, y) and g(x, y ) depend upon both productions. The second deals with the choice of consumptions x and y such that x+ y = w . Hence the subset U is defined by U = { { x , y } E X X Y such that x f y = w}. In this model, the loss functionfcan depend only upon x and g upon y . Finally, in Section 6.5, we introduce the very important special case of zerosum games, where g(x, y ) = -f(x, y). For such games, any pair of strategies is Pareto minimal and so the concept is not useful. One therefore looks for non-cooperative equilibria. These are nothing other than the “saddle-points” off.
Ch. 6, 5 6.11
SOME SOLUTION CONCEPTS
167
6.1. Some solutibn concepts
In this section we describe the main concepts of game theory as they apply to two-person games. 6.1.1. Description of the game
Let Xavier and Yvette be the (first) names of the two players. We describe the two-person game in the “strategicform” or “normal form” by the following items:
1
(i) the strategy set X of Xavier, (ii) the strategy set Y of Yvette, (iii) the loss functionf :X X Y R of Xavier, (iv) the loss function g :X X Y R of Yvette.
(1)
-. -
We associate with any pair {x, y } E X X Y of strategies the loss f (x, y ) allocated to Xavier and the loss g(x, y) allocated to YvetJe. We denote by F(x, y) = = { f ( x , y ) , g(x, y ) } E R2 the “biloss” associated with the pair of strategies {x,y } and call the map F : {x, y } E X X Y I-+ F(x, y ) E R2the “biloss operator”. It can happen that not all pairs of strategies can be implemented, but only those which belong to (2)
the subset U c X X Y of feasible strategies.
The description of the game in strategic form is then summarized by the notation (U,F) where U c X x Y and F maps U into R2. 6.1.2. Shadow minimum We write
Definition 1. We shall say that the game is bounded below if both ax and aY are finite. In this case, we shall say that the biloss a = {ax,aY}is the “shadow minimum” of the game. In the case where there exists {a,y } E U such that
= / ( a , J); {a, J } yields
ax
QY
= g(% y3
such il. pair the minimal loss to both Xavier and Yvette, i.e. F(2,jj) = a is the minimum (relative to the appropriate orderirigof R2)of F(x, y )
42
43
&
afl
168
TWO-PERSON GAMES : AN INTRODUCTION
[Ch. 6,
6.1
as {x, y } ranges over U.Such a pair (2,j } is reasonable as a solution concept but will obviously exist only in exceptional situations. This is why we call the vector a a “shadow minimum” or “virtual minimum”. We have that
F(U) c a+R$ Since in most cases the shadow minimum does not belong to F(U), we are led to introduce other solution concepts. 6.1.3. Conservative solutions and values
Consider the case where U = X X Y. We introduce the functionsf (fsharp) and g” defined by (4)
{
”
(i)f”(x) = suP,,yf(X, Y ) , (ii) g”(r)= SUPXEX g(x, u).
Since f(x, y ) is the loss for Xavier when Xavier chooses x and Yvette y , f ” ( x ) i s the “worst loss” possible for Xavier when he plays the strategy x E X . A consistent policy for Xavier whenever he has no information whatsoever about the strategy to be played by Yvette, is to associate with any strategy x E X the consequent worst lossf”(x). In the same way, when Yvette has no information about the choice of Xavier, she may associate with any strategy y E Y the consequent worst loss g # ( y ) . Thus, under these behavioral assumptions, Xavier will choose a strategy xQ which minimizesf” over Xand Yvette a strategy y x which minimizesg* over Y. Definition 2. We shall say that a strategy x” which satisfies (5)
f*(x”) = v$ = inf f # ( x ) X€
x
(resp. y” such that g”(y”) = v: = inf,,,,g#(y)) is a “conservativesolution” for Xavier (resp. Yvette). The vector v” = {v:, vy“} is called the conservatzve value of the game. Conservative values will be used as examples w of threats i n the following sense. Xavier Will reject any strategy x yielding a loss f(x, y ) larger than v$ = = inf,,,f* (x), since v? is the loss that he can obtain by unilateral action, whatever the other player,does. Most often these threats are not implemented. Usually, a “conservative solution satisfies neither the “individual stability property” nor the “collective stability property”. We discuss these important properties next.
Ch. 6, Q 6.11
169
SOME SOLUTION CONCEPTS
6.1.4. Non-cooperative equilibrium
For simplicity, we assume that U = X X Y. Consider a pair {x#,y ” } of conservative strategies for Xavier and Yvette, In some cases, Xavier has a strategy x E X for which
f(x, v#> -= fW, v#)* In other words, {x”, y”} does not have\t‘individualstabilita since Xavier, by acting alone, is better off by switching fromx#and playing x. We are led to introduce the following concept of “non-cooperative equilibrium” which does satisfy the individual stability property. Definition 3. We shall say that a pair {X, J } of strategies is a “non-cooperative equilibrium” if
(6>
I
( 0 f ( 2 ,J ) = min f ( x , J), X€X
(ii) g(2, J ) = min g(2, y). Y€Y
In other words, (2,J } is a non-cooperative equilibrium in the sense that when Yvette implements 7, the optimal strategy for Xavier in respect of the is T and, symmetrically, when Xavier plays Z, the loss function x t--f(x,J) optimal strategy for Yvette in respect of the loss function y F-- g(2,y)is 7. Remark. Suppose that there exists a map Cfrom Y into X (called the optimal decision rule of Xavier) satisfying
(7)
VY E y,
f(W, Y ) = minf(x, r) X€X
and a map I3 from X into Y (called the optimal decision rule of Yvette) satisfying
Then any solution {T, J } of the system (9)
C(J) = x’ and D(X) = J
is a non-cooperative equilibrium. More generally, if we denote by @ the correspondence fom Y into X defined by (10)
Vy E
Y, @(y) = x’ E X such thatf(x, y ) = minf(x, y)
{
X € X
I
170
[Ch. 6, Q 6.1
TWO-PERSON GAMES : AN INTRODUCTION
and by
athe correspondence from X into Y defined by
(1 1)
Vx E X, a ( x ) = ji
(
E Y such that g(x, j ) = min g(x, y)}, YEY
then a pair {Z, 7 ) is a non-cooperative equilibrium y a n d only i f (12)
i.e.
2 E
@(?) and jj E
a(?),
{a, y} is a fixed point of the correspondence {x, y } I-+
@(y)X m(x).
Remark. We introduce the functions~(j4at)and g" defined by (13)
fb(y)
= inf f ( x , y); X€X
$(x) = inf g(x, y). Y€Y
We set
t 14)
Fb(x,Y ) = {fb(Y), g"(x)l. Note that {X, j j } is a non-cooperative equilibrium i f and only if
(15)
F(I, 9 = P ( Z , P).
6.1.5. Pareto minimum
Consider a pair {x#, y " } of conservative strategies for Xavier and Yvette. In some cases, both Xavier and Yvette can find a pair of strategies {x, y} E U such that (16) f ( x , r) -=f(x+, Y') v) -= &", YX). In other words, {x", y " } does not have "collectivestability" since, by cooperating, both Xavier and Yvette can find strategies yielding both a smaller loss. We are led to introduce the following concept of "Pareto minimum" which does satisfy the collective stability property.
Definition 4. We shall say that a pair {Z, j j } E U of strategies is a "weak Pareto minimum" if there is no pair {x, y} E U such that both (17)
f ( x , r) -= f(%
7) and Ax, v) -= g ( Z a-
Remark. A ,,nice solution conceptq'is obtained when both the individual and the collective stability properties hold. Unhappily, such a requirement is too strong in most instances. The set of Pareto minima is usually large. The main problem is to find selection procedures for Pareto minima, i.e. to find solution concepts satisfying the collective stability property. We also notice that, if a pair {x, y} is not collectively stable, (in the sense of (17)), then the players wilt jointly reject {x, y } . This leads to the concept of the core of a gamc.
Ch. 6,
0
6.11
171
SOME SOLUTION CONCEPTS
6.1.6. Core of a two-person game For simplicity, we suppose that U = X X Y
Dehition 5. We shall say that a pair {x, y} of strategies is “rejected”’byXavier if f ( x , y) =- u$, is ‘‘rejected’’ by Yvette if g(x, y) > v$ and is “rejected” by both Xavier and Yvette if it is not a weak Pareto minimum. The “core of the game” is the set of pairs of strategies rejected neither by Xavier, nor by Yvette, nor by both of them. In other words, the core of the game is the subset of weak Pareto minima {Z, 7 )such that F(x, y) =s vU%. Fly, space of
Yvette‘s bilosses
of bi losses
I subset of
of Pareto minima
1
\
I I\
v
\
‘.
’.
bilosses of strategies of the core
conservative value
/
I
/
, /
/
Fig. 1. Image of$ game in the space of bilosses
6.1.7. Selection of strategy of the core The main problem remaining is the selection of a strategy which belongs to the core (whenever it is not empty). Chapter 10 is devoted to this problem in the case of n person games. We shall only mention here the following selection procedure. We assume that
172
TWO-PERSON GAMES: AN INTRODUCTION
(instead of a,
w$ and a y
[Ch. 6, 9 6.2
v,") and introduce the following ratios
These measure the distance to the shadow minimum relative to the distance between the conservative value and the shadow minimum. We assume that both players agree to minimize the worst ratio a x , Y) = m a (&(x, Y),d Y ( X , Y)).
(20) We set
It is easy to check that any bistrategy {Z,,j7} which minimizes d(x, y) belongs to the core and actually satisfis inequalities (22) f(% r3 =S (l-a)ax+av$, where at least one equality holds.
g(Z, p) =G (l-il)ay+dw$-
a
Fig. 2.
*6.2. Examples: some finite games
It is time to illustrate the above concepts with some simple examples with the strategy sets X and Y finite. We begin with an example for which non-coopera-
Ch. 6, 0 6.21
EXAMPLES: SOME FINITE G A M E S
173
tive equilibria do not exist but where the core consists of a unique pair of strategies. This pair can be regarded as the “solution concept” for the game. We introduce the “coordination game” as our second example. In one case, we shall describe a situation where the shadow minimum is achieved. In another case, we shall see that this game has non-cooperative equilibra which are Pareto minimal, i.e. which are both individually and collectively stable. In the’example of the “prisoner’s dilemma”, we have a situation where the conservative strategies are individually stable (i.e. form a non-cooperative equilibrium), but are not collectively stable. This example shows the need of communication for implementing a Pareto minimal strategy. In the “game of chicken”, we obtain a pair of strategies which is both a conservative solution, a non-cooperative equilibrium and a Pareto minimum. Finally, the “battle of the sexes” illutrates a case where all pairs of strategies are conservative solutions and where two of them are both individually and collectively stable. 6.2.1. Example In this first example, non-cooperative equilibria do not exist and the core consists of a unique pair of strategies. For this pair collective stability holds but no strategy pairs are individually stable. Let Xavier be the first player and Yvette the second player and let X = = {I, 11)and Y = { 1,2,3} be their strategy sets. We represent the biloss operator F defined on XX Y by the following “bimatrix”, each entry of which is a pair of scalars, the first being the loss to Xavier and the second the loss to Yvette. \ Xavier ( l
I
2
I
3
F:
irst, we notice that a, = - 6 and ay = -2. Then we check that f*(I) = 3, f*(II) = 4, g*(l) = 3, g”(2) = 2 and g*(3) = 7 and so (1) vx“ = 3 =f”(I); v; = 2 = g 7 2 ) Therefore (2)
a = (- 6, - 2) is the shadow minimum
174
TWO-PERSON GAMES : @4 INTRODUCTION
Fb :
I I1
(-6.0)
(-5.0)
[Ch. 6 , s 6.2
(-SO)
(-6, -2) ( - 5 , -2) (-3, -2)
By comparing the matrices of F and p ,it is easily seen that non-cooperative equilibria do not exist. In particular, the conservative solutions are not individually stable. The set of Pareto minima consists of the pairs {I, l}, (11, 2) and (11, 1). The core consists of the pairs (11, 2) and (11, 1).
I I I
I
--___.--I
Fig. 3.
Ch. 6, 0 6.21
175
EXAMPLES: SOME FINITE GAMES
6.2.2. Coordination gmne
Let Xavier and Yvette be the two players and let X = {I, 11) and Y = {I, 2) be their strategy sets. Let F be the biloss operator represented by the following “bimatrix”
O
F:
I
I
I
I
It is clear that f*(I) = b, f#(II)
Hence (4)
= SUP (a, x),
g*(l) = b, g+(2) = SUP (a,x).
..=(
(b, b) = (f*( I), g*(I)) if b e x, (x, x) = (f”(II), gX(2)) if a =S x 4 b, (a, a) = (f*(II), g*(2)) if x G a.
On the other hand f b ( l ) = a; f b ( 2 ) = inf(0, x),
?(I) = a, d(II) = inf (0,x).
The operator Fb is represented by the following bimatrix.
Interpretation. Xavier and Yvette are caught in a fire and have to push against a door to escape. But this involves exposing oneself to the flames for the time it takes the door to open. We denote by I, 1, the strategies: “do not push” and by 11, 2, the strategies “push”. If nobody pushes against the door, both stay in the fire and incurr a loss of b.
176
TWO-PERSON GAMES : AN INTRODUCTION
[Ch. 6, 8 6.2
If Xavier pushes against the door (strategy 11) and Yvette does not (strategy l), then Yvette escapes first (and looses nothing) and Xavier escapes second incurring a loss of a -= b. Let x be the loss incurred by each player when both push against the door together. This is treated as a parameter. Case 1: x =s0. In this case, strategy (11, 2) yields a gain to both, i. e.a neg-
ative loss and it is obvious that it is the optimal strategy. Indeed, in this case, the shadow minimum a = {x, x} = F(I1, 2) is achieved. Thus (11,2) is the smallest (in R2)of the bilosses (and thus, is no longer “shadow”).
l l
(Y
=
/
F(II, 2)
Fig. 4.
Thus (11, 2) is the best strategy.
Case 2: x z 0. In this case, bistrategy (11,2) (i.e. bistrategy push-push) yields a loss x a.0 to both players (because, for example, they are jamming the door way). The shadow minimum is no longer achieved. (a) Non cooperative equilibra. Strategies (11, 1) and (I, 2) satisfy the relations P(I1, 1) = F(I1, 1) and P(I,2) = F(I, 21, i.e. the relations
f(I1. 1) = a ef(1, 1) = b, g(I1, 1) = 0 -= g(II,2) = x
Ch. 6, 4 6.21
EXAMPLES: SOME FINITE GAMES
177
and (6)
f(I,2) = 0
{ g(I,2) =
a
-= g(1, 1) = b.
In other words, if Yvette plays strategy 1 (i.e. does not push), it is optimal for %vier to play strategy I1 (i.e. to push) and if Xavier plays strategy I1 (i.e. pushes), it is optimal for Yvette to play strategy 1 (i.e. not to push). (The same interpretation holds for {I, 2) by exchanging the roles of Xavier and Yvette). Inequalities (5) and (6) show that bistrategies (11, I } and {I, 2) are “individually stable”, i.e. non-cooperative equilibria. Neither player has an incentive to change his strategy by acting unilaterally. There remains the problem of distinguishing between these two non-cooperative equilibria. It is not clear, for example, that Xavier will accept a loss of a and let Yvette escape scot free. Such a choice requirt3 communication or collusion among the players. (b) Conservative solutions. If Xavier does not push against the door (i.e. plays I), he risks a loss of b if Yvette uses the same strategy. Otherwise, by pushing against the door, the worst possibility is a loss of sup(a, x). Hence, a cautious Xavier will naturally choose strategy I1 (i.e. push) if x =sb and strategy I (i.e. do not push) if b -s x. In the same way, a cautious Yvette will choose strategy 2 if x s i b and strategy 1 if b =sx. These choices are made independently by each player, by considering the worst situation. They do not require any communication. Naturally, if the game had to be repeated, neithkr strategy {II,2} nor {I, 1) is “individually stable” in the.above sense. Again, if x -= a and if Xavier plays strategy I1 (i.e. pushes against the door) and implements his strategy before Yvette, then it is optimal for Yvette to choose strategy 1 (instead of 2). (c) Pareto minima and core. If x a, the’conservative solution {II,2} is rejected by both Xavier and Yvette, since it is not collectively stable. Each of the players will set a loss (either 0 or a) by replacing the bistrategy {II,2} by either (11, I } and {I, 2). But this reason is no longer valid if 0 4 x =S a, since in this case no multistrategy yields a smaller biloss than F(II,2). (Notice that bistrategies {I, 2) and (11, I} share this property in this example). In other words, the set of Pareto minima coincides with the core, and it is equal to {{I, 2}, (11, I}, (11, 2 } ) if x a, to {{I, 2) (11, I}} if x a. 14
178
TWO-PERSON GAMES : AN INTRODUCTION
=
F:
I
l
l I
I1
[Ch. 6 , § 6.2
2
-
(a, a)
(b, 0)
(Qb)
(x,
4
O-=a
P:
I
(090)
(inf(b, x). 0)
I1
(0, inf(b, x))
(inf(b, x), inf(6, x))
Interpretation. Both Xavier and Yvette are prisoners, each of whom can either confess or not confess to a crime. If neither confesses (bistrategy {I, I}), they face moderate jail sentences (a years in prison). If &vier confesses (strategy 11) and Yvette does not, Xavier is set free (0 year in prison) and Yvette faces a hard jail sentence (b years in prison). The loss x (years of prison) when both players confess (strategy (11,2)) is treated as a parameter. Case x = 0 : This case is obvious, since the shadow minimum a = (0,0) is achieved by the bistrategy (11,2). Both p l a y a confess. Cuse 0 < x =sb. This means that they face a smaller number of years of
prison than b when both confess. The bistrategy (11,2) is the only bistrategy which satisfies the relation Fb(II,2) = F(II,2), i.e. f(I1,2) = x
180
[Ch. 6, 8 6.2
TWO-PERSON GAMES : AN INTRODUCTION
The case x > b is no longer interesting in the framework of prisoner’s dilemma. It is usually interpreted as the “game of chicken”.
F(II, 2) = v‘
if a < x < b
F(II, 2)= v*
if0txr;a
a
Fig. 6.
6.2.4. Game of chicken
Let Xavier be the first player, Yvette the second and let X = {I, 11) and Y = {I, 2) be their strategy sets. Let‘F be the biloss operator represented by the following bimatrix
It is clear that
f”(1) = b, f+(II) = X , g*(l) = b, g’(2) = X . Hence (8)
v+ = {b, b) = {fW, g+(l)}.
The pair {I, I} is thus the pair of conservative solutions of the game.
Ch. 6,o 6.21
181
EXAMPLES: SOME FINITE GAMES
Since fb(1)
= 0, fb(2) = b,
d(1)= 0, Y(I1) = b.
The values of Fb are given in the following bimatrix.
So, bistrategies {I,2} and (11, I} are non-cooperative equilibria. They are also Pareto minima.
I
U
/
b
a
Fig. 7.
Interpretation. Xavier and Yvette drive toward each other at high speed and the first one to swerve “loses”. We denote by I, 1 the strategies “swerve” and by 11,2 the strategies “don’t swerve”.
182
TWO-PERSON GAMES : AN INTRODUCTION
[Ch. 6, 0 6.2
If both swerve, they pay an amount of a units. If only one swerves, the winning player has no loss while the losing player pays b. If neither swerves, well, they crash and pay an amount x. The bistrategies where one player swerves while the other does not are noncooperative equilibria that are Pareto minima. The bistrategy where both swerve is a conservative solution that is a Pareto minimum. 6.2.5. The battle of the sexes
Let Xavier be the first player, Yvette the second and let X = {I, 11) and Y = {1,2} be their strategy sets. Let F be the biloss operator represented by the following bimatrix
F:
I
I
0<
It is clear that f " ( 1 ) = f *(II) = g"(1) = g+(2) = b.
Hence (9)
v+ = (b, b)
and the four pairs of strategies form conservative solutions! The values of Fb are summarized in the following bimatrix
6
Ch. 6, 0 6.31
EXAMPLE: ANALYSIS OF DUOPOLY
183
Hence the pairs {I, 1) and {II,2} are the non-cooperative equilibria and the Pareto minima of the game. Also, they form the core of the game.
Fig. 8.
Interpretation. Strategies I and 1 represent “attend a football game” and strategies I1 and 2 represent “go shopping”. XaVier prefers to attend the football game and Yvette prefers to go shopping. But in any case, they prefer to go together. 6.3. Example: Analysis of duopoly
We study in this section the basic example of duopoly in which both players are producers. The loss functions are the cost functions and these depend upon the productions of the two players. We begin by describing the set of Pareto minima, which yield a profit (negative cost) to both players. The conservative strategies amount to both players producing nothing. We afterwards construct the non-cooperative equilibrium (introduced by Cournot in 1838). This is not Pareto minimal. This non-cooperative equilibrium is constructed as a fixedpoint of the optimal decision rules. We then compute the productions which result if one player knows (or assumes) that his opponent will use the optimal decision rule. We will
184
TWO-PERSON GAMES : AN INTRODUCTION
[Ch. 6, 0 6.3
explain why he can implement a production whiih yields a smaller loss than the non-cooperative equilibrium to him and a larger loss to his opponent (Stackelberg equilibrium). But if each player guesses (wrongly) that his opponent will use the optimal decision rule, each of them will then implement the so-called Stackelberg disequilibrium, which .yields to both players a worse loss than the non-cooperative equilibrium. This example illustrates the danger of a priori guesses on the opponent's behavior. 6.3.1. The model of a duopoly In duopoly, Xavier and Yvette are producers of a given good, assumed to be homogeneous. Let x E R, and y E R, be the quantities of the good produced by Xavier and Yvette. We assume that (1)
the price p = a-p(x+y) is an affine function of the total production x+p(a 2 0, j3 =- 0)
and that the cost functions c and d of each producer are defined by (2)
c(x) = yx+d;
c(y) = yy+d,
y
>
0, 6 3 0
where y denotes the marginal cost, 6 the fixed cost. The net cost for Xavier is equal to (3)
f ( x , y ) = yx+.d-px = px x+y+- Y - " ] +
I
B
6
and the net cost for Yvette is also equal to (4)
g(x, y) = yx+b-py
= BY
Ix+r+--1 @ +6.
For simplicity, we will assume that & = 1, 6 = 0 and set a- y = u. This model of a duopoly is a two-person game whose strategy sets are X = Y = [0, u] and whose biloss operator is defined by (5)
m , r) = (x(x+y-u),
y(x+y-u)).
This model will be generalized in Section 9.2.5.
Ch. 6, 8 6.31
185
EXAMPLE: ANALYSIS OF DUOPOLY
6.3.2. The set of Pareto minima
Since f ( x , y)+g(x, y ) = ( x + y ) ( x + y - u ) = z(z-u) (where z = x + y is the total production), we see that the sum of the costs ranges from - u2 to 2u2. Thus the subset F ( X X Y) of the biloss space R2is the union of the set of vectors {a, b} such that - f u2 =s a+ b =s 0, -fu2 e a, b == 0 and of the square [O,
2421
x [O, 2421.
Fig. 9.
Actually, the square [0, u2]X [0, u2] is the image of the subset of XX Y such that x+y 3 u and the triangle is the image of the subset of X X Y such that x+y
=2
u.
We deduce that the set of Pareto minima is the set of bistrategies { x , y} such that
i.e. such that x+y = + u . Note that the losses to both players are positive as soon as the total production x f y is larger than u. 6.3.3. Conservative solutions It is clear that f * ( x ) = S U at x = 0 and that g x ( y ) = S U y = 0.
~
~
~ x(x+y~ , , ~u) ~= x2
achieves its minimum its minimum at
~y ( ~x + ~y - u~) , = , y2 achieves
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TWO-PERSON GAMES : AN INTRODUCTION
[Ch. 6,
6.3
Thus the conservative bistrategy is (0, 0}, i.e. both players produce nothing. Since TP= (0,0), we deduce that the core of the game is equal to the set of Pareto minima. 6.3.4. Non-cooperative equilibria
The concept of non-cooperative equilibria in the case of the duopoly game was introduced by Cournot (in 1838). Let y be Yvette's production. Then Xavier will implement a production x which minimizes over X his net cost function x x(x+y- u). The minimum is achieved at the point ~4
(7)
U-Y x = C ( y) = -
2
and is equal to
Simil'arly, Yvette will implement a production (9)
u- x y = D(x) = 2
which minimizes her net loss function when x is produced by Xavier. Thus (u- X)Z 9(x) = inf [y(x+y-u)l = 7 Y
The non-cooperative equilibrium {Z, y} is then the fixed point of the function {x, y } -I {C(y),D(x)} i.e. the point
and yields to each player a loss equal to
Ch. 6, 5 6.31
187
EXAMPLE: ANALYSIS OF DUOPOLY
We notice that the non-cooperative equilibrium is not Pareto-optimal. The difference between the total losses is equal to
But the total production is greater:
3u instead of f u.
6.3.5. Stackelberg equilibria
-
-
We assume now that players use “decision rules” C : Y X and D : X Y instead of pure strategies. We then seek “stable solutions” in this framework, {C(y), D(x)}. i.e. thejixed points of the map {x, y} -I Recall that in obtaining a non-cooperative equilibrium in pure strategies, we arrived a t unique optimal decision rules C and B. Assume next that Yvette chooses a fixed decision rule D : x E+ y = D(x) .and that Xavier knows it.
-
Then Xavier will choose a strategy x, which minimizes the loss function f (x, D W ) . For instance, if Xavier knows that Yvette will use the optimal decision rule g , he will implement the strategy x D = u which minimizes the function XI-+ f ( x , D ( x ) ) = x ( x - u + $ ( u - x ) ) = +x(x-u). Yvette will produce & x ~ ) = u. By doing so, the net cost of Xavier is equal to -$ u2 and that Xavier is then in a better situation than in the of Yvette is equal to original non-cooperative equilibrium (- $ u2 instead of - uz) while Yvette is in a worse situation (since she looses u2 instead of - u2). The result ( x D ,yb) is called the Stackelberg equilibrium for Xavier. In the same way, if we suppose that Yvette knows that Xavier will use the optimal decision rule C, she will implement the strategy yc = f u and Xavier the strategy C(y,) = u.
x
-A$.
-A
+
6.3.6. Stackelberg disequilibrium
We recall that the non-cooperative equilibrium is the fixed point of the map {x, y} t-+ {C(y), D(x)}. We call C and D the “Cournot decision rules”. Since they are a f i e , their derivatives are constant (equal to -+). Now, if Yvette uses the Cournot decision rule D, Xavier will solve the equation
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TWO-PERSON GAMES: AN INTRODUCTION
[Ch. 6,g 6.3
for finding the point 3which minimizesx +f( x, D(x)).This amounts to solving (14)
j j = D(Z)
and
The latter equation can be written (16)
x = C,(jj) = + ( U - - j j )
where Cs:y C- s u - y ) is called the “Stackelberg decision rule”. Thus the Stackelberg equilibrium for XaVier is obtained as the Gxed point of the map {x, Y ) +-I W,W, W)}. In the same way, the Stackelberg equilibrium for Yvette is obtained as the &xed point of the map {x,y} l- {C(y),D,(x)} where D, : x i-+ a u - x ) is the “Stackelberg decision rule” of Yvette. Now, suppose that the rules of the game neither force Yvette to use the decision rule b nor force Xavier to use C (and both players are fully aware of these facts). Then each player can incorrectly believe that the other will use the “naive” optimal decision rule. In this case, Xavier and Yvette will both use their Stackelberg decision rules. They implement the fixed point {xs, y,} of the map {x, y } -c {C,(y), D,(x)}, where x, = ys = u. Their net cost will be equal to -5 u2, and both of them are in worse situation than in the non-cooperative case. The bistrategy (x,, y,) is called the Stackelberg disequilibrium. If we restrict the available “strategies” for each player to either the Cournot decision rule or the Stackelbergdecision rule, we obtain a finite game whose bimatrix is given by
1-
Cournot
Stackelberg
It is apparent that this game has the same structure as the prisoner’s dilemma game. For both players the Stackelberg decision rules yields a smaller loss than the Cournot decision rule, but both players would be better off if both choose the Cournot decision rule than if both choose the Stackelberg decision rule.
189
Stackelberg equilibrium for Yvette
--% ' ~,
Set of bilorres of Pareto minima
.~ . ~
__
'\
-\
,\
i \\ \
\
\
-u2M
Fig. 10.
6.4. Example: Edgeworth economic game
We consider the basic model of the allocation of a commodity w among two consumers. With the aid of an ingenious graphical representation(the so-called Edgeworth box) due to Edgeworth, we can initiate a study of this model in the case of commodities of two goods. The main feature of this game (as opposed to the duopoly case) is that the set U = { { x , y ) E X X Y such that x+y =
190
TWO-PERSON GAMES : AN INTRODUCTION
[Ch. 6, 0 6.4
= w} is not a product, while the loss functions depend only on each player’s own consumption. In such a model the concept of non-cooperative equilibrium makes no sense. We indicate the subset of Pareto minima in the Edgeworth box. We then define the core. Since unilateral play by either player is impossible our previous definition needs to be extended by assuming that a player rejects an allocation if it yields him a loss larger than that he obtains with his initial endowment (instead of the conservative value). The core of this game proves to be quite large and the problem arises as to how to choose an allocation in the core. We shall indicate that all Walras equilibria are in the core.
6.4.1. The set of feasible allocations
In this game, Xavier and Yvette are trading two commodities. Their strategy sets are both equal to X = Y = [O, ulX[O, bl, which is the set of commodities x = {XI, xz} representing x1 units of a first good and x2 units of a second. We assume that Xavier’s loss function f depends only on his choice of a commodity x and that Yvette’s loss function g depends only on her choice of a commodity y. Thus
The new feature is that players cannot use the whole bistrategy set X x Y, but are restricted to a feasible subset U c X X Y representing the possible allocations of commodities that the players can achieve by trading. We assume that (i) Xavier owns the commodity fl = (a, 0), (ii) Yvette owns the commodity w y = (0, b). Then the set U is the subset of pairs {x, y } E X X Y such that x + y = wx+wy, i.e. such that (3)
x l + y l = a,
XZ+YZ
= b.
We say that U is the set offeasible allocations. 6.4.2. The biloss operator
Consider first the strategy sets X and Y and the loss functions for XaVier and Yvette.
Ch. 6 , $ 6.41
EXAMPLE: EDGEWORTH ECONOMIC GAME
191
i
192
TWO-PERSON GAMES
AN INTRODUCTION
[Ch. 6, 5 6.4
The curve passing through x (called an “indifferencecurve”) is the set of points = f ( x ) and the shaded area above represents the set of points 5 such that f ( 5 ) -= f ( x ) . Similarly, the curve passing through y is the set of points q such that g(q) = g(y ) and the shaded area above represents the set of points q such that g(q) -= g(y). We have also drawn indifference curves passing through Z and 7. In the biloss space R2,F(U) is indicated. The shaded area is the set of bilosses from F(U) which are strongly smaller than F(x, y).
E such thatf(5)
6.4.3. The Edgeworth box
Edgeworth (1 88 1) gave an ingenious graphical representation of the subset U.We superimpose the rectangle Y on the rectangle X after a rotation which maps 0’ onto A and wy onto d. Edyeworth’s box
.
- --
A 0’
Set of Pareta minima
ox--
- € -
x1
Y1
Fig. 12.
A point u = (5, q) represents a bistrategy u = {x, y } of U on the understanding that
x1 = 5, x2 = q, y l
= a-5,
y2 = b-q. We then have that x f y = wx+wy asnecessary. This amounts to saying that a given point u = {E, q} of U represents a commodity x for Xavier with respect to the basis {@wX, OxB} and a commodity y for Yvette with respect to the basis {OyB,0’ w’}. The diagram is called the “Edgeworth box”. The shaded area of the Edgeworth box represents the set of pairs ii = (2, jj} E U such that F(2 , jj) =s F(x, y ) , i.e. the preimage by F of the cone F(x, y)-R; (4)
A
A
-
d
Ch. 6, 0 6.41
EXAMPLE: EDGEWORTH ECONOMIC GAME
193
6.4.4. Pareto minima
A pair of strategies ii = {Z, j j } E U is a Pareto minimum if there is no other strategy ti = {Z,?} such that f(Z) < f ( Z ) and gdj -= gdj. This means that the cone F(Z,9-k: is disjoint from F(U),i.e. that the shaded areas of F(U) and U are empty. This happens if and only if the inmerence curves passing through ii = {j?, j j } are tangent. The set of Pareto minima is represented in the Edgeworth box by the dotted line. 6.4.5. Core
We regard the set of Pareto minima as the set of bistrategies U = {Z, j j }
which are not “rejected” by both Xavier and Yvette.
We now make the following behavioral assumptions. Since Xavier owns w”, he will reject any commodity bundIe x such that f ( x ) >f ( w x ) =f(a). In other words, he will only trade for commodities above the indifference curve passing through wx. Similarly, since Yvette owns wy, she will reject any commodity y such that g(y) =- g(wy) = b. In other words, she will only trade for commodities below the indifference curve passing through wy.
Fig. 13.
By definition, the core is the set of bistrategies ti = {Z, j j } which are rejected neither by Xavier, nor by Yvette, nor by both %vier and Yvette acting together. It is therefore the subset of Pareto minima rejected neither by Xavier nor by Yvette. 15
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TWO-PERSON GAMES : AN INTRODUCTION
[Ch. 6, 0 6.4
It is represented in the Edgeworth box by the part of the dotted curve lying between the indifference curves passing through wx and wy. In this context we call the core the “contract curve”. The problem arises as to whether it is possible to devise a selection procedure for an allocation in the core. The concept of Walras equilibrium provides such a selection procedure.
6.4.6. Walras equilibria Let u = {a, y } be a bistrategy in the core. Since the indifference curves for Xavier and Yvette passing through u are tangent, they have a common tangent (D).
Fig. 14.
Generally, this tangent does not pass through wx. Assume that U = {X, J } is a point of the core such that the common tangent (D) to the indifference curves passes through wx = wy.Consider the equation of the line (D):
( D ) : p15fp2rl = plu where PI,PZ
0.
We can regard p = { p l , p 2 } as a price system, p l being the price of a unit of the first good and p2 the price of a unit of the second good. The half-plane located below (b)represents Xavier’s budget set, i.e. the set of points x = {XI, XZ} such that the value (p, x) = plxl+pzxz of x is not greater than the value pla = =( p , w“) of fl.The half-plane located above (D) represents Yvette’s budget set,
Ch. 6, 9 6.51
TWO-PERSON ZERO-SUM GAMES
195
since it is the subset of points y = 0 1 , ya} such that pd+paq = pl(a-yr)+ +pa(b-y2) 2 pla, i.e. the subset of points y = @I, ya} such that the value (p, y ) = p49+pay2 of y is not greater than the value of wy. The fact that the indifference curve passing through ii is tangent to (D) at U means that 2 minimizesf(x) under the constraints (p, x ) 4 (p, w“) and that ji minimizes g(y)’under the constraints (p, y ) =s(p, w?. In summary, such a point ii = {Z, j j } has the following properties:
.
(ii)@,~)=(p,wx);(p,jj)=(p,wq (iii) Z minimizes f uoder the constraint (p, x) 4 (p , w”> and 7 (5) minimizes g under the constraint ( P , Y ) ==! (p, w 9 Such a triple {2,p, p } is called a Walrus equilibrium of the economic game. I
,
6.5. Two-person zero-sum games
We end this chapter by an introduction to the case of “zero-sum games” (also called ‘‘duels’’), where the loss function g = -f of Yvette is the opposite of the loss function of Xavier. For such games, any pair of strategies is Pareto minimal. Hence this concept is no longer useful. We are left with the concepts of conservative strategies and non-cooperative equilibria. We notice that the conservative value of Xavier is equal to v# = inf,,, supy,yf(x, y), whereas the conservative value of Yvette is equd to -vb = -sup inf,,, f (x, y). Note that it is always true that w b Y*. The interval [d’, w#] is called the dualitygap. If v* and w b coincide, the common value v = vu8= d’ is called the “valueof the game”. The main result is hat a p a r {Z, j j } of comenative strategies is a non-cooperative equilibrium if and only if the game has a value. In this case, {Z, J} is called a “saddlepoint” of the game. We end this section by studying perturbations of the loss fm&n by linear forms and by displaying exarhples of games with finite strategy sets for which a saddle point does or does not exist. 6.5.1. Duality gap and value
Let 11s consider a two-person game with strategy sets X and Y and loss functions f and g. Definition 1. A two-person zero-sum game is a two-person game in which the sum of the loss functionsf and g of Xavier and Yvette is equal to 0. 15’
1.96
TWO-PERSON GAMES: AN INTRODUcIloN
[Ch. 6 , s 6.5
In other words, Yvette’s loss function g(x, y ) = - f ( x , y ) is XaVier’s utility function f (x, y). Therefore, Yvette’s wotrs-loss function is g*(y) = - f b ( y )
(1)
wherefb(y) = inf f(x, Y ) %EX
and the conservative value vy“ is equal to (2)
vy” = - vb where vb = sup inf f (x, y). Ycr xcx
We always have that (3)
vx
E
x, V Y E y, f b ( AG f ( X , Y ) s f # ( x )
and thus (4)
vb
= sup inf f ( x , y ) -c v* = inf s u p f ( x , y). X€X Y€Y
Y € Y X€X
The interval [vb,v#] is called the “duality gap”. In other words, the inequality vx E
x,
vy
E Y, 0 sf”(x)-fbCv)
says that the sum of the two worst-loss functions is always non-negative (i.e. is a true loss). The inequality (5)
0
v#-
vb
= inf [ f # ( x ) - f b ( y ) ] XEX
YI Y
follows and may be interpreted as meaning that the sum
9-V* of the two
minimal worst losses is always nvn-negative.
Hence, in general, by playing “conservatively” (i.e. as though in the worst situation), the two-players (as a team) obtain a non-negative loss. We shall distinguish the case where the sum of the minimal worst losses is equal to 0, i.e. where
v* = the minimal worst loss of Xavier = vb = the maximal worst gain of Yvette.
-
Delinition 2. We shall say that a two-person zero-sum game represented by f:X X Y R has a ‘‘sddle-value” (in short, a “value”), if (6)
v+ = v b = 2,
and this common value v = v x = vb is called the “saddle value” of the game (in short, the value)
Ch. 6, 5 6.51
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TWO-PERSON ZERO-SUM GAMES
6.5.2. Saddle point
We introduce also the following definitions. Definition 3. A strategy X E X which minimizes the worst loss function f on X is said to be a “conservative”strategy for Xavier: (7)
h
f * ( Z ) = inff(x) = v+. X€X
If moreover the game has a value, then 2 is called a “minisup” off, i.e. (8)
supf(x’, y ) = sup inf f ( x , y ) = v. Y€Y
Y
X€X
A strategy y E Y which maximizesp(y) is Said to be a ‘‘conservative’’strategy of Yvette : (9)
fb(jF) = sup fb(y) = vb. Y€Y
If moreover the game has a value, then jJ is called a “ma-inf” off, i.e. (10)
inf f (x, y ) = inf sup f ( x , y ) = w
X€X
X € X Y€Y
A pair {i j? J}E , X X Y is called a “saddlepoint” off if (1 1)
supf(2, Y ) = f (2,r3 = inf f (x, 3.
Y€Y
X€X
We begin by characterizing saddle points. Proposition 1. A pair (2,j j } is a saddle point is a ma-inf
if and only if
iis a minisup and J
Proof. Assume that {Z, j j } is a saddle point. Then (1 1) implies that v*
4
f *(X)
=f ( X , 7) =fb(.pP
6 vb
and thus v x = d’ = v. Hence Z minimizes f # and J maximizes f”.Therefore X E X is a minisup and j j E Y is a max-inf. Conversefy, let 5 be a minisup and 7 be a max-inf. Then the game has a value v. Thus sup f ( 5 ,y ) = f + ( . ~ = ) v =fb0 = inf f ( x , 7). YE y
X€X
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Taking y = jj in the supremum of the left-hand side and x = 5 in the infimum ofthe right-hand side, we obtain f (Z, j ) 4 f *(Z) = 2, =f b ( J ) =sf (5,j). Therefore v = f(Z,
3.Hence {Z, y } is a saddle-point. 0
(2,p)off h nothing other than a non-cooperative equilibrium of the two person game described by the biloss operator F defined on X x Y by F(x, y ) = {f (x,y), - f ( x , y)}. In fact (11) reduces to
Remark. A saddle point
(12)
{ (ii)(ilf(5,-f@,J ) F)
gf
(x, 7)for any x E
-S
X,
- f ( f , y ) for any y E Y,
In other words, in a two-person zero-sum game, a pair {z, j j } of conservative strategies is a non-cooperativeequilibrium if and only if the game has a value. In a zero-sum game, the concept of Pareto minimum is not useful, since every bistrategy {x, y } is'Pareto-minimal. *6.5.3. Perturbation by linear functions Suppose that X and Y are wnvex subsets of vector spaces U and V. We shall perturb the loss functionsf by linear forms {x,y } i-+-(p, x)+(q, y). We set
Proposition 2. Zfyo is a max-inf of the lossfunction {xi y } -t f { x , y } - (PO,x)+ +(qo, y), then Y OE E W p o y .)(40) The converse is true whenever, V x E X, the function y !-+ f(x, y ) is concave und upper semi-continuous.
Roof. We shall prove that yo is a max-inf if and only if
(15)
E av*(po, { (ii)(0 infxcx [f(x, yo
(qo), YO)- (PO,4 0 )
1 = inf,
PO, 4)- (4, YO)]
and that the concavity and the upper semi-continuityoff whith respect to y implies (15(ii)).
Ch. 6,
6.51
TWO-PERSON ZERO-SUM GAMES
199
(a) Suppose that yo is a max-id. Then
and
Therefore (16)
V + ( P O , 40)-
W P O ,d
aG
(40-4,Yo)
(i.e. yo E av’(p0, -)(40)). Furthermore, we obtain that inf M x , YO)- (PO,x)l = PO, qd-
(40,
YO)
= inf tV*(PO, 4)- (q, Yo)]. 4
(b) Conversely suppose that y o satisfiesconditions (15). Since (15(i)) is equivalent to (16), we have that m p o , 40)- (qo, Yo) = inf [ W P O , d - (4,Yo)] 4
= inf x
V(x,YO)- (PO.x>l
and thus,
Q.e.
YO
is a max-inf).
(c) Suppose that the functions y + f ( x , concave. Write gQ = - f ( x , y). Then
are upper semi-continuous and
For any fixed x E X,we obtain that inf [““P [f@,Y)+ (4,Y>l- (49Yo)] = 4
Y
= inf [g*(q) - (q, yo)] = -g*+dyo). 4
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TWO-PERSON GAMES: AN INTRODUCTION
[Ch. 6, Q 6.5
Since g is convex and lower semi-continuous, -g**(yo) =f(x,.yo). Thereforc inf inf
(18)
SUP
P
X€X
= inf
Y
[ f ( x ,Y)-(Po, x)+(q, Y - Y O ) ~=
[ f ( x ,YO)- (PO,x)l.
Therefore (15(ii)) follows from (17) and (18).
Remark. If v#(p, q) is replaced by vb(p, q), the same proof shows that yo is a conservative strategy for Yvette if and only if 0 ) yo E 8vb(po, .)(!lo), (ii) inf [ f ( x ,YO)-((PO, 4 1 = inf [vb(po,q)-(q, YO)]. X€X
4
*6.5.4. Case of finite strategy sets: Matrix games
In the case where the strategy sets
X={l,
(20)
..., k}
Y={l,
..., I }
-
are finite, Xavier’s loss function f :X X Y R is characterized by its values =G j 4 1. This is the reason why such games ar, called ‘‘matrix games”.
f (i, j ) when 1 =s i =G k and I
{
1
2
. . .
t
j
.
.
1
. 1 Yvette selectscolumns
A
1
2
r“
Xavier selects rows
Once the game is represented by a matrix as above, we identify the ithstrategy of Xavier with the ithrow of the matrix and the j” strategy of Yvette with the j” column. In other words, one regards Xavier as selecting rows and Yvette columns. In this case Xavier’s loss (or Yvette’s utility) is the entryf(i,j) appearing in the corresponding row and column of the matrix.
Ch. 6,8 6.51
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201
For Xavier, the worst lossf"(i) is the maximum loss in the it"row, i.e.
(21)
f # ( i ) = maxf(i,j) lsjsl
and a conservative strategy is a row which has the smallest worst loss.
(22)
f * ( i ) = min
maxf(i.19.
1 S i S k ISjSI
For Yvette, the situation is symmetric. The worst gain f b ( J is the minimum gain in the "I column, i.e.
and a conservative strategy is a column which has the largest worst gain
(24)
f b ( j ) = max min f(i,j?. lsisl lsisk
For instance, consider the game
The worst losses of Xavier are 2 and 4 respectively and thus, Xavier's conservative strategy is the first row. In this case, Xavier expects Yvette to choose the second column and produce a minimal worst loss of 2. The worst gains for Yvette are respectively -6, -5 and -4 and thus her conservative strategy is the third column. In this case, Yvette expects Xavier to choose the second row and produce a minimal worst loss of 4. The sum of the minimal worst losses is equal to 4+2 = 6. By playing their conservative strategies (first row for Xavier and third column for Yvette), the loss is - 3 for Xavier (and +3 for Yvette). Neither player expected such a loss! Furthermore, if Yvette selects her third column and if we assume that Xavier is informed of this choice, then Xavier would do better selecting his second row (with a loss of - 4 ) instead of the first one. Similarly, if Xavier does choose his conservative strategy, then Yvette would do better playing her second row (with a gain of 2) instead of the third.
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[Ch. 6,§ 6.5
It is clear that the conservative strategies (1, 3) do not bring satisfaction to the players. The reason is that the game has no value since v b =-4 < v+ = 2. Now, consider the game
I
1
-2
2
1
-1
0
-4
-6
I
%vier’s worst losses being - 1 and 1 respectively, v* = - 1 and his conservative strategy is given by the fist row. In this case, he expects Yvette to choose the second column. Yvette’s worst gains are -2, - 1, -6. Thus, vb = - 1 and her conservative strategy is the second column. In this case, she expects Xavier to choose the first row. In other words, since the game has a value (equal to - l), the natural expectations of both players are fulfilled and the choices of conservative strategies are consistent. Given that Xavier implements his first row, it is optimal for Yvette to play her second column and conversely, given that Yvette select her second column, it is optimal for Xavier to choose his first row.
Remark. We shall see that the reason why a value does not necessarily exist in matrix game is that the finite strategy sets are “too small”. Another way of interpreting this phenomenon is to say that the description of the game is “too poor”. Where a saddle-point exists, individual stability can be attained even though the players both choose secretly and independently. Where no saddle-point exists and the game is played only once, the players do not have enough information on the choice of their opponent. But, by “repeating” the game, each player can gather enough information to reach an individually stable situation, even when their choices are made secretly and independently. This is the underlying idea leading to the use of mixed strategies. In this case, we replace the strategy sets X = {1,2, . . ., k } and Y = {I, 2, . . ., I } by the mixed strategy sets Mkand &‘(see Section 1.3). Theloss functionf is extended to MkX A”’ by f defined by k
I
f=1
j=1
f (n) ~ = C C aiSjf(iij> where m = Ifsl aiS(x,) and n =
‘&pjsis(yj).
Ch. 6 , s 6.51
TWO-PERSON ZERO-SUM GAMES
zi=l
203
If u; = p,/p where & pi = p and Pl, = qj/q where qj = q, are rational numbers, we interpret m as meaning that %vier’s ith strategy is to be played pi times out of p (i = I, 2, . . ., k). Similarly, n is interpreted as meaning that Yvette’sjth strategy is to be played q, times out of q ( j = 1,2, . . .,1).
CHAPTER 7
TWO-PERSON ZERO-SUM GAMES: EXISTENCE 'IIIEOREMS This chapter deals with the main existence theorems for solutions of twoperson zero-sum games. Some of these theorems play a key role not only in this book but in convex and non-convex analysis generally. For instance, many results in convex analysis are based on the minisup theorem. This states that if a function f satisfies the topological asmptions
(7
( x , y ) is lower semi-continuous, { (ii)(i) V3y0y E Y, xsucht . - fthat x ~ - f ( x yo) , is lower semi-compact, € Y
together with the concavity assumption
("1
V x E X, y I-
f ( x , y ) is concave
and the convexity assumption
(***I
V y E Y , x ~ - f ( x y, ) is convex
then there exists 1 E X such that supf(2, y ) = sup inf f ( x , y).
Y€Y
Y € Y X€X
Many results of non-convex analysis and, in particular, various fixed point theorems, can be deduced from the Ky-Fan theorem. This states that when X = Y, topological assumptions (*) and concavity assumptions (") imply the existence of 2 E X such that
These two results are proved in Section 7.1. In Chapter 13 we shall continue with a more comprehensive study of such results in which the continuity and convexity assumptions are relaxed. If we assume in addition to the three assumptions above that (i) V x E X , y F+ f ( x , y ) is upper semi-continuous, (ii) 3x0 E X such that y l--f(xo, y ) is upper semi-compact, 204
Ch, 71
TWO-PERSON ZERO-SUM GAMES
we may deduce the existence of a saddle point {Z, y}, i.e. sup f (2,y ) =f ( 2 , j j )= inf f (x,jj).
YEY
X€X
We have already described examples of finite games which have no saddlepoints. But, by embedding X X Y in the convex set of mixed strategies M ( x > X X M ( Y ) and by extending f to A ( X ) X M ( Y )by the function f“ defined by fA(m,n) =
2 2oliPjf(xi,y,) k
1
when m =
k
‘f aib(xi)
i =1
1x1 j=1
we obtain the existence of saddle mixed strategies since d ( X ) and &(Y) are convex compact subsets and f” is a bilinear continuous function (Von Neumann theorem). Hence, even if there is no saddle-point {X, j j } E XX Y among pure strategies, any (finite) game has saddle mixed strategies {fi,ii} E J?(X)Xc/n(Y). (This result will be extended to compact strategy sets.) The question arises as to whether this procedure is the only “extension” of games into “playable games”, i.e. extended games which always have a saddle point. This leads us to introduce a concept of game extension due to H. Moulin. The extension is described by the items {%, @, i, j , z} where i is an injective map from X into %,j is an injectivemap from Y into @, andnis a linear operator mapping a bounded loss function on XX Y onto its extension zf on %X@ (which is required to satisfy several reasonable properties). Besides standard mixed extension example (where $ = sn<X), (&# = c/n(Y), af =fA), we shall introduce in Section 7.2 a more general class of extensiins without exchange of information. These have the property that, if the extended game has a value v(nf), then this value is the mixed value v( f“). Among such extensions, we describe the sequential extension where 56 = XN is the set of denumerable sequences of pure strategies, (&# = Y N ,and af G,7) =
OD
C #.+1
aijf
(xi, vj) whenever 5 = {xi}icN and r] = {YI}im
The sequence {aii) iS assumed to satisfy Vi, j € N , aijaO and
C DD
i,j-1
aij = 1.
206
TWO-PERSON ZERO-SUM GAMES
[Ch. 7,
6 7.1
We also introduce the concept of extensions with exchange of information. The exchange of information is modeled by the use of “decision rules”. Instead of choosing a pure strategy x in X , Xavier is allowed to use a decision rule C in a subset 62 of maps from Y into X . Such a decision rule C for Xavier associates with any strategy y E Y played by Yvette the strategy x = C ( y ) . The rules of the game specify the subset @ of decision rules allocated to Xavier. Also Yvette is allowed to choose a decision rule D in a subset of maps from Xinto Y . Now, if Xavier chooses C in 62 and Yvette chooses D in @, the only pure strategies which are stable are the fixed points {X,J } of {C, D } : C(7) = X and D(X) = J .
ex
Hence, the exchange of information described by C D has a sense only if all pairs {C, D } E /2, do have k e d points. If such a fixed point {Z, J } is chosen, we assign to Xavier the loss Zf (C, D ) =
f(57)-
Extensions with exchange of information are obtained by taking 56 = @, C$i = CD and using the above map z. Finally, we devote Section 7.3 to the iterated extensions which combine sequential extensions and extensions with exchange of information. Let us describe an example. We embed X in the set 62 of sequences C = {C,} of nonanticipative decision rules C; mapping fljsi-l Yi into X (where Yi denotes the strategy set Y used at thejth step). In the same way, Y is embedded in the set CD of sequences D = {Dj} of nonanticipative decision rules Dimapping Xiinto Y (where Xi denotes the strategy set X used at the ith step). By choosing C = {Ci} and D = {Dj}, Xavier and Yvette implement the following sequence of pure strategies. At the first step, Xavier plays x1 E A’. Then they play in turns as follows; Yvette plays y l = Dl(xl), Xavier x z = Cz(yl), Yvette y2 = Dl(x1, X Z ) Xavier , x3 = = C ~ ( y 1YZ), , . . ., Yvette y p = Ddxl, . . ., xp),Xavier xp+l = C,+,(yl, . . ., yp). . . . and so on. aijf(xi,yj). This procedure defines Then we assign to Xavier the loss the so-called “iterated extension”. The Moulin theorem states that iterated games do have a value.
nisi
7.1. The fundamental existence theorems
Letfbe Xavier’s loss function, mapping X X Y into R. Suppose that (i) V y E Y , x I-+ f ( x , y ) is lower semi-continuous, f (x, yo) is lower semi-compact. (ii) 3 y o E Y such that x I--+
Ch. 7, !j 7.11
THE FUNDAMENTAL EXISTENCE THEOREMS
207
Then there exists a conservative solution ff E X,i.e. s u p f ( x , y) = inf s u p f ( x , y) = v # . X€X Y€Y
Y€Y
Furthermore, we shall prove that v# is equal to V Q = sup inf s u p f ( x , y ) K€J x € X Y € K
where S denotes the set of finite subsets of Y. (The inequality v u =StP is obvious). We use this result to prove that, under the further assumption, V x E X, y
(**>a
+-f ( x , y ) is concave,
we have that
where @(Y,X) denotes the set of continuous decision rules for Xavier. This equality means that, if Yvette is risk-averse (in the sense that (**)2 holds), then Xavier cannot improve his conservative value even if he is informed of the strategy to be played by Yvette. Thus the constant decision rule T achieves the minimal worst loss. In the general case, an “optimal decision rule” C is one which satisfies
VY E y , f ( C (Y), v) = inf f ( x , v). X€X
If such an optimal decisionrule is continuous, then inf,, (y, s~p,,,~f(C(y),y) = v # and thus v# = vb. But this latter identity can be obtained under another assumption. If we assume that V y E Y , x c--f ( x , y) is convex
(**>I
instead of assuming the continuity of C we obtain that v # = sup inff(x,y). YEY X € X
We also prove the Lasry theorem, i.e. if
(*Iz and
(**)I
(i) V x E A’, y I--- f ( x , y) is upper semi-continuous, (ii) 3xo E X such that y t-+ f ( x , y ) is upper semi-compact holds then
208
TWO-PERSON ZERO-SUM CANES
[Ch. 7, !j 7.1
Hence assumptions (*)I, .(*)a, (**)I, and (**)a imply the existence of a saddle point. We can also prove the existence of a saddle point when the convexity is replaced by the assumption that the correspondence C assumption which associates with any y E Y the minimal subset C(y) = {X E X such that f ( 2 ,y ) = infxcxf (x, y ) } is lower semi-continuous. We shall deduce most of the existence theorems of the following chapters from the Ky-Fan theorem. This asserts the existence of IE X satisfying supycYf(%,y ) 4 0 provided thatf : XX X I-- R satisfies assumptions (*)I, (**)2 and SUPY<,S(Y,Y) = 0 7.1.1. Existence of conservative solutions
Consider the %vier’s loss fiinction f :X X Y I- R.A conservative solution exists when the functions x F+ f(x ,y) are lower semi-continuous and some function x f- f ( x , yo) is lower semi-compact, since in this case the worst loss f # is lower semi-continuous and lower semi-compact. In fact, we shall prove a stronger result. We assume that Xis a subset of a topological space U and denote by fx(x, y) the extension off to U defined by
Also, we introduce the following “value”
(2)
vb = sup
inf max f (x, y ) (read v “natural”)
K € d x€X
y€K
where 8 denotes the family ofjlnite subsets
K = bl,..., y,} of Y. It is clear that (3)
vb
e V Q =sv#.
Because each y E Y forms a finite subset K = {y}, we obviously have v b = sup(yl inf,,, maxyc(v f ( x , y ) =s vu. On the other hand, since infXc, supycKf ( x , y ) == vx, inequality V Q G v # holds. Tbe~rem1. Suppose that
(4)
Vy
E Y , x I---f ,(x, y ) is lower semi-continuous on U
Ch. 7 , s 7.11
209
THE FUNDAMENTAL EXISTENCE THEOREMS
(assumption of continuity) and that 3 yo
(5)
E Y such that x
++f(x, yo) is lower semi-compact on X
(assumption of compactness). Then there exists a conservative solution 5 and
v# = VQ. In other words,
Proof. Let S(y) = { x E X such that f ( x , y ) v u } . Then the set of solutions of (6) is S = n,,,S(y). The subsets S(y)are closed by (4) and S ~ O is compact ) by (5). Therefore S is non-empty provided that the finite intersection property holds, i.e. SK = S(yO)n riel S(yJ # 0 for all finite subsets K = {yl, . . ,y,,} of Y. .,, , f ( x , y) is lower semi-continSince S(y0) is compact and x I--. uous, there exists X which minimizes this function. Therefore
.
,,
max f ( Z , yi) = inf
.. ., n
i=O,
max f ( x , yi)
x<X k 0 ,
...,n
e s u p inf supf(x,y) = vQ. K€C
XEX
YEK
This implies that 2 belongs to S,. We have proved that S is non-empty, i.e. that there exists a solution Z satisfying (8)
sup f (2,y ) =s VQ.
YEY
This implies that v# a supyEyf(~, y)
V'J 4
v#,
Thus, (6) holds and v Q = v#. Remark. More generally, we can replace assumption (5; by the weaker assumption (9)
3y1, ...,yn E Y such that x I-+ semi-compact on X.
ma^,,^, ...,J(x, y,)
is lower
n;-l
Together with assumption (4) this implies that the subset So = S(y,) is compact. It is also non-empty. The proof the Theorem 1 With S(y0) replaced by SOthen implies the following result. 16
210
TWO-PERSON ZERO-SUM GAMES
[Ch. 7, 8 7.1
Proposition 1. Suppose that assumptions (4) and (9) hold. Then there exists a conservative solution X and v# = vn.
This result is useful in the following example. Example. Consider the case when
n;=l
(i) X = X i where X i c Ui, (ii)f(x, y ) = C;xlf;:(xt,y ) wherefi is defined on X fx Y. Instead of assuming that there exists yo E Y such that the function x t--+ f ( x , yo) is lower semi-compact, we shall assume that V i = 1, . .., n there exists yi E Y such that
Proposition 2. Given (lo), the compactness assumption (11) and the continuity assumption (13)
.,
~y E Y , V i = 1, .. n,
on
u',
2 1-44.*,(xi, y ) is lower semi-continuous
then there exists a conservative solution X = {?,
. . ., T}and V Q
= vx.
Proof. We have to show that assumption (9) of Proposition 1 is satisfied. Taking K = { y l , ..-,y,}, we check that x I--+ maxlSis,f(x, yi) is lower semi-compact. If x = (XI, . . .,x,) satisfies maxlSi,,f(x, y,) == I, then, for any i,
YO
f;:(xi,
1- C f i ( x j ,Y;) jzi
2- C $ ( y i ) i#i
Therefore, for any i, xi belongs to a relatively compact subset. Remark. Examples of lower semi-compact functions are given in Section 3.1 Remark. We shall prove that the conclusion (6) of Theorem 1 holds under weaker conditions (see Theorem 13.1.1).
Ch. 7,§ 7.11
THE FUNDAMENTAL EXISTENCE THEOREMS
21 I
7.1.2. Decision rules Suppose now that Xavier is informed of the choice of Yvette’s strategies. Instead of choosing “pure” strategies x E X , Xavier can choose “Decision rules’’ C : Y F-+ X mapping Y into X . The worst loss associated with such a decision rule C is defined by
Therefore, Xavier will choose from a subset of feasible decision rules a decision rule C which minimizes C k--f *(C). Our results require that X and Y are topological spaces and that the set of feasible decision rules is the set @(Y,X)of contimom decision rules. Since any pure strategy x E X can be identified with the constant (continuous) decision rule C, :y k+ C,(y) = x it is clear that
This means that Xavier’s minimal worst loss is better when he is informed of Yvette’s strategy choice. However, we shall prove that this information is redundant (in the sense that equality holds in (15)) when the loss functionf is concave (or, even, quasiconcave) with respect to y .
Remark. In the case of zero-sum games, the concavity off with respect to y describes “cautious behavior by Yvette” (see Section 1.3.7). Before stating and proving the above theorem, we note that, as the topology of Y is strengthened, the set @(Y,X)of continuous decision rules for Xavier increases. Hence the minimal worst loss infcce(y,x , f * ( C ) should decrease. The identity w* = infcce(y,x,f’(C) is therefore most powerful when the topology of Y is strongest. We exploit the comment above by introducing “finite topology” on a convex set Y. This topology is stronger that any vector space topology. Any affine map from Y to a vector space U is continuous provided that both Y and U are equipped with the finite topology. 7.1.3. Finite topology on convex subsets
Suppose that Y is a convex subset of a vector space V. Associate with any finite subset K = {yl, . ,y n } of Y the affine map /lK from&(see Section 1.3.4)
..
16.
212
TWO-PERSON ZERO-SUM GAMES
[Ch. 7,
0
7.1
-tion 1.The ''fiite topology" defined on a convex subset Y is the strongest topology for which the maps PK are continuous when K ranges over the family 8 of finite subsets of Y. A map A from Y into a topological space 2 is continuous when Y is supplied with the finite topology if and only if the maps (17)
A& :A!" !-- Z are c o n t h o u s for all 1y c 8.
Proposition 3. Thefinite topology of Y is stronger than any vector space topology. Any affine operator L from Y into a vector space U is continuous when both Y and U are supplied with the finite topology. proof. Suppose that Z is the vector space V supplied with a vector space topology and that A is the canonical injection from Y into 2.We have to prove that A is continuous. This follows from the fact that, for all K = {yl, ...,yn}, the map 1 E An ApK(I) = A$i is continuous. We take now Z = U and A = L, where L is an affine operator. We have to prove that for any K = {yl, . . .,y,,}, the map L/?K: 1 E A'' F- L&(A) = L(xy=l = = A'L(yi) = &&) E U is continuous. But L& = BLcm is continuous by the very definition of the finite topology on U.
c;=l
1-4
Ah)
7.1.4. Existence of an optimal decision rule
Theorem 2. Let X be a subset of a topological space (Iand let Y be a convex subset of a vector space V,supplied with theJinite topology. Given the continuity and compactness assumptions (4) and (9,and the concavity assumption (18)
V x E X, y I--
f (y, x ) is quasi-concave,
there exist conservative strategies 1 which, viewed as decision rules, minimize
C I-+ f #(C) on @(Y,X),i.e. (19)
supf(2, y ) =
YEY
inf
sup f (C(y),y) = u+.
CEk(Y,x) YEY
Proof. By Theorem 1, the proof reduces to showing that
Ch. 7,s 7.11
i.e. for any K = 011, (21)
213
THE FUNDAMENTAL EXISTENCE TaulREMS
inf
...,y,,} and any C E @(Y,X),
max f ( x yyi)
XCX f-1.
....n
supf(C(y), U) = C.
Yer
In fact we shall prove that there exists 2 E X such that (22)
x-flax f ( J , Yi) =s c.
i=l. .... n
For this purpose, we introduce the function rp :A n X Y I-+
R defined by
and the sets (24)
Fj = {A E
AnSuch that q(A, yi) =S c}.
It is enough to prove that there exists 2 E Fi.If so, 2 = C ( ~ = l ~ ~ f > clearly satisfies (22). To prove that the n subsets Fi of Anhave a non-empty intersection, we use the fundamental Knaster-Kuratowski-Mazurkiewicz lemma (see Appendix B). We have to check that (25)
subsets Frof Anare closed, { (ii)(i) the En An, A UICAA V1
E
Fi,
where AA = {i such that L' =- 0). The subsets Fj are closed because A E &"F+ C(C;=,,AiyJ E X is continuous (for Y is supplied with the finite topology) and x i--fX(x, y) is lower semi-continuous. It remains to check that property (25(ii)) holds. If not, there exists LO E Mnsuch that, for any i E A0 = {i such that & > 0},
This means that, for any i E Ao, y j belongs to the open upper section of the function y t--.f(C(& A&, y), which is convex by assumption (18). There&,) =- c. fore '& A{y,belongs to such a section. HenceJ(C(& But this contradicts the definition of c = supYcyf(C(y),y). 0
Ah,),
7.1.5. The Ky-Fan inequality
We deduce from Theorem 2 the Ky-Fan inequality. This will play ajiuiui'amental role as a tool in proving most of the forthcoming existence theorems.
214
[Ch. 7, Q 7.1
TWO-PERSON ZERO-SUM GAMES
Theorem 3 (Ky Fan). Suppose that
(27)
X is a convex compact subset
++ R satisfies X,x +-+f (x, y ) is lower semi-continuous,
and that the function f :X X X
(28)
I
(i) V y f (ii) V x E X , y I--+ f(x, y) is quasi-concave, (iii) SUP,,XS(~, Y ) == 0.
Then there exists ? E X satisfying
(29)
sup f(5, y ) =s0.
Y€X
Froof. Since the identity map I is continuous from X supplied with the finite topology into X (which is compact for a vector space topology), we may deduce from Theorem 2 the existence of Z E X satisfying
(301
sup f(2,Y ) =
inf supf(C (Y), Y ) cEe(x,x) Y E X 4 sup.f(I (Y), Y ) = SUP f ( Y , Y ) -c 0. YEX
Remark. This theorem will be extended in Section 13.2 by replacing the continuity assumption (28(i)) by an assumption of pseudomonotonicity (see Theorem 13.2.1). 7.1.6. The Lasry theorem Theorem 4 (Lasry). Suppose that the continuity assumption (4), the compactness assumption (5) and the concavity assumption (18) of Theorem 2 are satisfied. Then
Proof. We know that infcE8(X,y) supYEyf(C(y),y) = v* by Theorem 2. Also v9 since inf,,,f(x, D(x)) ')2 for any map sup,,e(x, y ) infx,,f(x, QX)) D. It remains to prove that (32)
'U'6
SUP
hff(X,D(X))
D€Q(X,U) X€X
Let E r 0 be fixed. In the first place, there exists a map b E q ( X , Y) satisfying (33)
vu f
x,f "(u) = supf(u, y ) +-(u, YEY
D(u))++
E.
Ch. 7,
7.11
THE FUNDAMENTAL EXISTENCE THEOREMS
215
Secondly, since f is lower semi-continuous with respect to x, there exist open neighborhoods N(u) of any u E X such that
(34)
v x E N(u),
f(u, b ( u ) ) f(x,a u ) )
++ e.
We introduce the subset
(35)
XO = {x E X such thatf(x, yo) =sw # + E )
which is compact since it is a lower section of the lower semi-compact function
x k--f ( x ,Yo). Therefore, XO is covered by a finite number of open subsets Vi = N(uJ (i = 1, . . ,,n). If we set V = Cx Xo we deduce that
x c uVi. n
(36)
i=o
Let {p,}j-o, . ,, be a continuous partition of unity subordinate to this finite covering of X. This has the following properties. If po(x) > 0, then x E VO.Thus w* -= f(x, YO)- E -= f ( x , yo)+ E . If ( x ) > 0, then x E N(ui). Thus v" e f ' ( u i ) = s f ( u i ,b ( u , ) ) + a E e =sf ( x , &ui)) E by (33) and (34). In terms of the set ,
+
(37)
S = {y E Y such thatf(x, y) >- w+-
E}
these properties reduce to the following. yo E S when Po(x) > 0 and b ( u i ) E S when Bi(x) z 0.
But S in convex sincef is assumed to be quasi-concavewith respect t o y . Since 3 0 and ZL,, Bi(x) = 1, we deduce that
&(x)
i.e. that
(39)
vx E
x,
w#-
E
= = f ( x ,D C X ) ) .
Because the functions Pi are continuous, the map D : x I--We have proved that for all E z- 0,
D(x) is continuous.
216
[Ch. 7, 8 7.1
TWO-PERSON ZERO-SUM GAMES
7.1.7. The minisup theorem
If we could find among the optimal decision rules C satisfying f ( C ( y ) ,y ) = inf f(x, y ) for ally E Y X€X
one which is continuous on Y supplied with the finite topology, we could deduce the existence of x E X such that (42)
v++= supf(x, YEY
v) =z YEY SUPf(C0, Y ) = vb
i.e. the existence of a minisup. In fact this turns out not to be a good approach for proving the existence of a minisup. Instead, we shall proceed on the assumption that f is convex with respect to x and concave with respect to y. Theorem 5 . Let X and Y be convex subsets of topological vector spaces. Suppose that the continuity and compactness assumptions (4) and (5) and the foIlowing convexity and concavity assumptions
{ (ii)(i) VxV y EE X,Y , y t - +ff ( x , yy)) is concave, convex (x,
I-+
(43)
x
is
hold. Then there exists a minisup Z. Proof. By theorem 1, we have only to prove that K = {Yl, * Yn},
-
(44)
-s
vb i.e. that for any
- 9
vg = inf
max f ( x , y i ) s vb.
x € X I-1,
..., n
We can write n
and the concavity off whith respect to y implies that n
(Because @(x, y f ) f (x, A'yi) and hence inf,,, 6 infx,,f(x, C;=l 2 y f ) w'.) It is therefore enough to prove that (47)
vg
= inf
sup f ( x , yi) =sa when a
xEX I - 1 ,
. . .,n
4.
Gal Atf(x, y i ) -c
Ch. 7, Q 7.11
217
TRE FUNDAMENTAL EXISTENCETHEOREMS
[Let a converge to ."g, in these inequalities. Then v: 4 &. But (44) follows.] In fact, we prove the following stronger statement. 3xa E X such that sup f ( x a ,yi) e a when 01 =- 4 (48)
4 -s d and
lskn
For this purpose, it is convenient to denote by F the map from X into R" defined by (49) and to set (50)
F(x) = { f ( x , US,. . ., f ( x , yi),
{ (ii)(0 a
.
+
9 .f(x9
~ n ) }E
Rfl
F + ( X ) = F(X)+R:, = {a, . .)a}.
.
It is clear that (48) means that (51)
a E F+(X) when a
r
4.
(This is because a E F + ( X ) is equivalent to the existence of x, E X and c E R:. such that a = F(x,)+c, i.e. the existence of xu E X such that ( V 4 a) Fj(xu)= =f(xa9 yi).)
Now, in order to prove (51), we shall show that if a 4 F+(X), then a 6 vg. (52) For this purpose we use the separation theorem for a finite dimensional space. Convexity assumption (43(ii)) implies that F + ( X ) is convex (see Proposition 1.3.10). Therefore, if a ($ F+(X), there exists A E R"' (A # 0) such that
= ab(F+(X);A).
This implies that A E : R and that d ( F + ( X ) ;A) = #(F(X); A). Therefore, dividing by CSl A' w 0 and writing X = AIC;,l A, we obtain that (54) 7.1.8. The Nikaido theorem
We now state a corollary of Theorem 5 which will often be useful. Theorem 6 (Nikaido). Suppose rhat (i) X is a convex compact subset, (55) (C)V y E Y , x t - f ( x , y ) is convex and lower semi-continuous
{
218
TWO-PERSON ZERO-SUM GAMES
[Ch. 7, 0 7.1
and that 156)
subset, { (ii)(i) VxY isEaXconvex , y +--f (x, y ) is concave.
Then there exists a minisup 2 E X .
7.1.9.Existence of saddle points Applying the Nikaido theorem to the loss functions f and - f for Xavier and Yvette, we obtain the existence of a saddle-point under convexity assumptions. Theorem 2 and Theorem 4 applied to - f also yield the existence of a saddle-point under assumptions of “quasiconvexity ” only.
Theorem 7 (Sion). Suppose that (57)
[
(i) X and Y are convex compact subsets, (ii) V y E Y , x+ f (x,y ) isquasi-convexandlowersemi-continuous, (iii) V x E X , y F-+ f (x,y) is quasi-concave and upper semi-continuour
Then there exists a saddle-point (5p}.
Remark. We can replace the assumption that X and Y are compact by the assumption that x ~ - + f ( x ,yo) is lower semi-compact and that y F- f ( x o , y) is upper semi-compact for some {xo, yo}. *7.I.I0. Another existence theoremfor saddle points
We can replace the assumption of convexity off with respect to .x (i.e. of cautions behavior by Xavier) in the theorem above by the following regularity assumption. (58)
decision correspondence { The minimal E X which minimizes y ) } is lower semi-continuous. = {x
f(x,
y I--+ @(y)
(Such a regularity assumption may be regarded as describing “regular” behavior by Xavier.)
Theorem 8. Let Y be a convex subset of a locally convex vector space V . Suppose that (i) V x E X,y I- f ( x , y ) is concave, (ii) Q x E X , y t.- f r ( x , y) is upper semi-continuous, (59) (iii) 3x0 E X such that y +- f (9, y) is upper semi-compact
Ch. 7,s 7.21
EXTENSION OF GAMES
219
and that X is a topological space for which
(i) ‘dy E Y, x +-i f ( x , y ) is lower semi-continuous, (ii) the minimal decision correspondence Db is lower semi-continuousfrom Y supplied with the finite topology into X . Then there exists a saddle point.
Proof. Assumptions (59(ii) and (iii)) imply that there exists j E Y satisfying
Let Z E o”(j5) be k e d and let y be any element of Y. Since (1 -A)y+Ay converges to j5 when A converges to 0, and Db is lower semi-continuous on Y supplied with the finite topology, there exists TAE Db((l-A)y+Ay) such that XA converges to X when I converges to 0. We now use the concavity o f f with respect to y to obtain the following estimates. If A E 10, 1[,
+V(%¶ Y ) =G (1-4f(%,y3+Af(%,y) &f(% (1 - I)V+ Iy) =f *((I- Alp+ Ay) &Sb(V).
(1 - 4fb(u3 =5
Therefore, for any 3, =- 0, f (ZA,y ) 4 f ’(p). Sincef is lower semi-continuous with respect to x, we deduce that for any Y E y,
(62)
f(% Y ) =S fim inff h,v) e fb(j))a+o
Hence, s ~ p ~ , ~ f ’ (y)Z4 , fb(
Let X and Y be two strategy sets. An “extension” {%,@, i,j,z}of a game defined o n XX Y is specified by an injective map i from X into %, an injective map j from Y into g,a linear operator fi mapping bounded loss functions f defined .on X X Y onto bounded functions fif on %xQ satisfying
4 i ( X ) , I ( Y ) )= f (4v>
and other requirements. We shall investigate the extensions which are “playable”, i.e.such that, for some classof lossfunctionsf o n X X Y ,theextendedgame
220
[Ch. 7,
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zf on B X Q has a value. The first example is the “mixed extension” of a finite game. We take % = &(X), (7$ = M(Y)to be the convex compact subsets of mixed strategies, i = 6, and j = 6 , to be Dirac operator and fi to be defined by
ad (m, n) = ( m @ n , S ). Then Theorem 1.7 implies the existence of a saddle pair {rE, i i } of mixed strategies (Von Neumann’s theorem) for every loss function f. The second example is given by the “sequential extensions”, in which we take % = XN and Q = YN to be the subsets of sequences of strategies, and z to be defined by nnf
(5, T ) =
m
C
aijf (xi, Y j )
fJ=l
where Vi, j E N, afj 0 and Crj=,aij = 1 . We prove for instance that such an extension is playable when X and Y contain p and q strategies respectively and at, = (1/pq) ( ( p- l)/p)‘-’ ((4- l)/q)’-’. Furthermore, the value of the extended game is equal to the value of its mixed extension. This latter property is shared by all “extensions without exchange of information” (defined in Section 7.2.3). We also define “extensions with exchange of information” in the following way. We introduce pairs @ c @(Y, X ) and c @(X, Y) of subsets of decision rules for Xavier and Yvette respectively, which are consistent in the sense we can find a fixed point {Z, j j } of the map that, for any C E @ and D E {x, y } {C(y), D(x)}. By taking % = @, (7$ = fa and 3t to be defined by ?tf(C,D ) =f ( Z , J ) , we obtain the so-called extensions with exchange of information. For instance, the consistent pair in which @ = @(Y, X ) is the set of all deci= Y is the set sion rules for Xavier (who thus gets all the information)and of constant decision rules for Yvette (who has no information whatsoever} defines a playable extension. The value of the extended game is Yvette’s conservative value supvcy inf,,, f ( x , y ) for the initial game. The Lasry theorem implies the same result when wereplace @ = @(Y, X ) by the set @ = @(Y,X) of cortinuous decision rules, if we assume that f is convex respect to x and upper semi-continuous and semi-compact with respect to y . We shall devote the next section to the fundamental example of iterated extensions.
a
a,
-
a
7.2.1. DeJinition of extensions of games
Let us consider a two-person zero sum game defined by Xavier’s loss function R.
f :X X Y
-.
Ch. 7, 8 7.21
EXTENSION OF GAMES
22 1
Although saddle-points may not exist for the strategy sets X and Y,they may exist for larger strategy sets %and @with the loss functions extended to %X@. For instance, we can embed X in the set of mixed strategies, the set of sequences of strategies, a subset of decision rules or a subset of sequences of decision rules. It is worth introducing a general definition before studying specificexamples. Let U ( X X Y) a n d a (%x @), be the Banach spaces of bounded loss functions defined on X X Y and %X @ respectively (see Section 3.1.a). Definition 1. An extension {B,@, i, j, n} of the games defined on XX Y is defined by the five following items :
(i) and extended strategy set % for Xavier, (ii) an extended strategy set for Yvette, (iii) an injective map i imbedding X in %, (iv) an injective map j imbedding Y in @, (v) a linear operator mapping loss functions f E CU(XxY) onto functions nf E U(GX@). such that
Remark. In fact condition (2(i)) follows from conditions (2(ii), (iz) and (iv)). We shall say that rt is the “extension operator”. Assumption (2(ii)) means that fi is increasing and leaves the constant functions invariant. Assumption (2(iii)) means that if Xavier plays the “pure” strategy i(x) in the extended game, his WOIAloss in the extended game is better than his worst loss in the intitial game (he will not raise objections against the use of extended strategies by Yvette). Similarly, assumption (2(iv)) means that the play of a pure strategyj(y) in the extended game is no worse for Yvette than the play of y in the initial game. Proposition 1. The duality gap of the extended game is contained in the duality gap of the initial game, i.e.
(3)
sup inf f (x, y ) =S sup inf af (E, 7) Y€Y X € X ?I€@ €€a 4 inf sup nf ( 5 , q ) G inf sup f ( x , y ). hE% ?ICY xcx rcr
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Proof. This follows immediately from (2(i), (iii) and (iv)). 0 The most interesting extensioiis are those such that nf always has a value (even iff does not). Notice that this value belongs to the duality gap of$ Definition 2. An extension
{a, "$, i. j . a } is said to be '>layable" if
7.2.2.Mixed extensions As in Section 1.3, we can enlarge the strategy sets X and Y for Xavier and Yvette by embedding them in the sets M(X) and M(Y) of mixed strategies (i.e. discrete probability measures). We denote by 6 or 6,the Dirac operatorfrom X into M ( X ) a n d by 6 or 6, the Dirac operator from Y into &(Y). These will be the injective maps of definition 1. We extend Xavier's loss function f :X x Y -, R to a bilinear loss function nbf mapping M(X)x-/n(Y)into R in the following way. (5)
Whenever m = Ci ai6(xi) and n = Cj/PS(yj),we define nJ(m, n) = X I , j aiPf(xi, Y j ) .
It is clear that the items {M(X),M(Y), , ,a 6,, ZO}define an extension of the games defined on X X Y . Conditions (2) are satisfied because (6)
up,,~f(x, { (i)(ii)Vf,Vf, VY EE x,sinfXcxf(x, vx
y,
Y ) = S U P , ~ M ~4, Y)~O~(W, Y ) = infmEmx)~of(m, 60).
This extension is called the (discrete) mixed extension. (a) Case offinite strategy sets. If X is a finite set of k elements and Y a b i t e set of I elements, then M(X>= -/nk and A ( Y ) = -/n' are convex compact x i = l a v ( i ,j ) is a continuous subsets and nof defined by nof(m, n) = bilinear form. Thus Theorem 1.7 implies the Von Neumann's theorem. Proposition 2 (Von Neumann). Suppose that X and Y areJinite sets. Then the mixed extension {M(X),-/n(Y), ax, ,a, ZO) is playable. (b) Case of compact strategy sets. In the case of (infinite) compact strategy sets X and Y,the sets M ( X )and A ( Y ) are convex, but no longer compact. We have seen in Section 3.1.6 that the subsetsM(X) and J ( Y )of Radon prob-
Ch. 7, Q 7.21
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223
abilities are both convex and compact. The Dirac operators 6, and S, imbed Xand Y in d(X) and d(Y) respectively. It remains to define zofwhen f E E @ ( X XY ) is a continuous function on X X Y . To define mf, we notice that we can biunivocally associate with f E @(XX Y) continuous linear maps F E J ( @ * ( X ) , @(Y)) and F* E 2(@*((Y),@.(X))defined by (7)
V m E @*(XI, VY E Y , F(m)OI) = (m,f(-,u))
and (8)
'in E e*(Y), Vx E X, F*(fi)(x) = (n,f(x,-)).
Fubini's theorem implies that F and F* are transposed operators. Hence we define nof(rn,n) = (F(m),n) = (myF*(n)).
-.
We denote by m @ n E @*(XX Y ) the linear f o r m f t - ( m @ n,f) = zof(m, n) defined on @(XX r). It is clear that {m, n } n ~ f ( mn), is a separately continuous bilinear form which coincides with the bilinear form defined by ( 5 ) when m and n are discrete measures. Hence the items define an
{M(X>,-m>, d,,
d,, zo}
extension of games, also called the mixed extension. Thus Theorem 1.7 implies the following result.
Proposition 3. Suppose that X and Y are compact spaces. Then the mixed extension {&'(X),-@(Y), d,, d,, SO}is playable.
In other words, these two results imply the existence of a saddle mixed strategy for any two-person zero-sum game whose loss function f is continuous and whose strategy sets are compact. 7.2.3.Extensions without exchange of information
The mixed extension { & ( X ) , d ( Y ) , d x , d,, ing property.
obviously satisfies the follow-
(zogh)(W ,n) = g(x) ~ Z h(q) Y and ( z o gh) (m, N Y ) ) = (flxg)(m) h(Y) [where (nxg)(m) = Caig(xi)and zuh(n) = CBjh(yj) when m = Caid(xi) and n = Cpjd(yj)]. Vg E
(9)
@(x),V h E @(Y),
?to}
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We shall distinguish the extensions of games which satisfy an analogous property.
Definition 3. We shall say that {%,@, i, j,n} is an extension “without exchange of information” if there exist f i x E J(G?l(X),a@)) and Zzy E P(GY(Y),G?l@)) such that V g E M ( X ) , V h E (z1(Y), Vx E X, V y E Y , V5 E %, Vr] E Q, (10)
(W { (ii)(i) n(gh) n(gh) (5, Av))
r ] ) = g ( 4 nrh(r),
= ( n x d (5) h(v).
Interpretation. If the initial loss function of the game does not depend on y (i.e., iff = gor), this means that Yvette is a “dummy” in this game. Then we obtain that ( z f )(E,j(y)) = zxg(5), i.e. that Yvette remains a “dummy” in the extended game. In other words, Yvette has no “influence” on the Xavier’s extended loss function. If we agree that any “influence” of one player on the other comes necessarily from an “exchange of information”, we motivate the above terminology. Proposition 4. Let X and Y be compact strategy sets. Let {%,@, dx, 8y, fi} be an extension without exchange of information. Then, i f oo(f) denotes the value of the mixed extension nof of a continuous function f,we obtain:
Remark. In other words, if an extension without exchange of information is playable, then the value of the extended game .tf is the mixed value wo(f).
Proof. Denote by a*(t,q)E @*(XXY>,z:(5) E @(X) and .t;(q) E @*(Y) the Radon measures f I-+ zf(5, q ) , f t - nxf (5) and g I-+ Jzyg(q).Properties (10) imply that n*(i(x),q) = 8(x)@&(r]), Z*(t,j(Y)) = n2(5)63 4 y ) .
Thus, we obtain the following inequalities
We deduce that
Ch. 7, 0 7.21
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EXTENSION OF GAMES
Therefore, the above inequalities imply that sup inf ~f (5, r]) VEQ €€a
v0Cf)-
The other inequality is proved in the same way. 7.2.4. Sequential extensions
Let X and Y be two strategy sets and let N be the set of integers. We take (12)
{ (ii)(i) 55
= XN = set of sequences 5 = { x I } I E ~ ,
@ = YN = set of sequences r]
=~J},€N.
The sets X and Y are canonically embedded in XN and YN by the maps s and t defined by (13)
sx = {x,
..., x . . . } , ry = cy, ..., y,
...}
mapping x and y onto constant sequences. Let a = {u,,}~, be a summable sequence on N X N satisfying (14)
Vi, j E N,aij
1 00
0,
aij
= 1.
iJ=1
I f f € @(XX Y), any sequence V ( x , ,y,)} is bounded. The convergence of the series (15)
nuf(E,17) =
5
aijf ( X I ,
YJ)
l.J=1
follows. It is obvious that {X”,YN, s, r, na} defines an extension, called the “sequential extension” associated with the sequence a. Taking loss functions of the form gh, we obtain that
17
226
TWO-PERSON ZERO-SUM GAMES
[Ch.7,5 7.2
Therefore, if we define zx and styby
c aiXg(Xi), d 4 r l ) c aTh(r,)ra
m
ns(Q=
=
i=l
j=l
we see that sequential extensions are extensions without exchange of information. Interpretation. Sequential extensions may be interpreted in terms of the initial game as follows. Xavier secretly chooses a sequence 5 = of “pure” strategies. Yvette secretIy (and therefore independently) chooses a sequence q = {y,XENof “pure” strategies. The game is then played repeatedly as specified by the sequences of pure strategies chosen by the players. The players evaluate the results of this infinite sequence of plays by discounting over time. In particular, Xavier’s extended loss function is the “generalized discounted” s u m nJ((E,q) of the individual lossesf(x,, y,) at each play.
We give below an example of a playable sequential game in the case of finite strategy sets. Proposition 5. Suppose that X and Y are$nite subsets containing p and q elements respectively.Let a = {aij}be dejined by
Then the associated sequential game is playable.
Therefore, we can write
Ch. 7,s 7.21
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227
By Proposition 1.3.4, is a surjective map from XN onto &(X) = dp and z r is a surjective map from YN onto &(Y) = dq. We can therefore write inf sup nJ(5, v) = inf €Em q € r g €€XU =
inf
mEJn(X)
SUP TKqp
sup (m@n,f) = V
(d(t)@$(~),f) O ( ~ .
nEJn(Y)
Remark. This result is false when one of the strategy sets is infinite. In fact, one can prove +t, when P and Y are inhite, there is no sequence a = {%) satisfying (14) such that the sequential game is playable. Remark. When X and Y are both finite, F’rop0sit;on 5 gives a &terministic way of playing the initial game which yields the same value as the mixed extension. Remark. We can approximate a saddle-point { {zf}&N, {j,},EN} by the @ of finite sequences { Z f } l s f G K , {jj,}lsjsp It easy to check that the error is estimated by
7.2.5. Extensions with exchange of informatfon Decisioa rules. We denote by V ( Y ,X)(resp. @(X, Y)) the set of maps from Y into X (resp. from X into Y). We shall interpret a map C E @(Y, X ) as a “decision rule” for Xavier. Such a decision rule can be implemented under the behavioral assumption that Xavier is informed of the choice of the strategies y played by Yvette. %vier’s strategy set X can be imbedded in the set V ( Y ,X) of decision rules by the map c from X into V ( Y ,X) defined by (21)
x E
x t-+
c(x) :y .-I
c(x) Q = x.
-.
Similarly, we denote by d :Y @(Y, X ) the canonical embedding from Y into the set of decision rules for Yvette defined by (22)
Y E Y+d(y):x+d(y)(x)=Y.
These maps are injective. By embedding X and Y in q ( Y , X) and V(X,Y) respectively, we identify “pure strategies” x and y with constant decision rules. 17’
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How can both Xavier and Yvette simultaneously use a decision rule C E V ( Y ,X ) and D E P(X,Y)? Suppose that there exists a fixed point {Z, 9 ) of {x, y } k+ {Cb), D(x)}, that is a pair {Z, y } satisfying (23)
C ( 3 = Z and D(2) = 7.
If %vier and Yvette implement strategies P and j , then these will be consistent with their chosen decision rules if and only if (23) holds. Such a pair ( 2 , j ) is “stable”: none of the players have an incentive to move from such a fixed point. We are led to introduce the following definition. Definition 4. A pair {@,m} of subsets of decision rules is called “consistent” if (i) x c @ c m y , m, (24) (ii) Y c @3c V ( X , Y ) , (K) V {C, D } E @X@, 3 {P,y} E X X Y such that (23) holds.
[
If the pair {@,/a} is consistent and if a map associating with {C,D } E @ X1z) a fixed point {Z,y} < XX Y satisfying (23) is selected, we extend a loss function f for %vier to the loss function zddefined on ex /a by
s(25)
fiqf(c, 0 ) = f ( % 9).
It is clear that
(26)
w-(c(x), 4 Y ) ) =f ( x , Y )
since {Z, 9) is the unique e e d point of the map {x, Y }
+
{c@)
cv), d m <x>}= {%9 )
It is easy to check that the map nd:f I-+ ndf is linear, positive and leaves the constant functions invariant. Also, we notice that
(27)
{ (ii)(i)
vx E tly
X,sup,cYf (x, Y ) Z= supmnf ( x , ax)),
E y, infxcxf ( x , r) 6 infccrf (C
a}
Therefore, (26)and (27)imply that, if (62, is consistent, then {Q, ‘13, c, d, nd} defines an extension, called an extension “with exchange of information”.
Counter-example. The pair {@(Y, X),@(X, Y)} is not consistent, since there are maps {x, y } F--. {C(y),D(x)} which have no fixed points (when Xand Y contain more than one element).
Ch. 7, 0 7.21
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In other words, both players cannot have full information about the other player’s choice and agree on a “stable” pair of strategies. Example 1. The pair { p ( Y ,X ) , Y} is consistent. For any C E @(Y, X) and p E Y, {C(p),p) is the unique fixed point of the mep {x, y} I-+ {C(y), y}. %vier’s extended loss function is d e h e d by (28)
G2!f(C,v) =f(C(Y),Y).
The worst loss associated with C is supyryf ( C ( y ) ,9).By associating with any E z 0 a decision rule C, satisfying f ( C, ( y) ,y) r inf,,,f(x, y)+ e, we deduce that Xavier’s minimal worst loss is equal to
Hence (27(ii) )and (29) imply that this extension with exchange of information is playable and that the value of the extended game is equal to supyfr inf,,, f (x, y), i.e. the conservative value of Yvette in the initial game. Interpretation. The choice of the pair { q ( Y ,X), Y} can be interpreted as meaning that Xavier has full information about Yvette’s choice of strategies, while Yvette has no information at all about Xavier’s choice. In other words, Yvette first chooses her pure strategy y E Y and Xavier, knowing this choice, chooses the pure strategy C(y) according to the decision rule C (which is chosen in advance).
Example 2. The Lasry theorem yields another example of an extension with exchange of information. We take @ = e(Y, X ) and 11) = Y which obviously form a consistent pair. Suppose that (30)
1
(i) b’x E X,y t - c f (x, y ) is upper semi-continuous, (ii) 3 x o E X such that y I-+ f ( X O , y) is upper semicompact. (iii) V y C Y, x + f ( x , y ) is quasi-convex.
Then the Lasry theorem implies that (31)
sup inf f ( x , y ) = Y € Y X€X
inf sup f ( C ( y ) ,y). C€e(y,x) Y f Y
Hence (27(ii)) and(31) imply that this extension with exchange of information is playable when it is restricted to the cone of functions satisfying (30). The value of the extended game is equal to supucyinf,,, f (x, y), i.e. the conservativevalue of Yvette in the initial game.
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Remark. If we assume that X and Y are convex compact subsets (32) then the pair (@(Y,X),@(X,Y} where @ denotes the set of continuous functions is consistent since the map {x, y } E XX Y .+ {C(y),D(x)} E XX Y has a k e d point by the Brouwer theorem. (See Theorem 9.3.5 below.)
$7.3. Iterated games By combining extensions without and with exchange of information, we can model the concept of extension with partial exchange of information. We shall illustrate this concept only with the example of iterated extensions. The appropriate scheme of exchange of information depends upon two subsets S and T of N. %vier chooses X I . Yvette knows x1 if the index 1 belongs to T (otherwise she does not know XI). In the former case, she chooses yl = Dl(x1) according to a decision rule D1. Xavier is informed of the choice of y1 if 1 E S,and not informed if 1 fl s. In the former case, he implements x2 = Cs(y1). Now Yvette is aware of this choice if 2 E T and not aware otherwise . . . and so on. Thus S is the set of those “stages” i E N such that, before he chooses xi+1,Xavier is informed of the strategies played at stage i. In the same way, T is the set of “stages” j E N such that, before she chooses yj, Yvette knows 3.Assuming that the players have perfect recall, they choose sequences of non-anticipative decision rules. If S = T = N, the information is “perfect”. If S = R = 0, there is no exchange of information and we recover the sequential extensions. In any case, once a pair of sequences { x , } and ~ ~ b,},6N ~ is implemented by using sequences of decision rules as above, the extended loss function is defined by &ri=latj f (3,Yj). The Moulin theorem states that the length of the duality gap of the iterated game is bounded by
This bound confirms the intuitive feeling that the “larger” the exphange of informatian, the smaller the corresponding duality gap. In particular, iterated games with perfect information are playable. The proof of the Moulin theorem is quite involved. We begin by assuming that a convenient system of functional equations is solved. We then prove the existence of a solution of such a system by using a fixed point theorem which generalizes the theorem of “successive approximations”. With this apparatus, we can then prove the Moulin theorem.
Ch. 7,
5 7.31
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231
7.3.1. Iterated extensions
An Iterated game may be regarded as the “product” of a sequential extension with an extension with pure exchange of information. To see this, let a = { u ~ , } ~be , ,a~sununable ~ sequence satisfying (1)
aij
0,
C aij = 1. 00
I, +I
Let {X”, Y”, s, t, n,} be the associated sequential extension. In order to construct its extension with exchange of information, we have to introduce consistent pairs of subsets of decision rules {@,@} where @ c @(YN, XN) and CZ) c @(XN, YN).The subsets @ and @ which we choose are associated with the subsets S c N and T c N as follows. A decision rule C = {C,},,, belongs to @ if and only if CE = {C1&. . .,C,f, . ..}, where (2)
Vi, Ci maps
n
Icsn[o,1-11
Y,into X.
(We denote by Y,the strategy set Y used at the jth stage). A decision rule D = = {D,}leN belongs to fD if and only if Dq = (017, . . .,D,q, . . .}, where (3)
Vj, 0,maps
Xiinto Y.
fc ~ n i o , i i
(We denote by X, the strategy set X used at the ith stage.) Interpretation. Condition (2)means that at the ith step, Xavier is informed of the choice of the stategies y j ( j E S n [0, i- 11) previously played by Yvette and im‘plements the strategy C,(y,, ...,yI-J according to the decision rule C,. In the same way, at the jth step, Yvette @ informed of the strategies xt(i E T n [0, I]) previously played by %vier and implements ,the strategy Dj(x,, ..., x,). according to the decision rule D,. Note that at the first step i = 1, the set of Xavier’s decision mules is the set X of consistent decision rules. Proposition 1. The pair {$ sistent.
fD} of subsets of decision rules defined above is con-
Proof. Let C and D be fixed. We shall prove that there exists a unique fked point E = { 4 } l E N and 4 = {jj,}IEN of the m p {E, rl} --c {C(d, D(5)). Since C1 = Cxl is a constant decision rule, we take I1 = X I . Since D1 depends only on xl, we take jjl = Dl(ll). Since CZdepends only on yl, we take ZZ= = C2(jj1), and so on. Then Z,, is equal to C,,(jjl, . . .,j,,n-l) and jj,, is equal to
232
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8 7.3
D,,(Zl, ...,3J. It is clear that the sequences 5 and 1 form the unique fixed {C(d, W)}. point of {E, rl}
-
We denote by i and j the maps from X into @ and from Y into fa defined by (i) V q E YN, i(x)(q) = {x, x , (ii) V5 E XN,j ( y ) ( I ) = { y , y ,
. . .,x, . ..}, ...,y, ...}.
We can now define the iterated extension. Definition 1. Let a = be a summable sequence satisfying (1) and let @ and fabe the consistent pair of subsets of decision rules associated with subsets S and T of N. Let fi,, be the map associating with any loss function g E @ ( X X Y ) the loss function
where 5 = {Zj}I and 3 = {F,}, form the fixed point of {& r} -c (C(r1,D(E)}. Then {@,fa, i, j,5,,} is called the “iterated extension”.
Example. If S = T = 0, then @X@can be identified with XNXYN and we obtain the sequential game (without exchange of information). Example. If S = N and T = 0, then @ is the set of “non-anticipative” decision rules for Xavier and fD= YN.In other words, Yvette chooses yl, .. ., y,,, ... without information about Xavier’s choices while Xavier chooses xa after yl, xs after y l , y~ and so on, knowing at each step the strategies implemented by Yvette. Example. If S = T = N, then the iterated game is called a “game with perfect information”. The subsets @ and /I) are the subsets of all “non-anticipative decision d e s ” for Xavier and Yvette. Example. We consider the case where S = T = 2N are subsets of even numbers. Thisis an example in which each player is only partially informed. Xavier and Yvette choose to keep their odd choices secret and inform their partner only of their even choices.
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ITERATED GAMBS
7.3.2. The Moulin theorem
Theorem 1. The duality gap of an iterated game associated with S,T and a = {ai,,} satisfying ( I ) satisjes the following estimate.
I sup
-
I
inf fi,,g(C, 0)- inf sup %g(C, 0) cce
DE'D CEQ
DED
where llgll = s w , c ~ , y E Y l g ( x y y ) l Note the following fundamental consequence.
Theorem 2. Iterated games with perfect information are playable. Proof. Since S = T = N, we have that
We also prove a further consequence.
Theorem 3. Suppose that X and Yare compact spaces and that the lossfunctiong is continuous. If
then there exists a saddle-point (C,B) C
ex@.
7.3.3. Proof of playability of iterated extensions
If 5 = (XI, ..., x,,, ...) and r) = 011, ...,y,,, ...) are sequences of X N and yN respectively, we write F" = (XI, . . .,x,) E n i c T ; is,, Xi and by
7'") = ( Y I , . Y J E 7
n j c s ,j e n
Yi.
Suppose for the time being that the following lemma is true.
Lemma 1. There exist two sequences {A,,},, and {p,,),, of non-negative numberswhich converge to 0 and satkfy (8)
11 2-
12
...
1, z- p,
An+l
10
234
[Ch. 7 , 6 7.3
TWO-PERSON ZERO-SUM GAMES
und two uniformly bounded sequences of f ictio n s
.and
+An+lwn+,(E(");
?j(n-1),
On(,!("), q(n-1))).
We define the map if = (Cl, . . ., C,,, . . .) E @ as follows. Let q = (yl, . . ., yn, . . .) E YN be any sequence of Yvette's strategies. For n = 1, we define C1 = Cxl to be the constant decision rule satisfying (13)
A l V l = 7 &l+PlWl(Xl)
For n = 2, we define (7, by (14)
= SdXl, y1).
C&l)
Given C1,Cz, ..., p-11
= (Xl,
we write x, = C,(q('-l)) for j *
.
.7
xn-1).
=sn-
1 and
235
isn
Adding these inequalities for n = 1, . . .,p, we obtain that
isp jai
Since the functions vp.are uniformly bounded and Ap converges to 0, we deduce that
= &+%lg(w?)94-llgll 5
where a
1
I - Zicr;jsi aij-
C j E s ;i s j + l
aij.
Because this inequality is true for any sequence q E YN,we obtain that AlWl
(19)
&+SUP
?EYN
% & % ? r)) ,
II g II0:
= & + s u p ?i,g(C,D)-llglIa. D€m
This implies that 120)
2 1 ~ 1 3inf
cce
supzug(C,D)-Ilg(la.
DEQ
Similarly, we define decision rules B,,by setting (21)
Dn(p0) = O n ( p ' ,q("-l)),
where E = (xl, . . ., xn, . . .) and y = ( y l , . . .,yk, . . .) with y k = Bk(5'k').
236
TWO-PERSON ZBRO-SUM GAMES
[Ch.7, 6 7 . 3
We deduce from inequalities (12) that PnWn(P),
-G
qb-'))
& + a d 5 3 DO))+ IIg I I a-
This implies that (24)
Ilvl
=G
sup inf %&C, D ) ~ c mcce
+ 11 g 11 a.
Therefore, (20) and (24) imply that
inf sup s,,g(C, D) -sup
(25)
cce 4
DEYD
inf Sag(C, D)
D € O CEI
2 IIg 11 a-
7.3.4. A s y s t m of functional equations
In order to solve systems (9) and (lo), we notice that they can be written in the following form
.Ch. 7,s 7.31
It is obvious that
134)
237
ITERATED GAMES
11
2
ila a
...
2 An 3 p,
a Aa+l*
...
since the coefficients aU are non-negative. Furthermore, An and pn are nonnegative. To prove this, add equations (32(i) and (6)).We obtain that
Hence, adding these equations for n
nsp
=
1, . . .., p ,
nsp
Therefore, z= 0 and ,up Ap+, 0 for all p. Furthermore, both {A,,} and {pp}converge to 0. Suppose that one of the s2: (or one of the pis) is equul to 0. Assume for instance that
Then we can solve the system (9), (10).
238
[Ch. 7, 0 7.3.
TWO-PERSON ZERO-SUM GAMES
We make an arbitrary choice for the functions w, for n a no and v,, for n zs no+ 1, since, in this case, eqs. (9) and ,(lo) can be written 0 = 0. This follows from the fact that, if A,, = 0, then eqs. (32(i)) and (32(ii)) imply that jsn-l ad = 0 and CiET; ,an a,,, = 0. Hence anj = 0 when j E 5 ' and j =sn- 1; a, = 0 when i E T and i =sn. For n = no, eq. (9) shows that
Let us set
niET;iIp njcs; Yj R the subset offunctions wp:nicT;isp-l X X njcs; jsp--lYj+R
(37)
UZ,= the subset of functions wp : such that tlw,ll llgll,
(38)
U2p-1= such that I I wpII
XiX
llgll.
Then inequalities (31(i) and (ii)) imply that (39)
{ (ii)(i) h2, =
y: maps Uzp+l into Uap, hzp-1 = 4; maps UaPinto U2,-1.
Inequalities (36(i) and (ii)) show that the maps h,, satisfy (40)
II Md)- M u 3 I I
an
I 14- u ~ I I ,
jsp-l
+
Ch. 7, Q 7.31 where (41)
239
ITERATED GAMES
u2p =-A,+, ., u2p-1.= PP
nZl
-.P P AP
ny=l
u, = AP+, and T]i$' ui = ,up+l.Hence aiconverges We have that to 0 because the sequences {A,} and {p,} converge to 0. Now, any sequence {u,} of elements u, E U, satisfying
(42)
un = hn(un+S
defines a solution of the system (9), (10) by setting up = u2p-1and wp = u2,.
Such a sequence {u,} exists as shown by the following lemma.
7.3.5. A lemma on successive approximations
-
Lemma 2. Let {U,} be a sequence of complete uniformly bounded metric spaces and let {h,} be a sequence of maps h, : U,,, U, satisfying (43)
{ (ii)(i) nzl
d(h,(x), hn(y)) u f 1 4 x Y, ) Vx, Y E Un+l, u i = 0.
Then the exists a unique sequence {Z,}, satisfying
(44)
{
(i) 2, E un, (ii) Z, = hn(Zfl+l).
Proof. I f n a p , we write
(45)
K]: = closure (hpohp+lo . . . 0hn[U,,+1])c Up.
Let 6 be a common upper bound of the diameters of the subsets U,. Then we can check that (46)
diameter (K;) ==
(fi ui)S. I=p
Thus {Ki},,3p is a decreasing sequence of closed subsets of Up whose diameters converge to 0. Since Upis complete, there exists Zp E Upsuch that (47)
nK]:
=
{xg}.
n=p
Therefore, since Zp+l E K;+' whenever n a p+ 1 , (48)
V n Z= p + 1, hp(Xp+l)E hP(K]:+l) c K$.
240
TWO-PERSON ZERO-SUM GAMES
[Ch. 7,
6 7.3
Hence hp(Zp+l)belongs to the intersection of the decreasingsequence {Kz}nsp+l, which is equal to {Zp?,). Thus h,,(Zp+l) = Xp for all p, i.e. there exists a sequence (5)satisfying (44). It is unique because, if f i p }is another sequence satisfying (M),then if p < q, d(X,,jjq) rg a,) d(Zq,jjq) == a,)& Letting q tend to we deduce that d(Zp,jjp) = 0 and thus that Zp = jjp. 0
(m$
(nzj
00,
7.3.6.Proof of existence of saddle decision rules Proof of Theorem 3. Note that, if X and Y are compact and g is continuous, then we can take E = 0 in inequalities (11) and (12). Since a = 0, inequalities (19) and (23) with E = 0 imply that
These inequalities show that the pair point of the game. 0
{C,D} constructed above is a saddle-
CHAPTER 8
THE FUNDAMENTAL ECONOMIC MODEL: WALRAS EQUILIBRIA This chapter is devoted to the description of a fundamental economic model. We define Walras and pre-Walras equilibria and prove several existence theorems (using Ky-Fan Theorem 7.1.3). Briefly, the problem is this. There are n consumers (labelled i = 1,2, . ..,n) who have to choose xi E R' c R' such that the sum C;=l x i of their consumptions belongs to a given subset Y c R' of available commodities. They use a selection mechanism called "demand correspondence" which associates with any price p E R" and income r E R a subset D,(p, r) of the budget set Bi(p, r ) = = {x E R' such that (p, x ) e I } . When the total income r ( p ) = supucr (p, y ) is appropriated by allocating to each consumer i an income ri(p)such that ri(p) = r(p) for all p, the problem arises as to whether it is possible to find a price and consumptions F' such that (i) 2' E Di(p, ri(p)), i = 1, 2, . . ., n. (ii) C;=l-i?E Y .
xzl
{
When these conditions are satisfied, we shall say that p and 2,(i = I, 2, . ..,n) constitute a Walras pre-equilibrium We shall prove an existence theorem in Section 8.2. The main assumption is that the demand correspondences are upper semi-continuous. In Section 8.3 we study the case when the demand correspondences are determined by the minimization of a loss function x t + f ( x , p ) (which can depend upon the prkep). Thus
.
Di(P, r) = { x E M p , r) such thatfi(x, p ) = minyc~(p.,)fi:O1,PI}. Finally, in Section 8.4, we introduce explicitly rn producers (labelled j = 1,2, . . .,m)into the model. These are described by their cost functions gj : R' R. An existence theorem for a suitably defined Walras equilibrium is proved. These existence theorems are deduced from a general result about the surjectivity of correspondences which follows directly from the Ky-Fan theorem. The result concerns an upper semi-continuous correspondence S mapping elements -+
18
241
242
THE FUNDAMENTAL ECONChiIC MODEL
[Ch. 8 , s 8.1
p of a convex compact subset P c U*onto convex compact subsets S(p) c U which satisfy the general "Walras law" v p E p , O"S(P),P) = inf (P,Y ) 4 0. Y €S(PI
The conclusion is that the problem of finding
z E s(jj)n-p+ admits a solution {Z, jj}.
8.1. Description of the model In this section we define the model and the Walras and pre-Walras equilibria. We describe several examples of appropriations of the subset of available commodities. In the case of quadratic demand fmctions, we prove the existence of a Walras equilibrium of an exchange economy. 8.1.1. The subset of available commodities
Consider the commodity space R'. We begin the. description of an economy with (1)
the subset Y c R' of available commodities.
Consider a set N = (1, . . ., n } of n consumers (households) denoted by i = 1, ..., n. With each consumer i f N we associate (2)
its "consumption subset" R' c R', which is assumed once and for all to be convex, closed and bounded from below.
The basic problem is to allocate the available commodities y E Y among the consumers. An "allocation"among consumers in N is an element x = {XI, . . .,2) belonging to RN = R' such that the corresponding aggregate consumption Cy=lx' is an available commodity. We denote by X ( N ) c RN the subset of allocations defined by
n7xl
(3)
X ( N ) = { x E R such that
&N
xi E Y } .
The upper support function of Y will be denoted by (4)
r(p> = a"(Y; P) = SUP b y ) . Jt€ y
Ch. 8, 0 8,1]
DESCRIPTION OF THE MODEL
243
We interpret r(p) as the “pro$t fmction” of the economy, where p E R” is understood as a price (which associates with any y E Y its value ( p , y ) E R). We denote the barrier cone of Y byP(Y”) = { p E R’* such that r(p) -= } (see Definition 1.4.1). In most examples, we assume the existence of
+
(5)
a non-zero vector 5 E Int( -P,(Y))
CQ
which we call the numtraire.
In such a case, the cone P ( Y x ) is spanned by the subset
(6)
P = { p E P(Y”) such that (p, 5 ) = 1) of normalized prices.
Let us recall that in this case, Theorem 1.4.3 implies that
(7)
P is a convex compact subset of R’*
and that
(8)
Pf = -P,(Y)
is the negative of the recession cone.
Recall that r is lower semi-continuous, convex and positively homogeneous. The property that
(9)
Y is closed and convex and satisfies Y = Y+P,(Y)
implies that Y is characterized by its maximum profit function, i.e. (10)
Y = {y E R’ such that (p, y) -z r ( p ) for all p E P}.
If (1 1)
O E Y,
then r is a non-negative function on P.
8.1.2. Appropriation of the economy In order to devise a decentralized mechanism which allocates an available commodity y E Y , we shall assume that the subset Y of available commodities is “appropriated”. This means that consumers i E N are endowed with subsets Y(i) of commodities such that (12)
Y =
i€N
Y(i) where Y ( i )c R‘
In other words, any available commodity y = EiEN y‘ E Y is assumed to be appropriated in the sense that each consumer i E N is entitled to an available commodity yi E Y(i). .8*
244
[Ch. 8 , s 8.1
THE FUNDAMENTAL ECONOMIC MODEL
We shall say that Y(i) is the initial endowment of available commodities. Such an appropriation of Y allows the profit function to be shared among the consumers, i.e.
(13)
vP E p,
c4P) n
r(p) =
I =1
where (14)
ri(p) = ~ + ( Y ( i ) ; p=)
SUP
Y EWl
( P ,Y )
denotes the maximum profit that consumer i obtains from his initial endowment Y(i). This profit I-,&) is used as income. If we define the budget set Bi(p, r) of consumer i as
Bi(p, r) = {x E Risuch that ( p , x ) =Sr } (15) the budget set of player i defined by the appropriation Y = Br(P9
m)).
xtl Y(i) is
8.1.3. Demand cwresponciences
Once the the set of available commodities has been appropriated, each consumer knows his budget set Bi(p,ri(p))when the price p prevails. We shall assume that each consumer i devises a selection procedure which enables him to consume a good in a subset Di(p, r) of his budget set B,(p, r). Definition 1. We shall say that a correspondence Di :P X R correspondence” if (16)
-
R’ is a “demand
V p E P,Vr E R, Di(p, r) c Bi(p, 4-
(We shall construct examples of demand correspondences later (see Section 8.3 for instance).) An economy is therefore defined by the items
(17)
{R*,Y ( 0 , D*}IEN.
We can now state the basic problem. If we assume that (18)
1
(i) Y = & Y(i) is appropriated, (ii) each consumer i chooses a commodity according to his demand correspondence Di,
is it possible to find a price jj such that there exist commodities 2;satisfying (19)
{ (ii)(i) V i E N,2‘ f‘Y E &N
E
Di(jj, rdj)),
Ch. 8, 0 8.11
245
DESCRIPTION OF THE MODEL
8.1.4. Walras equilibrium
.
Definition 2, We shall say that a pair {Z, jj} = {Zl,. . ,Z", P } E R N X P satisfying (19) is a Walras pre-equilibrium. We call Z = {Zl,. . ., Z"} a Walras allocation a n d p a Walras price. We shall say that {Z, jj} is a Walras equilibrium if, in addition, (20)
v i E N, ( P , 2,)
= ri(F).
8.1.5. Examples of subsets of available commodities and of appropriations
Example 1. The simplest example is when
(21)
Y = W-R;
is the set of commodities less than or equal to an initial holding w. If each consumer i is assumed to hold an initial endowment wi of goods such that w = wi, then the appropriation of Y is defined by
(22)
Y=
C Y(i)
where Y(i) = w'-R$.
&N
The functions r(p) and r,(p) are equal to
(23)
rdp) = ( P , w3, r ( p ) = (P,w).
Example 2. Let us consider M firmsj , described by their production subsets Zj. If w is the initial holding, then
(24)
Y = W+
m
C Zj-R:.
j=l
The usual way to appropriate Y is to assume that n
(25)
w = C d I-1
is the sum of initial endowments and that each consumer i owns a fraction 0; E [0, 11 of firm j and shares to that extent in the production. The system 6 = {s',> of shares satisfies obviously
(26)
vj = I,
..., m, C Oj = 1. i€N
In other words, Y = ~S-Cirn,~ 2j-R:
is appropriated in the following way
246
THE FUNDAMENTAL ECONOMIC MODEL
[Ch. 8, 0 8.1
The profit functions r(p) and ri(p) are defined by
{ (ii)(i) rdp) r(p)
(28)
=
(P,wf)+& Ojo*(Zj; p ) , ) - a+(Zj; t ~ p). ~ = ~
= (p, ~
The income of consumer i derives from two sources, i.e. the sale of his initial endowment wi and the shares O~*(Zj,p)held in the profits of firmsj.
Proposition 1. Let {Z,? } be a Walras equilibrium and let jj = CIcN 2 be the "aggregatedemand vector". Thenfor any allocation {Zl, . . .,i"} of theproduction satisfying m
we have that (30)
V j = 1,
.. ., m,
{p,9) = max (F, z ) ZEZl
i.e. Zi maximizes the pro$t over the production set Zj when the Walras price @ prevails.
xicN
Proof. Since j = Zi belongs to Y, there exist 9 E Zj for all j = 1, . . ., m such that (29) holds. Hence
( p , w) + C (F, 9 ) s r(j). n
s
j=1
Therefore, since r(p) = ( j ,w ) + ~ ~ a*(Zj, ! = ~ j ) , we obtain that
2 ( ( p , 9)-o"(Zj;p)) = 0. m
j=l
But ( p , Zj)-a"(Z';p) all j. 0
=s
0 for all j. We deduce that (@, ,ifi)= a*(Zi; ji) for
*8.1.6. Example; Quadratic demand functions
We take R' = R' for all consumers i. Consider the case where the demand correspondences Di(p,r)are the "quadratic demand functions" Di defined by
Di(p, r) = ul- (P,4 - r J-lp ( P . J+P)
ifp # =0
Ch. 8, Q 8.11 where
(32)
247
DESCRIPTION OF THE MODEL
{ (ii)(i)
ui E R' is a consumption objective, J E R(R! R'*)is a self-adjoint positive definite linear operator.
(Recall that Di(p, r) minimizes the distance I I x- ui I I to the objective ui subject to the budget constraint (p, x) = r. See Proposition 2.3.1.) Definition (3 1) satisfies the requirements for a demand function since
(33)
Vp
z 0, (p,Di(p,r)) = r-
Consider the case of an exchange economy, where
(34) We regard
(35)
Y ( i ) = { w i } ; Y = { w } with w =
{ (ii)(i) v i
n
C wi.
i=l
= ui- wi as the "net demand" of consumer i,
v = '&N
vi =
NU^- CiENwiasthe"aggregatenetdemand".
Proposition 2 . Suppose that the aggregate net demand v a Walras equilibrium (2,p ) satisfying (36)
{ (ii)(i)
f
0, then there exists
Vi, Xi = Di(p, ( p , wi)), C i E N 2=
w,
where X = {?I, ...,?'} is unique and p is unique up to the multiplication by a scalar a # 0. The quantities and p are given by
(37)
(i) 2 = ui-((Jv, vi)/(Jv, v)) v, (ii)p = a h , a # 0.
Proof. Since w = C i E N D i ( p( p, , wi)) = ~ i E N u i ~ ( ( j.)/(p, j, J - l p ) ) ~ - l p ,we have that p = aJv. Thus Xi = Di(aJv, a(Jv, w')) = Di(Jv, (Jv, wi)) = ui-((Jv, vi)l(Jv, v)) v. 0
Since ((x, y ) ) = ( J x , y ) is a scalar product, we deduce from (37) that, for any player i,
where the Hilbertian cosine is defined by cos (x, y ) = ( ( x 9 y ) ) E [ - 1 ,
llxll llvll
+1].
248
THE FUNDAMENTAL ECONOMIC MODEL
If the entries of the matrix J are denoted by dk(l-s h, k ponent of jj is equal to (39)
jih
=u
1
C
4
[Ch: 8, 0 8.2
l), then the com-
ahkvk.
k=l
This implies the “law of supply and demand”: We have that dph/dvh= aahhz 0 since the entries dhof J are positive. Hence p,, increases with respect to the aggregate net demand vh and the aggregate objective of consumption u,, = = Cj=I~f,, while it decreases with respect to the aggregate endowment (supply) wh = Z i E N d h . I f 5 # 0,E E R‘plays the role of a “num6raire7’, i.e. the price is normalized by the condition ( p , E ) = 1. Then the Walras price p is unique and define. ‘ ’
8.2. Existence of a Walras equilibrium
In this section we prove the existence of a Walras equilibrium when the demand correspondences are upper semi-continuous. For this purpose we require an abstract theorem on the surjectivity of correspondences. 8.2.1. Existence of a Wdras pre-equilibrium
Theorem 1. Suppose that the consumption sets R‘ satisfy
R‘ is closed, convex and bounded below that the subset Y of available commodities satisfies (1)
(2) and
Y is closed, convex and -R$
(3)
3 w E Y such that (Y-
W)
n:R
is its recession cone, = {0}
be a numkraire and P = { p E R: such that ( p , E ) = 1). Suppose Let E E that the demand correspondences D’ and the incomefunction ri satisfy (4)
-
Q i E N, p E P D,(p, ri(p)) is upper-semi-continuous with non-empty closed convex images.
Then there exists a Walras pre-equilibrium {T, p } E RT X P.
Ch. 8,
5 8-21
249
EXISTENCE OF A WALRAS EQUILIBRIUM
Remark. Assumption (3) is, by Proposition 3.2.7, a sufficient condition for the (5)
{xl,
..., P , y } E RNXYk--cCx'-yER'tobeproper i W
We shall actually prove that Theorem 1 holds when assumption (3) is replaced by assumption (5).
Proof. It is clear that p is a Walras price if and only if 0 E S(p), where (6)
S(P)=
C Di(P, ~,(P))-Y*
i€N
Since the subsets Di(p,ri(p)) and Y are closed, we deduce from ( 5 ) that S(p) is closed. It is convex (being the sum of convex subsets) and satisfies S(p) = = S(p)+R: since Y = Y-R: . Thus property ( 5 ) implies that:
(7)
-
V p E P, S ( p ) = S(p)+R: is closed and convex.
Furthermore, the upper semi-continuityof the correspondencep Di(p,r,(p)) implies that the functionsp !-+ aX(Di(p,ri(p)),q ) are upper semi-continuous (see Proposition 2.5.1). Therefore,
(8)
(i) V q E R'*,p I-+ c"(S(p); q) is upper semi-continuous, (ii) S(p) is non-empty closed, convex and satisfiesS@) = S(p)+R:.
Let p E P and z be any point of S(p). 2--y where x' E Di(p, ri(p)) and y E Y, we deduce that Since z -= (P,2 ) 6 C i c N ri(P)-(p, Y ) ~ ( P ) - ( P , Y). Then
xicN
Therefore, we have proved that
(9)
VP E p , ab(S(p>,p ) =s 0.
Finally, since 5 E implies that (10)
and -R!+ is the recession cone of Y, Theorem 1.4.3
P = { p E RT such that (p, E ) = 1) is convex and compact.
The existence of p E P such that 0 E S(p) follows from the following abstract existence theorem.
250
THE FUNDAMENTAL ECONOMIC MODEL
[Ch. 8, 8 8.2
8.2.2. Surjectivity of correspondences: the Debreu-Gale-Nikaiido theorem Theorem 2. Let U and U* be two paired spaces. Suppose that P is a convex compact subset of U* which does not contain 0
-
and Pi c U is its polar cone. U be a correspondencesatisfying Let S :P .(12)
1
(i) V q E u*, p F-+c r " ( ~ ( p ) q; ) is upper semi-continuousl, (ii) V p E P, S(p)+P+ is non-empty, closed and convex, (iii) V P E p , ab(S(p);p ) = infy&P, Y>.s 0.
Then there exists p E P such that (13)
S ( p ) n -P+ # 0 (i.e., 0 E S ( p ) + P + ) .
Proof. We introduce the function Q, defined on P X P by (14) d P , 4 ) = .*(S(P), 4 ) . This functiod is concave with respect to q, lower semi-continuous with respect to p (since the correspondence p S(p) is upper hemi-continuous) and satisfies p(p, p ) s 0 for all p E P. Since P is convex and compact, the Ky-Fan theorem implies the existence of p E P such that
-.
(15) vq 6 p , ,(seeTheorem 6.1.3).
d F , 4 ) = a*(S(F),4) 6 0
This implies that 0 E %c ( S ( p ) + P + ) = S(ji)+P+ (since S(p)+P+ is closed and convex). This means that S ( p )n -P+ = 0.0
*Remark. We can relax the assumption that P is compact and assume the following property instead. (16)
There exists 40 E P such that, for any I, the subset of elements p E P such that S ( p ) intersects the half-space H($,A) = {z E U such that (qo, z ) =sA } is relatively compact.
Proposition 1. Suppose that
(17)
-
P is closed convex subset of Ub
.and that S :P U is a correspondence satisfying assumption (12) of Theorem 2 and property (16). Then there exists a solution p of (13). W e recall that such a correspondence S satisfying (12(i)) is said to be "upper hemicontinuous" (see Definition 2.5.2.).
Ch. 8, 3 8.31
DEMAND CORRESPONDENCES
25 1
Proof. Property (16) amounts to saying that the function p i-+ q(p,q") = - infzcs~,,l(qo,z) is lower semi-compact, since the subset of elements p E P such that S(p)nH(qO, A) # 0 is just the subset { p a P such that ab(S(p),qo) = ~ ( pqo) , =S A}. Furthermore, Q) is lower semi-continuous with respect to p and concave with respect to 4. Therefore, Theorem 6.1.2 implies that there exists f P such that
This implies that 0 E S(jj)+P+. 0 8.3. Demand correspondences defined by loss functions
In this section we investigate the case when the demand correspondences are defined by
D i b , r) = {X E Bi(p, r) such thatfi(x, p ) = m i n y ~ ~r ~) f p; 4, ~ PI). , To prove the upper semi-continuity of such a demand correspondence, we need the compactness of the consumption sets R'. Therefore, we have to "compactify" the economy. For this purpose we introduce an assumption implying 2 - y is proper. Hence we construct a new that the map{xl, . . .,Y?, y } ++ economy in which the consumption subsets Ri are compact and the assumptions of the existence Theorem 2.1 hold. We check that these two economies are equivalent in the sense that their subsets of equilibria coincide.
E=l
8.3.1. Statement of the existence theorem
Consider the demand correspondences Di obtained by assuming that consumer i chooses a commodity in the budget set Bi by achieving his minimal loss. Specifically, we assume that each consumer chooses a commodity according to a loss function (1)
fi
: R'XP ++R
which can be indexed by the price prevailing in the economy. Then we define DI by (2)
Di(p, r) = {x E Bi(p, r) such thatfi(x, p ) = min,,B,(,,
d ( V , PI)-
THE F U h ’ D M N T A L ECONOMIC MODEL
252
[Ch. 8, 8 8.3
In such a situation, we shall summarize the features describing the economy by
(3)
{ R’, Y(i),f i } i E N
since demand correspondences Di depend upon loss functions A. We begin by stating an existence theorem for a Walras equilibrium.
Theorem 1. Suppose that (4)
the consumption sets R‘ are closed convex and bounded below
that (5)
the initial endowments Y(i) are closed, convex and -R!+ is their recession cone.
and that (6)
vi,
0 E Ri- Int Y(i).
Suppose also that (7)
the map {x,y } E RN X Y th C x‘- y E Rl is proper i EN
and that the lossfunctions-4 satisfy (8)
(i) ‘dp E xi t-+h(xi,p ) convex { (ii)fi continuous on R’XP. P,
is
is
Suppose furthermore that the loss functions f satisfv the non-satiation property (9)
E and ’dxi [ V-=pmi, PI. P
E Ri there exists yi E Ri such that fi(yi, p)
Then there exists a Walras equilibrium {Z, p}. The proof of this theorem Will follow from several preliminary results. To apply Theorem 2.1, we first need to investigate under what conditions the demand correspondences Di are upper semi-continuous. We will see that this requires compactness properties (i.e. that Ri be compact and that Y(i) = = Yo(i)-R: where Yo(i) is compact). We therefore need to reduce the initial economy to an equivalent “compact” economy (in the sense that the sets of Walras equilibria coincide). We begin by studying the budget and demand correspondences.
Ch. 8, 5 8.31
253
DEMAND CORRESPONDENCES
8.3.2. Upper semi-continuity of the demand correspondence
Proposition 1. Suppose that (i) the incomefunction riis continuous on P , (ii) V p E P,3 2 E Ri such that (p, 2)-= r'(p).
Then the budget correspondence p k-+ B(pi,ri(p))has non-empty closed convex images and is closed and lower semi-continuous. If we assumefurthermore that (i) R' is compact, (ii) the lossfunctionfi : R i X P then the demand correkpondencep
-
+-+R is continuous,
Di(p, ri(p)) is upper semi-continuous.
-
Proof. (a) The budget subsets Bi(p, ri(p))are non-empty by (lO(ii)). The graph Bi(p,ri(p))is closed since it is the subset of pairs of the correspondence p (p, x } satisfying(p, x)-ri(p) s 0 and p F-- ri(p) is upper semi-continuous by assumption (lO(i)). (b) The budget correspondence is lower semi-continuous because, if po E P , xo E Bi(po, ri(po))are fixed and N(x0) is a neighborhood of XO, we can construct Xe E N(xo) and a neighborhood N(p0) of PO such that xe E B,(p, ri(p)) for all
P E NPO).
For this construction, let 5' E R' be defined by assumption (lo@)). Then
(12)
(PO, 5')- r'(p0) = - co < 0.
Also, there exists 8 z 0 such that X, = 85'+(1 -O)xo belongs to N(xo). Since (PO,xo)--r,(p~)e 0, we deduce that (PO,xe)-ri(po) =s -coo. Take E = %COO. Since r, is lower semi-continuous, there exists a neighborhood N(p0) of po such that, for, all p E N ( ~ o ) , ( p , x>- ri(p)
(PO,
xe)--i(po)+
E
+( p
-PO,
xe)
4-i
~0
0
< 0.
This implies that x, belongs to B(p, r,(p)) whenever p E N(po). (c) If we assume furthermore that R' is compact, then B,(p, ri(p)) stays in the compact set R' and thus, is upper semi-continuous by Proposition 2.5.3. Thus it is continuous. Since the loss functionf' is continuous, we deduce from Theorem 2.5.3 that the demand correspondence is upper semi-continuous. 0 Note that the income function r, = u#(Y(i);.) is continuouson P whenever Y(i) = Yo(i)-R$ and Yo(i) is compact. We therefore obtain the following existence result.
254
[Ch. 8, !j 8.3
THE FUNDAMENTAL ECONOMIC MODEL
Proposition 2. Suppose that (13)
subsets Ri are compact and convex, {(ii)(i) theY(i)consumption Yo(i) - : R where Yo(i)are compact,
(14)
the loss functions with rmpect to xi,
=
: R‘XP
-+
R are continuous and convex
and (15)
V i , 0 E Ri-Int Y(i).
Then there exists a Walraspre-equilibrium, Proof.This is a straightforward consequence of Theorem 2.1 and Proposition 1, since (15) implies that thereexists 1’ E R‘ such that ( p , 5‘) < ri(p) for alI pEP.0 We shall say that an economy {Ri, Y(i),fi} satisfying (13) is a compact economy. The question arises as to whether any economy is equivalent to a compact economy in the sense that the subsets of Walras equilibria coincide. 8.3.3. Compactification of an economy
Suppose that the map {xl, Then the subset (16)
...,xn, y} E
n R‘XY n
I---
i=1
n
C 9-yE
R’is proper-
1=1
{ n R‘ such that C xi E Y n
X(N) = x E
i=1
i:l
is compact. Therefore, the projectionsn l X ( N )of X(N) onto the factor spaces Ri
are compact. If B is a closed ball of positive radius, then n,X(N)+B is also compact. We set
Ri = Ri n (niX(N)+B); p(i) = Po(i)-Ri;
yo(i) = Y(i)n (niX(N)+B); = Yn C ( n , X ( N ) + B ) .
P = C P(i) 10‘
i€N
Definition 1. Suppose that (7) holds. Then we shall say that the “compactifiedeconomy” of {R‘, Y(i),f;>}rcw We denote the “compactified budget sets” by &(p, r) = {x E Ri such that ( p , x ) 4 r} (18)
{R‘, P ( i ) , A } i E N is
Ch. 8, 0 8.31
DEMAND CORRESPONDENCES
255
The “cornpactified incomefunctions”
(20)
*
F i b ) = x (P(9;P ) =sr i b )
are continuous since yo(i)is compact for all i E N. It is clear that any Walras pre-equilibrium {X, p } E nI;=,R‘X P of the economy {X‘,Y(i),fi}ieNis a Walras pre-equilibrium of the compactified economy.
Proposition 3. Suppose that (7)holds and that (21)
the subsets Ri,Y(i)and the loss functions xi t--A(xi, p ) are convex.
Then any Walras equilibrium of the compactijied economy {& f(i), Walras equilibrium of the initial economy {Ri, Y(i),f i } i c N .
is a
Proof. Let {X1, ...,P,p } be a Walras equilibrium of the compactified economy. Then 2 “ E l ? c R f f o r a l l i a n d j ? = C ; = l i ? € f c Y . T h u s 2EX(N). This implies that (22)
V i E N , X i E niX(N).
Furthermore, we know that (p, J) = F ( f i ) . (a) We prove that (p, jj)=r(p3. If this is false, there exists y E Y such that @, y) > @, 3).Let ye = (1 - O ) J + Oy. Since Y is convex and j E Y,it follows that ye E Y. Since y = LEN# where # E Y(i)and 2‘ E zi X(N),2 + O ( f - X i ) E E Y(i) (n,X(N)+B) = Fo(i) for 8 small enough. Hence ye = LEN (xy’+ - 3)€ P.On the other hand, @, ye) = (1 - 0 ) (& A+ 8 @, y ) =- @, j j ) . This is a contradiction of the fact that @, j j ) maximizes (p, -) over P (i.e. that
Ow-
(P, V ) = P(P>).
(b) W e deduce that V i E N , ri(p) = Pi@). Since (p, 2‘) s P,(p) and ri(p), we obtain that = P(p) = r(P) = n
C (Pi(F)-r&j))
I= 1
= 0.
But &(p) e r,(p) for all i. This implies that
~i(p)= ri(p) for all i.
( p , J) =
256
THE FUNDAMENTAL ECONOMIC MODEL
[Ch. 8, Q 8.4
(c) Therefore 3' E b'(p, ri(j5)) = b i ( p , P,(p)). It remains to prove that Ti belongs to Di(p,ri@)). To prove this, suppose that 2' 4 D,(p; r,(p)). Then there exists xi E Di(p, ri(p))such that (23)
mi, a mi,F). -c
Therefore, for all 8 E 10, 1[, x: = (1- 8)
(P, 4 sz P i ( P 3 .
' + Ox' satisfies the budget constraint
For 0 small enough, xk belongs to f i i X ( N ) + B because I'E f i J ( N ) . Since 4 also belongs to R'. Therefore 4 E E Bi(p, ri(p))for 8 small enough. But, sinceJ;: is convex with respect to 2, we deduce the following contradiction
x' and 2' belong to the convex set R',
(24)
f i ( x ~p,)
(1 - e)fi(xi,PI+ ej;.(xi,
p) -= fi(zpi.
Proposition 4. Suppose that (7) holds. If the property (25)
V i E N , O€R'--IntY(i)
holds in the original economy, then it also holds in the compactified one, i.e. (26)
vi % N,
o E Ri-
Int l(i).
Proof. Property (25) means that, Vi, there exists 2' E R'n Int Y(i) c R'n Y(i). Thus Cy.=l2 ' E Y,i.e. 2 = (9, . . ,,P) E X(N).This implies that d € fiiX(N). Hence 2' E 8'. On the other hand, 3'+B(q) c Y ( i ) where B(q) is a ball of sufficiently small radius q. Since Z'+B(q) c z,t,X(N)+B if q is small enough, then 3,+B(q) c l(i), i.e. 15.' belongs to the interior of f ( i ) . 8.3.4. Proof of the existence of a Walras equilibrium Assumptions (4), (9,(8(i)) and (7) and Proposition 3 imply that we can replace the initial economy {R', Y(i), A} by the compactified economy {A,, p(i),fr}. Assumptions ( 6 ) and (8) and Proposition 2 imply the existence of a Walras pre-equilibrium of the compactified economy {Ai, f(ci),A}. Assumptions (8) and (9) imply that 5' E Di(2, F,(p)) satisfies (p, 2) = r,@) by Proposition 2.1.8. Hence there exists a Walras equilibrium of the compactzed economy, which is a Walras equilibrium of the initial economy by Proposition 3.
Ch. 8, § 8.41
ECONOMIES WITH PRODUCERS
257
In this section we introduce m producers (labelledj = 1,2, ...,m) who are described by proper lower semicontinuous convex cost functions g, defined on R’. The (maximum) profit functions are dehed by g*(,7) = SUP [(P. z)-g,WI. zERI
+c/”-
Hence the income r,(p) = @, .c>! 0; g;@) of i is the s u m of the value of his endowment w’ and his shares 0; g, (p) in the profits of firms j . We shall prove the existence of (3, .., 2,9, ., P,jj} such that 2 € D,(jj, r,(p3) for all i, 9maximizes the profit (jj, z)-g,(z) of the J” iirm for aU j and ZI196 s w+ xy-l 5’. For this purpose, we begin by “compactifyhg” the economy and apply the abstract Theorem 2.2 to the equivalent compctifkd economy.
-
..
8.4.1. Description of the model
We now introduce into the model m firms (or producers)j described by their cost functions
= {z E R’ such that &) whose domains 2.’ empty. For anyp € R’*,we regard
<
-} are assumed to be non-
as the profit fmction of producer j . In this case, if each consumer i is endowed witha vector w, the income h e tion r, of consumer i is d e h e d by (3)
rxp) = @, w‘)+
;qiF;(P)>
1-1
i.e. r,(p) is the sum of the value af his initial endowment w’ and his shares efg;cp) in the profits of firmsj . We summarize the description of an economy With producers in the notation I9
258
THE FUNDAMENTAL ECONOMIC MODEL
Definition 1. We shall say a triple {X, I, jj} E Walras pre-equilibrium if (4)
1
n:=l
[Ch.8, 0 8.4
R'X&'=1 ZjXR'* is a
. .
(i)Vi = 1, . ., n, j?' E Di(3, ri(p)), (ii) V j = 1, . ., n, 3 maximizes the profit ( p , . ) - g i ( . ) . (iii) j;i =sC Tj+w.
Eel
We shall say that the triple is a Walras equilibrium if, in addition, (5)
Vi,
(p,xi)= ( p , wi)+
c Oj g;(p). m
j=1
Remark. The usual models deal with the case where the cost function is (6)
gj = yz,.
the indicator of Zj
and thus, where (7)
gj(p) = a v j ; p).
Otherwise, gj can be regarded as special fixed costs, anti-pollution taxes, etc. 8.4.2. Statement of the existence theorem
Theorem 1. Suppose that (8)
the consumption sets R' are closed, convex and bounded from below,
and that the map
is proper. Suppose also that the cost functions g j satisfy (10)
{
(i) g j is a convex lower semi-continuousfunction defined on the closed set 27, ($0 = gj(0) = rnin,EZj gj(z).
that the endowments w' satisfv (11)
V i , there exists ii Ri such that 1' -=xW'
and that the lossfunctionsA satisfy
(i) Q p E P , xi ++h(xi, p ) is convex, (ii)fi is continuous on R'XP.
Ch. 8,
0
8.41
259
ECONOMIES WITH PRODUCERS
Assume also that the 1ossf;inctionsfr satisfy the non-satiationproperty (13)
V p f P and V x' E Ri, there exists yi E Ri such that fi(yf,p ) -=A(xi, p).
Then there exists a Walras equilibrium {%2,jj).
Remark. Recall that the compactness assumption (9) is satisfied if (14)
or, if (15)
the production sets Zj are bounded above
.
(i) V j = 1, . ., m, 21 is closed and 0 E Z', (ii) Z = Elll 21 is closed and convex, ( i i i ) Z n - Z = (0)andZnR: = (0).
(See Proposition 4.2.9.) Remark. Assumption (10) means that producing nothing is costless (g,(O) = 0) and that all costs g,(z) are non-negative (i.e. are true costs). Assumption (10) is satisfied in the case where there are no costs, i.e. in the case where
Remark. Before proving Theorem 1, we state several preliminary results. We construct a compactified economy and show that it has a Walras equilibrium which is also a Walras eqdibriwn of the .initial economy. 8.4.3. Compactification
Suppose that the map m
n
m
I
(9)
is proper. Then the subset (17)
E
fi R f X n such that i x ! m
i=l
'=1
i=1
m
C z'sw
J=l
is compact. Thereforewe can replace the initial consumptionsetsR'and produc tion sets 2' bytheprojections niUwand njUwonto R'and Zjrespectively. If B is a closed ball of positive radius, then qUW+B and n,Uw+ B are also compact. 19.
,260
"Efi FUNDAMENTAL EcoNohrIIC MODEL
[Ch. 8,8 8.4
Definition 2. Suppose that (9) holds. We shall say that (Ri,fi, wi, {ej}, g,}r, is the compacti$edeconomy of {Rl, 5,wi, {ef),gj}i, I.
Roposttaon 1. Suppose that (9)and (13) hold and that (23)
-
the cost fmctions gj and the loss j k d o n ~ the subsets Rl, XI fr(xf,p ) are convex D
O
,
Then any Walrasequilibrium of the compacfified economy is a Walrusequifibrim of the initid economy.
Proof. Let {Z, 2, j i } be a Walras equilibrium. Since
Ch. 8, 0 8.41
261
ECONOMIES WITH PRODUCERS
We prove that 2’ maxhbm the profit @, *)-gk-) over 2’(instead of %& If not, there exists d E 2’such that
(3)
(F, +g,(i9
-c
(P, 4-4&9.
For any 8 E 10, I], zi = (l-fl)%+8z‘belongs to 2’. For 8 small enough, 4 = z’+ e(9-9)belongs tox,Uw+B since 9 E rt,~,,,, Therefore 4 belongs to 2{ for e small enough. The convexity of g, implies that
(26)
(W9-g’m
-c
@,4)-g/<.c>.
This is a contratIiction of the fact that ~j maximizes the profit over 9. his implies in particular that for all j , = g;@) and thus, that r,(p3 = r@) for all i = 1, ..., n. We deduce that 13‘ E D&, &)) as in Proposition 2.3. Proposition 2. Suppose thut (9)h&. Then the compaczijkd cost functions are continuous. If we suppose that inficz,g~z) 0 for all j = 1, .., m. then inequalities
.
(27)
V P E R’*,
m) D 8 0 (PI -€
hold for all j . Proof. First, observe that
since g,(z) z- o for all z.since 2{ii compact, a+(& -1is a continuousfunction, Therefore, $ being a convex function dominated by a Continuous function, is a continuous function. Hence it is differentiable from the right and satisfies g;(p) eD&*(p)(p)+&*(O) (see Proposition 4.3.2). But $(O) 6 d(0)=-g,(O) = = 0. Thus (27) is satisfied. 0 Proposition 3. Suppose t h t (i) V i , there exists 2‘ E Ri such that f i e wi,
Then (29)
V j = 1,
...,m,V p E P,
V i =j ,
...,n, V p E P ,
and
(30)
g’(p)
0,
( p , 2) < r(p).
262
THE FUNDAMENTAL ECONOMIC MODEL
[Ch. 8, Q 8.4
Proof. Since E=l2'- '&0 e w, we have that, for all j = 1, . . .,m, 0 E njUwc 2;. Therefore, g,*(p) (p, 0)-gj(0) for all p. Since 9 << w', we deduce that, for all p E P , ( p , zi)
< ( p , wi) ==
( p , w')
+ c ejg;(p) m
j=
= qp).
1
8.4.4. Proof of the existence of a Walras equilibrium
Proposition 4. Suppose that (9)and (28) hold. .Then there exists a Walras preequilibrium of the cornpactiJed economy.
Proof. Indeed, p is a Walras pre-equilibrium price if and only if
Let S be the correspondence defined on the convex compact subset P by m
The subsets S ( p ) are non-empty, convex and compact. Since the compactified profit functions g; are continuous, the compactified income functions Pi are also continuous. But there exists 2' such that (p, 9)-c -= P,(p) for all p and 8' is compact. Hence thecompactified demand correspondences are upper semi-continuous (see Proposition 2.1). Since the functions .$ are continuous, the correspondences a$ are upper semi-continuous (see Proposition 4.1.8). Thus the correspondence S is upper semi-continuous. Finally,
Proof of Theorem 1. By Proposition 4, there exists a Walras pre-equilibrium of the compactified economy. It is actually a Walras equilibrium thanks to the non-satiation property (13) (see Proposition 2.5.8). Thus Proposition 1 implies that it is also a Walras equilibrium of the initial economy..
CHAPTER 9
NON-COOPERATIVE *PERSON GAMES
This chapter is devoted to the study of non-cooperative n-person games with and without constraints and their applications to the noncooperative Walras equilibria of an economic model with externalities. In passing, we prove the fundamental Brouwer and Kakutani fixed-point theorems (by means of the Ky-Fan theorem) and some selection theorems for fixed points. We begin by X‘ is a describing a game { X ( N ) , F } in strategic form, where X ( N ) c subset of multistrategies and F: X(N)I- R”is a multiloss operator. After definitions of the shadow minimum, the conservative values and multistrategies, we introduce the concept of a non-cooperative equilibrium x E X(N). For any player i E N,this satisfies
flel
where f;: is the ith component of F and n! is the projector from flyel X i onto j z i Xi. It “models” the individual stability of a non-cooperative equilibrium (i.e. no player can obtain a smaller loss by switching choices under the assumption that the remaining players make no change in their strategies). We prove the Nash theorem for the existence of a non-cooperative equilibrium very simply by means of the Ky-Fan theorem. We do not need the Ky-Fan theorem when the loss functions are quadratic. In this case, we are even able to exhibit explicit lbrmulas when X ( N ) is defined by linear equality constraints. We use these to obtain the Walras-Cournot equilibrium (thus generalizing the analysis of duopoly of Section 6.3). We then assume that the choice of the ith player is cmstrained by those of the remaining Xj,player iis constrained to choose his strategy in a players. Given n’x E Xiinto Xi.Hence we set Si(dx), where S, is a correspondence mapping generalize the concept of non-cooperative equilibrium by requiring that, for all piayers i, x‘ E ~ ~ ( z ’ and x ) that ~ ( x =) min9cs,cntxl;dyrdx,fi(y).We notice that this problem amounts to finding x satisfying for some function p:
n
njgl
fl,,,
(i) x E S(x) (ii) SUPJ€S(X)V b , 7 ) S Z 0 263
264
NON-COOPERATIVE n-PERSON GAMES
[Ch. 9 , § 9.1
i.e. a selection of afixedpoint x of a correspondence S mapping X into itself by a function g~mapping X X X into R. In particular, on taking q~ = 0, this problem reduces to finding a iixed point of S. Hence we shall prove the Brouwer and Kakutani theorems. We end this section by applying the existence theorem for a non-cooperative equilibrium to find a non-cooperative Walras equilibrium of the fundamental ecanomic model where the loss functions f i depend not only upon the choice of the ith consumer’s own consumption x’, but also on the consumptions of the remaining consumers (which are sometimes called “externalities”). 9.1. Existence of a non-cooperative equilibrium
We begin by describing n-person games in strategic form (or normal form) and by defining the conservative values and multistrategies. We then introduce non-cooperative equilibria and prove the Nash theorem. This asserts that, if the multistrategy subset X ( N ) c X i is convex and compact and the loss functions A are continuous and convex with respect to the ith projection x’ of x, then there exists a non-cooperative equilibrium. We also study the stability properties with respect to perturbations of the loss functions by linear forms. Finally, we characterize the non-cooperative equilibria as solutions of variational inequalities when the loss functions are convex and differentiable with respect to the ith projection x’ of x.
nrx1
9.1.1. Games described in strategicform
We denote by N = {1,2, ..., n } the set of n players. We shall describe a n-person game in the strategic (or normal)form by strategy sets and loss functions for each player. Specifically, we associate with each player i (1)
a strategy set X i
and denote by XN =
n Xi the set of multistrategies x n
1=1
= {xl,
...,x”).
When needed, we write 9= {gS1, ...,pen} E XN instead of using the notation x = {xl, ...,2‘)in order to avoid confusion between elements of XN and elements of a given factor X i of the product.
Ch. 9,§ 9.11
EXISTENCE OF A NON-COOPERATIVE EQUILIBRIUM
265
The context will make clear whether the notation x denotes a multistrategy or a strategy. We begin the description of the game by specifying a multistrategy set X(N)c X N of feasible multistrategies x.
(3)
-
Next, the behavior of the ith player will be described by a loss function R associating with any multistrategy x a real number f;(x) measuring his loss. He prefers multistrategies yielding smaller losses. It is useful to regard R” = RN = R 8s the mltiloss space. (4)
fr :X(N>
n
I W
-.
The behavior of the n players is then described by the multiloss operator F :X(N) R”defined by x E X(N)l--
(5)
e(x) = Vi(x),
.- .,fn(x)} E R“.
Definition 1. An n-person game {X(N), F} is described in the strategic form by a multiloss operator F mapping a multistrategy set X(N)c XNintothe multiloss space R”. We shall meet two important particular cases:
-
(a) when X ( N ) = XN is a product of strategy sets; (b) when the loss functions fr :X‘ R depend only upon strategies implemented by player i.
Definition 2. We shall say that the game is “bounded below” if, for any i, aI = inf,.xcN,fr(x) =- In this case, we shall say that the multiloss a = = {a1, .,a,,}is the “shadow minimum” of the game.
..
00.
Note that a game is bounded below if and only if
(9
F ( X ) c a+R;.
Ifa = F ( f ) belongs to F(X), the multistrategy f achieves the minimum of the loss function for each player i. In this case, ff is the natural solution concept. This is seldom the case and we have to investigate other solution concepts.
9.1.2. Conservative values and multistrategies
We shall denote by I’ = N- i the coalition adverse to player i. From the point of view of player i, the set X N of multistrategies is split as follows: X N = X 1 x X r where Xz= XI. (7)
n
J#i
266
NOM-COOPERATIVE n-PERSON GAMES
[Ch. 9 , s 9.1
If n! and lz’ denote the projectors from XN onto X i and X’,we set x’ = nix and x f = n’x. Player i will use the decomposition x = {d?x’} E X’XX’ for a multistrategy x E X”,i.e. he will distinguish the component xi E X‘on which he can act alone from the components !A E X’ over which he has no control. We regard
as the worst loss achieved by player i when he implements x‘ E Xi. Suppose that each player i has no information about the choice of the other players. If cautious, each player i will choose a strategy xi# which minimizes the worst loss function I; over X: . Definition 3. We shall say that the vector v* of components v#
(9)
v? = inf f i x ( x i ) dEX‘
is the “conservative value” of the game and that any strategy x+ such that (10)
V i E N , h#(x* ‘) =
vp
is a “conservativemultistrategy” (or a “minimax multistrategy”).
Remark. In fact, we use the conservative value as a “threat” functional, the understanding being that player i will reject any multistrategy x yielding a loss J;(x) larger then .:v The set of multistrategies accepted by all players (i.e. rejected by no players) is the set of x E X ( N ) such that F(x) =s8. Proposition 1. Suppose that each strategy set X i is a subset of a topological space U‘. If the extended loss functions J;.,x(N)are lower semi-continuous and lower semi-compact, then there exists a conservative multistrategy x* . Proof. This is left as an exercise. 0 9.1-3.Non-cooperative equilibria
To define conservative strategies, we assumed that each player can select a specific strategy xi independently of the remaining players. In doing so, he cannot take advantage of the choice xt of the complementary coalition 2. We shall consider another type of behavior in which any player i may vary his strategy as a function of the complementary coalition’s choice.
Ch. 9 , s 9.11
EXISTENCE OF A NON-COOPERATIVE EQUILIBRIUM
267
It is as though a player i announces his intention to play, forces the complementary coalition {f} to movejrst, and then responds. With such a rule, a multistrategy will be in “equilibrium” if no player i can obtain a lower loss by making an alternative choice under the assumption that the remaining players make no change in their strategies. In other words, given the complementary coalition’s choice 2,player i responds by playing a strategy x‘ which minimizes yi f;(y‘,x’), i.e.
-
Therefore, a multistrategy x will be an equilibrium if eqs. (11) hold for any player i. Definition 4. We shall say that a multistrategy x E X ( N ) is a “non-cooperative equilibrium” (or a “Nash-equilibrium”)if (12)
V i = 1, ..., n, h(x) = min f i ( y ) .
In other words, non-cooperative equilibria are the solutions of the equation (13)
F(x)-Fb(x)
=
0
where Ff : X ( N ) -, R“ associates with any x the multiloss Fb(X) = {.Ab(x),* * .,f,b(x)}. Note that F(x)-Fb(x) E : R for all x . 9.1.4. The Nash theorem
We introduce the function q~ : X ( N ) X X ( N ) (14)
c [m)-W, 81.
-
R defined by
n
V(X3 Y )
=
i=I
Proposition 2. Suppose that x E X ( N ) satis@ (15)
SUP
YEXO
d x , Y) =z 0.
Then x is a non-cooperative equilibrium. The converse is true when X ( N ) is a product of strategy sets.
268
NON-COOPERATMi
n-PERSON G
M
[Ch. 9,# 9.1
Proof. Let x f X(N) be a solution of (15) and let y f X(N)satisfy d’ = 9 Theny’=dfoi allj#iand~y=y’.Thuswecanwritetheinequality &,r) -G 0 as
c (fi(x)-fi(x*,
J d
x?)+(fXx)-AW, x?)
=fi(x)--f(y!
x? =f;:(x)-A(y)
=s 0
Thereforem) = i”fucx(N);nfu=#f;(Y) =Lb(x). This being true for each i, x is a non-cooperative equilibrium. Conversely, if X(N) is a product of strategy sets, we obtain y(x, y) =G 0 by adding the inequalities J;(x)-f(#, 2)=G 0 (whichhold by the very definition of a non cooperative equilibrium). 0 Proposition 2 suggests the use of the Ky-Fan theorem to prove the existence of a non-cooperativeequilibrium. Theorem 1 wash). Suppose that the mltistrategy set (16)
X ( m is a convex compact subset
and that,for each player i, the lossfunction (17)
fi is continuous
-,2)is convex for all xt E Xt.
Then there exists a non-cooperativeequilibrium. Proof. We check that the assumptionsof the Ky-Fan theorem 7.1.3 are satisfied. In the first place, X ( N ) is compact and convex. Secondly, tp is lower s e d continuous with respect to x, concave with respect to y and satisfies y ( y , y ) = = (f;(y)--f;(y))= 0. Thus there exists x E X ( N ) satisfying supucx(N)Hx, y ) 4 0. Thus x is a non-cooperative equilibrium by Proposition2.
z-t
+9.1.5.Stability
Suppose that X i is a subset of a topological vector space U! We introduce the “conjugate” functions defined by (18)
Vpi E
ui*,fiz(pi; x3 =
SUP
ntp#
[(pi,yi)--f,(y’,x?I
which are lower semi-continuous convex functions with respect to pi.
Ch. 9,g 9.11
EXISTENCE OF A NON-COOPERATIVE EQUILIBRIUM
269
Froposition 3. r f x E X ( N ) is a non-cooperative equilibrium, then (19)
V i E N, xi E a f ( 0 ;fi.
The converse is true when X(N)= XN and the extendedfunctions 9 E 0 t---
++A, xi(yi, 4 are convex and lower semi-continuous.
Proof. If x E X ( N ) is a noncooperative equilibrium, then
m o ;x3 =- m ' ,
x3 = - (41, x')+(q1, X')-fi(X) (O-ql, xi)+fl(qi; 8
i.e. x' C afr'(0,x'). Conversely, we use the fact that x i E afi'(0; x') if and only if x' minimizes the function y' 2)over X' (seeProposition 4.1.3). 0
+Av,
*9.1.6. Associated variational inequalities
Suppose that (20)
V x ' E Xr, the functions yi t--h(yi, able.
are convex and differenti-
Then any solution x E X(N) of (15) is a solution of the variational inequalities
(21)
(W, x-Y)
c (am), n
=
X'-Y?
0
1-1
where we denote the derivative of fr at x = {x', }'.
in the direction b',0) by
(see Section 4.2). Therefore, Proposition 4.2.2 implies the following result.
Proposition 4 . Suppose that (20)holdr. Any solution x E X(N)of the variatwd inequalities (21) is a non-cmperative equilibrium. The converse is true if X(N) is a product of strategy sets. We will use this proposition for proving the existence of a non-cooperative equilibrium in the case of quadratic loss functions.
270
[Ch. 9, 5 9.2
NON-COOPERATIVE n-PERSON GAMES
*9.2. Case of quadratic loss functions; applicationto Walras-Cournot equilibria
We do not have to use the Ky-Fan theorem to prove the existence of a noncooperative equilibrium when the loss functions
are quadratic (1 1 -1 Ii denotes the norm of a Hilbert space U',ui E U' are objectives, Mj E L?(Uj, U')).A non-cooperative equilibrium exists if we assume that the operators Mfsatisfy
for all x E and that
n Ui j= m
(where c =- 0)
1
X(N)is a closed convex subset of
nb, U'.
For this purpose, we use the Lions-Stampacchia theorem which states that a solution x E X of the variational inequalities (Gx-p, x - y ) e 0 for all y E X
exists if G E d ( V , V') is V-elliptic and X c V is a closed convex subset. We even obtain explicit formulas when X(N)is an affine subspace defined by linear equality constraints, i.e.
In this case, we can write x = M-1 (u-.J-lL*p) where
J x = {Jixi}i,J being the duality operator of
n U'J . n
i
i=1
We apply this result to compute the Walras-Cournot equilibrium of an (economywith n consumers and m producers. We assumethat consumerschoose their consumptions using their demand correspondences (which are assumed t o be quadratic) and that producers choose their production non-cooperaitively.
Ch. 9, 0 9.21
CASE OF QUADRATIC LOSS FUNCTIONS
27 1
9.2.1. Non-cooperative games with quadratic lossfunctions Suppose that the strategy sets Ui are Hilbert spaces for the scalar products ((xi,y i ) ) i = (Jixi, y i ) where J i E P(Ui, Ui*).
(1)
We consider the case when the loss functions f; are quadratic loss functions defined by
where (3)
Vj #
k, M j E Oe(Uj, U k ) (we let Mf be the identity).
Interpretation. By choosing x’, the adverse coalition transforms the original objective ui into a perturbed objective uiMjx’. The aim of each player i is to get as close as possible to this perturbed objective. 0
cjzi
Denote by (4)
M = {M/}j.k &UN, UN)
the “matrix” of operators Mi”. We shall make the following assumption about M. There exists a constant c =- 0 such that for any x E UN
c n
(5)
j , k=l
n
((MfX’7 X k ) ) k
Z= c
i=l
11 xi I I?
which is a kind of “consistency” for the perturbation operators Mf. Suppose now that the multistrategy set satisfies (6)
X(N)is a closed convex subset of UN.
Theorem 1. Suppose that assumptions (5) and (6) hold. Then there exists a unique non-cooperative equilibrium x E X(N)which is defined by the following variational inequalities: For a l l y E X ( N )
c I1
(7)
((Mfxj-
Ilk, X k - p ) ) k
=s0.
j , k=l
Proof. Inequalities (7) can be written n
(8)
( s ( x ) ,x - y ) =
k=l
(s(x)ky x k - y k ) =S 0
272
NON-COOPERATIVE
n-PERSON GAMES
[Ch. 9, 0 9.2
.where
is the derivative of& at x in the direction {yb 0). 0 We set
Assumption (5) amounts to saying that G iS V-elliptic, in the sense that (1 1)
Vx
E V (G(x), x ) 2 ~ l I x 1 1 ~where c =- 0.
9.2.2. Existence of solutions of variational inequalities
Theorem 1 follows from the following
Theorem 2 . (Lions-Stampacchia). Let G E 2 ( V , V*) be a V-elliptic operator and let K be a closed convex subset of V . Then there exists a unique solution x =.G-' ( p ) E K of the vGiational inequalities (12)
Vy
E K, (G(x)-P, x - y )
The map GZ1 : V* (13)
-
4
0.
K satisfies
II G2(p)-GZ1(q)II
-s
1
y IIp-qII*-
Proof. Suppose that x i s a solution of (12) and that y is a solution of (12) With p is replaced by q. Adding the inequalities (G(x)-p, x - y ) =XS 0 and (GO-q, y - x ) 4 O, we deduce from the V-ellipticity of G that (14)
cIlx-vll"~(G(x)-GQ, (P-4, X-Y>< llP-!?ll*
X - Y b
Ilx-vll-
On taking p = q, this implies that there exists at most one solution x = = GF1(p)of (12). If y = G$(q) exists, we have proved inequality (13). We now prove the existence of y = G&) by Writing (12) in the form Vy E
, K, ( ( X - X + A J - ~ ( G X - P )x-Y))
=S
0
Ch. 9 , § 9.21
273
CASE OF QUADRATIC LOSS FUNCTIONS
i.e. x = f(x-U-l(Gx-p)), where t denotes the projector of best approximation onto K. We consider the iteration (15)
X ~ + I=
xo E K.
t(Xn-AJ-l(Gxn-p));
Since the projector t satisfies (1 tx- tyll e 11 x - y l J , we obtain
We estimate II ( 1 - AJ-lG) I I in the following way.
since (Gx, x) c ( 1 = I1(l-ilJ-lG)II -4 Thus (18)
~ 1 1 ~ . Taking 1 = c/ll
IIxn+l-Xnll
we
deduce
that
8=
-= 1. -C
.
~llxn-xn-1ll
and the (Cauchy) sequence converges to an element x of the Hilbert space U. Since x t-+ t(x- AJ-l(Gx-p)) is continuous, we deduce that the limit x satis0 fies x = t(x-AJ-l(Gx-p)), i.e. that x = G&). Remark. We can replace the assumption of V-ellipticity by the weaker assumption of K-ellipticity
(19)
V x , y E K, (Gx-Gy,x-y)
2
cllx-yll2
where c
=- 0.
Proposition 1. (Lax-Milgram). Any V-elliptic operator G E B(V, V*) is an isomoprhism. Its inverse G-l E B(V*, V ) is also V-elliptic; Proof. Take K = V in the above theorem and note that G-l = G;l is nothing other than the inverse of G since the variational inequalities.reduce to (Gx-p, z ) = 0 for any z E V, i.e. Gx-p = 0. It is clear that G-l E B(V*, V) is Velliptic since we can write
-
= (Gx, 4 2 c II x I I (ell I G I12)1 I P Il2 (because llPll* = IIGxll* =s IIGII*lIxlI). 0
@, G-lp) 20
.NON-COOPERATIVE n-PERSON
274
[Ch. 9 , s 9.2
GAMES
9.2.3. Examples Proposition 2. Conside the case when X ( N ) = U N .If assumption (5) holds, then the unique non-cooperative equilibrium is equal to M-lu = G-l Ju.
Proof. The non-cooperative equilibrium x is the solution of Gx = Ju, i.e. of J M x = Ju since G = J M , where J f 2 ( U N ,UN*)is the duality mapping defined by J x = {Jlx’, . . ., J,,X”}. 0 Now, consider the case when
(20)
X ( N ) = X N is the product of n closed convex subsets Xi.
Denote by ti = txi the projector of best approximation from U i onto Xi. Proposition 3. Suppose that assumptions (5)and (20) hold. Then there exists a unique non-cooperative equilibrium x € X ( N ) which is defined by the following equations
Proof. This is left as an exercise. 0
cjzl
t,(t/M;xi) for solvRemark. The “natural” iteration defined by ing (21) does not necessarily converge. Theorem 2 shows that if ;1 = c / /I GI 12, the iteration dejned by
converges to the non-cooperative equilibrium. In other words, at the (n+ 1) th step, player i minimizes over X iits distance Mjx’,+(l-L) ( x k - d ) . to the objective d-1 9.2.4. Mulristrategy sets de$ned by linear constraints
We introduce n continuous linear operators Lj f A?(U’, V ) such that the operator
(23)
L =
2
~j
E -P(UN,V ) is surjective.
j= 1
We define the multistrategy set
(24)
X(N) =
{X
€ U N such that LX = x y = , L j X i = w}.
Ch. 9,$ 9.21
275
CASE OF QUADRATIC LOSS FUNCTIONS
Interpretation. The space V is regarded as a resource space, w E V as a scarce resource, Ljxj as the resource needed by player j to implement 2.Then X ( N ) is the set of multistrategies such that the sum Lx of resources Ljxj needed to implement them is equal to w. Then a non-cooperative equilibrium x is defined by
(i) Ljxj = w, (ii) V i E N , xi minimizes I I y f - (ui straint Liyi = w- Cjzt Ljxj.
(25)
c
jz
Mjx J )I I I under the con-
Theorem 3. Suppose that (5)and (25) hold. Then there exists a non-cooperative equilibrium defied by
(i) x = M-l(u- J-lL*p), (ii) where p = (LM-lJ-lL*)-l (LM'lu-
w).
Proof. This is analogous to the proof of Theorem 2.3.1. Note that y E X(N)if and only if y = x - z where z E Ker L.Then inequalities (7) imply that
(27)
(Gx- Ju, z ) = 0 for any z E Ker L,
i.e. Gx-Ju
E (Ker L)I
(28)
Gx- JU = -L*p,
= Im La. Thus there exists p E
Y* such that
i.e. such that x = G-lJu-G-lLp = M-lu-G-lLp since G-IJ = M-l. Applying L to both sides of this equation, we obtain that L x = w = L M - k -(LG-~L*)P. Since G is UN-elliptic, G-I is also U**-elliptic. On the oher hand, since L is surjective, L* is an isorr,orphism from V* onto its closed range in U". Thus LG-lL* is V*-elliptic since (LG-lL*q, q) = (G-lL*q, L'q) cllL*qll&
*C'l1~II~*.
Therefore LG-IL* is an isomorphism from V* onto V (by Proposition 1). We can write
I
p = (LG-lL*)-' (LM-lu- W ) = (LM-'J-'L*)-' x = M-lu- G-lL*p = M-lu- M-1 J-lL*P-
( Lit4 - l~ -w),
Example. Consider the case when (29) 20*
V
= R, L x =
c (p), n
j= 1
XI) where pj
E Uj*, w
=r E
R.
276
NON-COOPERATIVE II-PERSON GAMES
[Ch. 9,8 9.2
A noncooperative equilibrium is defined by (30)
[
(1) ==I (PJ, x J ) = WI (ii)ViEN,x'minimizes ~~~-(u'-~J#~M'~'~J)[~ under the constraint (pi, xi) = r- x J # i (pi, xi).
The solution x is defined by
Consider an economy With n consumers and m producers. The behavioral assumption of the model is that consumers choose their consumptionsaccording to the Walrasian model (by using their demand function) and that producers choose their production in a non-cooperative way. For simplicity, we use a model with quadratic demand functions (Section 8.1.6) and with production processes described by linear production operators. Specifically, we assume that the behavior of consumers i is described by their .
_.
.-.
-.
(ii) demand functions Di &lined by D,(p, r) =
= ui- ((P, u i ) - M ( p 9 J - ~ P ) J-4, ) (iii) initial endowment wi E R',
(32)
By Proposition 8.1.2, we obtain the equality m
n
Ex'= C w t + C z J I
(33)
f=1
j-1
i=1
if and only i f p is equal to
Normalize the price system by setting 6 = 1 and denote by v = E S 1 ( d the aggregate net demand when there is no production. Then
(35)
(
P(Z) = J V-
m
C J-1
.
a 5 5 9-21
CASE
OF QUADRATIC LUSS P U N ~ O N S
277
We now describe the behavior of producers j . If each producer j produces then the pro$t for j is
zj,
The loss functionfi = -gj is of quadratic type, i.e.
On the other hand, we assume that the input space is equal to V = Rk,that the production process of each producer j is described by (37)
a produdion operator LJ E 1 ( R f ,Rk)
and that (38)
JJO E Rk is the scarce resource.
We assume that producers will choose a non-cooperative equilibrium (i.e. Cownot equilibrium)dehed by (i) Zm& LjisI = 9, (ii) ~ j Zj ,maximizesits profit zi @ti?, . ,zj, .- .,P), zj) (39) under the constraint L@ = yo- &+j L&k.
1
-.
Proposition 4. Suppose that
(40)
L
rn
C Lj E B(R1,Rk)
= 1-1
is surjectiive.
Then there exists a unique Walras-Cournot equilibrium. Specijically, i f we set
the production zk are equal to
the equilibrium price is equal to
..
278
[Ch. 9,s 9.2
NON-COOPERATIVE a-PERSON GAMES
and the consumptions 2 are equal to
Proof. The matrix M = {MF} of operators MF E B(R', R') is defined by (45)
M; : - 1 (= identity), Mf =
1 i f j # k.
Assumption ( 5 ) is satisfied since we can write
The inverse M-l of M is defined by the matrix {Njk}j,k of operators iVf E f J(R', R') defined by (46)
2m -2 N; = ___ 1, N; =1 ififk. m+l mfl
Since the objectives x i are equal to &v, we have that (47)
(M-lx,,)j = v-
1 m v=m+ 1 m+ 1
2).
The matrix LM-IJ-lL" is defined by (48)
L&felJ-ILL+ =2
m
Theorem 3 implies that, if'we set (m+ 1 )
C LkJ-lLI - f rn
k=l
LjJ-'Li]-'
j , k=l
then we can write
The equilibrium price can then be written
(( 5 Lk)v-(m + 1)yO k=l
Ch. 9 , § 9.31
CONSTRAINED NON-COOPERATIVE GAMES
279
Therefore, the consumption xi is equal to m
9.3. Constrained non-cooperative games and fixed point theorems
This section deals with the fundamental Brouwer and Kakutani theorems and with the problem of selecting a Iked point of a correspondence. Curiously enough, such results in non-convex analysis are nothing other than existence theorems for equilibria of non-cooperative games with constraints. In the framework of game theory, the problem amounts to each player finding x' E &(xi) such thatfi(x) = minyics,cxr,f,(y', xi), where Si is a correspondence mapping each multistrategy x i E X c of the complementary coalition onto a constrained subset S,(x') of strategies of the ith player. We shall notice that this amounts to solving the following problem:
(where S is a correspondence from X into X)and
(**I where q~ is a function mapping X X X into R. If q~ = 0, then the problem reduces to (*) alone and a fixed point x of S exists under the assumptions of the Kakutani theorem. If S is the constant correspondence defined by S(x) = X,then the problem reduces to (**) alone and the Ky-Fan theorem implies the existence of a solution. We shall prove the existence of a solution of (*) and (**) under the assumptions of the Kakutani and Ky-Fan theorems and a consistency assumption between p and S. We continue the study of fixed points of a correspondence in Section 15.1. in the rest of the section we state and prove a theorem on the existence of an equilibrium of a constrained game. 9.3.1. Selection of a fixed-point
Theorem 1. Suppose that
(1)
X is a convex compact subset.
280
NON-COOPERATIVE
n-PERSON
GAMES
[Ch.9, 9.3
Suppose that a function y :XX X t-- R satisfis
fa
1
(i)V y , x t- ~ ( xy ), is lower semi-continuous, (ii) V x ,y I-+~ ( xy ), is concave, Ciii) SUPY€XycY, v) =G 0.
and that a correspondence S :X (3)
X satisfis
S is upper herni-continuous with non-empty closed convex images (see Definition 2.5.2).
Suppose also that thefmction the sense that (4)
-
(p and
the correspondence S are consistent in
thefunction x t-- a(x) = SUP,~S(~) p(x, y) is lower semi-continuous.
Then there exists x E X satisfying (5)
xE { (ii)( 0 SUPY€S(%) S(4,
T(X.9 Y ) =s0.
Remark. We recall that upper semi-continuouscorrespondencesare upper hemicontinuous (see Proposition 2.5.1).
Remark. Recall that (5(i)) amounts to saying that x is a “fied-point” of the correspondence S. Hence any solution of ( 5 ) can be regarded as a selection of a fixed point of S. Recall that Theorem 2.5.2 implies that assumption (4) is satisfied if we assume that R is lower semi-continuous, (i) cp :XX X (ii) S :X -c X is lower semi-continuous.
{
-
Hence we can state the following theorem. Theorem 2. Suppose that assumptions (I), (2), (3) and (6) are sati&ed. Then there exists a solution x of (5). Proof of Theorem 1. We have to prove that
(7)
3 x E X such that x E S(x) and a(x) = S U P , ~ S ( ~~ ( xy ), =s0.
Suppose that this conclusion is false. Then, for any x E X,either x 6 S(x) or a(x) z- 0. But x ff S(x) means that there exists p E U* such that (p, x)-U+(S(X),p ) > 0.
Ch. 9, 0 9.31
28 1
CONSTRAINED NON-COOPERATIVE GAMES
In other words, if (8)
{ (ii)(i) V,VO
= {x = {X
then (9)
xc
vou
u
PE
E X such that a(x) > 0}, E X such that ( p , x ) - u * ( ~ ( x ) , p ) > 0},
v p
U*
when the conclusion of the theorem is false. But, YOis open (by (4)) and the V> are open because the correspondence is upper hemiantinuous (see Definition 2.5.2). Since X is compact (by (I)), there exist n elements p, E U*such that (10)
Xc
vou rj VP'. 1=1
We now introduce a continuous partition of unity {PO,PI,.. .,Bn}subordinate to this covering. We consider the following function u defined on XX X by
We can apply the Ky-Fan theorem (see Theorem 7.1.3) to deduce that there. exists Z E X such that
To see this, note that aCy, y ) -G 0 since @, y) 0 by assumption (2@)).The functionsy +a(x, y) are concave because the functions y ~ ( xy), are concave by assumption (2(E))and the functions x I-+ a(x, y ) are lower semicontinuous because the functions x F-D~ ( xy), are lower semi-continuous and the functions Pi (i = 1, . . ., n) are continuous. A contradiction to (12) is obtained by proving that there exists an element. jj E X such that
+
(13)
u(Z,y3
=- 0.
We take (14)
E S(Z)i f a ( 4 =s0 and J E S(Z) satisfying cp(Z, fl- a ( 3 - e where e = +a(%) if a@) > 0.
j
282
NON-COOPERATIVE n-PERSON GAMES
[Ch. 9,p 9.3
Since X E X , either Po(?) =- 0 or Pi(I) =- 0 for at least one index i. We prove that (15)
{ (ii)(i)
Po(2) > 0 implies that p(3,y) z 0,
=- 0 implies that ( p i 2-7)
=- 0
from which it follows immediately that a(?, jj) > 0. Suppose that PO(?) > 0. This implies that j z E VO and thus, that a(?) Therefore, by (14),
=- 0.
4- %(I)= +a(%) =- 0.
~ ( 39), 3 ~ ( 2 ) -
Suppose that Pi(2) =- 0. This implies that I E Vp,, i.e. that (pi, 3 ) > u*(s(Z~, pi) Z= (pi,9)
(sinCe jj E S(X)). Thus (pi, 2 - y )
> 0.
Remark. We shall extend this theorem in Chapter 15 (see Theorems 15.2.1 and 15.2.2) by relaxing the continuity assumption (2(i)). 9.3.2. Equilibria of constrained non-cooperative games
Associate with the multistrategy set X ( N ) c XN the correspondences S mapping X’ into Xiwhich assign to the choice xt of strategies of players j # i the subset l(16)
Si(xq = { y i E X i such that {yi,x’} E X ( N ) }
of feasible strategies for the ith player. We can restate the definition of a non-cooperative equilibrium in the following way : (i) Vi E N , xi E Si(xf), (17) (ii) Vi E N,fi(xi,8 = min,r,S,(xP)f;.(yi, 2).
{
In this section, we shall study the existence of a solution x of problem (17) when the correspondences Si are no longer defined by a multistrategy set X(N). Consider n correspondences Si : X i - X‘.
-
Definition 1. A “constrainedn-person game” {S, F } is described by n correspondences Si : X’-+ Xiand a multiloss operator F :XN RN. A multistrategy x E XN is said to be a non-cooperative equilibrium if
Ch. 9, 8 9.31
CONSTRAINED NON-COOPERATIVE GAMES
283
Write
Proposition 1. A muftistrategy x E XN is a non-cooperative equilibrium if and onfy i f
Proof. This is analogous to the proof of Proposition 1.2. 0 We are now able to prove the following existence theorem.
Theorem 3 (Arrow-Debreu-Nash). Suppose that (22)
the strategy sets X i are convex and compact
and that the lossfunctions satisfy (23)
J;:is continuous andf;:(-,8is convexfor all x f E X!
Suppose also that the correspondences S, satisfy (24)
Si is a continuous correspondence ,from X iinto X i with non-empty
closed convex images.
Then there exists a non-cooperative equilibrium. Proof. It is clear that if we set X = XN and take q and S to be as defined by (19) and (20), assumptions (22), (23) and (24) of Theorem 3 imply assumptions (l), (2), (3) and (6) of Theorem 2. Thus there exists a solution x E XN of (21), which is a non-cooperative equilibrium by Proposition 1. 0 9.3.3. Fixed-point theorems Note that, if we take for S the constant correspondence S(x) = X , assumptions (3) and (4) of Theorem 1 are satisfied. We therefore recover the Ky-Fan theorem.
284
NONCOOPERATIYE
n-PERSON
GAMB
[Ch. 9,s 9.3
If we take (p = 0, assumptions (2) and (4) of Theorem 1 are satisfied. W e then obtain the Kakutani theorem.
Theorem 4 (Kakutani). Suppose that (1)
X is a convex compact subset
and that a correspondenceS :X (25)
-
X satisfis
S is upper hemi-continuous with non-empty closed convex imqges.
Then there exi%tsafixed point 2 of S. In the case where S is a function, we obtain the Brouwer theorem generalized by Schauder in the infinite dimensional case.
Theorem 5 (Brouwer-Schauder). Suppose that (1)
X is a convex compact subset
and that (26)
f is a continuous map from X into itself.
Then there exists a fixed point x o f f , i.e. a solution of x =f (x).
*Remark. We have deduced the Brouwer h e d point theorem from the Ky-Fan theorem. Actually they are equivalent. If we assume that (10) and (11) hold and that any continuous map D from X into X has a jixedpoint, then there exists 3 E X such that sup ~ ( 3y ,) c 0. The proof is as follows. By the LasIy theorem (Theorem 7.1.4), there exists Z E X such that suprcy &T ., y ) 4 supDcecx,x3 inf,,. (p(x, D(x)). Now. since any continuous map D from X into X has a fixed point 2 by assumption, then, ~ (D(2)) i , 4 @,2) e 0. Hence suprcy d3,y ) =s0. inf,c, ~ ( xD(x)) ,
In other words, we can either, as in our development, assume Ky-Fan theorem (via the Knaster-Kuratowski-Mazurkiewicz lemma) and deduce the Brouwer theorem or assume the Brouwer theorem and deduce the Ky-Fan theorem. Note that Knaster, Kuratowski and Mazurkiewicz proved the Brouwer theorem directly using their lemma (see Appendix B). Remark. We will give another proof of the Kakutani fixed point theorem in Section 15.1, as well as some other fixed point theorems (see Theorem 15.1.3).
Ch. 9,$ 9.41
285
NON-COOPERATIVE WALRAS EQUILIBRIA
9.4. Nonaoperative W h s equilibria
We devote this section to the fundamental economic model in which the loss functionsf i of the n consumers depend not only upon their own consumptions, but also on the consumptions of the other consumers. We are led to define a noncooperative Walras preequilibnum (3,.. .,P,p} by the following requirements
[
(i) 2‘ E Y, (ii) V i E N,f E 3&, r f i ) ) , (iii) v i E N,f l ( ~ ‘ 22; , F) = min9Eq(>.rm)flCY‘,F). $9
We shall prove the existence of such an equilibrium under assumptionsanalogous to the ones which imply the existence of a Walras equilibrium. To prove
such a result, we construct an (n+l)-player noncooperative game in which the (n+ 1)th-playeris the “market”. Its role is to choose a price p which minimizes p I--- r(p)-(p, EP1 2). We prove that any noncooperative equilibrium of such a game is a noncooperative Walras equilibrium of the economy. Then we check that the assumptions of the Arrow-Debreu-Nash theorem are satisfied. In the last part of this section, we introduce producers in the model and so extend the former results. 9.4.1. Description of the model
We consider in this section an economy {R‘, Y(0,&N
where the loss functions f ; :R N X P I-+ R depend upon the allocation x = = {xl, . ., 2’)instead of on the consumption x’ of the ith player only. Consider the budget correspondences
.
where
&(p, ri(p)) = { x E R‘ such that ( p , xi) r i b ) = SUP (P9Y). YE
no
4
ri(p)}
286
NON-COOPERATIVE n-PERSON GAMES
[Ch. 9,§ 9.4
We shall say that it is a non-cooperative Walras equilibrium if also
V iE N
(jj,?i}
= ri(p).
Theorem 8.2.1 on the existence of a Walras pre-equilibrium can be generalized in the following way.
Theorem 1. Suppose that (1)
the consumption sets Ri are closed, convex and bounded below.
Suppose that (2)
the initial endowments Y(i) are closed, convex and -Rrf is their recession cone
and (3)
V i , 0 E Xi-Int Y(i).
Supposefurther that the map (4)
{x,y} E R N X Y t-+
C 2-y
E R’
1€N
is proper and that the loss functionsf, : R N X P
-R satisfv
(i) V p E P = Mi, V x ‘ E R’, xi l--h(xi, ($5 is continuous on RNXP.
2,p ) is convex,
Finally, we require the non-satiation property
(6)
V i E N , V p E. P , Qx E RN, there exists yi E Risuch that hW, x’; p ) xf, p).
-=mi,
Given all these assumptions, there exists a non-cooperative Walras equilibrium. 9.4.2. Existence of a non-cooperative Walras equilibrium: the Arrow-Debra
theorem To prove this theorem, we shall construct an (n+ 1)-person constrained noncooperative game in which player 0 is called the “market” (or the “invisible hand”) and where the n other players are just the n consumers. Consider the set
Ch. 9, 9 9.41
NON-COOPERATIVE WALRAS EQUILIBRIA
287
of allocations and its projections z i X ( N ) onto the factor spaces R'. Let B denote any closed ball of positive radius. We introduce the following strategy sets (8)
{ Xi
fori=O = R i n ( 7 t i x ( N ) + B ) for i = 1, . . ., 12.
XO=P=Al
We consider the subsets (9)
Po(i) = Y ( i ) n ( z i X ( N ) + B ) ; P(i) = po(i)-R$
The compactified income functions are defined by n
(10)
Fi(p) = o*(f(i);
p)7
~ ( p=)
C Pi(p) i=1
and the compactified budget correspondences by (1 1)
Bi(p, Pi(p)) = {x E X i such that ( p , xi) =sFi(p)}.
We define the correspondences Si :X'(12)
{ (ii)(i)So(x) Si(xf;
Xi by
for i = 0, p) = Bi(p,r i ( p ) ) for i = 1,
=Xo =P
. . ., n.
Finally, the loss functions of the (n+ 1)-persongame are defined by (13)
(i)fO(x,p) xi> { (ii)fi(x, are the initial loss functions. = P ( P ) - ( P , xi,,
p)
To prove Theorem 1, we have to check that, any non-cooperative equilibrium of the constrained non-cooperative game is a non-cooperative Walras preequilibrium and that there exists such a non-cooperative equilibrium. Proposition 1. Suppose that the subsets R' and Y ( i ) are closed and convex, that the loss functions are convex with respect to xi and satisjy (6). Any non-cooperative equilibrium of the constrained (n I)-persongame defined by (8), (22) and (23) is a non-cooperative Walras equilibrium.
+
Converse&, any nun-cooperafive R'dras equilibrium is a nun-coopea five
equilibrium of the (n+ 1)-person game.
Proof. Let {p, X} E Xoxny=lX i be a non-cooperative equilibrium. Since 2 E @ ( p , Fi(jj)) and Zi minimizes A( .,2,p ) on B(p, Fi(p)),we deduce from (6) and Proposition 2.1.8 that (14)
(p,2)= F i ( j j )
=sri(p).
-288
NON-CooPERATIVE ?Z-PERSON GAMES
[Ch. 9,g 9.4
Summing these inequalities, we obtain that
This implies that
Hence f E X(N)and thus .(16)
V i E N, 2
E niX(N).
We prove that r,@) = P,@) for all i E N as in the h t two parts of the proof of Proposition 8.3.3. Now, since {p, Z} is a noncooperative equilibrium of the constrained (n+ 1)person game, we deduce that, for all i E N, (17)
fiW, 2,P ) =
min d u x h
Jot', 9,P),
G3)
Since B,(p, Pi@)) = B,(jj, r,(p3), we deduce that
as in the proof of the third part of Proposition 8.3.3. The proof of the converse statement is obiious. 0 Propition 2. Suppose that the assumptions of Theorem I hold. Then there exists a non-cooperative equilibrium of the constrained (n+1> person game defined by (8),(12) and (13). Proof. We shall check that the assumptions of Theorem 3.3 are satisfied. (a) The strategy set XO is convex and compact. The strategy sets X'= = R'n ( I ~ J ( N ) + B )are also compact since the map {x, y } 2 - y is proper on R NxY (in this case X(N)is compact, hence IZ,X(N)is compact and thus,It,X(N)+B is also compact). They are obviously convex. (b) When i E N,the loss functionfr is continuous and convex with respect to x' by assumption. The function fo is obviously convex with respect to p and continuous, since the compactified income functions P, are continuous (for they are the support functions of the compact sets fo(i>).
+
Ch. 9,
9.41
289
NON-COOPERATIVE WALRAS EQUILIBRIA
(c) The correspondences S, are continuous with non-empty convex compact images. This is obvious for i = 0. Let i E N , S,(.', p ) = B f ( p ,P,(p)). Since the assumptions of Proposition 8.3.2 are satisfied, the result also follows for i = 1, 2, ..., n. Thus there exists a non-cooperative equilibrium. 0 *9.4.3. Non-cooperative Walras equilibria of economies with producers
We introduce m producers described by their production sets Z' c R'. We consider an economy {Ri, f j , w', {$}, in which we assume that the loss functions
qf,/
depend upon {x, z, p } instead of x' and p . We introduce rn
rf:PXnZj--R I= 1
defined by (19)
ri(p, Z) = ( P , w')+
m
C @(;P, 4-
j= 1
Definition 2. We shall say that {Z, z-,I? } E ative Wdraspre-equilibrium" if (20)
{
R'X
n:
ZjX P isa nnon-cmpr-
z~~ +sm=l
(i) zi wf 3, (G) V i E N,Z' E Bi(p, ri(p, 5))and hi*, $3 3, = &$€B&, r&%@),,frot', 2 3 % $), (iii) V j E M , 9 maximizes the profit (p, -)on 2,.
We shall say that it is a "noncooperative Walras equilibrium" if also (21)
V i E N, (p, i?)= ri((W 2).
Theorem 2. Suppose that
(22)
the consumption sets R' are closed, convex and bounded below
and that the map (23) is proper. 21
{x, Z , U } E
n R'Xn ZjXRi n
rn
1-1
j-1
I-+
R
m
1-1
/=l
2 x i - C Zj+U
E R'
290
[Ch. 9,p 9.4
NON-COOPERATIVE n-PERSON GAMES
Suppose also that the production sets Zj satisfv (24)
Zj is closed and convex, contains 0
and that the endowments wi satisjy (25)
V i E N, there exists ZiE R' such that 4'
Finally, suppose that the lossfunctions satisfy (i) V p E P, Q x E RN,Vz E Z', xi l-+h(xi, x?,z , p ) is convex, (ii)fi is continuous on R ~ X ~ Y Z ~= X~ P , (iii)f, satisjies the non satiation property (6).
Then there exists a non-cooperative Walras equilibrium.
To prove this theorem, we shall construct an (n+m+ 1)-person constrained noncooperative game. We shall prove that any non-cooperative equilibrium of this game is a non-cooperative Walras equilibrium and that there exists such a non-cooperative equilibrium. Consider the subset (27)
{x, z } E
n It
i=l
m
R i X n Zjsuch that j-1
n
m
i-1
j=1
C xi- C zj==w
of allocations and its projections n$JW and njUwonto R' and Z respectivel! We defined the strategy sets by
(28)
XQ=P=Jp xi = R' n(aiuw+ B) Xj =Zjn(i%(niUwU{O})+B)
if i = 0, if 1 e i ~ n . ifn+l 4jen-t-m.
Write
and (30)
&(p, Pi(p, z)) = {x E X i such that ( p , x ) =sPi(p, z)}.
The correspondences S are defined by (31)
(i)SO(X,z, p ) = X o = MI if i = 0, (ii) Si(xr,z, p ) = Si(p , pi(p, 2)) if 1 6 i e n, (iii>Sj(x,z,?p) = X j i f n + l s j = s m + l .
Ch. 9,s 9.41
NON-COOPERATIVE WALRAS EQUILIBRIA
291
The lossfunctionsA are defied by
I
xi) for i = 0, (i)fO(x, 2, p ) = (p, w + xy!l zj(ii)&(x, z, p ) are the original loss functions for 1 =Si =S n, (iii)fi(x, z,p ) =- (p, zj) for n+ 1 ==je n+m.
Proposition 3. Suppose that the subsets R' and 2.'are closed and convex and that the functions fi are convex with respect to 2 and satisfy the non satiation property (6). Then any non-cooperative equilibrium of the (n+m+ 1)-person game defined by (28), (31) and (32) is a non-cooperative Walrus equilibrium.
Proof. Let {& Z, 2) E X o X ~ b l X ' X f l ~ , Xbei a non-cooperative equilibrium. We obtain that, for any producer j ,
(p,3) = max (a,)'2
(33)
B
ZfEXJ
0 (since 0 E S)
and thus, that m
(34)
Pi@,
3 = (p,w')+ C 8j(p, 5'). /= 1
Since Ti E B,(jj, Pi(jj)), we deduce that
Therefore
Hence (36)
Since
i=l
and that
21*
w+
c m
sj.
i-1
{ P , 1,Z} is a non-cooperative equilibrium, we know that
(37)
(38)
2" zi
trj = 1, .. ., m, Zj maximizes (p, -)over X i V i = 1, . . .,n, 1' E S,(p, n
( ~23), minimizes xi i-xi, 9,2, j j ) over ~ ~ (pi(jj, p ,23).
292
[Ch. 9 , § 9.4
NON-CooPERATIVE n-pBRsoN GAMES
The proofs of Propositions 8.3.3 and 8.4.1 imply that, in fact, (39)
V j = 1,
...,m,
2, maximjzes (p,
0 )
over 2,
and
W)
v i = 1, ...,n,
3 minimizes XIF--fdxf,2,5,~)over &(p, r@, z)).
Thus { j j , X, 2) is a noncooperative Walras equilibrium. 0 Proposition 4. Suppose that the assumptwm of Theorem 2 hold. Then there exists a non-cooperativeequilibrium of the (n m 1)-persongame deFed bY (24, (31) and (32)-
+ +
Proof. We shall check that the assumptions of Theorem 3.3 are satisfied. (a) The strategy sets P,X' and''A are clearlyconvex. The strategy set X o is compact and the strategy sets X' and XI are also compact since we assumed zi+u is proper. 2that the map {x, z, u } -I (b) The loss functions fo and fi are clearly continuous and the loss fundons I; are continuous by assumption. Obviously, fo is convex with~espectto p, ft with respect to z, and fi is convex with respect to xi by assumption. (c) The correspondencesSoand S, are constant. Since ( p , 29 -= ( p , d)6Fr(p,z) for all p E P,z E ng1Z ' , we deduce that the correspondencesS, are continuous from Proposition 8.3.1. Thus there exists a noncooperative equiiibrium. 0
z-l
CHAPTER 10
MAIN SOLUTION CONCEPTS OF COOPERATIVE GAMES
In this chapter we define cooperative games and introduce the main solution concepts studied in this book except for the concept of the Shapley value (see Chapter 11). We prove here only easy results. The di5cult results dealing with the existence of sukh solution concepts and the equivalence which hold among them will be proved in Chapter 11 in the particular case of games With sidepayments and in Chapter 12 in the general case. Consider a game {X(N), F) described in strategic form. We begin by considering the case when the players subordhte their interests to those of the whole set N of prayers to the extent that they cooperate in excluding all but collectively stable strutegies, ie. they exclude all but weak Pureto muItistrategies. This means that no other multistrategy yields a smaller loss to each player. For example, the players may use A = (A1, ..., A”) E A”to b‘pool” their loss functioasf,, i.e. they play a strategy x, which minimizes x I-+ on X(N). Then x, is a weak Pareto minimum. The converse is true if wz assume that X(N) and F are convex: Any weak Pareto strategy can then be implemznted by m b h i d n g a weighted loss function c;I1dyk-) for a convenient 1 E &’. This result also shows that the set of Pareto strategies is “too large”. Since our aim is to devise procedures yielding as small a subset of strategies as possible, the problem of selecting Pareto strategies arises. We have to set aside the concept of a “strong equilibrium”, which is both a noncooperative equilibrium and a Pareto minimum i.e. which satisfies both individual and collective stability because such a strong equilibrium exists only in exceptional cases. We therefore weaken the individual stability requirement by introducing the concept of a “threat functional” ti. This ~ssociateswith any loss function f, a value tf(A),which sets a maximum to the amount of loss that player i will accept for himself. We will say that a strategy f is an imputation if it is a weak Pareto minimum (i.e. it is not rejected by the whole set N of players) and if, for all players i, it satisfies h(Z) EG tfU)(i.e. it is not rejected by player i).
z&(-)
293
294
MAIN SOLUTION CONCEPTS OF COOPERATIVE GAMES
[Ch. 10
An example of an imputation is the “best compromise” K defined by
where,
Vi E N,
a=
(I-d)ai+dti(jJ
inf
max
and ai = infxcx(mf i ( x ) . However, the set of imputations can also be too large. In order to define a subset of the set of imputations, we assume that the behdvior of player i alters when he participates in a coalition A of players (i.e. a subset of the set N of players). We describe the behavior of a coalition A by a pair { X ( A ) ,FA} whsre X ( A ) represents the multistrategy set of the coalition A and FA :X ( A ) I-+ RA represents the multiloss operator of coalition A. The family { X ( A ) ,FA}Acd describes a “cooperative game”, which involves the description not only of the behavior of the whole set of players, but also the behavior of each coalition of players. We say that a coalition A rejects a multistrategy 2 E X ( N ) if it can find a multistrategy !x in its own multistrategy set X ( A ) which yields to each player i of A a smaller l o s s f p ( ~ ) . The core of the cooperative game is the subset of multistrategies 2 E X ( N ) which are not rejected by any coalition A . If we take ti(f;) = infxrcx((i),J;.(i)(x’) as threat functional, we see that the core is contained in the subset of imputations. When the core is not empty (this happens for a rather large class of games, as we shall see in Chapter 12), it is again too “large”. We can “shrink” the core by asking “more coalitions” to reject strategies. Hence we have to “enlarge the family of coalitions”, i.e. to embed it in the family of the so-called “jiuzzy coalitions”. A fuzzy coalition is an element z = {zl, . . ., zn} E [0, l]“, where the ith component z, E [0, 11 represents the rate ofparticipation of the ith player in the fuzzy coalition z. Clearly, any coalition A c N is identified with the , the rate of participation t: of player i in A is either 1 fuzzy coalition t Awhere when i E A or 0 when i B A . If we adequately define the behavior of fuzzy coalitions and, in particular, how a fuzzy coalition rejects a given multistrategy 2 E X(N), we can introduce the fuzzy core of the game. This is the subset of multistrategies 2 E X ( N ) which are no rejected by any fuzzy coalition z E [0,1]”. Is it possible to identify a useful subset of the fuzzy core? The answer is positive. We introduce the concept of a canonical cooperative equilibrium X E X ( N ) for which we require the existence of E Jn“such that, for any coalition A ,
C Xi(?)
i€A
e
inf
x A c x ( A ) IcA
XifA(xA).
Ch. lo,§ 10.11
PARETO STRATEGIES
295
We shall see in Chapter 12 that under quite reasonable assumptions, the fuzzy core and the set of equilibria coincide and that they are not empty. Throughout this chapter we study the example of the economic game constructed from an economy {Z?, Y ( i ) , A } l E N .
*10.1. Behavior of the whole set of players: Pareto strategies After defining Pareto strategies, we interpret elements A E R“*of the dual of the space of multilosses as “rates of transfer” of side-payments. Then, after noticing that any strategy which minimizes A x ( .)is a weak Pareto minimum we prove that the converse statement holds under convexity assumptions. We generalize this theorem in the case of an economic game. Under convenient assumptions, an allocation X = {Zl, .- .,Z”} E X ( N ) is a weak Pareto minimum if and only if there exist a rate of transfer 1 E R**and a price j j E R”*such that (i) V i E N, x i J ( Z 9 + ( P , Z i ) = min,ER [xrh(y)+(F,y)l, (ii) ELl (p, xi) = S U P ~ ~( PY , Y ) -
ZEl
10.1. I . Pareto strategies
Although a non-cooperative equilibrium is “individually” stable, it lacks “collective stability”. Indeed, in many instances, the players can find a multistrategy yielding to each player a smaller loss than a non-cooperative equilibrium. Thus, by adopting such a multistrategy, aNpIayers will improve upon a noncooperative multistrategy. Such a move requires a minimum of cooperation among the players because each player has nothing to lose in making the change. When this small amount of cooperation is assumed, it is natural to set aside any strategy which can be improved upon by the whole set N of players and to keep the others. One is then left with the Pareto strategies. These are defined as follows. Definition 1. Let {X(N),F } be a game described in strategic form. We shall say that x E X ( N ) is a weak Pareto strategy if and only if
there is no y E X(msuch that, V i E N,f;(y)
there is no y E X(N)such that F(y) -scF(x). (2) Usually, a somewhat stronger concept is used.
296
MAIN SOLUTION CONCEPTS OF COOPERATIVEGAMES
[Ch. lo,# 10.1
Definition 2. We shall say that x E X(N) is a (strong) “Pareto strutegy” if and only if there is no y E X(N)such that F(y) < F(x).
(3)
It is clear that any (strong)Pareto strategy is a weak Pareto strategy.
Remark. To study Pareto strategies, we do not need to assume that X(N) is a subset of a product flyml X iof strategy sets. Remark. The concept of Pareto minimum is very often used in modeling “decision problems under several criteria”. The set.X(N) is regarded as a “decision set”, the loss functionsf; as “criteria” to be minimized by the decision maker. A (weak) Pareto minimum is usually called a (weak) eficient decision in this framework.
Remark. Preordering on X associated with the multiloss. A Pareto strategy is nothing other than a maximal element of X ( N ) for the partial preorderingrelation on X defined by (4)
x
y
if and only if F(x)
Fb).
The associated indifference relation is given by (5)
x
y
if and only i f F ( x ) = F(y).
*y
if and only if F(x) e F(y)
Y
We shall set
(6)
x
so that x E X ( N ) is a weak Pareto strategy if there is no y y x.
*
E X ( N ) such that
Remark. The concept of a Pareto minimum depends only upon the partial
preordering and not upon its representation by the multiloss operator F from
X(N)into R”.
It is clear that two multiloss operators F and G define the same preordering if and only if there exists a injective strictly increasing map Q, from F(X(N)) into R”such that G(x) = @(F(x)).
Proposition 1. A strategy 2 E X(N) is a weak Pareto minimum if and only i f
W) (4 k * ( w o ) .
Ch. lo,§ 10.11
PARETO STRATEGIES
297
Proof. To say that F(2) belongs to k+(X(N)) means that there exists y E X(N) and c E I&+ such that F(Z) = F(y)+ c 4 F(y),i.e. that 2 is not a Pareto minimum. 0 Proposition 2. The set of weak Pareto minima is F-l[(F(X(N)) nCE+(X(N)))]. It is closed whenever F is continuous.and F(X(N))is closed. 10.1.2. Rates of transfer The problem arises as to whether losses “can be transferred” from one player to another and if so, how this can be done. The key to this is to use linear functionals ilE R”*which associate with any multiloss c E R”a number of “units of account” (A, c) called a “side-payment”. The linear functional A defines “rates of transfer of loss” among the players.
Definition 3. We shall interpret the dual of the multiloss space as the space of “systems of rates of transfer” (in short, “rates of transfer”) of side-payments. We shall say that two multilosses b and c are “transferable” (under the rate 1 E R”’) if and only if (A, b) = (A, c). In another words, rates of transfer play the same role vis A vis the multilosses as the prices vis il vis the commodities. The use of the label “rate of transfer” can be justified in the following way: ciunits of loss of the ith player and cj units of loss of thejth player are transi.e. if ci = (A’/Ai)cj (if Ai # 0) or ci = (Ai/Ai)ci (if # 0). ferable if Aici = We can say that ilj/A’(resp. 1’/1j) is the rate of transfer of losses from player j to player i (from player i to player j ) . We can either use elements of R supplied with a unit (side-payments) or a given multiloss c E R” (numtraire) for implementing exchanges of multilosses among players. Usually, we shall choose the unit vector 1 E R” as, such a numtraire. Thcn (A, c)/(A, 1) = (A/(A, l),c} is the number of units 1 which can be exchanged with c under the rate A. Therefore, if we are dealing with non-negative systems. of rates of transfer, we shall use rates belonging to the n-simplex A!” of R”*. 10.1.3. Pareto multipliers
Let {X(N), F } be a game defined in strategic form. If a rate of transfer 1E A!”prevails, the players can “pool” their loss functions and use the collective loss function x (A, F(x)) = 1x(x). Proposition 3. Let 1E &” and let x, E X(N)be a strategy minimizing the fittion x +-.
298
[Ch. lo,§ 10.1
MAIN SOLUTION CONCEPTS OF COOPERATIVE GAMES
Proof. If x, is not a weak Pareto strategy, there exists x E X ( N ) such that F(x) e F(x,). Since A E A?,this implies that (A, F(x)) -= (A, F(x,)), i.e. thatx,does not minimize x I- (A, F ( x ) ) over X ( N ) . 0 The converse is true under convexity assumptions. Proposition 4. Suppose that (7)
{
(i) the strategy set X ( N ) is convex, (ii) the loss functionsf;:are convex.
Then any weak Pareto strategy I E X(N) minimizes over X ( N ) a collective loss function (A, F( .))for a convenient rate of transfer A E _M".
Proof. Write X = X ( N ) . By Proposition 1.3.10, convexity assumptions (7) imply that (8)
k + ( X ) = F ( X ) + k ; is convex a subset of R".
Let I E X be a weak Pareto strategy. Hence 1 F ( 2 ) does not belong to the open convex subset e ( X ) . Therefore, by the separation theorem, we can find A E R"*, A # 0, such that (9)
(A, F(x)) -s inf E
(A, c)= ab(F(x);A)+ub(R;; A)
EK ( X ,
This implies that ob(R: ; A) is bounded below. Hence A E RY and d(R; ;A) = = 0. Dividing A by ELl Ai =- 0, we obtain that (10)
( A , F ( 2 ) ) = ab(F(X); A) = inf (2, F(x)) where X
=
X€X
A
~
i:Ai
E
&.
Y
i=l
Definition 4. We shall say that a rate of ransfer A E Ansuch that (A, F(I)) = = inf,,, (A, F(x)) is a "Pareto multiplier" of X . Proposition 5. Let respectively. Then (1 1)
(A-
A and p be Pareto multipliers of weak Pareto minima x and y n
p, F(x)-F(y)) =
E (Ai-
pi)( f i W - f i ( Y ) )
i=l
Proof. We have that and
(A,
m)) =z (2, F(Y))
(Pel, F(Y)) =s
FW).
We obtain (1 I) by summing these two inequalities.
=s0.
Ch. lo,§ 10.11
299
PARETO STRATEGIES
We denote the lower support function of F+(X(N))by (12)
w(A) = inf (A, F(x)) = ab(F+(X(N));A). xEXP"
Proposition 6. A strategy Z E X is a weak Pareto minimum associated with a Pareto multiplier 1 if and only i f F(2) belongs to the super-dgerential of the function w at X, i:e.
(13)
F(Z) E Sw(1).
Proof. Proposition (13) means that
(A- p, F ( 3 ) 4 4 ) W(d. If (13) holds, then (1,F ( i ) ) = w(x) (by taking
( 1 9 F(*)>.
,u =
0) and thus, 1 minimizes
(x,
Conversely, if 1 minimizes F( -)), then (1,F(2)) = ~(1).Otherwise, for any p E RY, (p, F(x)) z= w(p). Thus (13) holds. 0
-hn
*Remark. Let A E = Jnnk'; be strongly positive. We can easily check that, if 3 minimizes (A, F( .)), then 1 is a (strong) Pareto strategy, i.e. that (14)
(F(X)-F(Z))
n -R'&
=0
where R; = R$\{O}.
It is even obvious that (15)
(F(X)-F(x))++
n -R%
= 0,
since the fact that I minimizes (A, F ( . ) ) amounts to saying that A E ( F ( X ) - F ( f ) ) + = (F(X)-F(Z))+++. The converse is also true. Proposition 7. Let 2 be a strong Pareto strategy. If the closed convex cone spanned by F(X)- F(1) does not intersect -R", then there exists a stronglypositive Pareto multiplier A E h n .
Proof. Suppose that there is no strongly positive Pareto multiplier, i.e. that (F(X)- F(Z))+ n:"i = 0. By the separation theorem, we obtain the existence = - R I and c E (F(X)-F(Z))++, i.e. of c E R", c # 0 such that c E (*,*)such that c E (F(X)-F(Z))++ 17 - R t . This is a contradiction of the assumption. EI We can devise sufficient conditions for a strong Pareto strategy to have a strongly positive Pareto multiplier. For instance, we have the following result.
300
MAIN SOLUTION CONCEPTS OF COOPERATIVE GAMES [Ch. 10,
10.1
PropositionS. Let x belong to X(N). Suppose that, for any non-empty subset A # N of players, we can find y E X(N)such that
fib)-=jW V t E A
f/O* f / ( x ) V j d A. If x minimizes (A, F( .))over X(N),then I belongs to A@. (16)
t d
Proof. If A E Mt-A', take A = {i E N such that 1' w 0) and y E X(N) satisfying (16). We obtain the following contradiction :
10.1.4. Pareto allocations
Consider the set of allocations defined by
where
(i) Ri is subset of a vector space v', (ii) Y is a subset of V = Rk, (iii) Lf is a map from U' into V.
(18) Consider also (19)
n loss functions
:R'
-R
and the profit function r defined by (20)
tlp E Rk*,r@) = sup ( p , y ) = q*(Y; P). Y€Y
Denote by P the barrier cone of Y, i.e. the domain of r.
.
Proposition 9. Let 1E &' andg E P b e m d . rf2 = {3,.. ;X} E RN satisfis the foIIowing properties
then I is a weak Pareto alIocation.
Proof. If x' is not a Pareto multistrategy, there exists y E X(N)such that fi&) -==f(i?)for all i. Thus
a.10,g 10.11 Since y E
PARETO STRATEGIES
301
x(W, (3,LEN L9')=s r@). Using (21(ii)), we deduce that
Adding inequalities (21(i)), we obtain that
This is a contradiction. 0 We shall prove the converse statement under convexity assumptions.
Theorem 1. Suppose that (25)
(i) The subsets R' and Y are convex (ii) The operators L,from Ui into V = Rk are linear, (iii) Thefunctionsf , are convex.
and that
Let 2 E X ( N ) be a weak Pareto allocation. Then there exist 1 E dn and p E P such that properties (21) hold.
Proof. Introduce the map 9 :R N x Y
-
R"XRk defined by
Write (28)
Q =: R X (0).
It is clear that 3 E X ( N ) is a weak Pareto allocation if and only if
(29)
{F(Z), 0) 4 4(RNXY)+Q.
(To prove this, observe that the fact that {F(%),0) belongs to +(RNXY)+Q means that there exist x E RN and y E Y such that F(Z) s F(x) and sENLfXl= = y E Y i.e. that F(x) z- F(x) where x E X(N).) It is also clear that convexity assumptions (25) imply that the subset t$(flX Y)+ Q is convex.
302
MAIN SOLUTION CONCEPTS OF COOPERATIVEGAMES
[Ch. 10,s 10.1
Therefore, the separation theorem in finite dimensional spaces implies the existence of {A,p} E Rn*XRk*,(1,~)f ; 0, such that
This implies that A E RY and that d(R: ;A) = 0. On the other hand, assumption (26) implies that 'A # 0 [To prove this, assume that A = 0. We deduce from (30) that 0 = infXERN; y c y (P,x i € N Li2-r) = B(&N U R ' ) - K P). Since 0 belongs to the interior of CiENLi(Rf)-Y,we deduce that ob(&N L,(R')- Y ;p ) < o whenever p z 0. Since {A, p } = (0,p} z 0, then p # 0. Thus 1 cannot be equal to 01. Therefore, XIENA' =- 0. Writing = = ;I/& A' E M nand p = P / & N Ai, we deduce from (30) that
x
This implies in particular that r(F) -= m, and thus, that E P. Taking x = 2 in (31), we obtain that 0 =S( p , & & + - r ( p ) .since 2 E x(N),we know that ( F , &ENLf$er(F). Thus r(p) = x i E N ( pLi*. , Inequality (31) can be rewritten
Thus property (21) holds. Remark. In the case when n = 1, we obtain Theorem 5.3.1 which asserts the existence of a Lagrange multiplier. We shall say that E A'i"' is the Pareto multiplier and jj E P the Lugrange multiplier associated with the weak Pareto allocation 3 E X(N).
Remark. The following definition is often used. Definition 5. We shall say that {z, 3) is a "~s~do-equilibrium" if the following properties hold for all i: (32)
fi(x') e f i ( 2 ' )* ( p , L i X i )
2
( p , Liz').
Then the above theorem implies the following result.
Ch. 10,
10.21
SELECTION OF PARETO STRATEGIES AND IMPUTATIONS
303
Proposition 10. Suppose that (25) and (26) hold. We can associate with any weak Pareto allocation 2 E X ( N ) an element E P such that {Z,p} is a pseudo-equilibriwn. Proof. We deduce from (21) that (jj,L#)-
(p,& x i )
=sAi(h(xi)-J(?i))
from which we deduce statements (32). 0 10.2. Selection of Pareto strategies and imputations
The concept of Pareto optimality is useful in dismissing strategies which can be improved upon by the whole set of players acting as a unit under the behavioral assumption that it is good to make someone better off provided noone else is made worse off. This cooperation requirement among players is too weak. In general, the set of Pareto minima may be rather “large”. This is quite a drawback for a solution concept. We therefore have to find selection procedures enabling us to discriminate among Pareto minima. We shall see that the concepts of imputations, core and cooperative equilibria are just such selection procedures. Before defining them, we shall begin by surveying examples of a simple kind of selection procedures, namely, procedures R -c R which are increasing. We select Pareto using “selection functions”~:: strategies by minimizing s[FT(x)]on X ( N ) , where the multiloss operator FT is deduced from F by the formulas
with
(0 aU) = infxcxcn,h(x), In particular, by taking s(c) = supiENc,, we obtain the concept of the “best compromise” 5 defined by: V i E N , J ( Z ) e ( 1 - d ) a(fi)+dt,(J) where (7 = = infxcX(N) maxieNfr,(x)E [0, 11. We require that the functionals ti :fi -, r,(J) satisfy ti(af+b) = at,(f)+b whenever a E R,, b E R. They are interpreted as threatfunctionuls, in the sense that they set a maximum ri(f,) to the amount of loss that player i accepts.
304
MAIN SOLUTION CONCEPTS OF COOPERATIVE GAMES
[Ch. lo,§ 10.2
We see that a “best compromise” Zis a weak Pareto strategy such that, for any i C N,A(Z) e t,(f;).This is a particular example of an imputation. We end this section by using selection functions to choose particular imputations. We show that the Nash bargaining solution is an example of such a selection procedure in the case of two players. 10.2.1. Normalized games
We shall require that a selection procedure does not depend upon positive affine transformations of the loss functions. For this purpose, we introduce functionals t : 8 ( X ( N ) ) R satisfying the following property:
-
(1)
V a > 0, VP E R, t(af(.)+B) = atCf)+B.
For instance, the following functionals defined by
c W(j?k), A’
(2)
to =
(3)
t ( f ) = inf f ( x ) or t ( f ) = sup f ( x ) ,
(4)
t ( f ) = inf sup f&i, $€m > € X I
n=l
x€X(N)
X € W o
8
satisfy the above property. Definition 1. Let {X(N),F } be a game bounded below and let T = (?I, be n functionals satisfying (1) and (5)
{ (ii)(i) V3xi E N,X(N)such=- thatf;:(x)infxEx(N)j;(x), ti(fj)for all ti(f;)
E
. .., 1,)
a(&) =
=S
i E N.
We shall say that the game {X(N),FT} defined by the loss functions
-
is the T-normalized (or, normalized)game. We shall say that s :: R R is a “selection function” (resp. “strong selection”function) if
(3
Vc, d E R$ such that c x- d, then s(c) that s(c) =- s(d)].
w
s(d) [resp. c
=- d implies
Ch. lo,$ 10.21
SELECTION OF PARETO STRATEGIES AND IMFWTATIONS
305
Remarks. Notice that the sets of Pareto multistrategies of the games {X(N), F} and { X ( N ) , FT} are the same and that the shadow minimum of the normalized game {X(N), FT}is 0.Also
then FT = G,. In other words, if we say that two games {X(N),F} and {X(N), G) are strategically equivalent i f (9) holh, the normalized game is an equivalence class o f mutually strategically equivalent games. 10.2.2. Pareto strategies obtained by using selection functions
Proposition 1. Let T be a set of n functionah satisfying ( I ) and (5)and let s be a selectionfunction. Then any Z E X ( N ) which minimizes thefunction x -c s(F,(x)) over X ( N ) is a weak Pareto strategy.
Proof. We have that FAX) c RY since 0 is the shadow minimum of {X(N), FT). Thus the function x -c s(F(x))is defined on X ( N ) . If Z is not a weak Pareta minimum,there exists x E X(N)such that FAX)e &(Z) Therefore s(F,(x)) -= s(FT(Z))b y (7)which is a contradiction. 0 Remark. If s is a strong seIection function, X is a strong Pareto minimum. Remark. Functions (A, are selection functions if A E A?"and strong selection functions if 1 E Proposition 1 generalizes Proposition 1.3.
h".
0 )
Proposition 2. Suppose that (10)
a convex subset, { (ii)(i) Xthe( Nloss) functionsf, are convex is
and that (1 1)
U
the selection function s is non-decreasing and dyerentiable.
1 minimizes s[F,(*)], then the rate of transfer A of components li = = D,s( F&t))/( tiV;) - aU;)) is a Pareto muhiplier of 1.
22
306
MAIX SOLUTION CONCEPTS OF COOPERATIVE GAMES
Proof. Assumptions (10) imply that F,(X(N)) +R:
[Ch. lo,§ 10.2
is convex by Proposition
1.3.10.
Then FT(2)+O(FT(x)-FT(2)) is larger than or equal to F,(y) for a suitable y when 8 E [0, 11. Thus s(F,(Z)) s(F,(y)) =S s(F,(Z))+B(F,(x)-F,(Z)). This implies that (DS(FT(2)), FdX)- F T ( 3 ) =
for all x C X(N). Since s is not decreasing, Ds(FT(2))E RT. Therefore, 1' = D,s(FT(Z))/( r,(fr) - a(&)) is non-negative. Thus (12) implies that 37 minimizes (1, F( where X = {I1,...,An}. We give sufficient conditions of existence in the following proposition. a))
Proposition 3. Suppose that X ( N ) is a subset of a topological space U N and (13)
I
(i) the extended lossfunctionrj&, : UNIare lower-semi-continuous, (ii) a lossfunction fi is lower semi-compact.
1- -, + -1
Suppose also that (14)
the selection functions is continuous and its lower sections are bounded above.
Then there exists X E X ( N ) which minimizes s(F,( -)). Proof. We defineg byg(x) = s(F,(x)). Then g is clearly lower semi-continuous. It is also lower semi-compact because, if g(x) -c 1, then FT(x)6 c since the lower section of s is bounded above. Therefore, sincef, is lower semi-compact, the inequality
implies that x belongs to a relatively compact subset. Therefore g achieves its minimum. *10.2.3.Closest strategy to the shadow minimum
-
Simple examples of strong selection functions are the functions sp : c ++ ;--s,(c) = (IicN) ci )1' where 1 e p < (forp = 1, we have the linear selection studied in Section 10.1.3).
+
Ch. 10, Q 10.21
SELECTION OF PARETO STRATEGIES AND IMPUTATIONS
307
In this case, Z E X(Zf) minimizes the function x I-+ +,(FAX))over X(N)if and only if F(X) is the projection of the shadow minimum onto FT(X(N)) for the norm
where we set ai = a(fr). The simplest case is that when p = 2.
Proposition 4. Suppose that assumption (13)of Proposition 3 hold and that (i) X ( N ) is convex, (ii) the lossfunctionsf, are convex. Then there exists a (strong) Pareto strategy x' whose multiloss FT(X) is closest to the shadow minimum. Its Pareto multiplier C RY is defined by
Proof. It is clear th&t assumptions (13) of Proposition 3 imply the existence of
x minimizing
Since F,(X(N))+R$ is convex, by (15), we deduce that for all x E x(N)
10.2.4. The best compromise
The function , s defined by sm(c) = I I c I I o D = supiENI ci I is clearly a selection function.
a
Proposition 5. Let = infxEx(N) s,(FT(x)). Then % E X ( N ) achieves the minimum of s-( FT( .)) if and only if (18)
V i E N,
fi(x) 4 (1 - a) a(&)+ati(f;,)
where the equality holdsfor at least one i. Further, 2 is a weak Pareto strategy. 22.
308
mnu SOLUTION
CQN~JTS OF
COOPE'RATIVE GAMES [Ch. lo,$ 10.2
Proof. We know that iis a weak Pareto minimum by Proposition 1. Since
we deduce (18). Conversely, if Z satisfies (18), it is clear that s,(Fdz)) 4
a.
Definition 2. A n y strategy R satisfying (18) is called a best compromise. It is both as close as possible to the shadow minimum and as far as possible from the vector {~,cfr)lrcw 10.2.5. Existence of Pareto strategies
We can use this proposition to prove the existence of Pareto multistrategies.
Proposition 6. Suppose that X(N)is a subset of a topological space Uw and that ( i ) V i E N , fl.x(W:UN+]--, +-]~lowersemi~-conti~ (ii) 3 i E N such that ft is lower semi-compact Then,for any xo E X(N)such that F(xo)w a, there exists a weak Pareto strategy I E X(N)such that F(Z) Q F(xo).
Proof. We take the functionals defined by t&) =I;(xo). Then the function
is lower semi-continuous and lower semi-compact. Therefore there exists i which achieves its minimum. By Proposition 5, Z is a weak Pareto strategy satisfying A(?) ( 1 4 aCfr)+aI;(xo)e f ; ( x 0 ) since inf,cx(Ws,(FT(x) E
f r0911.
a=
*10.2.6. Interpretation: threat functionah
We can interpret the functionals t, :I; -I t,(fi)as threat functionals in the following sense. If the n players are reluctant to come to an agreementfor choossing a Pareto strategy, each player can threaten to implement his "threat" value ziV;). Once these threat functionals are chosen, the normalized loss functions take their value between their minimum equal to 0 and their threat value equal to 1.
Ch. 10,s 10.21
SeLECTlON OF PARETO STRATEGIES AND IMPUTATIONS
309
Therefore, a selection procedure takes threat-functionals into account if it yiela3 to each player a loss not larger than This is the case when we use the selection function ,s and we choose a best compromise. The choice by players of a given multistrategy xo can define a “threat functional” by setting t i ( j J = f ; ( x ~ ) . It is agreed that xo is the worst strategy which must be implemented if the players do not come to an agreement to play a strategy yielding to each of them a smaller loss. In other cases, we can take for the threat multistrategy either a conservative strategy or a noncooperative equilibrium (if they exist). Such a choice describes the following two-stage procedure. Players begin by playing as individuals. They may choosea conservativestrategy or, if it is possible, amve at a noncooperative equilibrium. Secody, they use their individually chosen strategies as “threat strategies” and look for a Pareto minimum better for all the players than the result obtained when the players implement their “threat strategies”. Another example of a threat functional is given by the functional
flu,).
e* : f i e w*(fi) = inf
sup f(xi,
*C€M xfEXf
a
which describes the minimum loss that players i can obtain by unilateral actions, whatever the other players do. We can interpret w* as a particular example of an “implicit” threatfunctional,in the sense that v*(fi)sets a maximum to the amount of loss that player i will accept for himself.
*10.2.7.Imputations: the Nash bargaining solution The above comments motivate the following definition.
Definition 3. Consider a game {X(N),F} and n threat functionals t, satisfying (1) and (5). We shall say that 2 E X(N) is an ‘‘imputation” if
(a)
(
(i) Z is a weak Pareto multistrategy,
(ii) Vi C N, f;:(2)s tiV;.).
Property (20(ii)) is often called “individual rationality”. We can give a systematic way of selecting an imputation. Let us introduce the subset (21)
XT(N> = {X E X ( N ) such thatfi(x)
=s ti(fi)Vi
E N}
and a selection function s. We set 1 = {I, . .., I} E R”,which is the threat vector of the normalized game.
310
MAIN SOLUTION CONCEPTS OF COOPERATIVE GAMES
[Ch. 10, p 10.3
Proposition 7. Consider n threat functionals ti satisfying ( I ) and (5) and a selectionfunction s. Any 2 E X A N ) which maximizes thefunction x k- s( 1-FAX)) is an imputation. Proof. The proof is left as an exercise. 0
Example. The best compromise. The best compromise is obviously a selection of an imputation. Example. Nash bargaining selection. Let us consider the case of n = 2 players. We take for selection function the function s defined by s(c) = clca. Then
The solution x’ C X,( { 1,2}) which maximizes g over X,( { 1,2}) was introduced by Nash and is called the solution to the “Nash bargaining problem”. If we asume that (22)
the strategy set Xis convex, the loss functionsf1 and f a are convex,
we deduce that the rate of transJer
(23)
21
= t2(fa)-f2(2),
X
A2
= (11, 2 2 ) defkted by
= tl(f+fl($
is a Pareto multiplier of the imputation 2.
Remark. The question arises as to how to choose a “threat strategy” x = = {xl, x2} (i. e. ti(f;)= A ( x ) for i = 1,2). The “threat stategies” can be obtained as the non-cooperative solution of a new game, whose loss functions are the losses assigned to each player by the Nash bargaining solution obtained from this threat. Other methods can be devised.
10.3. Behavior of coalitions of players: the core
We can regard a weak Pareto multistrategy x’ E X ( N )as a multistrategy which is not “rejected” by the whole set of players provided it is understood that x is “rejectedby N (or “blocked”) only if there is another multistrategy y E X(N) such that f;(y)
Ch. 10, 5 10.31
BEHAVIOR OF COALITIONS OF PLAYERS: THE CORE
31 1
If we introduce threat functionals ti(i E N) for which there exists at least one x E X ( N ) such thatA(x) ES rt(f;) for all i E N , we can regard an imputation as a multistrategy which is neither rejected by the whole set N of players nor by the individual players i. It is to be understood that x E X ( N ) is rejected by player i only if f;(x) > ti(fi). In order to extend this selection procedure for Pareto multistrategies, we shall broaden the description of the game by including assumptions about the behavior of a family of coalitions A. We define the multistrategies “rejected” by coalitions A. The multistrategies not rejected by any coalition constitute the “core” of the game. 10.3.1.Coalitions
We shall regard a non-empty subset A c N players. We denote the “adverse coalition” by (1)
=
{I, . . ., n} as a coalition of
d=N-A.
If coalition A forms, it “splits” the multistrategy set xN in two parts (2)
XN
= XAXXA where XA =
fl X i
and
XA =
t€A
n XI.
JW
X A(resp. X’) is the set of multistrategies “controled” by A(resp. A). We denote the projector from XN onto X A by
(3)
ZA
:X N
-
XA
and write (4)
x =
{dX,Z‘X}
=
{XA, XA}
E X-4XX.c = X N
whenever coalition A is involved. We shall identifr any coalition A with its characteristic function zA : N (0, 1) associating with any player i his rate of participation z: defined by zf = 1 if i participates in coalition A and by zf = 0 if i does not participate in A. We shall denote by .the projector of R” defined by
-
We notice that tN.is the identity and that z0 = 0. We set (6)
and also
RA = zA .R, RA,
= ZA
.R;,
=
+.*+
312 (7)
MAIN SOLUTION CONCEPTS OF COOPERATIVE GAMES
RN = z N * R ” = R”, RT =R;,
R+ =*+, ON
[Ch. 10,g 10..
Re = {0}
By a family d of coalitions,we mean a family of non-empty subsets of N such that
(8)
d -C C d C d
where (9)
- - ., {n}} is the minimal family of the (n+ 1) coalitions N (whole set of players) and {i} (individual players)
d?= {N, {l},
and where
2 is the family of the 2”- 1 non-empty subsets of N.
(10)
In this section, we shall assume that a family d!is given once and for all. 10.3.2. Cooperative game described in strategic form and its core
Definition1. Let d be a family of coalitions. A cooperative game {X(A),FA}AEd is described in strategic form by associating with each coalition A E d (1 1)
{
-
(i) a multistrategy subset X ( A ) c X A ,
(ii) a multiloss operator FA :X ( A )
RA
defined by P(x) = {f;A(X)}icA.
We shall say that a coalition A E d “rejects” (or “blocks”, “can improve upon”) a multistrategy x E X ( N ) if there exists a multistrategy 2 of the multistrategy set X(A) controlled by coalition A yielding to each player i E A of A a 10ssf;~(x“)smaller than the lossf;(x), i.e. (12)
3 x A E X ( A ) such that, V i E A, LA(xA)-=f;(x).
We shall say that the “core of the game” is theset @({X(A),F A } A E d satis) fying, (9 x E X ( W , (13) (ii) V A E d, x is not rejected by A,
{
i. e. the core is the set of multktrategies x E X(N)which are not rejected by any coalition A E d.
Remark.We notice that if d i s the family d of the(n+ 1) coalitions N, {l}, .. ., {n}, then the core is nothing other than theset of imputations when the threat functionals t, are defined by tl(f;)= infx’EX((r),f;{‘)(2).
Ch. lo,§ 10.31
BEHAVIOR OF COALITIONS OF PLAYERS: THE CORE
313
Remark. The core of the game is contained in the subset of imputations, which is itself contained in the subset of (weak) Pareto multistrategies. Actually, the larger the familyd of coalitions which is allowed to form, the smaller will be the associated core @({X(A),FA}AEd). 10.3.3. The multiloss operator FA* of the coalition A We shall give a canonical way of constructing a multiloss operator for the coalition A. As a member of a given coalition A, a player i E A will modify his loss function, by taking into account the fact that he is cooperating with other players j E A and not cooperating (in fact, anti-cooperating) With players of the adverse coalition d. Therefore, if cautious, players i of A will assume that players j E d will choose a multistrategy d x which maximizes their loss function. Therefore, they associate with a multistrategy x" E X ( A ) the worst losses (14)
fi"'(%A)
= sup fi(y) = Y E X(N)
nAy=xA
sup
w. J } € X ( i V
fi(x",d).
In other words, we can say thatf;'" is the loss function for player i as a member of the coalition A in opposition to the adverse coalition d. Notice that when fr :X i -, R depends only upon the strategies of player i,. AA#(x')=A($) when i E A. Consider the game {x", We can reformulate the definition of the core in this case. A multistrategy Z E X ( N ) belongs to the core if and only if for all coalitions A E ui?,for all E =- 0 and for all multistrategies # E X ( A ) feasible for A, there exists at least one player i E A and a multistrategy 2 of the complementary coalition d such that (15)
&(XI e J ; A + ( x A ) G f i ( % A ,
XA)+E.
10.3.4. Examples of multistrategy sets X ( A ) An important class of examples consists of those in which we have to allocate scarce resouTc3es among n players. We define (16)
c of available resources to the coalition { (ii)(i) nsubsets resource operators L,E &u', 0. Y(A)
V
The subsets X(A) are defined by
(17)
X(A) = {xA E RA such that
C t QLr(XA, ~ 3 E Y(A)}.
A,
314
MAIN SOLUTION CONCEPTS OF COOPERATIVEGAMES
[Ch. lo,§ 10.3
In other words, X ( A ) is the subset of multistrategies x A = { X A i ‘ } i c A such that the sum of the resources needed to implement them is available to the coalition A. 10.3.5.Economic games and core of an economy
Consider an economy {R‘, Y(i),A}iEN where R’ is the consumption set for i, Y(i) c R‘ is his initial endowment in available resources and fi is his loss function (see Section 8.1). We associate with this economy the following game {X(A), FAJAEd. We assume that members i E A of coalition A place at coalition A’s disposal the sum (18)
Y(A)=
1 Y(i)
icA
of their endowments in resources. Therefore, the subset X(A) c RA of allocations of coalition A is defined by (19)
X ( A ) = { x A E RA such that
1
E Y(A)}.
i€A
We thus define FA by (20)
fiA(x) = f ; ( x i ) whenever i
E A.
Definition 2. Se shall say that the game {X(A), FA}AEddefined by (19) and (20) is the economic game associated with the economy {Ri, Y(i),A}iEwThen a coalition A rejects m allocation x E x ( N ) if it can find an allocation J? E X(A) yielding to each member i E A a loss f;(p* ‘) smaller than the loss A(x). We shall say that the core @({X(A), FA})of the game is the “core of the economy {R‘, Y(i),fiJiEN”. 10.3.6. Cooperative game described in characteristicform and its core Let {X(A), FA}Acd be a game described in strategic form. Denote by
(21)
J(A) = F“(X(A)) the set of feasible multilosses of coalition A.
i
We shall say that (i) J+(A) = J(A)+R$ is the set of “admissible multilosses” of of the coalition A, (ii) .?+(A) = J ( A ) + k $ is the set of multilosses “rejected” by the coalition A.
We can prove the following proposition
Ch. lo,$ 10.31
BEHAVIOR OF COALITIONS OF PLAYERS: THE CORE
315
Proposition 1. A coalition A “rejects”the multistrategy x E X ( N )ifand only ifthe multiloss c = F(x) satisfies (23)
Z ~ * E C
j+(A).
A multistrategy x E X ( N ) belongs to the core of the game multiloss c = F ( x ) satisfies
is and
only
if the
0) c E J ( N ) , (ii) v A E d,ZA -c 4 j + ( ~ ) . Proof. This is left as an exercise. 0 Since the concept of the core depends upon the strategy subsets X ( A ) only via their images $(A) by FA, we can get by with “poorer” description of the game. Definition 3. We shall say that a correspondence (d, J ) mapping coalitions A E & into subsets J ( A ) c RA of multilosses of the coalition A “describes a cooperative game in characteristicform’’. The subset J + ( A ) = J ( A ) + R t is called the subset of admissible multilosses of the coalition A and the subset j + ( A ) = J(A)+R$ is called the subset of multilosses “rejected” or “blocked” by the coalition A. We shall say that the subset e0(d, J) of multilosses satisfying (25)
{ (ii)(9 cv A J(N), A, E
E
ZA
-c 4j + ( ~ )
is the “feasible core” of the game (d, J ) and that the subset @(&, J ) of multilosses c satisfying (26)
[
(i) c E J + ( N ) , (ii) V A E d,Z A - C
6j + ( ~ )
is the “admissiblecore”. Remark. Since @o(&, J ) is obviously contained in @(d, J), the non-emptiness of &(&, J ) implies the non-emptiness of @(&,J ) . The converse is also true. Proposition 2. The two following statements are equivalent
( a ) the feasible core @o(ui?, J ) is non-empty, (b) the admissibie core @(&,J ) is non-empty.
316
MAIN SOLUTION CONCEPTS OF COOPERATIVE GAMES
[Ch. 10.8 10.4
Proof. We have actually to prove that, ifc € J+(N)and V A € d,#*c 4 ]+(A), then there exists d E J ( N ) such that, VA E uf, #ad 4 J+(A). There exist d E J(N) and b E : R such that c = d+b. We have to check that d is rejected by no coalition. If A rejects d, then # - d belongs to )+(A) and thus, s h e # * c = #a+# *b zdd, c is rejected by A. But this is impossible. 0 Remark. In view of the above result, since the debition of the admissible core is more symmetric, we shall use the admissible core instead of the feasible core in most instance. 10.4. Behavior of fuzzy coalitions: the h z y -re
By extending the minimal family d?= {N,{I}, . . .,{n}} of coalitions to a larger family uf, we “shrank” the setof imputations to the core of the game. In many instances, the core remains too large a set. Since we would like to have solution conceptsyielding a set of solutions as small as possible, the question arises as to whether it is possible to “shrink” the core again by “enlarging” the set of “coalitions”. This is done by embedding the set of coalitions (identified with (0, l}”)into the subset 17i = [O, 11” of “fuzzy coalitions”. We define a fuzzy game (TJ, J) as a correspondence mapping any fuzy coalition t onto its subset J ( z ) of multilosses. Its core is the set of multilosses which are not rejected by any fuzzy coalition. It is obviously contained in the core of the restriction of (5, J)to the set of usual coalitions. We end this section by constructing fuzzy economic games and by associating canonically a fuzzy game with a game {X(A), FA}AEddescribed in strategic form. 10.4.1. Fuzzy coalitions
Recall that we identified a coalition A with its “characteristic function” tA E (0, 1)” defined by its rates of participation 1 ifiEA, b = {0 i f l B A . In other words, a player i either participates wholly in the coalition A or does not participate in it at all. A little thought shows that it is too strong a restriction to allow only such “crude” coalitions to form. A “ h e r “ analysis should allow “fuzzy coalitions” to form, i.e. coalitions in which a player i can participate with a “rate of participation” t,E [0, 11 (instead of (0, I}).
Ch. lo,$ 10.41 BEHAVIOR
OF FUZZY COALITIONS: THE FUZZY CORE
317
Definition 1. We shall say that any t E [O,lY is a ‘Yuzzy coalition” and that its ith component t,is the rate of participation of player i in this fuzzy coalition. A subset Zi of [0, lp which containsA is d e d a family of f m coalitions.
Remark. The concepts of fuzzy coalitions t E [0, 11” and of rates of transfer 1 E M”are different. We recall that a rate of transfer A is a rule for sharing a “side payment” among players i participating in coahion A. Naturally A also enables a fuzzy coalition t to share a given Side payment w among players by means of the rule (1)
{C
f RN Such that
D-1 Aitjc, = (A, Z-C) = w}.
Remark. From now on we shall associate with z = {TI, map t from RN into RN defined by (2)
(Z’C),
...,
7*}
E [O, l]” the
= zicr
We shall denote the support of the fuzzy coalition t by (3)
A, = {i E N s u h that t i > 0},
i.e. A, is the set of activeplayers in thefuzzy coalition.We shall set (4)
R7 = z-R” = RAT; R‘+ = z.RT
6; =z.lil+
E
+,
= P Ai7-
10.4.2. Extension of a f m l y of coalitions
Let d be a family of coalitionsA c N , identified with the subset of (0,l)”C c R”of their characteristic functions @. With G+?we shall associate its convex
hull Zi = co d which consists of the fuzzy coalitions (5)
z=
C
m(A)zA
A€&
where {rn(A)JAEA ranges over the set of probabilities M ( d ) . The rates ofparticipation of such coalitions are equal to
(6)
V i E N, zt =
C m(A).
A €4 ,€A
Such a formula speaks for itself. r f m ( A ) denotes the probability of coalition A forming, the associatedrate of participation of player i is the sum of the probabilities of the formation of coalitions A to which i belongs.
318
MAIN SOLUTION CONCEPTS OF COOPERATIVE GAMES
[Ch. 10,s 10.4
For instance, we can check that
zi - = co (a&) is the set of fuzzy coalitions t satisfying ElcNt i I
(7)
and, V i E N,t i
(&CN
Ci-
l)/(n- 1).
It is clear that t = mo@’+C;,l rn,t(’) satisfies these properties. Conversely, suppose that z E [0, 11” satisfies the properties. We set
mo = (C;sl zi- l)/(n- 1) 3 0 and mi = zi-mo a 0. Since
ti = mo+mi,
mo+
C
we can write that z = mofl+C;,,
n
rni
= mo+
1=1
n
C zi-nrno
mid‘). Finally,
= 1.
I=1
Thus z E co(d) - = - 0 In the same way, if check that (8)
2 denotes the family of all non-empty coalitions, we can
Z= co 2 = {z E [0, 11” such that XIENz1
Z-
1)
and that (9)
d = (0, 1)”-
(0) is the set of extremal points of
z.
*10.4.3. Debreu-Scarf coalitions
We shall say that a fuzzy coalition z E % is a “Debreu-Scarf” cfuzzv) coalition if its rates of participations z, = pJq E Q are rational. This implies thatqz=pp,Vi=1, ..., n. Debreu-Scarf fuzzy coalitions are usually interpreted in the following way. We regard i as a “type of player”, pt as the number of players of type i and q total number of players of each type. Then the rate of participation of “type” i is the proportion of players of’ type i involved. Recall that any fuzzy coalition can be approximated by a Debreu-Scarf coalition. If 8 = nQ“ is the set of Debreu-Scarf coalitions, we have that (10)
8 is dense in %.
*10.4.4. Fuzzy coalitions on a continuum of players
Suppose that the set of players N is no 1onge.r finite, but infinite and is in fact a set Q of players homeomorphic to [0, I] is said to be a “continuum of players)”. We denote by ui! the a-algebra of Bore1 subsets A c 0,interpreted
(a
Ch. 10, $ 10.41 BEHAVIOR
OF FUZZY COALITIONS:THE FUZZY CORE
319
as the “family of coalitions” of players. It is identified with the subset (1 1)
o f = {z” E L1(sZ) such that zA(o) E (0, 1) almost everywhere}.
Now, we can regard the subset (12)
2; = {z E L1(sZ) such that z(o) E [0,1] almost everywhere}
as the family of “fuzzy coalitions” of players o E Q. Let us supply L1(sZ) with the weak topology a(L1(sZ), L”(sZ)). We can prove the following result. Proposition 1 (Castaing). The family oi? of coalitions is dense in the family 5of fuzzy coalitions, which is a convex compact subset of L1(sZ).
Proof. Since 2; is clearly a bounded subset of Lw(sZ),it is relatively compact in L”(f2) supplied with the weak topology a(L”(Q), LYQ)). Since L”(sZ) is contained in L1(sZ)(for sZ is compact) and since a(L”(sZ), Ll(S1))is stronger than a(L1(Q),Lw(sZ)),then 2; is relatively compact in U ( S )supplied with the weak topology. On the other hand, 2; is clearly convex and closed in U(0) supplied with the initial topology. Hence 2;is closed in L1(sZ) supplied with the weak topology. Therefore, 2; is compact. We prove that d is dense in 5, i.e., for any seminorm PK(.) = ..., , ]{A,-)I (K = { f1, . . .,f,} c L”(sZ)) of the weak topology and for any E > 0,we can find A E uf such that
For this purpose, consider the vector measure F associating with any A E ol the vector F(A) = {jAj;(~)dw}iof R”. By the Lyapunov convexity theorem (see Appendix C) we know that F ( d ) is a convex subset of R”,which is equal to F(Q. Hence, we can associate withany z a coalition A € oi? such that PK(Z-z“) = 0 =sE. 0 Remark. The fact that the family of Debreu-Scarf coalitions of a finite set N of players is dense in the family of fuzzy coalitions (or the fact that the family of coalitions on a continuum of players is dense in the family of fuzzy coalitions) shows that under convenient continuity assumptions, properties which are true for the Debreu-Scarf coalitions (or the coalitions on a continuum o f players) remain true for fuzzy coalitions.
320
MAIN SOLUTION CONCEPTS OF COOPERATIVE GAMES
[Ch. lo,$ 10.4
10.4.5. Fuzzy games described in characteristicform
Definition 2. Let 2, c [0, l]” be a family of fuzzy coalitions. We shall say that
(G, J ) describes a fuzzy game in characteristic form if J is a correspondence associating With any fuzzy coalition t E ‘zi its subset J ( r ) c R’ of multilosses. We shall say that a fuzzy coalition t “rejects” a multiloss c if
(14)
z * c E &i+(z)( = J(z)fk+).
The core @(.is, J ) of a fuzzy game is the set of multilosses c E J + ( N ) which are not rejected by any fuzzy coalition z E 53, i.e. such that
These definitions are the natural extensions of cooperative games and their core. The following statement is obvious. Propition 2. The core @(%,J) of a fuzzy game is contained in the core of its restriction to & = ‘zin2.
@(d, JI
Proof. This is left as an exercise. 0 10.4.6. Characterizationof thi core of a ( f m z y )game
Let 7 j be a family of fuzzy (or usual) coalitions and (53, J ) be a fuzzy (or usmQ game. We use the lower support functions d(J(r),A) = infc,,(r, (A, c) of the se’ J ( t ) of multilosses to characterize the core of the game. For this purpose, we denote by &(t) the set of probability measures on tl support A, of z. We introduce the-functiona defined by
Tbeorem 1. Let c E J+(N). If aa(c) 4 0, then c belongs to the core @(%, Conversely, if (17)
Vz E ‘i5, J(z)+R: is convex,
then a&c) is non-positive when c belongs to the core @(-,
J).
Ch. 10, Q 10.41
321
BEHAVIOR OF FUZZY COALITIONS: THE FUZZY CORE
Proof. Suppose in the first place that a,(c) -s 0. Then, for any fuzzy coalition z, inf,,,,, [(A, t-c)-d'(J(z), A)] 4 0. Since M ( z ) is compact and A F+ (A, z - c ) - d ( J ( z ) , A) is lower semi-continuous, there exists 1C x ( z ) which achieves the minimum. Therefore, Qz E
czi, 3 h E M(c) such that (1,z
c)-
ub(J(z); 2)
-= 0.
This implies that c is not rejected by z (otherwise there would &t such that tic, =- d, for all i E A t . Then,
d E J(z)
(A, r.c) 2- (A, d)a ub(J(z);A ) which is a contradiction.) Conversely, suppose that c belongs to the core @(.is, J). Since z-c 6 j+(z), we deduce from (17) and the separation theorem that there exists A E (R')*, 1 # 0 such that (18)
(A,
Z-C) s
inf (A,
d€
d)= ub(J+(z);A).
Since ab(J+(z);A) =- - m, this implies that ilE ( R ' ) : . Dividing both sides of 0, we can say that (18) holds With A E A?!@). (18) by z { E A r A' Therefore, we have proved that
and thus that aa(c)
=s0.
10.4.7. Fuzzy economic games and fuzzy core of an economy
Consider an economy {P, Y(~),A},~~ We assume that members i of a fuzzy coalition z place at the disposal of fuzzy coalition z the weighted s u m (20)
Y(z) =
n
C ztY(i)
f=1
of their endowments Y(z3 of available commodities (i.e. by participating in 7 witb a rate of z,, player i b$gs the z,th part z,Y(i) of his endowment). We define the subset X ( z ) c RATof allocations of the fuzzy coalition z by (21)
X ( z ) = {xr E RAr such that
x;-l
ztx'J E Y(z)}
and the rnultiloss oper'ator F' o f t by (22) 23
S ( x ) = z i f ( x f ) whenever zt > 0.
322
MAIN SOLUTION CONCEPTS OF COOPERAT~VE GAMES
[Ch. 10,s 10.4
Definition 3. We define the “jiuzzy economic game” (B, J) by
(23)
J(Z)
= P(x(z)) for all z E
B.
Therefore, a fuzzy coalition rejects an allocation x E X(N)if it can find x‘ E E X ( z ) yielding to each player i E A, (participating in z) a loss A(X‘*‘) smaller than the loss f;(x). Notice that, for any coalition A, we obtain (24)
Y(A) = Y(zA), X(A) = X(zA), FA = FA.
Hence the economic game (2, J) is the restriction of = FA(X(A)) = F ” ( X ( # ) ) = J ( @ ) for all A. Therefore, (25)
(s, J), since J(A) =
J), called the 6‘jiuzzycore ofthe economy”,iscontained the core @(%, in the core of the economy.
Let $ = that
fQ“ l be the subset of Debreu-Scarf coalitions. We clearly have
(26)
@(%,J ) c @($,J)c
@(a, J).
Since 6 is dense in 3,we obtain that the fuzzy core @(%,J) is the same as the J) of the economy. “Debreu-Scarf core” @(8, Froposition 3 (Ekeland). Suppose that (27)
V i E N, the lossfunction f ; is continuous.
Then thefuzzy core and the Debra-scarf core of the economy coincide.
Proof. It remains to prove that @(B,J)c @(aJ). , Suppose that c = F(x) (where x E X(N)) is not rejected by any fuzzy coalition CT of $. We prove that c belongs to @(%,J). If not, there exists a fuzzy coalition 7 E ‘Z and y E XA7such that
Now, if we set q Z- 0 such that
E
=
4 mhiEAr(h(x’)-J;(y‘)) =- 0, we know that thereexists
Il#-z’II
4
q implies
Ch. lo,$ 10.41 BEHAVIOR OF
FUZZY COALITIONS: THE P'JZZY CORE
323
It remains to showthatwecanfindsucha E 8 andz' E X'satisfying ~ ~ ~4 - z ' ~ eqand
But since 8 is dense in 'if,we can find u E 8 such that
Taking such a fuzzy coalition a E 8 and writing
we check that
(33)
V i E A,,
llyi-zill =ST.
Also,by (28(i)) and (32),
c w- c ziY(o+ c aLv(l? = c aiJ@E c atY(i).
c
=
fE-47
I€Ar
l€Ar
i€Ar
if&
G47
We have therefore proved that the fuzzy coalition u E 8 rejects F(x) which is a contradiction.
* Interpretation of Debra-Scarf
economic games
Consider a Debreu-Scarf coalition
:{
a = -, -,
...,&I 4
where q is the number of players of each type. We assume that each player of type i has the same characteristics, i.e. the same consumption set R', the same endowment Y(Q,the same loss h c t i o n A. We also assume that the allocation of each player of the same type i is ths same. Therefore, we can write the set of commodities dvailable to u as
324
MAIN SOLUTION CONCEPTS OF COOPERATIVE GAMES
[Ch. 10, 0 10.4
i.e. n times the average of the sum of initial endowments brought by the players involved. The set of allocations x E X(u) is defined by X(a) = { x E RAusuch that pix' E CTp1piY(i)}. (35) We can interpret A! as the (common) commodity allocated to each player of type i and p'x' as the commodity allocated to the ith type of player. Therefore, an allocation 2 = {Zi ..., Y }E X(N)(where x' is the common commodity allocated to each player of type i) is rejected by a Debreu-Scarf coalition of p l players of type 1, . . .,p,, player of type n if this coalition can allocate to each player of type i a commodity x' such that
*10.4.8. Fuzzy games described in strategicform and fuzzy core
We introduce fuzzy games { X ( z ) , F }described in strategic form and, in particular, extend a cooperative game {X(A), FA} described in strategic form to a fuzzy game {X(z), F}.Since the loss functions f;' of players i of the coalition A can depend upon strategies of all players i, we are led to use a multilosa correspondence F :X' R' of the fuzzy coalition 'G instead of a multiloss operator. This slight modification does not complicate the situation too much. We begin by defining
-
137)
M(z) = {m E R$ such that CAcdm(A)cA = z}
i.e. M(z) is the set of {m(A)}Acdwhere
(38)
V A E d,m(A) B 0 and Vi E N,
Notice that (39)
C m(A) = zi. A31
if rn C O'l(z), then A c A, whenever m(A) > 0.
Canonical multistrategy sets XO('G)of a fuzzy coalition
W e assume that (40)
V A E d,X(A) is a closed convex subset of
Ch. 10,s 10.41 BEHAVIOR
OF FUZZY COALITIONS: THE FUZZY
cow
325
where we define x‘ = CAE& (m(A)/z)xAby
Debition 4. We define the “canonicalmulthtrafegy mbJeet” Xo(t) c UA‘of
the fuzzy coaLi~onz to be the set of multi‘strategies xt
as m ranges over m(r)and
=
over X ( A ) for all A E d.
CAE& (m(AjdJ{z
Construction of the multiloss correspondence F of a fuzzy coalition Definition 5. The multiloss correspondence F from UA7into R‘ is defined by (43)
F(X‘)
=
}C,
d rn(A)xAlr=x*
Definition 6. We shall say that {Xo(t). F},,,describes the “canonical extension” of the cooperative game {X(A), FA}AEdinto a fuzzy game described in strategic form. We set
(44)
{ (ii)(i) Ff,(xr)
+
= F ( x z ) R;
f i ( x r ) = Fqx‘)+k;.
Consider now a fuzzy game {X(z), FyEa described in strategic form, where, for any F E 53,
(45)
is multistrategy set, { (ii)(i) XEt( z X) c( z )UAr -Rr is a multiloss correspondence. a
:
Dehition 7. We shall say that a multistrategy 2 E X(N)is rejected by a fuzzy coalition t if there exists x’ E X ( r ) such that z-F(Z) E c ( x . > . The core @(X(r),F) of the fuzzy game is the subset of multistrategies Z E X(N) which are not rejected by any fuzzy coalition z. ‘The notion of a canonical extension {Xo(r),F} leads to the following terminology. Definition 8. Let {X(A),FA} be a game described in strategic form. We shall say that Z E X ( N ) is “canonically rejected” by a fuzzy coalition t E ‘z; if there exists x‘ E X o ( t ) such that zoF(2) E ?+(xr), i.e. if there exist rn E &(t) a n d p E X ( A ) such that (46)
V i E A,,
ziJ(Z) =-
C m(A)f“(xA).
IEA
326
MAIN SOLUTION CONCEPTS OF COOPERATIVE GAMES
[Ch. 10,s 10.4
The "canonicalfuzzy core'' of the game {X(A), FA} is the subset of multistrategks ff E X(N)which are not canonically rejected by any fuzzy coalition 7. The characteristic form (Z,J ) of the fuzzy game {X(z), F} is obviously defined by (47)
J ( z ) = P ( X ( z ) )=
u P(x).
xr€ XLlW
We next show how convexity, continuity and compactness properties of a game {X(A), are carried over to its canonical extension.
Proposition 4. Suppose that for any A E d (48)
convex and compact, { (ii)(i) XV i( AE )A,is hA is convex, lower semi-continuous.
Let {Xo(t), F},be the canonical extension of {X(A),F A } A E dThen, . for anyfuzzy coalition z, (49)
[
(i) XO(Z)is convex and compact, (ii) Vxr E XO(Z),F+(P) is closed and convex, (iii) F+(Xo(t)) is closed and convex.
an4 V l E ATn
(50)
XI
t-.-f'(x';
A) = d ( P ( x r ) , A ) is convex.
a(%)
Proof. (a) We begin by noticing that is a convex compact subset of R$. (b) Therefore, if x', = (mk(A)$)/t E Xo(z) for any k = 1, .. ., K, we can write any convex combmation x' = akxi in the form
cAcf
cf=l
where m(A) = Ifsl akmk(A). Furthermore Xo(t) is compact, being the continuous image of the compact subsets m(z) and X(A). (c) We prove that F+(x') is convex. It is sutficient to prove that
belongs to F+(x'), whenever m(A) = apk(A).
Zs1
Led( m k ( A ) 4 ) / t = x'
for all k, where
Ch. lo,§ 10.41 BEHAVIOR
OF FUZZY COALITIONS:THE FUZZY CORE
327
Since the functionsAAare convex, ProposiGon 1.3.10 implies that
ckK,l
where y A = (akfflk(A))/m(A)4 belongs to the convex subset X ( A ) . Therefore, since CAE& ( m ( A ) / t y" ) = x', inequality d z- Ledm ( A ) F A ( f ) implies that d E F+(x7). (d) We prove that F+(xy is closed. Since the subsets m(z) and X ( A ) are compact and the maps
{m, {X"}"} t--+
C m(4AA(xA)
i€A
are lower semi-continuous, Proposition 2.1.4 implies that F+(x9 is closed. (e) A proof analogous to the proof of the convexity of F+(x') shows that F+(Xo(t))is convex. (f) F+(Xo(t)) is closed because the subsets X X ( A ) are compact m(A)fiA(#) are lower semi-continuous and the maps {m,{ / } A } F+ is closed). (Proposition 2.1.4 implies that F+(Xo(r)) a,& be a convex combination. For any E > 0 there (8) Let x' = exists mk E and 4 E X ( A ) such that
m(t)
m(t)
Remark.
because
nACd
328
MAIN SOLUTION CONCEPTS OF COOPERATIVE GAMES
[Ch. lo,§ 10.4
Remark. Consider the case when, for all i E N, (52)
1
(i) Ri is a closed convex subset of a vector space U', (ii) Y ( i )is a closed convex subset of a vector space V, (iii) L~E &(ui,V ) is a continuous linear operator
and the multistrategy sets X ( A ) are defined by X ( A ) = { x A E RA such that '&AL~X~.' E Y(A)}. (53)
Proposition 5 . Suppose that (52) and (53) hold. Then the canonical multistrategy set Xo(z) of the fuzzy coalition z is contained in the multistrategy sets X ( z ) defined by x(Z)= {X' E R A T such fhaf ~ , E A ZiLiX'" , E Y(t) = x i c A T ZY(i)}. (54)
Proof. Any x' defined by x',' = fies
&., ( m ( A ) / t , ) P * 'where 2 E X ( A ) satisn
n
Remark. Consider the case where X ( A ) = X A and (55)
the lossfunctionsfiA = fi are convex functions from R' into R.
Then (56)
p+(xr) = {zifr(Xi)}IEA7+ R'+
To prove this, we observe on the one hand that { ~ , f i ( . ' ) } ~ ~ ~belongs , to
m(t)
where m E is defined by m(A) = ziif A = {i} and m(A) = 0 otherwise. On the other hand,
+
p+( x ' ) = { ~ i f i ( X ' > } iE A% R ', since, for all i € N,
because the the loss functions are convex. 0 We shall extend this remark to the case when f i A 11.2.5 below.
=AAx. See Proposition
Ch. 10, $ 10.51
SELECTlON OF ELEMENTS OF THE CORE
329
10.5. Selection of elements of the core: cooperative equilibrium and nucleolus
-
Theorem 4.1 characterizes the elements of the core of a game (Z,J ) as the subset of c E J ( N ) such that
a&c) = sup 7-e-
inf [(A, z - c ) - u ~ ( J ( z ) ;A)]
A€A(7)
0.
We are led to look for elements c E J ( N ) such that /?,(c)
= inf sup [(A, z . c ) - u b ( J ( t ) ; A)] =s0. a E m
7p.z
(called canonical cooperative equilibria) and/or for elements which minimize the function a,, which form the “least-core” of the game.
10.5.1. Canonical cooperative equilibrium We begin by defining the concept of a canonical cooperative equilibrium. Definition 1. Let (Z,J ) be a fuzzy (or usual) cooperative game. We shall saythat a multiloss c € J ( N ) is a “weak canonical cooperative equilibrium” if there exists a Pareto multiplier X E Ansuch that
We shall say that it is a (strong) “canonical cooperative equilibrium” if there exists a Pareto multiplier X of c which is strongly positive. The following statement is obvious. Proposition 1. Any (strong) canonical cooperative equilibrium of a game (Z,J) belongs to its core @(Z, J).
Proof. Let 1 E &’be a Pareto multiplier of the canonical cooperative equilibrium c. Then (X/CIEArlI) E M(z) for any fuzzy coalition z. Therefore,.. for any t E 53, inf [(A, z . c ) - d ( J ( t ) ; A)] a w w 4
-S
1
7 [(I,z*c)-a(J(z);X)] 8s 0
c I’
fEAr
by (3). Hence aa(c)
4~
0 and thus, c belongs to the core @(G, J) of (21,J). 0
330
MAIN SOLUTION CONCEPTS OF COOPERATIVE GAMES
[Ch.10,s 10.5
Example. (Games in strategic form.) Consider a game {X(A), described in strategic form. Then c = F(3) (where 3 E X(N)) is a weak (resp. strong) equilibrium if there exists 1 E &'(rap. E &')such that, (4)
Weshall say that 3 C X(N) is a weak (resp. strong) canonical cooperative equilibrium if(4) holds with 1 E &'(resp. E &).
x
We can prove a stronger result than Proposition 1.
Proposition 2. Any (strong)canonical cooperativeequilibrium3 E X(N)of a gmne {X(A),FA}Acdbelongs to its canonicalfuzzy core. Proof. Suppose that x' E X(N) satisfying (4) is canonically rejected by a fuzzy coalition z. We deduce a contradiction. There exists x' = CAEd m(A)p/z such that, for all i E A,, t f A ( Z ) Z- E A 3 i m ( A ) f f ( # ) .Hence
'Therefore, by (4), we obtain that
=
c PzJ@). n
i=1
This is a contradiction. 0 We shall prove the converse statement in Theorem 12.1.2. Example. (Economic game.) Consider an economy {Ri, Y(i)f}fEwAny Walras allocation T = {a, . ., Z"} belongs to the fuzzy core.
.
Proposition 3. Any Walras allocation Z = {Z, . its canonical fuzzy core.
. .,Z"} of an economy belongs to
Proof. Let {%p} be a Walras equilibrium. Suppose that a f a y coalifion z rejects 2. Then there exists an allocation x' E X ( z ) such that ji(x'*') -=fi(Z') for all i E A,. Since Zi minimizes f;( -) over B,@, ri(p)),these strict inequalities imply that x'.' B. Bi@, r,(p)), i.e. (p, x'") =. ri($) for all i E A,. Hence
Ch. 10, $i 10.51
SELECTION OF ELEMENTS OF THE CORE
@, &,zlx"')
'&cAzziri(j7).This is a contradiction: In fact x'
331
E X ( z ) , i.e.
Remark. We shall see in Chapter 12 that, under additional assumptions, the fuzzy core is (almost) the set of Wdras allocations (see Theorem 12.3.1). '10.5.2. Least-core
We now describe a procedure for selecting a Pareto minimum which belongs to the core whenever it is not empty. Consider a game(&, J).We introduce the lower support functions of the subsets J(A) and put w(A,A) = d ( J ( A ) ,1). Then Theorem 4.1 implies that, if a multiloss c E J ( N ) minimizes over J ( N ) the function a = ad defined by
then E belongs to the core whenever a(E) =s0. It is quite natural to select from the elements of the core those which minimize a( .). Furthermore, we can define elements which minimize a( .) even when the core is empty. Definition 2. The "least-core'' is the subset of elements c E J(N) minimizing over J(N) the function a = ad defined by (5). , Proposition 4. Suppose that (6)
The subset J ( N ) is closed and bounded below
Then the least-core is non-empty and compact. It is contained in the set of Pareto minima. If the core is non-empty and if the subsets J(A) R$ are convex, then the least-core is contained in the core.
+
Proof. The function a is lower semicontinuous. To see this observe that, since {& c} F-- (A, P .c} - w ( ~ ,1) is lower semi-continuous, the function c -I
8 (A, c) = inf [(A, P - c } - W(A, A)] l € A A
is also lower semicontinuous because M Ais compact (see Theorem 2.5.1). Hence a is lower semi-continuous. The function a is also lower semi-compact. If a(c) 4 g, then for any individual coalition i, Oi (c) = c, - w(i, 1) =Sq.
332
MAIN SOLUTION CONCEPTS OF COOPERATIVE GAMES
[Ch. lo,§ 10.5
Therefore, c 4 w + 11 1 where w = {w(i, l)}, N. Since c is larger than the shadow minimum, we deduce that c lies in a relatively compact subset. Therefore, there exists an element c which belongs to the least-core. Let c belong to the leastcore. It is Pareto minimal. If not, there exists e z 0 such that d = c-el belongs to J(N>. In this case, for any A E d?and A E J?!(A), we obtain (A, 8d) = (A, #c)-e. Hence B(A, 6) = B(A, c)-e and a(d) = GC(C)--E. This is a contradiction. Now, if the core is not empty, then infdcJ(N)a(d) 0 (by Theorem 4.1) and thus, if c belongs to the leastcore, a(c) =s0 and c belongs to the core. 0
Remark. By replacing a family UI?of usual coalitions by a family .Zj of fwzy coalitions and the function a by %, we can define the concept of “@zy leastcore” as the mirlimal set of aa. This is obviously contained in the leastcore and in the fuzzy core whenever it is non-empty. Also, we can prove that assumption (6) of Proposition 4 implies the non-bmptiness of the fuzzy leastcore. To investigate further properties of the least-core, we shall make an assumption analogous to assumption (16)of Proposition 10.1.8. Let Jo(N) be the subset of multilosses c E J ( N ) satisfying ci 4 w(i, l)+infdc,(ma(d).
(7)
For any c E Jo(N) and any non-empty coalition A f N, we can find e0 =- 0 such that, for any e > 0 less than eo there exists d E J ( N ) satisfying di 4 ci- es for any i E A and ,C -si d, e c,+ e for any j E A (where p =- 0).
Proposition 5 . Suppose that (7) holds. Let Q3 be a family of coalitions satisfying
Then,for any c belonging to the least-core, we have (9)
a(c) = max BE a
B(B, c).
Proof. Let c belong to the least-core and let @ be the family of coalitions A such that B(A, c) = a(c). If @ = d,then (9) holds for any family B of coalitions. If not,
b = rnax 8(A, c) 4 8
< a(c).
Write e l = 1/3(a(c)- b) and E = min(e0, E ~ where ) EO appears in assumption (7).
Ch, lo,§ 10.51
SELECTION OF ELEMENTS OF THE CORE
333
By assumption (7), there exists a multiloss d E J(N) such that di = c,- e& for any i E Bo, c, 6 a,’ == c,+ E for any j 6 Bo,where BO is the coalition defined by (8). Jf A c Bo, we have 8(A, d)=S 8(A, c)- ee and, if A Q Bo, we have - ee =S e(A, 4- e(A, c) 6 &. We now assume that property(9) is false,i.e. j3 = max,,, O(B, c) -= a(c). Hence, when 8(A, c ) w p, the coalition A does not belong to CB and thus, is contained in Bo by (8). Such coalitions A exist and satisfy 8(A, d) 8(A, c)- ee 6 a(c)- QE. On the other hand, if 8(A, c) UG B, then either 8(A, d)e = ~ 8 ( A , c ) - e e = s a ( c ) - or ~ ~8 ( A , d ) ~ 8 ( A , c ) + E ~ B + & ~ b +=Ea1( c ) - 2 ~ 1 . Hence a(d) = maxAcd8(A, d)ES a(c)-el min(2, e) < a(c). This is impossible since c belongs to the least-core. 0 Corollary 1. Suppose that property (7)holds. For any non-empty coalition C # N , and for any multiloss c of the least core, we have that (10)
a(c) = max O(A, c). AUC
In particular,for any player i E N,we have that (1 1)
a(c) = max f€A
8(A, c).
Proof. This is left as an exercise. 0 ‘ *10.5.3. Nucleolus
A drawback in using the least-core as a selection .procedure is that it can be large. We are therefore led to define a finer selection procedure which yields a subset of the least-core called the nucleolus. Interpret (12)
8(A, c) = inf [(A, &,MA
zA.c)-w(A,
A)]
as the “complaint of the coalition A” associated with c. Then the elements of the least-core minimize the “loudest complaint” a(c) = sup,,& 8(A, c). Schmeidler suggested a more sophisticated selection method, which first, minimizes the loudest complaint (i,e., constructs the leastcore)
second, minimizes the second loudest complaint in the least-core, 0
and so on.
334
MAIN SOLUTION CONCEPTS OF COOPERATIVEGAMES
[Ch. 10,s 10.5
To formalize this idea we need to define a “classification map” o which arranges the components of 8 in decreasing order. Having so re-classified the complaints, we then sekk the smallest of these with respect to the lexicographic ordering.
Definition 3. Let lo?I be the cardinal of a Gnite set d?. We shall say that the map into Rld I associating with any 8 = {€J(A)IAEd E Rd the vector oB
w from Rd
defined by
is a classijication map.
In other words, the components of 0 6 are the components of 8, but classified in a decreasing order. It is clear that o is a continuous map. The map e(c) : A F+ e(R, c) (14) can be identified with the vector in R” of complaints associated with the multiloss c. Then o8(c) is the vector of complaints classified in a decreasing *der.We then introduce the lexicographic ordering on RIdI defined by 3j such that (15) x -s y This is a total preordering.
xi
= yi V i ==j- 1,
XI > yj.
Definition 4. We shall say that Z E J(N) belongs to the “nucleolus” of the game if the classified vector of complaints we(.?) is smallest with respect to the lexicographic ordering. By the very construction of the nucleolus, it is a subset of the least-core (and thus, is a Pareto minimum). It is constructed in the following way. We define the least-core, i.e. (16)
JI(W= {cl E J ( N ) such that (rn8(cl))1 = infcEJ(N)(oe(c))l}
Secondly, we define
J d N = {CZ E (17) and, in the same way,
JW)such that (wB(c3)t = inf,,CJ,(N) ( 4 c d ) d
Jk(N) = {Ck E Jk-i(1v) such that ( O e ( C k ) k = minq-,EJi-l(N)( W @ k - l ) k ) ) . It is clear that the nucleolus is nothing other than the set JIdI(N)-
(1 8)
Ch. lo,$ 10.51
SELECTION OF ELEMENTS OF TRE CORE
335
Proposition 6. Suppose that
(6)
J ( N ) is closed and bounded below.
Then the nucleolus is a non-empty compact subset of the least-core.
Proof. The maps c I-- B(A, c) are lower semi-continuous (see the proof of Proposition 4) and the maps c F+ (wO(c)), are also clearly lower semi-continuous. We also know that assumption (6) and Proposition 4 imply that the least-core J @ ) is non-empty and compact. Therefore, the subsets J,(N) are also non-empty and compact. In particular, the last of these, JId,(ZV), which is the nucleolus, is compact and non-empty. 0
Remark. We shall prove in the next chapter that the nucleolus of a game With side-payments consists of a unique element (see Proposition 11.4.5).
CHAPTER 11
GAMES WITH SIDEPAYMENTS
rn this chapter, we study games and fuzzy games with side-payments. We consider conditions under which the core is non-empty and introduce the concept of the Shapley value and some related ideas. We begin by defining a game (d?, v) (resp. a fuzzy game (Z,v)) with sidepayments to be a game (&, J ) (resp. (T, J ) such that V A E d, J(A) = {c € RA such that
(resp. Vz E ‘77,
J ( t ) = {c E R‘ such that
ci = v(A)}
C;=lci = ~(z)}).
In other words, the function v, called the characteristicfunction of the game, is a lossfunction which associates with any coalition A (resp. fuzzy coalition z) its loss v(A) (resp. v(z)) measured in (monetary) side-payments. The rules of the game require that the loss w(A) is shared by players i E A (resp. that the loss v(z) is shared by players participating to z proportionally to their rates of participation). This implies that J ( N ) (or J(P)) is the set of Pareto minima.
The core @(&,v ) is easily expressed. It is the subset of multilosses c satisfying (9 Z L l CI = W), (ii) V A E A, &A ci .=s v(A). Hence, an element c E @(&,v) displays rather strong stability properties since, for any coalition A, the loss CEAq it receives is not larger than the initial loss v(A) assigned to A. For instance, we can associate with a cooperative game {X(A),FA}AEddescribed in strategic form, the game (04, v ) with side-payments defined by
336
337
In the case of fuzzy games (55, v) where the Characteristic function v is assumed to bepositively homogeneous,the core e(55,v), which is the set of multilosses c E R" such that
I
ci = v(zN), (i) (ii) Vz E 55, zici == v(z),
is nothing other than the subdifferential of v at z", i.e. @(Z, v) = av(.">. Hence, if v is also convex, the core of the fuzzy game is non-empty. Note that if v is convex and positively homogeneous, it satisfies inequality V(Zl+ z2) =s
v(r1) + V(%)
which means that "l'union fait la force" (unity is strength). We note also that @(%,v) c @(of, ~ 1 ~ ) . i.e. that the core of a fuzzy game is contained in the core of its restriction to the family d of coalitions. The converse problem arises. We can associate with any characteristic function v E Rd its convex covernv defined by
nv(r)
= inf
m€dW
C
m(A) v(A)
where d ( z ) = {m E R,d such that CAEdm(A)? = r}. The characteristic function nu is clearly convex and positively homogeneous. We shall prove that the core @(d, v) = @(%,nu) = am(?) is non-empty if and = v(N). We shall say that a characteristic fqction v is balanced only ifnu(.") if nv(zN)= v(N). Examples of balanced characteristic functions are given. We mention here only the following result due to Scarf. The characteristicfunction v defined by
is balanced when the functionsA are convex. Several objections can be raised against the concept of core. For instance, the fact that the core can be empty for certain types of game is a drawback. Another approach, introduced by Shapley, is to evaluate a game a priori, by assigning to a characteristic function v a multiloss v,,v E R".This is required to obey the following axioms. (1) ynv is Pareto minimal. (2) v,,v does not depend on 24
338
GAMES WITH SIDE-PAYMENTS
[Ch. 11, Q 11.1
how the players are ordered. (3) The loss assignedto a coalition of any partition is the sum of the losses assigned to each player of the coalition. (4) q,,is linear. We shall prove that there is a unique sequence of such linear operators p,, (called fuzzy values) which associate with any positively homogeneous characteristic function differentiable at 7?‘ the multilosses. q,,v = Dv(zN) (gradient of v at zN).
Therefore, e(5,v) = {p,,w}whenever the characteristic function is positively homogeneous, convex and digereentiable at zN. Of course, we can apply the fuzzy value qnto the convex cover fiv of a characteristic function v E Rd. This requires first that zv is differentiable at (this is not always true). Another drawback is that the map v t - g p v is no longer linear, since is not a linear operator. This suggests introducing the Cornet linear extension operator w , associating With any v E Rd a positively mv(t”) = v(A). We shall prove homogeneous function wv such that, V A E d?, that the map yn = pnw : v q,,,,ov = D[ow](2”) is the Shapley value. This is the unique linear operator y,, E &Rd, R”)mapping v into a Pareto minimal multiloss ynv, which does not depend on how the players are ordered and which assigns 0 to each dummy pIayer i (in the sense that w(A U { i } ) = v(A) for any coalition A).
-
11.1. Core of a fuzzy game with side-payments
We define games and fuzzy games with side-payments and describe several examples. Then we prove that the core of a fuzzy game @(%,w) is equal to av(.“) when v is positively homogeneous and thus, that it is non-empty whenever v is also convex. We investigate the case where v(z) = suppcpw(z, p). Under convenient assumptions, the core is the closed convex hull of the gradients D w ( P ; p ) as p ranges over the maximal set of w(zN;.). We apply this result to characterize the core of a “‘marketgame” in terms of “sidepayment competitive equilibria”.
Definition 1. Let 1 E (1)
k:.
We shall say that a (fuzzy) game (5, w) defined by
VZ E 5,J ( T ) = {C E
RAr
such that (A,
C) = w(z)}
is a game with side-payments (or with transferable losses) and that the function w :t E 5 ++ w(z)is the “characteristic function” of such a game, which associates with each coalition z its loss ~ ( 2 ) .
Ch. 11,s 11.11
CORE OF A FUZZY GAME WITH SIDE-PAYMENTS
339
The subset of feasible multilosses of the fuzzy coalition z is nothing other than the set of multilosses transferable with the amount w(z) under the rate of transfer A E & .: Such a game describes the case where the players of a fuzzy coalition agree to share a given side-payment allocated to the coalition with respect to the same system of eights A.
Remark. Even if it entails replacing c by Ac = { A i ~ , } I E N ,we can always take A = {I, .. ., 1). In this case, we set (77, w ) = ~ (Z,w). Due to the special form of the sets of admissible multilosses of a game with side payments, a nice description of its core is available. Proposition 1. Let A E p : bejixed. The core @(%,w) of a(fuzzy)game with side payments (53,w ) is~ the set of multilosses c satisfving
0) (A, c)
= w(N),
(ii) v z E 5, (A, z - c ) 4 w(z). Proof. It is enough to prove that for any t E ‘zi (3) and
J+(z) = {c E RArsuch that (A, c )
(4)
j+(z) = {c E RAr such that (A, c ) r w(z)}.
w(z))
(If (4) is true, to say that “z does not reject"^ amounts to saying that (A, z-c) e 4 w(z). In particular, (A, c) w ( N ) since c is not rejected by N and (A, c) 2 Z= w(N) since c belongs to J+(N).) Statement (3) is quite obvious. We prove statement (4). Observe that c E 1, (z) if and only if there exists d c J+(z) and b E fi: such that c = d+b. Therefore, (A, c) > (A, d ) a w(t), Conversely, if s = (A, c)- w(z) > 0, the vector b with components b, = = s/Ai where i E A,, belongs to and satisfies (A, b> = s. Therefore, the multiloss d = c- b is feasible for the coalition z since
*+
(A, d) = (A, c)-s = w(z). Hence c E
I+(t).0
Example. (Games with sidepayments associated with cooperative games). Let { X ( A ) ,FA}Acctbe a cooperative game described in strategic form. Let A E &’ bs fixed. 249
340
[Ch. 11, 0 11.1
GAMES WITH SIDE-PAYMENTS
It is natural for the coalition A to pool their losses. We introduce (5)
which measures the smallest transferable side payment of the coalition A associated with the rate il E &. We can therefore 'forget about the underlying multistrategies and just consider a game where coalitions A choose utransferable multilosses" instead of multistrategies or feasible multilosses. The core @(d, W ( A ) ) ~of this game with side payments is the set of multilosses satisfying
(6)
{ (6)
(i) &N Aici = infx,Zx(mC Aff;(x), V A E d,&A AiCi 4 infxEX(A)&A
ASA($.
Remark. If there exists x E X(N) such that c = F(x) belongs to the core @(&, w(A)), of the associated game with side-payments, it is clear that c is a canonical cooperative equilibrium of the game (d, J ) (see Section 10.4.1). We shall use this remark in the next chapter. 11.1.2.Linear games
W e shall say that a game with side-payments(d, w ) (resp.a fuzzygame(5, w)
is "linear" (or "inessentifl) if
(7)
C w(i> [resp. v z E 'i5, w ( t ) = C ziw(i)] V A E of, w(A) =
i€A
i EN
It is clear that the core @(%,w ) of a lineargame contains the unique vector
+m)i€ N
Example. (Games with side-payments associated with an economy). Consider an economy {Ri, Y(i),h}icN(see 8.3). If a price system p E R'* and a rate of transfer il E RC prevail, we measure the smallest transferable.side-payment of a fuzzy coalition z E [0, l]" by
where
is the smallest net loss of the it" player. (Recall that r,(p),= sup,,cy(f)(p, y ) is the income allocated to the ith player.)
Cb. 11,s 11.11
CORE OF A FUZZY GAME WITH SIDE-PAYMENTS
341
We set (10)
w(A; A, p ) = w(zA,A, p ) =
C w(i, A, p). f€A
Therefore, the fuzqgarne (53, w(A, p ) ) with side-payments associated with the economy is a linear game. Its core contains the unique vector {w(i, l,p/Ai)}fEw
Remark. If there exists Z E X(N)such that E = F(%)belongs to the core @(d, w(1, p ) ) = @(Z, w(1, p)) of the game with side-payments, we notice that the pair {Z, p } is a Walrus equilibrium because, writing pi = 1/1‘, we have that
, x ) - w(i, 1,
f . ( ‘
p i F ) -s
inf
f;:(x?.
a€B#(F,QG))
Also, for all i E N ,
A(%’)==j;:(??+(p, 2)-.ri(p).
cZl
But, since 3 E X(N), ((p, ?‘)-r,(F)) e 0. Therefore, (jj, 2’) = ri(p) for all i. In particular, Zi E B,(jj, ri(jj))minimizes f I ( .) over B,@, rf(jj)).This remark will play an important role later.
Remark. Notice that w(i, A‘, $) is the dual utility function o f the problem (1 1)
inf ;liJ (xi). XWJ~G. .to)
11.1.3.Non-emptiness of the core of fuzzy games with side-payments
We take A = 1. We consider the class of characteristic functions v defined on R: which are positively homogeneous. In other words, the characteristic functions v are defined on the subset (12)
{Z
E
[o, I]”
such that
&N
Zi
= 1)
R of fuzzy coalitions. They are extended to :
v(0) = V(Z@) = 0
by setting
ifz = 0.
Proposition 2. Suppose that the characteristicfunction v of the game is positively homogeneous. Then the core of the game is equal to the subdverential av(.”) of v at P.
342
[Ch. 11, 0 11.1
GAMES WITH SIDE-PAYMENTS
Proof. If c belongs to the core, we deduce that for any z E :R (2, c)-
v(z) =s(ZN, c)- v(zN) = 0
and thus, that c E av(."). Conversely, if c E av(.">, we deduce from inequalities ,".(-)w c) == v(z)-(z, c) with z = 0 and z = 2". that w(z") = (.", c}. Therefore, (z, c) -G V(T) for all z E.:R 0 In particular, if v is a positively homogeneous function defined on R", we obtain the following result.
Proposition 3. Suppose that the characteristicfunction v satisy7es
t 14)
into R.
v is convex and positively homogeneousfrom
Then (15)
the core of the fuzzy game is non-empty, convex and compact.
Furthermore, if we assume that (16)
v is di@'erentiable at zN = {I,
__. ., l},
then the core consists of the single vector D v ( ~ ) . Proof. Since zN belongs to the interior of R", the convex function v is continuous at ."and thus, subdifferentiableat p (Proposition 4.1.7) and the subdifferential av(.") is convex and compact (Proposition 4.1.7). Therefore, statement (15) follows from Proposition 2. The core av(Z") contains a unique element if and only if v is differentiable at and, in this case a v ( P ) = { D v ( p ) }(Proposition 4.2.3). In fact, Propo3ition 4.3.4 implies that the derivative from the right of v at zN, namely (17)
Dv(zN) (2) =
cc
sup (z, c} = o*(@(ris,91); e m , U)
c),
is the upper support function of the core of the fuzzy game. Interpretation of positively homogeneous convex characteristic functions Since any positively homogeneous convex function is subadditive, (18)
v(z f 0)'G v(z) f 2(u)
we can say that coalitions gain by joining forceJ. In particular, if A and B are and we obtain disjoint coalitions, #+9= P U B (19)
U(+B)
=sv@A)+v(zB).
cl
Ch. 11,g 11.11
343
CORE OF A FUZZY GAME WITH SIDE-PAYMENTS
Consider the case when z is no longer differentiable at zN,but is the pointwise supremum of functions differentiable at p.
Proposition 4. Suppose that the characteristicfunction v is defined by (20)
V t E R$, ~ ( z )= SUP w ( z , p )
where
[
(21)
PEP
(i) P is compact, (ii) Vz E R$, p -t w(z, p ) is upper semi-continuous, (iii) V p E P , z F- w(z, p ) is convex, positively homogeneous and differentiable at zN.
Let P(N)be the subset of P defined by (22)
P ( N ) = { p E P such thar v(zN)= w(zN,p)}.
Then the core is the closed convex hull of theset { D w ( P ;p)}pEp(N, of gradients at 8'of the functions w( p ) as p ranges over P(N).
.,
Proof. The convex positively homogeneous function v is finite on : R (i) and (ii))). Therefore, (23)
(by (21
Dv(zN) (z) = fJ"(@(Z, v); 2).
On the other hand, Proposition 4.3.6 implies that
Therefore, @(Z, v) = CO [{Dw(zN,P ) } ~ E P ] . 11.1.4. Core of fuzzy market games
Associate With an economy {R', Y(i),A},cNthe 'Ykzzy market game" defined by the characteristic function
where the set X ( z ) of allocations of the fuzzy coalition z is defined in Section 10.4.7.
Consider the saddle points {Z, p} of the Lagrangian
344
GAMES WITH SIDE-PAYMENTS
[Ch. 11, g 11.2
of the minimization problem v ( N ) = infx,xcw cf;(x’), i.e. the pairs {Z, P } E E X(N)XRY satisfing v ( ~= ) Z(x, p) = min Z(x, li> x€RN
and
We shall say that such a pair is a “side-payment competitive equilibrium” of the market game. Associate with any side-payment competitive equilibrium {Z, PI the vector,
(27)
4%
= {fr(Z‘)+
(P, 2‘)- ri(p)}i€N.
Proposition 5. Suppose that assumptions (9, (9,(6), (7),(8) and (9)of Theorem 8.3.1 for the existence of a Walras equilibrium holds. Then the core of the fuzzy market game is the closed convex hull of the set of vectors c(X, jj) as { X , j7} ranges over the set of siakpayment competitive equilibria. Proof. Since the subsets R’ and the loss functionsf, are convex and since 0 E Int (Ri-Y(i)) for all i, we deduce that there exists a Lagrange multiplier p , E RY of the minimization problem
(28)
v(z) =
inf
C zi(ff(x7-1) +(p7, xT.i)-
x~€Rr 1
ri(p,)).
We prove later that these Lagrange multipliers p7 remain in a fixed convex compact subset p of RT (Proposition 12.3.1). We notice that P ( w is nothing other than the set of Lagrange multipliers of the minimization problem w(.“), i.e. the set of side-payment competitive equilibrium prices. On the other hand, the characteristic functionsz I-+ w(z; p ) are linear. The core of the associated game consists of the single vector {w(i,p)}iEN‘Proposition 4 implies that the core of the fuzzy market game is the closed convex hull of the subset of vectors {w(i, p)},,, as p ranges over p E P ( N ) . But w(i, p) =A(:’)+ +@,2!)-ri(P) where x’E X(N) ,minimizes over XN. Hence the proposition is proved. 0
zeNf,(x’)
11.2. Core of a game with side-payments
Since the core of a fuuy game is contained in that of its restriction, we can use the existence result, Proposition 1.3, whenever we can in some sense “extend” a game (&,v) into a fuzzy game(-, sw)where z w is convex and positively homogeneous. This can be done by using the “convex cover operator” 7c de-
Ch. 11, 5 11.21
CORE OF A GAME WITH SIDE-PAYMENTS
345
fined by
where M(z) = { m E Rf such that’&,d
m(of)rA = r } .
We then prove that @(&, v) = @(%,nv) = anv(zN)# 0 if and only if’ v ( N ) = ZW(.“). (In this case we say that w is “balanced”). On the assumption that the family {X(A)IAEdof multistrategy sets is “balanced”, i.e. t/m E a(zN),
C
m ( 4 X(A) c X ( W ,
A€&
we shall prove that the characteristic function v defined by w(A) =
inf
CAA*(xA)
xA€X(A)
i€A
is balanced whenever the functionsfi are convex. 11.2.1. Convex cover o f a game
We have proved that the cores of fuzzy games (21, w) defined by convex positively homogeneous characteristic functions are non-empty. This implies that the core of their restriction ( o f , wid) to families of usual coalitions is also non-empty. The converse problem arises. Can we in some sense extend a game (Oe, v) with side payments into a fuzzy game (21,nw) in such a way that the cores of the two games coincide.
Definition 1. Let of be a family of coalitions and let z be a fuzzy coalition. We shall denote by (1)
M(z) = { rn E Rf such that
m(A)zA = z}
the subset of non-negative weights m(A) such that (2)
b’i EN,
ri
=
C
m(A).
A€ d A3i
We shall say that M(.”) is the set of “balances”.If v : A I-+ R is a characteristic function, we shall say that the function nw :% t-+ R defined by
is its “convex cover” and that n is the “convex cover operator”
346
GAMES WITH SIDE-PAYMENTS
[Ch. 11,§ 11.2
Remark. Notice that (4)
If m E (En(z), then m(A) =- 0 implies that A c A, = {i E N I zt > 0).
The following proposition summarizes the obvious properties of the map a. Proposition 1. For any w, IIV is a positively homogeneas, convexfunction d e m on,:R satisfying (5)
tfA E d, nw(zA)=sw(A).
Proof. This is left as an exercise. 0
Remark. The map II ,does not have the interpolation property. Therefore, nw is not a true extension of 2’. This fact motivates the following definition. Dellnition 2. We shall say that a characteristic function v from d into R is “balanced” if
and that it is “totally balanced” if (7)
VA€
04, w(A) = IIV(Z”).
If v is totally balanced, the convex cover IIU of w is an extension of w. With these definitions, we can characterize the games having a non-empty core. 11.2.2. Non-emptiness of the core of a balanced game
Theorem 1. The core @(d, w) of a game with sidepayments(d,w) is non-empty ij-and only if w is balanced. In this case @(d,w) = @(/is, IZW). Proof. We begin by proving that, if there exists c E @(&, v), the function v is balanced. Let m E be a balance. We obtain that
m(.“)
v(N) =(+“c)=
C
C E d
m(A)tA, c )
Ch. 11,
0 11.21
347
CORE OF A GAME WITH SIDE-PAYMENTS
Conversely, suppose that w is balanced. Then @(Z, n!w) c @(d?,w) because, @(Z,m), then (.“, c ) = w ( N ) = v(N) (since v is balanced) and, for any A E d?, (tA, ;I)==3tv(P) =s v(A). Hence c E @(d?, v). Now, @(d?, w) # 0 since @(.zj,rcv) is non-empty by Proposition 1.3 (because m(.is )convex and positively homogeneous).
if c E
Finally, it remains to prove that, if c C @(of, w), then c belongs to @(Z, m). We know that (T”, c ) = v ( N ) = z v ( N ) . Since ( r A ,c ) < v(A) for all A , we deduce that for any m E CaZ(z), (7, c ) = CAE& m(A) (.”, c ) s z A E d m ( A v(A). ) Hence ( t ,c ) m(t)for all fuzzy conditions. 11.2.3. Balancedfamily of multistrategy sets
We now give examples of “balanced” characteristic functions associated with strategic games { X ( A ) , F A # } .For this purpose, we require the following definition. Definition 3. We shall say that a family { X ( A ) } A E dof multistrategy sets X ( A ) c X A is “balanced” if
Proposition 2. Suppose that the strategy sets X i are convex. Then the family {XAIAE is “balanced”.
Proof. Let m E m(zN) be a balance and let xA belong to X A for all A E d. Therefore, the ith projection of = CAE& m(A)xAis equal to fl = CAEd;,EA m(A)x?‘. Since X‘is convex and z A c d ; i E A m ( A=) 1, wededuce that fl,’ E E Xi.Therefore, belongs to X N .
‘
Example. Consider the subsets X ( A ) defined by
where (i) Y(i)c V is a convex subset of resources allocated to player i, (ii) Lr E 2 ( U i , V ) is a resource operator. Proposition 3. Suppose that (10) holds. Then the family { X ( A ) } , of multistrategy sets dejined by (9)is balanced.
348
GAMES WITH SIDE-PAYMENTS
[Ch. 11, 6 11.2
Proof. We consider multistrategies 2 E X ( A ) and check that, for and balance m E m(.”>, x N = C m(A)xA belongs to X ( N ) . (1 1) A€&
Since P9j= x A 3 j m(A)xASJ, we deduce that
11.2.4. Balanced characteristicfunctions and convex lossfunctions
We are now able to state the main result of this chapter. Theorem 2. (Scarf). Consider the game with side-payments associated with a cooperative game {X(A), FA*}AEd by the characteristic function v defied by
Suppose that (13)
sets X i are convex, { (ii)(i) the strategy lossfunctionsf’ are convex the
and that (14)
thefamily { X ( A ) } A €is~balanced.
Then the core of the game with side-payments is non-empty. By Theorem 1, we have to prove that the characteristic function v defined by (12) is balanced. This fact will follow from the following lemmas.
Lemma 1. Let m be a balance. Then for any pair of different players i and j , we have
Proof.
1 =
C m(A) = AN
A3i A Bi
m(A)+
C m(A)
A31 A 31
Ch. 11, 0 11.21
CORE OF A GAME WITB SIDE-PAYMENTS
349
Lemma 2. Suppose that the strategy sets X i are convex. Let m be a balance and
2 E X(A). Then the multistrategies y(f) defined by belong to X’=
njZ,xi.
.
-
Proof. By (15), we can write
which is a convex combination of elements p*’of the convex subset Xj. 0 Lemma 3. Let m be a balance and let {x”} be a family of elements # E X ( A ) . Let y ( i ) be defied by (16). Then m ( ~{ )x ~ , d y ( i )=} m(~)x~. V i E N, (17)
c
c
A3i
A€&
Proof. For any j E N , j # i,
I f j = i, then
CA3I
m ( ~{ X) A , nAy(i))i = C A 3 i m(A)xAB1.
Lemma 4. Suppose that convexity assumptions (13) hold. For any balance m E E OZ(d”)and for any family {PIAEd, we have that
c~~
c&i
Proof. Since we can write m ( ~ ) #= m ( ~{#, ) d y ( i > ) ,and since CAalm(A) = 1, we deduce from the convexity offr that
350
[Ch. 11, fi 11.2
GAMES WITH SIDE-PAYMENTS
Therefore,
Proof of Theorem 2. We can associate with any E M(z") multistrategies !A E X ( A ) such that
E
r
0 and any balance m E
Let d" = CAE& m(A)x".Then 2"longs to X ( N ) since the family { X ( A ) } ~ ~ & is balanced. Lemma 4 implies that
V) -s c A( i€N
6
C
c
m(4xA)
A€d
"FA4 9 c &A
f,"*(x">
m(A) ~ ( A ) + E .
A € d
Hence v(N)
4
CAE& m(A)v(A)for all balances m,and thus, w(N)
e n~(p).U
*Example. Consider the case when X ( A ) is defined by (9). If the subsets R' andY(i)areconvexand if the loss functionsf, are convex, Theorem 2 and Proposition 3 imply that the characteristic function v defined by
is balanced. It is also clear that the characteristic functions w( -,p) defined by
are balanced. Proposition 4. Suppose that
(a)
1
(i) the subsets R' c U' and Y ( i ) c V are convex, (ii) the operators Li E J?(v', V ) are linear,
(iii) the lossfunctions& are convex.
For any p , the core of the game associated with the characteristic function w( p ) defined by (21) is hon-empty.
.,
Ch. 11, 0 31.21
CORE OF A GAME WITH SIDE-PAYMENTS
Proof. Let E =- 0 and m E such that
35 1
a(.")be fixed. There exist multistrategies # E R A
*II.2.5.Further properties of convexfunctions and balances
We can extend Lemma 3 to positive sequences m E
m(t).
Lemma 5. Let m E m(t)and # E X A be given. For any i E A,, we can f i n s ~(z') f &Ar;lZi Xi such that (24)
V j € A,,
C m(A)xA" = c
A31
Proof. This is trivial i f j = i. I f j # i, we can write,
We define y ( i ) by
zj
A3i
m(A) {xA,nAy([))i. zi
Therefore,
Consider a game {X(A),FA"},, and its extension {Xo(z), F"}rEa (see Definition 10.4.6). Proposition5. Suppose that convexity assumptions (13) hold. Then, for any x' E XO(T), V A E A$
(28)
2AiZjJ(X')
4
CTb(W(xz);
A),
i-1
where d ( P ( x 3 ; A) is the lower support function of F'(x')
Proof. Writex' = CAEd(m(A)x')/t, where m E M(z) and x' E X ( A ) for all A E d. By Lemma 5, there exists y(r")such that
Hence, the convexity of the loss function
implies
Therefore, for all representations of x' = CAE& (m(A)p ) / z , we have that
Ch. 11,8 11.31
353
VALUES OF FUZZY QAMES
Thus, by taking the infimum, we obtain
2
A i Z i f , ( X ) .=s a"(F"*(x');
A). 0
i=1
We also mention the following property of balances.
Proposition 6. Let c E R" and z E
Proof. Since tl =
1
iCN
Ti
5.W e can write,for any m E m(r),
cA3i m(A), for all i E N,we deduce that C c m ( 4 = c C m ( A ) = c m(A)IAl.
=
i € N A31
AEd
A€ d
i€A
Therefore,
Interpretation. For any f u z y coalition t and m E center
as a burycenter of the Ia l l barycenters (1 /I A I)
m(t),we can write the bary-
c,.
11.3. Values of fmzy games
We define another solution concept for a game with side-payments different from the core. The concept involves associating a multiloss c E V(N)satisfying "a priori" axioms with any characteristicfunction v :'Ti R. These axioms are apparently quite weak and intuitivelyreasonable. We shall see that for fuzzy games, the new solution concept and the core coincide when they are both defined. -L.
25
354
GAMES WITH SIDE-PAYMENTS
I I .3.1:The diagonal property
[Ch. 11, !j 11.3
-
Definition 1. We denote by Vn the vector space of functions v : [0, l]” R vanishing at 0 and continuously differentiable on the diagonal {t.“}O == t e 1 of [O J]”. We supply V” with the scalar product 1
[
((v, w ) ) = ~ (DV(tzN),D w ( t z 9 ) dl J 0
1
for which V” is a pre-Hilbert space. Remark. Notice that if the function v E V” is positively homogeneous, then
(2)
Dw(tz9 = Dv(zN)
since -- (Dw(tzN),z) = lim v(tzN+ 6z) - ?J(tz”) 6 e+o
= lim 8-0
8
In particular, if v and w are positively homogeneous, (3)
((v, w ) ) n = (Dv(+”,D w ( z 9 ) .
Interpretation. The scalar product ((v, w)),, takes into account the following “diagonal property”. For a given collective rate of participation t E [0, 11, we consider the diagonal fuzzy coalition t p = {t, . . ., t } in which each player participates equally. The ith partial derivative a v ( t p ) / a z i measures the marginal loss imposed on player i when he leaves the diagonal fuzzy coalition t8”. In other words, Dv(&”)can be regarded as the “marginal multiloss” imposed on the players when they want to modify a diagonal coalition. In the case where w is positively homogeneous, we take into account only the marginal multiloss for the whole set of players p.
Ch. 11,
0
11.31
355
VA!.,UEs OF FUZZY GAMES
11.3.2. Sequence of fuzzy values The aim is to define maps q,,associating with any characteristic fundon
v
C V" a multiloss qnv E R" satisfying the following axioms.
Pareto optimality. This requires that
Symmetry axiom. Let 8 be any permutation of the set N of n players. We define (5)
{
-
(9 @*v) (z1, . - 9 Zn) = f+e-W, (ii) (8*c)i.= Cecil for all i E N.
--
- 9 S1(l)),
The symmetry axiom requires that (6)
Vn, V v E V",V permutation 8, qn(8*v) = 8*q,v.
In other words, the symmetry axiom requires that the value does not depend on "how the players are named". Or, equivalently, that the value does not depend upon the order in which players play. Atomicity axiom. This requires that the loss imposed on a coalition of players in a game between coalitions is the sum of losses imposed on each player of this coalition in the initial game, Namely, let Cp = {Al, ..., A,} be a partition of the set N of players in m non-empty "types" A, of players. We assume that each fuzzy coalition a = {UI, ..., } , a of types of players induces a fuzzy coalition t = Cp .u = {GI, ...,zn} of players where the rate of participation ti of the i& player is equal to the rate of participation a, of the type A, in which he belongs, i.e. (7)
t i
= OJ whenever i E A,.
This implies that we can associate with any game (5,w) of n players and any partition Ip in m types of players a m-person game defined by the characteristic function Cp* v (8)
(P*r)(a13 . . ., a,)
=V
( P 4
where t i = ajwhen i E Aj).
( = V h , . . -,Zfl)
The atomicity axiom states that V n , V partition Cp in m types of players, V v (9) V j = 1, . . ., m, (p,(Cp+v), = ~ f E A j ( q n 2 r ) l . 25
<
V",then
356
GAMES WITH SIDE-PAYMENTS
[Ch. 11,s 11.3
Definition 2. We shall say that a sequence {q~"},~ of maps from V" into R" is a "sequence of fuzzy values" if the maps Q),, are continuous linear operators which satisfy the Pareto optimality, symmetry and atomicity axioms. 11.3.3.Existence and uniqueness of a sequence of fuzzy values Theorem 1. The sequence of continuous linear operators vnE &(V", R") defined bY (10)
1
V v E V", 'pnv = $ Dv(tzN)dt E Rn 0
is the unique sequence ofluzzy values. For m y linear operator A E L?(Rm, R") satiflying A P = t", we obtain (1 1)
V V E V", cpm(voA)= A*~,,v.
If we assume that v E
V" is convex and positively homogeneous, then the value q,,v is the unique element of the core of the fuzzy game (5, v).
Interpretation. Formula (10) shows that ~ , , vis the average marginal multiloss imposed when players leave diagonal fuzzy coalitions. Proof of existence. It is obvious that v,, is a continuous linear operator. Since
we obtain by integrating this equality from 0 to 1 that 1
1
= V(ZN)-v(0) = v(zN).
Hence the Pareto optimality axiom holds. To prove property (1 l), we recall that D(voA)(z) = A*Dv(Az) when A E 2(Rm, R"). Since D v ( t A p ) = = OH#), we obtain 1
1
pm(wo A) = J D(v o A) (@) dt = J A*Dv(trN)dt = A * ~ v . 0
0
In particular, we may take m = n and associate with any permutation 0 of = (4where 4 = 1 if j = 8 (i) and 0 if j # €J(i).We then obtain the symmetry axiom, since (vo A) (z) = (u43) ( t )and A*c = O w .
N the matrix A
,
Ch. 11, Q 11.31
357
VALUES OF FUZZY GAMES
In the same way, by associating with a partitition 9 of N in m types A, the matrbi9€B(Rm,R")efined by(8),we havevoQ=Qwand(Q*c), = &,c,. Therefore the atomicity axiom follows from (11) with A = 9.
Proof of uniqueness. To prove the uniqueness of the sequence of operators p,,, we use the fact that, for each n, the set of polynomials (in n variables) is dense in V" (by the Stone-Weierstrass theorem). Therefore, since q,, is assumed to be linear and continuous, it is suf6cient to prove that any sequence {v,,}" of fuzzy values associates with polynomials v, :z vk(z)= @ +the same multiloss p,, vk. For this purpose, we check that the Pareto optimality and symmetry axioms imply that
-.
ynv = --I?v(zN>
(12)
n
-
whenever v is symmetric.
[If v is symmetric, then v,,v = p,,(ew) = e*p,,v for any permutation 8 of NSuch a vector p,,v satisfies pnv = d = {a, , . ., u}. By the Pareto optimality axiom, we deduce that an = (gp), = v ( p ) . Therefore (12) holds.] We now use the atomicity axiom in the following way. We regard v&) = = $ . * * e a s v k =(p*vlkl, where lkl =kl+-**+k,,,pdenotesthepartition of I k I players into kl players, k2 players, ...,k,,players and where 2 vlkl(al, * GI&$ = ~ 1 '~ OIkl. Since vlklis symmetric, then (q&l"lkl)j = 1/1 k I. Therefore,
EZl
--
We have proved that any sequence {p,,} of fuzzy values satisfies
(We can check that formula (10) yields the same result!) This proves the uniqueness. Proof of equivalencewiththe core. When v E V" and w is positively homogeneous and convex, we have av(Z") = {Dv(CN>}.But av(t") = @(G, v ) by Proposition 1.3 and 1
1
Dv(+" = J DV(tN)dt = f Dv(tZN)dt = VnV 0
0
since v E V" and is positively homogeneous. Therefore @(%,v ) = {p,,~}.
358
GAMES WITH SIDE-PAYMENTS
[Ch. 11,s 11.3
11.3.4. Relations between core and fuzzy value
We saw that the concepts of core and fuzzy value for fuzzy games coincide when they are both defined. We can even relate these concepts when v is only the supremum of a family of differentiablefunctions,by reformulating Proposition 1.4 in the following way.
Proposition 1. Suppose that the characteristicfunction v is defined by
where the functions w satisfy assumption (21) of Proposition 1.4. Then the core @(Z,, v) is the closed convex hull of thef u z y values ynw(.,p ) asp ranges over
P(N) = { p E P such that v(zN)= w(zN,p>}. *11.3.5. Best approximation property of f u z y values
We can also characterize fuzzy values by best approximation properties. v) amounts to replacing the charFinding a "solution" c E R" to a game (5, acteristic fmction v :z € Z, +v(t) by a linear characteristic function c :z € E Z, + (z, c). The question arises as to whether it is reasonable to propose the best approximation to v E V N by linear characteristic functions c < R" as a solution concept.
Proposition 2. The fuzzy value y , , ~of a characteristic function v best approximation to v by linear characteristicfunctions.
< V" is the
Proof. Such a best approximation exists since ((w, w)), induces on R" the scalar product ((c, 6))" = E=1ci4. The best approximation c to v is defined by ((v-c, 6)) = 0, for all d E R". Since c and d are linear functions, this can be written
ci =
ari
v(tzN) d t = ( ; P ~ v for ) ~ all i. 0
Remark. We also mention some other properties satisfied by fuzzy values.
Ch. 11, 5 11-31
VALUES OF FUZZY GAMES
359
Dammy property. We say that player i is a “dummy” or a “null player” in a fuzzy game (55,v ) if the characteristic function v does not depend on zr(i.e. does not depend upon the participation of player i). Then the following “dummy property” (16)
If i is a dummy player in (55, v), then (qp) = ,0 is satisfied by the fuzzy value.
Positivity property. Consider a non-decreasing characteristic function v. Since the loss becomes greater when a fuzzy.coalition increases, it is reasonable that the marginal multiloss is non-negative, i.e. the fuzzy values: satisfies (17)
If v is non decreasing, then qnv a 0.
*11.3.6. Convex values of fuzzy games
The question arises as to whether we can extend the concept of fuzzy value to games.whose characteristic function is no longer differentiable on the diagonal, but only differentiable from the right. For this purpose, instead of replacing a characteristicfunction z E Zi I- v(z) by a linear characteristic function, we shall only look for replacements of v by convex positively homogeneous fuctions +#)no :z E Zi k+ ($5”~) (z). We can prove the following result. Proposition 3. Let be the cone of characteristicfunctions v defined on [0, 13” vanishing at 0 and diyerentiable from the right on the diagonal {td“},E[o,ll: The maps @,,frompn into the cone of convex, positivei) homogeneousfunctions defined by 1
(18)
(+nu) (2)
=j 0
wm (r)dt
are additive, positiveIy homogeneous and satisfy the. Pareto optimaIity axiom
(19)
(@nv)(zN) = v(zN).
Furthermore, for any operator A E &(Rm,R7satkfying A+ = z”, they satisfy
(20) rf v E
@,,(PI o A ) = (cp,,~) o A.
t“ is positively homogeneous and convex,
is the lapper supportfunction of the core @(.is, v).
Proof. This is left as an exercise. 0
the function z +-I
(@,,v)(z)
360
GAMES WITH SIDE-PAYMENTS
[Ch. 11, 5 11.4
11.4. Shapley value and nucleolus of games with sidepayments
Let OC = P(A9-0 be the set of all non-empty coalitions. Since we can v ) into a fuzzy game (5, 7tv) via the (non-linear) map 7t extend any game (d, defined by (2.3), we can define thefuzzy value qn7tv of thegame (d, v ) whenever the extended function z v is differentiable. This is the first drawback, since there exist functions v E Rd such that 7tv is not differentiable. Secondly, q,p is not linear. The problem arises as to whether it is possible to find linear operators satisfying three analogous axioms (Pareto optimality, symmetry and dummy axioms). There exists such a unique operator ynycalled the “Shapley value”. It is related to the fuzzy value qnby the formula yn = (pno where w is the Cornet aA(v) ti)ulAl where aA(v)= extension operator defined by (wv)(t)= Led = (- l)IAl-IE1v(B). It can also be written y,, = q , where ~ x is the Owen extension operator defined by
(niEA
LCA
(XV) (z) =
c 44fl n (1 -zJ ti
AEA
i€A
iQA
We end this section by proving that the nucleolus of a game with side-payments consists of a unique element. 11.4.1. The Shapley value
Let yn :Rd -, Rn be a map associating with any characteristic function v E Rd a multiloss yn v E R”.We shall say that yn satisfies the Pareto optimality axiom if (1)
vv
E Rd,
c n
f=l
(ynv)r = v ( N ) ,
the Symmetry axiom if V v E Rd,for any permutation 8 of N,
(2)
yn(e*v) = e*ynv
where O*v is defined by (O*v)(A) = v[O(A)] and the Dummy axiom if
(3)
V v E Rd, (y,,~),= 0 whenever i is a dummy player of v
where we define i to be a “dummy player” of v if (4)
VAE
d, v(A U {i})
= v(A).
Ch. 11, !j 11.41
SHAPLEY VALUE AND NUCLEOLUS
36 1
Definition 1. We shall say that a map yn is a “Shapley value” if it is a linear operator from Rd into R” which satisfies the Pareto optimality, symmetry and dummy axioms. We shall prove that such an operator yn exists and is unique. 11.4.2. Existence and uniqueness of a Shapley value
Theorem 1. (Shapley). There exists a unique linear operator yn E &(Rd, R”) satisfying the Pareto optimality, symmetry and dummy axioms. This operator can be written yn = v,,w where w E &(Rs-d, V”)is the linear extension operator dejned by
where I A1 = card ( A ) denotes the number of elements of A and
The following formulas hold
where 0 ranges over the set of permutations and where (i) Ae(i) = { j E N such that 0 ( j ) =s 0(i)}, (ii) &(i) = { j E N such that O ( j ) 4 O(i)}. Definition 2. We shall say that y, is the “Shdpley value” and that w is the “Cornet extension operator”.
Proof. The proof of the Shapley theorem consists of the five following lemmas. Lemma 1. The 2“- 1 characteristicfunctions p A dejned by 1 ifBxA, (9) PA(B) = 0 otherwise.
{
form a basis of the (2”- 1) dimensional vector space Rd. Any function v can be written 21 = C aA(v)pA where aA(v) = C (- 1)I~I-IBI v(B). (10) A€d
BCA
362
GAMES WITH SIDE-PAYMENTS
[Ch. 11, 5 11.4
Proof. The functions pA are linearly independent. [If not, there exist coef8dents a, # 0 such that a A p A = 0. Let A0 be a coalition with minimum cardinality among the coalitions A such that aA # 0. Then we can write
c
(1 1)
PA#=
c
Aa4
PAPA
(since if A $ Ao, we should have IA 1 < IAOI). Since A Q Ao, we have pA(A0)= = 0. On the other hand, pA,(Ao) = + 1. Therefore (11) implies that 1 = 0. This is impossible.] We prove (10). It follows from (9) that, for any C E d,
c
A€&
aA(v) pA(C) =
c
aA(2))
ACC
=
c c (- l)lA1-lB1 = V(B)
ACC B C A
coalitions A between B and C having a given car-
Since there are dinality IA I, w-e deduce that
0 i f I B ( < ICl, 1 if)BI = ICl.
Therefore, the two above equalities imply that A€&
aA(2))p A ( c ) = v(c>- 0
Lemma 2. For any coalition A E
d,any “Shapley value” satisjies
=I
Proof. If4 6 A , i is a dummy player of p A . (Because, if B 2 A, then B U {i} A and pA(B)= pA(B U {i}) = 1. If B 3 A , B U { i } 3 A since i 6 A. Thus UA(B) = pA(B {i}) = O.1
u
Therefore, the dummy axiom implies that
= 0 whenever i
6 A.
Ch. 11,
3
11-41
363
SHAPLEY VALUE AND NUCLEOLUS
Let i a n d j belong to the codition A and let 8 be the permutation which exchanges i and j and keeps the other players invariant. Then f3*pA = p A since 8(A) = A and the symmetry axiom implies that (Y)npA)i
=vn(e*pA)i
= (e*vnpA)I = ( y n p A ) j .
Finally, the Pareto optimality axiom implies that (ynpA)j f€N
=
c
(ynpA)i
=pA(w
1.
i€A
Since (ynpA)idoes not depend on i, we obtain (ynpA)= 1/(A I. Therefore, since y,, is assumed to be linear, there exists a unique Shapley value yn defined by
Lemma 3,. The Shapley value yn can be written yn = q~,,owhere o is defied by (5) and (6)Proof. We introduce the functions PA :z
-.PA(.) (n,,, =
zi)l/lAl:
Since : is a positively homogeneous (concave) function differentiableat we deduce that
Therefore, for any player i, (16)
(g)nW@i
= (yJnv)i. 0
Lemma 4. W e can write
Proof. We deduce from (13) that
d”,
364
GAMES WITH SIDE-PAYMENTS
[Ch. l l , § 11.4
Write
If B does not contain i, then C = BU {i} does contain i. It is easily checked that
Therefore, we can write
since any B which does not contain i can be written B U {i}= C- {i} where C does contain i and conversely. We now compute y,(B) when i E B = BU {i}. coalitions A between B and N, i.e. contain-
There are exactly ing B. Therefore, we can write
1
=
C
braen
( ) [ta-ldt 1
(-l)a-b n-b a-b
0
Ch. 11,$ 11.41
SHAPLEY VALUE AND NUCLEOLUS
365
Hence
J
0
Lemma 5. We can write
Proof. A and i E A are fixed, the number of permutation 8 such A = Ae(i) = = { j E N such that O(j3 4 O(i)} equals (IA I - l)!(n- IA I)!, since the members of A- {i} and N- A can be permuted at will. Therefore (22) follows from (17). 0
Remark. This formula may be interpreted in terms of the following procedure for allocating a loss to a given player i. We choose an order (i.e. a permutation) 8 at random, with equal probability l/n!. For a given order 8, the loss of player i is the amount he adds to the loss of the coalition of the preceding player since
Then (yp), is nothing other than the expected loss under this random procedure.
Example. It may be useful to, carry out the calculation in a few simple cases.
We &gin wi&ha two-person game. We obtain that
and we deduce the other components by circular permutation. For the threeperson game, we obtain
366
GAMES WITH SIDE-PAYMENTS
[Ch. 11,§ 11.4
Remark. It is easy to check that if a characteristic function v E Rd is symmetric, then its convex cover is also symmetric. If 8 is any permutation of N.
(O*v)(z) = zv(ze-- 1(i), inf
. . ., re- 1cnl)
~ 1 3 m(A)=ze+o d
= =
inf
C13jm[B(A)I-q
inf CA3jMA)=‘3
=
inf zA3jm(4=rt
A
c m(A)v(A) E
~
c m(A)v(A)
AEd
C
~ I [ B - ~ ( Av(A) )]
C
m(A) v[O(A)]= @*v)
ACd
(z).
ACd
*Rematk. The Owen extension operator. We can also write y,, = qa where x E B(Rd, V”)is the ‘‘Owen extension operator” defined by
The function xu is multilinear and interpolates
(26)
(xv)( d )= v(A) for any A E d.
If we denote by v~ the function
we deduce that
In other words, using (21), we find that
Y
since
Ch. 11, 6 11.41
367
SHAPLEY VALUE A N D NUCLEOLUS
*11.4.3. Properties of the Cornet extedon operator
We mention some properties of the extension operator w.
Proposition 1. The extension operator o maps characteristicfunctions into positively homogeneous fuzzy characteristicfunctions. It is an interpolation operator, i.e.
(27)
V coalition A, (021) (ZA) = v(A).
Therefore (28)
@(%,mu) c @(d, u).
Proof. The first statement is obvious. Since
c 4B)
we deduce from Lemma 1 that (mu) (4=
BB(ZA)
=
B€d
Since (d, u) is the restriction to ui? of
c
a(B) p ( A ) = v(A).
B€d
(a, wv), we ob&
(28). 0
In particular, we deduce the following corollary.
Proposition 2. Suppose that the characteristicfunction v satisfies
Then the Shapley value y,,v belongs to core @(d, v ) of the game (d, v). Proof. Since p A is linear if 1 A 1 = 1 and concave if I A I
Z- 2, aswmption (29) implies that mu is convex and positively homogeneous. Hence {Dy,p} = = @(%,mu) by Proposition 2.3. 0
We shall give an example of a game satisfying assumption (29).
Proposition 3 (Comet). Let C E :R and p E N, p tion v dejned by (30)
w(A)
=-
(6
Cj)’ for all A E
== 1.
The characteristic func-
d
satisfies property (29). Its Shapley value belongs to its core.
368
GAMES WITH SIDE-PAYMENTS
[Ch. 11, 0 11.4
This result is true when 1 A I = 1 =s p, since, when A = {i}, we can write
Suppose that it is still true for IA I B a- 1. Consider a coalition A satisfying I A 1 = a. Withoat loss of generality we may take A to be (1, .. ., a}.Hznce, we obtain that
If u s p , the induction hypothesis implies that (34)
v(A) =
On the other hand, we know that v(A) =
cBCA aB(v).Hence we obtain
We have proved (31(i)). Since the components C, are non-negative we deduce that aA(v)6 0. If a =- p, we can write again that @(A)= CBCA aB(u). The induction hypothesis implies that aB(v)= 0 whenever p < I B 1 -s a- 1. Hence (36)
@(A)=
ae(+t-aA(v). BCA
PISP
On the other hand, we deduce from (34) that
Hence aA(v)= 0.
Ch. 11, 0 11.41
SHAPLEY VALUE AND NUCLEOLUS
369
*I1.4.4. Simple games A game (d, pr) is said to be simple if for each coalition, we have either u(A) =0 or w(A) = 1. It is a game in which every coalition either Wins (value 0 ) or loses (value l), with nothing in between. As such, simple games are applicable to political science, as they include voting “games” in elections. In this case, the formula giving the Shapley value becomes especially simple. In fact, the terms v(A)- v(A- {i}) will always take the value b or 1, taking the value 1 whenever A is a losing coalition but A- {i} is not. Hence we have
where the summation is taken over the set of,of all losing coalitions A such that A- i is winning. *11.4.5. Nucleolus of gmes with side-payments
In the case of games with side-payments(&, w), we can prove that the nucleolus contains only one point. The complaint becomes: 6(A, c) = ci- v(A) (39)
c
,€A
and depends linearily on c. Proposition 4. The nucleolus of a game with side-payments consists of a unique multiloss c.
Proof. Assume that c and d are two different multilosses in the nucleolus. We shall obtain a contradiction by proving that
where < denotes the 1exiCOgraphicordering. Consider the first component. Then
6
26
1 2
1 2
-max 8(A, c) +- max B(A, d) AEd
AEd
370
GAMES WITH SIDE-PAYMENTS
[Ch. 11,s 11.4
If the inequality is strict, this implies (40). If not, let AIE a? satisfy O(+(c+d>)l = +(&A1, c)+e(Al, d)). Then we deduce that O(A1, c) = m u A c d 8(A, C ) = wO(c)l and that 8(A1, d) = m a A , &8(A, d) = w8(d)l. Assume that there exist A1, ..,Aj-, such that, for any k j - 1,
.
Taking the supremum over the set of coalitions A # A1, that
. ., A,-l
we obtain
1
1 rnax ~ ( A , c ) + ~max 8(A,d) =S7 AzA,, ..., Aj-, A#Az, .... 4 - 1
If strict inequality holds, then the contradiction (40) holds. If not, we deduce the existence of a coalition Aj such that (41) holds for all k =sj. W e have proved that either o($c+d)) < we(c) = oe(d) or that for any A d 1 4 k 6 1 1) (42)
e(Ak, c) =
= @e(qk= e(Ak, d).
This implies that e(c) = B(d), and thus, that c = d. 0 Symmetry property of the nucleolus Let C be the nucleolus of the game (d, v). We associate with any pair G,k} of players, j
f
k, the number
We can regard sk, as the maximum complaint that player k can raise against player i. Indeed, to say that sw =- sjk amounts to saying that player k is “stronger” than player j. We shall prove the following result.
Ch. 11,s 11.41
SHAPLEY VALUE AND NUCLEOLUS
37 I
Proposition 5 . Let C be the nucleolus of thegume (d, v). Ifsu > sik, then Zj
=
= v(o’>>.
Interpretation. If player k is stronger than player j, then he forces player j to submit to his worst possible loss ~ ( { j } ) . This property can be viewed as a symmetry property of the nucleolus.We or c k = ~({k}). have sk] = s]k except in the case where either El =
~(u})
Proof of Proposition 9. Assume that there exist two players k and j , k # j , such that (44) s k j Z s j k and ? j < V ( { j } ) . Choose S small enough that (45) S k j - 6 z- S]k+ 6 and Z j Z V({i})8. Consider the multiloss c defined by (i) ci = Zi if i f k , j , (ii) ck = Zk-8, (46) (iii) cj = C j + 6 . We have that
I
Also we obtain that
C ci-
v(A) = C
Ei-
v(A) if A 3 k, A 3 j ,
i€A
i€A
ci-w(A)=
C Zi-v(A)
ifA3k,A3j,
Therefore, in Rd,we obtain O(c) from O(Z) by keeping the components A such that either A or A^ = V- A contains b o t h j and k, and by modifying only the components which contain either k or j. Since Skj = max B(A, Z) sjk = max B(A, Z) A 3k
A BI
A 31 A Bk
the largest component which has to be modified is the one such that A contains k and not j . It will be decrcased and thus, o8(c) < wO(C). This is impossible since Z yields the smallest element for the lexicographic ordering. 0 26.
CHAPTER 12
GAMES WITHOUT SIDEPAYMENTS
We shall prove in this chapter the main theorems dealing with the concepts of the core, the fuzzy core and equilibria of games without side-payments. We use two fundamental tools for this purpose. The first of these is the “representation“ of a game without side-payments by a family of games with sidepayments. The second is the “cover” of a game without side-payments. Such a cover is a f i u q game without side-payments constructed by “extending in some sense the family of games with side-payments of the representation of the original game into a family of fuzzy games with side-payments. In other words, each representation of a game without side payments defines a cover which is a f i z y game (without side payments). This allows the definition of thefuzzy core of the representation of the game as the subset of admissible multilosses of the whole set N of players which are not rejected by any fuzzy coalition. We may also define an equilibrium of the representation as an admissible multiloss of the whole set N of players which belongs to the core of a game with side-payments of the representation. The equivalence between these two concepts is proved by applying the minisup theorem. We then prove that the fuzzy core is non-empty (i.e. that an equilibrium exists) by using the Ky-Fan theorem. In particular, we shall prove that the canonical fuzzy core and the set of canonical equilibria (almost) coincide and that they are non-empty under assumptions slightly stronger than the Scarf theorem for the non-emptiness of the core. Also, for a convenient representation of the economic game, we shall prove that the fuzzy core of an economy (almost) coincides with the set of Walras allocations. 12.1. Equivalence between the fuzzy core and the set of equilibria
We say that a family w(M*X P ) of characteristicfunctions w(A, p) (A E
p
M‘,
E P)represents a game ( u t , ~if)for any A E ~ 4 B(J(A); , A) = suppcpw(A, A, p). 372
Ch. 12, 9 12.11
313
FUZZY CORE A N D SET OF EQUILIBRIA
Then, we define a multiloss c E J+(N) to be a weak equilibrium of such a representation if there exists { I , j j } E J P X P such that VA E
d, (1,+ - c )
=sw(A, 1,p3.
k.
It is said to be an “equilibrium’yif there exists such a A which belongs to We shall say that a multiloss c E J + ( N ) belongs to thefuzzy core of such a representation if,for any fuzzy coalition z E 21, there exists A E JZ(A,) and p E Psuch that (A, z-c) = ~ 3 t W ( t A,p), , i.e. i f t - c 4 zJ(z)+&+, where
aJ(z) = { c E Rnsuch that (1,2 - c ) =saw(z, A, p ) for any {A, P} E M nx P} is a subset of multilosses of the fuzzy coalition z constructed with the representation w(M“XP).This defines a fuzzy game (21,zJ) which will be called the cover of a game (d, J). This extension, of course, depends upon the representation w(M”’P), as also do the concepts of the fuzzy core and equilibria. We shall prove that the set of equilibria is contained in the fuzzy core and that the latter is contained in the set of weak equilibria. 12.1 .I. Representation 01a game
Definition 1. Let (d, J ) (resp. (‘Ti, J)) be a game (resp. fuzzy game) without side-payments. Let w(JZnXP) be a family of characteristic functions w(A,p ) : :A E ot w(A;A, p ) (resp. z E ‘Ti ~ ( 1 t,p, ) ) of games with side-payments obtained when A ranges over M nand P a set of parameters p ) . We shall say that the family (d, w(JPXP)) (resp. (%, w(&’XP))) “repreJ) (resp. (%, J))-if V 1 E sents” the game (d,
-.
-t
VA E (1)
d, sup w(A,A, p ) = ab(J+(A);A) PEP
= inf (A, c ) c€ J(4
1.
(resp. Vz E 5,sup w(z, A, p) = o*(J+(z);A) = inf (A, c ) c E J(7)
PEP
Remark. If one is looking for minimal requirements, one may replace condition (1) by the weaker condition
(9 SUPPEP W ( N , 1,P ) = d(J,(N); A), (ii) V A E
d?, supPEpw(A,A,
p ) e ab(J+(A);1).
Example. Canonical representation. It is always possible to represent a game (d, J)of a fuzzy game (‘Ti, J) by the functions
WqA) :z E ‘Ti -.+w(Z, A) = ab(J(z);A).
314
GAMES WITHOUT SIDE-PAYMENTS
[Ch. 12, 0 12.1
Definition 2. We shall say that the representation(d, w"(M'')) of a game (d, J) (resp. (Z,d ( M * ) )of a fuzzy game (Z,J ) ) is the "canonical representation". For instance, the canonical representation of cooperative games {X(A), FA}Acd described in strategic form is defined by (3)
since J ( A ) = F ~ ( x ( A ) ) . e Lscribei by the multiConsider its canonical extension {XO(Z),F}Tc strategy sets (4)
{
Xo(A) = xr =
c7
A€&
and by the multiloss correspondences (5)
C AE&
m
(see Section 10.4.8). We shall show that it is represented by the convex cover d ( z , A) of wb(A, A).
where we set
Ch. 12,g 12.11
FUZZY CORE AND SET OF EQUILIBRIA
375
where d ( A , A) is defined by (3). In other words, the functions d ( z , A ) and (A, A) are related by
d
12.1.2. Equilibrium of a representation
Dewtion 3. Let w(A?”’P’’)be a representation of a game (a?, J) (resp. fuzzy game (53,J)).We shall say that a multiloss E is a “weak cooperative equilibrium” of the representation if there exist I E Anand € P such that (9)
1
(0 E E J+(W (ii) VA E of!(,I , zA.E)=sw(A, p) (resp. Vz E 55, (I, Z-Z) e w(z, 1,ji)).
x,
We shall say that E is an “equilibrium” (or a “strong equilibrium”) if there exists A E &’satisfying (9(ii)). Remark. To say that E is a weak cooperative equilibrium of the representation w ( A n X P )amounts to saying that
(10)
z E J ( N ) n@(4 4,a),
i.e. that Z is an admissible multiloss of the whole set N of players which belongs to the core of one of the side-payment games defined by a characteristic function w(I, jj) of the representation W(&XP). Remark. It is clear that any (weak) cooperative equilibrium Z of the representation (Z,w(MnX.P))of a fuzzy game is a (weak)cooperative equilibrium of its restriction (d,w(M”’P)). Example. Canonical cooperative equilibrium. By using the canonical representation of the game (d?,J), the definition of (weak) cooperative equilibrium of the canonical representation coincides with the definition of canonical cooperative equilibrium introduced in Section 10.5.1.
Propositionl. Let (d,w(&XP)) be a representation of a game (d, J). Then w ( A n X P ) ) is a any (weak) cooperative equilibrium of the representation (d, (weak) canonical cooperative equilibrium. Proof. Since wb(A, A) = ab(J(A), A)) = suppEpw(A, A, p), then conditions (X,I?-E) G w(A, 1,p) imply conditions ( I , 9 . 5 ) .S wb(A, I ) . 0
376
GAMES WITHOUT SIDE-PAYMENTS
[Ch. 12, Q 12.1
12.1.3. Cover associated with a representation
Let ~ ( M x Pbe) a representation of a game (d, 4. We associate with the characteristic functions A k--+ w(A;A, p ) the characm(A) w(A,I , p ) of a fuzzy teristic functions z i-4 xw(z, 1,p ) = inf, J ) into a fWzy game. In some sense, we can therefore “extend” the game (d, game (57,zWJ).The extension depends on the choice of representation of the game (4 4-
xAcd
Definition 4. Let w(A?!”XP)bea representation of a game (d, J).Its “cover’’ associated with the representation w ( A ” X P ) is the fuzzy game (2;,z,,,J) defined by
(I1)
Vz E ‘7i, nwJ(z)= {c E Rnsuch that (A, c> 3 nw(z, A, p ) for any
A E A!“,p E P}.
Notice that (12)
Vz E ‘zi, n,J(z) is a closed convex subset such that n J ( z ) = = ~ , J ( z ) R‘,
+ .
The lower support function of z J ( z ) is defined by
Also notice that we always have (14)
V A E d, J+(A) c
Example. Canonical cover of a game. By using the canonical representation d(M)of a game (d,J), we obtain its “canonical cover” (55,nbJ) defined by (1 5 )
Indeed, we
by Proposition 1.4.4.
Ch 12, Q 12.11
FUZZY CORE AND SET OF EQUILIBRIA
377
Dehition 5. We shall say that the fuzzy game (5, d J ) defined by (15) is the canonical cover of the game (d, J). Proposition 2. Suppose that (16)
V A E of, J+(A) is closed, convex and bounded below.
Then,for any fuzzy coalition,
(17)
C
d'J(z) = (J
m(A)J+(A).
m€Ol(r) A € d
UmEOl(r)
Proof. We begin by checking that K = CAE& m(A)J + ( A ) is convex. We can write any d = uk CAE& &(A) P w k E co K in the forni d = = ~ , , , m ( A ) d A where m(A) = CfZl.&(A) and where rk' = = (a&(A)Psk/m(A)) E J + ( A ) when PSk E J+(A). Since CAjlm(A) = ak C A j i m k (= ~ Cbl ) u k t t = z,when mk E OZ(z), m belongs to @l(z). We next prove that K is closed. Since m(z) is compact and the subsets J+(A) are bounded below, the map {m,{c%&} E ~(~)xnAc-=t J+W C A E & m ( ~ ) is proper. Hence its image J ( z ) is closed. Thus K = Z(K)= d J ( z ) . 0
zfzl
-
Consider a game { X ( A ) ,FA}Acd described in strategic form and its canonical (see Section 10.4.8). These definitions are consistent. extension {Xo(z), F}rEq Propmition 3. The canonical cover (%, d J ) of a game {X(A),FA}Acd described in strategic form is akjined by (18)
VZ E
5,d'J(z) = CO (F(Xo(t))+R$).
In particular7 if we assume that for any A, (19)
and compact, { (ii)(i) X(A)A , hconvex A is convex and lower semi-continuous, is
ViE
then
(20)
V t E Z, 7tbJ(t) = P(Xo(t))+R$.
Proof. We proved in Section 10.4.8 (formula (51)) that (21)
WXO(.>> =
u
C
m€Ol(r) A € ,
m ( 4FR(W)).
Therefore, the fist statement follows from (15) and the second statement from. Proposition 10.4.4. 0
378
GAMES WITHOUT SIDE-PAYMENTS
[Ch. 12,s 12.1
12.1.4. Fuzzy core of a representation We now introduce the concept of the fuzzy core of a representationw(APXP) of a game (d, J).
Definition 6. The fuzzy core @,,,(a J)?, of the representation w(M"XP) of a game (d, J)is the subset of multilosses c satisfying (22)
{ (ii)(0 cv z J+(N),zE
E '7j,
c
6 n , ~ ( z )+R; .
In other words, the fuzzy core @,,,(&, J)is the subset of available multilosses of the whole set N of players which are not rejected by any fuzzy coalition z. Since J+(A)+&$ c 7zJ(zA)+fi: for all coalitions A E d,we deduce the following result.
Proposition 4. Let w(Ji?"'P)be a representation of a game (d, J). The fuzzy core @,,,(&, J) is contained in the canonicalfuzzy core @(&, J ) , which is contained in the core @(d, J). Remark. The fuzzy core @,,,( J)d and , the core of the fuzzy game (Z,3 t J ) are related by formula (23)
@,Ad, J ) = e(Z, nwJ)nJ + ( N ) .
12.1.5. The equivalence theorem
Theorem 1. Let w ( M X P ) be a representation of a game (d, J ) without sidepayments. (a) The subset of strong cooperative equilibria of w(M"XP ) is contained in the fuzzy core @J&, J) of w(M" X P). (b) Conversely, if we assume that (24)
(i) is convex and compact, { (ii)VAE&, {1,p}+ w(A, 1,p)isconcaveandupper semi-continuous P
then the fuzzy core @,(d, J) is contained in the subset of weak cooperative equilibria. Proof. Let E E J + ( N ) be a strong cooperative equilibrium, and 1 E &' and
p E P the associate multipliers. Since (1,p-I?) .s w(A, 1,p) for any A E &, we deduce that (1,z-c) =~7zw(z, 1, p) =G B(n,J(z); 1)for any fuzzy coalition
Ch. 12,$ 12.11 ?:
379
FUZZY CORE AND SET OF EQUILIBRIA
E 5.Since 1 E &, then (X/&,,Xi)
(25)
q ( c ) = sup ZE'Zj
inf
IEM(A*)
E _/n(AJ. Therefore, [(A, z*c)-ub(zwJ(z);A)] e 0.
Hence Theorem 10.4.1 implies that E belongs to the core of the fuzzy game
(95, nJ) which is represented by w(M" X P ) by construction. Conversely, suppose that Z belongs to the fuzzycore @,(d, J) c @(%,nd,J). Since (26)
Vz
C 5,nwJ(z)+R', is convex
Theorem 10.4.1 implies that a&)
s 0.
We introduce
We check that inf [(A, z-c)- d(zwJ(z);A)]
B(Z) = sup
ZEZ, I E J m
s sup rE'Zj
inf
AE./1I(Ar)
= .a(Z)
4
[(A, z-c)-a6(nwJ(z);A)]
0.
On the other hand, 2; and M " X P are convex compact subsets (by assumption (24(i))) and (A, z.F)--7tw(z,I, p ) is convex lower semi-continuous with respect to {A, p } by assumption (24(ii)) and concave with respect to z (since mw( A, p ) is convex with respect to z by Proposition 11.2.1). Then the Nikaido Theorem 7.1.6 implies the existence of (1, p} E M X P such that
.,
(28)
B ( E ) = sup [(I,z.E)-zw(z, 2, p)] s 0. 7
m
This implies that E E J ( N ) is a weak equilibrium. 0 In particular, the canonical representation of a game obviously satisfies conditions (24). Theorem 1 becomes the following.
(d, J ) be a game. Any strong canonical cooperative equilibrium belongs to the canonical fuzzy core. Conversely, the canonical f m z y core is contained in the subset of weak canonical cooperative equilibria. Theorem 2. Let
The strategic formulation of the above theorem is the following.
Theorem 3. Let {X(A),FA}Acd be a game described in strategic form by multistrategy sets X ( A ) and multiloss operators FA satisfying (i) X ( A ) is convex and compact, (ii) V i E A,AA is convex and lower semi-continuour.
380
GAMES WITHOUT SIDE-PAYMENTS
[Ch. 12,s 12.2
Any strong canonical cooperative equilibrium f E X(N) belongs to the core @({&(z), F},(%) of its canonicalcover. Conversely, the canonical fuzzy core is contained in the subset of weak canonical cooperative equillbria.
We shall now give a sufficient condition for the fuzzy core to be contained in theset of cooperative equilibria. We denote by w+ the multiloss with components (30)
w,?+= inf c. c € JW
It is clear that the core @(of,J ) of the game is contained in w*-R:. Proposition 5. Suppose that assumptions (24) of Theorem I hold. Suppose that for any c E J+(N)n(w*--R;) and for any non-empty coalition A E d A E d , we can find d E J ( N ) such that
(31)
V i E A , di
-= ci.
Then the fuzzy core is equal to the set of (strong) cooperative equilibria. Proof. If C belongs to the fuzzy core, there exist 1 E (19) holds. In particular, (32)
and p E P such that
{ (ii)(i>(1,E F) .?
= mind,J(m (X, d), J ( N ) n(w* - It?).
If 1 did not belong to&, we would obtain a contradiction because we may take A = {i E N such that 1‘ Z- 0} and d E J ( N ) satisfying (31). Then inequality is a contradiction of (32(i)). 12.2. Non-emptiness of the fuzzy core of a balanced game
We shall prove the fundamental theorem for the non-emptiness of the fuzzy core. This is also an existence theorem for a weak equilibrium by the equivalence theorem. Roughly speaking, the fuzzy core of a representation w(A”’XP) is nonempty whenever the cores d ! , w(1, p ) ) , of the associated games with sidepayments are non-empty, i.e. whenever the characteristic functions A +w(A,
e(
Ch. 12,g 12.21
NON-EMPTINESS OF FUZZY CORE OF A BALANCED GAME
381
-
A, p ) are “balanced”. We also need the upper semi-continuityof the correspond@(&, w ( 4 PI),. ence {A,P } Then the non-emptiness of the fuzzy core is proved using the Ky-Fan theorem. There is nevertheless a difficulty which comes from the fact that {A,p} @(of, A, p ) is not necessarily upper semi-continuous when A belongs to the boundary of An.We overcome this difliculty by an approximation device, which lengthens the proof of this theorem. We apply the theorem to the canonical representation of a game and obtain an improvement of the Scarf theorem about the non-emptiness of the core.
-
-+
12.2.1. Statement of theorems OJ-non-emptiness of the fuzzy core
Theorem 1. Suppose that the game (d, J ) satisjks (1)
J + ( N ) is convex, closed and bounded below.
W e also suppose that the game is represented by a family w ( A n X P )of characteristicfunctions sathfying
(2)
i
(i) P is convex and compact, (ii) {A, p } I- w ( ~ A,, p ) is continuous on (iii) V A E d,{ I , p ) t-+ w(A, I , p ) are upper semi-continuous and concave on MnX P.
A~xP,
Finally, we assume that (3)
V {A, p } E &XP,
A I-+
w(A, I , p ) is “balanceff’.
Then the fuzzy core of the representation w ( M nX P ) is non-empty (and thus, there exists at least a weak cooperativeequilibrium of the representation w(MnXP)).
Remark. Recall that assumption (3) is equivalent to the assumption (4)
V {A, p } E J h P , the core @(of,w(I,p))I of the associated game with side-payments is non-empty.
(See Theorem 11.2.1). If we use the canonical representation of a game (d, J), Theorem 1 becomes the following.
Theorem 2. Suppose that the game (d, J ) satisfis (1)
J + ( N ) is convex, closed and bounded below
382
G-ES
WITHOUT SIDE-PAYMENTS
[Ch. 12,s 12.2
and that (5)
f o r any balance m E M(zN),
C
A€&
m(A)J+(A) c J + ( N ) .
Then the canonical fuzzy core is non-empty and thus, there exists E E J ( N ) and E &"such that
Proof of Theorem 2. Take P = 0 and w(A, A) = ab(J(A);A) which is clearly upper semi-continuous and concave on.:R Since J ( N ) is bounded below, A w(N, A) is defined on : R and thus, continuous on its interior k'+.Hence assumptions (2) of Theorem 1 are satisfied. Finally, assumption ( 5 ) implies that the characteristic functions A w(A, A) are balanced, since, V m E m(zN)
-
1.e.
w ( ~a),
=W(ZN,
-
A).
Hence Theorem 2 follows from Theorem 1. 0 We now apply Theorem 2 to games {X", FA"} defined in strategic form. Theorem 3. Consider a game {X", FA*}AEd described in strategicform by stratxi and by loss functions f;'"(.") = sUp,A,,,,,d A ( y ) egy sets xA= where (i) V i E N , J;: is convex on XN, (ii) V i E N , fi is lower semi-continuous and bounded below on X N , (7) (iii) 3io E N such thatfi, is lower semi-compact.
Then the canonical fuzzy core is non-empty and thus, there exists x' E XN and 1E M"such that,
Proof. We deduce Theorem 3 from Theorem 2 as follows. Assumptions (7) imply that J + ( N ) is convex, bounded below and closed. Also, assumptions (7(i)) and Theorem 11.2.2 imply that the characteristic functions are balanced. Hence the fuzzy core is non-empty and Theorem 1.3 implies that there exists a weak canonical cooperative equilibrium. 0
Ch. 12,o 12.21 NON-EMPTINESSOF
FUZZY CORE OF A BALANCED GAME
383
Theorem 3 can be extended to the case of multistrategy sets X ( A ) c XA which form a balanced family.
Theorem 4. Consider agame {X(A), FA’}AEd. Suppose that hypotheses (7) and
(9)
the family {X(A)},iCd is balanced
hold. Tlren the canonical fuzzy core is non-empty and thus, there exists a weak canonical cooperative equilibrium.
Proof. This is analogous to the proof of Theorem 3. 0 Remark. Theorem 2 implies in particular that the core @(A,J) of a game (d, J)satisfying assumptions (1) and ( 5 ) is non-empty. It is a slightly weaker result than the following Scarf theorem, which we will not prove. Theorem 5 (Scarf). If the game (a2,J)satisfies (10)
J + ( N ) is closed and bounded below
and (1 1)
n
for any balance m E m(zN),
(‘cA>-lJ+(A)c J+(N),
(A/m(A)-0)
then the core, of the game is non-empty.
Notice that assumption ( 5 ) implies assumption (11). To see this, let c E (@)-l J+(A). For any A such that m(A) =- 0, there exists d“ E E r)(,, E J+(A) such that z”c d“. Hence c E CAE& m ( A ) P c z- x A E dm(A) d“. But CAE& m(A) dA belongs to J + ( N ) by assumption (5). Hence c belongs to J+(N). Also Theorem 3 implies another Scarf theorem, which states that the core of the game {X”,F A X } A E dis non-empty (Scarf called the elements of such a core “cooperative equilibria”).
Remark on the proof of Theorem 1. There is a small technical difficulty which complicates and lengthens the proof of the theorem. Several assumptions and several properties of the associated side-payment games will fail when 1 does not belong to &’. We overcome this difficulty by using an approximation device. We replace the set of multipliers A n X P by M;XP where E -= l / n and (12)
A:= { A E Ansuch that 2,
E
for all i E N } .
384
We define the polar cone by the larger subset (13)
(m+ of A!:, which contains :R
and replace J + ( N )
J,(N)+(-m+.
We shall prove that, for any fying (14)
[Ch. 12,# 12.2
GAMES WITHOUT SIDE-PAYMENTS
E
=- 0, there exist (2, p'} E &:XP
and E satis-
c E @(4 w W , P3 ) a s n ( J , ( N ) +( M 3 + ).
This implies that c" is rejected by no fuzzy coalition. By letting e converge to 0, we shall prove that cE converges to an element c of the fuzzy core.
-
We shall prove successivelythat the subsets J+(N)- @(&, w ( l , p ) ) a + ( a + are closed and convex and that the correspondence { 1 , p } @(&,w(1,p))is upper semi-continuous. In the third lemma, we prove the existence of c, satisfying (14). We deduce finally that c, converges to an element c which belongs to the fuzzy core of
(4J ) .
*12.2.2. Upper semi-continuity of the amociated side-pqyment games
We begin by proving the following lemma.
Lemma 1. If U c R" is closed and bounded below, then the map {c, d) E U X X (M3+I-+ c+ d E R" isproper. In particular, assumption ( I ) holds, V 1 E f M : , V p E P , Q E=- 0, J + ( N ) - @ ( d , w(A, p))~+(&)+ is closed and convex. (15) Proof. (a) Since A42 is the convex hull of the vectors e p with components
is spanned by the n vectors E; with compothen its positive polar cone (Ma+ nents
-(17)
'd,k =
&
1-ne
I-& I-ne
if k # i, if k = i.
'To see this, one has only to check that 0 ifi # j, (ete, 4)= I i f i = j.
{
Ch. 12, 0 12.21
385
NON-EMPTINESS OF FUZZY CORE OF A BALANCED GAME
Ezl
(m87+
Therefore, any p" E can be written p ; = Afet where A; = 0 for all i E N. (b) Now let U c a+Rn be a closed subset. We prove that if C" = b"+ +pm E U+(J4!3+ converges to c, we can extract convergent subsequences from {b"},,,and {p"},. Let p m = Ayef,, For each k E N, we have c~ = bkm
+
= Ek+--
c:pl
Aye:,
k
ak+ i=l
i=l &
1-ne
___
1-ne
i=l
A$.
Summing these inequalities, we obtain that
t 18)
n
n
i= 1
k=l
(ak-c?):).
-
Since A; a 0 for all i, we deduce that the sequences A? are bounded. Thus subsequences of Ayconverge to Ai 0 for all i. Then a subsequence ofp" converges to p E (M3+and a subsequence of bm = P-p" converges to b E J+(iV). (c) Let w x be the multiloss with components w? = infctGlc(,)) c,.
Then (19)
va E &n,
v p E P , Q((&,
W ( A , p)lAc
since, V c E @(&, w(A, p)),, V i E N, we have (20) Aici 4 w(i, 2, p ) d ( J ( i ) , Ai)= Aiwp;.
w#-R;
-
On the other hand, the map {c; d, p } E (a+R:)X(-w*+R:) X A: c+ f d + p E R" is proper. Therefore, J + ( N ) - @ ( d , w ( A ,P))~+(&:)+ is closed sincethesubsets J + ( N ) c a+R: and -@(A, w(A,p)), c -wx+R: are closed. It is ako convex since these subsets are convex. (7 We shall prove that the assumptions of Theorem 1 imply that the correspondence associating with any {A, p} E M;XP the core @(&, w(A, p)) of the associated game with side-payments is upper semi-eontinuow.
Lemma 2. Suppose that (i) P is compact, (ii) {A, p } F-+w(N, 1,p) is continuous on M ~ x P , (21) (iii) V A E d,{A, p } I--- w(A, A, p ) is upper semi-continuous.
1
Then the correspondence {A, p} E rinuous. 27
X P -e @(of,w(A, ?))A is upper semi-con-
386
GAMES WITHOUT SIDE-PAYMENTS
[Ch. 12,s 12.2
Proof. To prove that the correspondence {A, p } +@( d,w(A, p)), isupper semi-
continuous on@XP, we use Proposition 2.5.3 and check that its graph i.. closed and its values are contained in a fixed compact subset. The graph of @(A, w(A, p))nis the set of elements {c, A, p } E R"X A:XP such that 0) (1, c)-w(N, 4 P ) = 0 (22) (ii) V A E a?,(A, A c ) - w(A, A, p ) =s0.
{
Since {A, p } +w(N, A, p ) is continuouson A!: X P and the functions w(A,A, p)' are upper semi-continuous with respect to {A, p } , the graph is closed. We prove that it r n a p s X ~ Pinto a compact set. Let (A, p ) E ~ x P . By (20), c -s wy whenever c E @(&, w(A, P ) ) ~Hence, if A E Mz,
(A, w + - c ) = (A, w+)-
= =S
sup wr-
1sisn
inf
(2, P) €Jn:xP
w(N, A, p ) w(N,A,p)=sM#<+-,
-
since E =sA, for all i, (A, c ) = w(N, A, p), [ 11II = 1 and since {A, p } w(N, A, p ) is continuous on the compact set 2: XP. Thus EelI w# - c, I 4 c1 M, and hence, @(&, w(A, p)), is contained in the ball of center c and radius. E-lMe. 0 *12.2.3. Existence of approximate cooperative equilibria
The Ky-Fan theorem implies the existence of approximate cooperative equilibria P, i.e. of elements c" satisfying (14).
Lemma 3. Suppose that, for any e r 0, the sets (23)
J+(N)-@(&w(A, P ) ) ~ -(A!:)+ are closed and convex when A E J Z F andp E P.
Suppose also that (i) P is compact and convex, (24) (ii) the function {A, p ) F+ w(N, A, p ) is concave.
{
Supposefurthermore that
(i) the cores @(d, w(A, P ) ) o~f the associated side-payment games are non-empty for any (A, p ) €M:XP. (ii) the correspondence {A, p } I-+ @(a?, w(A, p))n is upper semicontinuous on A::xP.
Ch. 12,s 12.21
NON-EMPTINESS OF FUZZY CORE OF A BALANCED GAME
Then there exists {A', p"} E
387
4X P such that
@(4 w(Ae, P9),.n(J+(N)+(A:)+) z 0.
(26)
Proof. Write @(A, p ) = @(a?, w(A, p)),. We have to prove that there exists (A',py € Z X P such that (26) holds, i.e. such that
0 E J*(N)- @(n.,p8)+(A%)+
(27)
Since this set is closed and convex, this amounts to proving that
d((J,(N)-@(A', P")+(A:)+); 2) (2, 0) = 0
(28)
for any A E R"*.But d((J+(N)-@(A', p?+(m+); A) = - m when A and, if A E A:
~ * ( ( J + ( N ) - @ ( A E , p " ) + ( A ~A)) +=) ; = ob(J+(N)-@(P, P"); A) = d(J+(N): A)-a'(e(A', = sup [w(N,A, p)-u*(@(Ae, PI;41. PEP
Therefore, we have to prove that V{A, P} E JeXP, (29) where we set
PI;A )
**
dp,P% (4 PI)
!d{AC, PI, (2, PI) = WJ,A, PI-
(30)
-
4@
I
Q
0
ww,PI,A).
For this purpose, we shall apply the Ky-Fan theorem (Theorem 7.1.3). In the first place, X = A?:xP is a convex compact subset. Secondly, since the correspondence @ which associates the core @(A",,p9with {A', p'} E X : X P is upper semi-continuous, then the functions {2,p'} -, + o*(@(fp,p9; A) are upper semi-continuous (see Proposition 2.5.1). -Thus, v({A*,p'}, (A, p}) is lower semicontinuous with respect to (2,p'}. Thirdly, w(N, A, p ) iS concave with respect to {A, p} by assumption (24(ii)) and A a*(@(Ae,p'), A) is convex. Thus, q({Ac, p'), {A, p}) is concave with respect to {A, p}. Finally, we have to check that
-.
(31)
V{A, P I ,
Q)(@,
PI, {A, PI) e s 0.
Since u*(@(A,p); A) = S U ~ , ~ ~(A,( ~c), and ~ ) (A, c) = w(N,A,p) when c belongs to @(A, p), we obtain that v({A,p}, {A, p } ) = w(N, A, p)-a*(@(;l, p ) ; A) = 0. Therefore, by the Ky-Fan theorem, there e h t s a solution ,{A', p'} of (27). i.e. of (26). 0 27.
388
GAMES WITHOUT SIDE-PAYMENTS
[Ch. 12, 0 12.3
"12.2.4. Proof of the non-emptiness of the core
We can now prove Theorem 1. Since the assumptions of Theorem 1 imply the assumptions of Lemmas 1, 2 and 3, we know that there exists a sequence { c " }of ~ elements (32)
F E (J+(N)+(&)+)
n@(A8, pel
where we set @(A, p ) = @(&, w(A7P ) ) ~ . Let E == E < l/n. Since the map {c, d, p}E { a + R " , ) x ( - w w # + R " , ) x ( ~ ~ ) + c + d + p E R" is proper, we deduce that (J+(N)+(Mt)+n(wx-RR",) is compact. Therefore c" stays in a compact subset and we can extract a subsequence which converges to an element c E R". (a) Since c" E J+(N)+(&'t)+ c J+(N)+(M;)+,it can be written C" = d'+p" where d" E J + ( N ) andp" E (M:)+. By Lemma 1, subsequences of d" and p" converges to d E J + ( N )and p E respectively. Actually, since pE E (@)+, it is easy to check that the limit p E R+. Hence c = d f p E J + ( N ) . (b) On the other hand, since c' E @(Ae, p"),we have
-
(e)+
(33)
(P7z e )
=szw(z, As, pa)
for any z E '6.
This implies that z-c' 6 z,,,J(z)+&+ = zws+(z)= Int ( z w J + ( z ) ) . Therefore, ce E z-l(c~w~+(z)). z-l(Cz:,j+(r)). In other Since this set is closed, the limit c belongs to words, for any coalition z, z - c 6 Z,,,~+(Z), i.e. c i not rejected by z. We have proved that c belongs to the fuzzy core.
nrEs
nrEZ
12.3. Qmvalence between the fuzzy core of an economy and the set of Walras allocations
We shall show that the fuzzy economic game is obtained from the economic game by a convenient representation and that the equlibria of this representation are the Walras allocations. We Will then deduce from theequivalence theorem that the fuzzy core of an economy and the set of Walras equilibria (almost) coincide. 12.3.1. Representation of economic games
Consider the game (04, J)and the fuzzy game (Zj, J ) associatedwith an economy {R', Y(%A}lEN defined by (1) J ( t )= P ( X ( 7 ) ) ; F*(x) = { z i J ( x ' ) } i c A r
Ch. 12,
5
12.31
FUZZY CORE OF AN ECONOMY
389
where
(3) and X(z) is represented by the budgetary constraints
~ ~ it ~ is natural to inas p ranges over R" (and where ri(p) = S U ~ , ,( p~, y)), z,(A%(x')+ (p, 2)- ri(p))and the utiltroduce the Lagrangian defined by ity function of the dual problem defined by
LEN
(4)
W(Z,
A, P ) = C ziw(i, 2,p ) {EN
where (5)
w(i, 2,p ) = inf [Alt;,(x')+(p,~ ' ) - r i ( p ) ] . xr€Rt
Therefore duality Proposition 5.3.4 implies that tation of the fuzzy economic game.
(zw, ( d * X P ) ) is a represen-
1
Proposition 1. Suppose that, for all i E N, (6)
(i) Ri and5 are convex, (ii) Y(i) = Y(i)-R: is convex, (iii) 0 E Ri-Int Y(i)
and that (7)
the functionsf, are bounded below on X(N).
Then the functions w(A, p ) represent the economic fuzzy game. Furthermore, there exists a convex compact subset P of R" such that, V ilE A",
(8)
vz E Z, max W(Z, PEP
a, p ) = ab(~(z);A).
Proof. Since the functionsfr are convex and since 0 E R'- Int Y(i>c lnt (R'-Y(i)) for all i, we deduce that 0 E Int (CIcN z , d - Y ( t ) ) for all fuzzy coali-
390
GAMES WITHOUT SIDE-PAYMENTS
[Ch. 12, 0 12.3
tions z. Then Proposition 5.3.4 implies the existence of elements pa, E R" such that (9)
d ( J ( z )A) ; = w(z, 1,pa, 7 ) = SUP w(z, A, PER'*
PI-
Since RT is the common barrier cone of the subsets Y(z)(since Y(i) = Y(9-R: for any i), pk belongs to RT. It remains to prove that the Lagrange multipliers stay in a bounded set B. In this case, we take P = G(B),which is convex and compact. For this purpose, it is sufEcient to prove that, for any z E R', there exists m(z) -= such that
-
We use again the fact that 0 E Int(Y(i)- R,). We can associate with z a scalar 8 > 0, elements x' E R' and yi E Y(i) such that 8-lz = y - 2 . Thus
Therefore, since XIENz,
1 (for z E
s),we deduce that
(by definition of w(z, 1,pA,J). Hence (9) implies that
Since x' depends only upon z, we deduce that
(13)
c lzz'fi(x')
I€N
6
sup i€N
[A(.+)[
= mdz).
Now, we use again the convexity of f r and the constraint qualification assumption (6(iii)), which implies the existence of w' E R'n Y(i). Indeed if x' E X ( z ) , then t,x'.'+(l-z,) w' E Y(i) = Y ( N ) . Furthermore, y' = Z,X"'+ +(1 - z i ) w' E R' for all i. Hence the allocation y = {yl, ...,y,} belongs to X(N).Assumption (7) implies that V i E N,a, 6~i(y')6 z,f;.(x'")+(l-zi))(d).
xicN
XIEN
Ch. 12,§ 12.31
FUZZY CORE OF AN ECONOMY
391
'Therefore, for all 2 E X ( z ) and A E A?!",we have
Finally, we deduce from (12), (13) and (14) that for any z f R'
PA,^,
(15)
Hence
f)[ml(z)+sup I€N
(fiCw>-~*)].
stays in a bounded subset. 0
12.3.2.Fwzy
core and
Wdrns dhc&iom
In particular, the characteristic functions
represent the economic game. Since they are linear, their convex cover consists ofthe characteristic functions z F+ w(z, A, p),which representthe fuzzy emnomic game by Proposition 1. Therefore, the cover of the economic game by means of the representation defined by (16)is thefuzzy economic game and thefuzzy core of this representation in Section 10.4.7 by (25). is thef w z y core of the game &+d Now,we check that the Cooperative equilibria of the representation defined by (16) are the multilosses of Walras allocations. hopsition 2. Let F = F(2) (whereZ E X ( N ) ) be a (weak)cooperativeequilibrium of the representation (16)of the economic game. Then, i f (1,F} E d n X P is an associated pair of multipliers, we have, V i E N,
(17)
(F, zi) rdP), { (ii)(0 ViEN, 2z(,t1) =
= m i n d c ~rm) ~ , X$(xi).
In particular, if i? = F(2) is a "strong equiiibrium", then {%@}is a Walras equilibrium. Proof. For all coalitions {i}, w e
obtain that
;"EI = x%(zl) w(i, J~, Q
6
I$<~+I- ( j ~ 5, )- r,(jj).
Then (jF, .">-r(P) a 0 for all i E N. Since x' f X(N), we deduce that &(@, - r,(p)) -S 0. Therefore (17(i)) holds.
2)
392 Jf
GAMES WITHOUT SIDE-PAYMENTS
x’ E Bi(.E, ~ ~ ( 3then ) ) .(ii.x’)-r;(p) J$&?)
=S
4
[Ch. 12, 5 12.3
0 and
w(i, 1:~)== T ~ ( x ’ ) + ( p2)-ri( , jj) s Xx(xi).
Thus (17(ii)) holds. If 2‘ =- 0, we deduce that (18)
j & ? )=
niin
xd E B i ( p , r d p ) )
fi(d).
We are led to introduce the following definition. Definition 1. We shall say that {Z, p } E X ( N ) x P is a “weak Walras equilibrium” if there exists A E &?’such that (17) holds. 12.3.3. The equivalence theorem
The above results imply the following equivalence theorem. Theoreml. Any Walras allocation belongs to the fuzzy core of an economy. Conversely, if assumptions (6) and (7)of Proposition I hold, the fuzzy core is contained in the set of weak Walras allocations.
Proof. The first statement follows from Proposition 10.5.3. The converse statement is a consequence of Proposition 1 and equivalence Theorem 1 . 1 . 0
CHAPTER 13
MINIMAX TYPE INEQUALITIES, MONOTONE CORRESPONDENCES AND +ONVEX FUNCTIONS In this chapter we shall improve the minimax type inequalities of Section 7.1 (which played a fundamental role in game theory). Werecall that both the existence theorem for an optimal decision rule (and thus, the Ky-Fan theorem) and the minisup theorem were proved by estimating V Q = sup inf sup p(x, y ) KE-3 X € X y € K
(where 8 denotes the family of finite subsets K of Y) and by using the existence theorem for a conservative solution in the particular form
The problem arises as to whether it is rossible to weaken and/or replace assumptions implying (*). For this purpose, we devise another proof of (*). We first observe that there always exists a generalized sequence {x,,} of elements x,, of X such that (* *)
Vy
E Y, limsup P
9(xP,y ) =S TI+.
Then we point out that convenient compactness ummptions imply that there exists a sequence x,, satisfying (* *) and (***)
x,, stays in a compact set and converges to X.
Finally, convenient continuity assumptions imply that any sequence {x,,},, satisfying (**) and (***) converges to solution i? of (*). In the first section of this chapter, we prove (*) and analogous inequalities according to the above scheme. We also devote our attention to the relaxation of the compactness assumptions. Recall the compactness assumption of Section 7.1, i.e. that there exists y o such that
396
MINIMAX TYPE INEQUALJTIES
[Ch. 13
This is replaced by the much weaker assumption 0 E Int Dom q?(-,y).
U
Y€Y
In the second section, we relax the continuity assumption of Section 7.1 [Vy E Y,x qx(y, y ) is lower semi-continuous] in order to generalize the Ky-Fan theorem. We define pseudo-monotone functions p: XX X --c R to be functions such that for any compact generalized sequence { x i } converging to X and satisfying limp sup q(xp, a) =s0, we obtain -+
Vy
-
E X, & yi ) ,lim inf y(xP,y). cc
This condition is obviously (much) weaker that the lower semi-continuity of q with respect to x. Nevertheless, we will prove a result due to Rrdis-Nirenberg-Stampacchia. This asserts that, if x -c p(x, yo) is coercive for some yo, if the functions y -c q(x, y ) are quasi-concave, if sup,,,p(y, y) = 0 and if is pseudo-monotone and lower semi-continuous with respect to x for thefinite topology, then there exists 2 E X such that
-
sup q(2,y) =s 0. Y€X
This result generalizes in a useful way the Ky-Fan inequality. Indeed, we apply it to the case where q(x, Y ) = (L(x), x-Y>
(where L maps a closed convex set X of U into the dual U*). We notice that a completely-continuous map L (continuous from X supplied with the weak topology into U* supplied with the strong topology) implies the lower semi-continuity of q with respect to x whereas amonotoneoperator L, i.e. satisfying vx, y
E
x, ( L ( 4 - L(y), x-
y>
=a
0
implies the pseudo-monotonicity of q. This is important since the gradient of a convex function is monotone, but not completely continuous. Such results imply in particular that any coercive bounded monotone andfinitely continuous operator L from U into U*is surjective. We shall extend such results to the case of correspondences and end Section 13.2 with a very short introduction to (maximal) monotone correspondences. In the third section of this chapter, we replace the convexity assumptions by other conditions to prove the estimates
Ch. 13, Q 13.11
RELAXATION OF COMPACTNESS ASSUMPTIONS
397
and
(*****) More precisely, we observe that, in our previous consideration of these inequalities, the convexity assumption was largely made use of via the barycentric inequalities Vm E
MV), f ( P 4
==
(mf)
using only the fact that the barycentric operator maps M(X) into X (and using neither the convexity of X nor the explicit definition of Pm). We are led to associate with any map y from M(X)into X the cone of yconvex functions f defined by V m E -h'(X>,
f (ym) <m,f).
We prove in particular that if the componentsf, of a map Ffrom X into R" are y-convex, then F(X) f R: is convex. This latter property played a fundamental role in game theory. We will prove the converse result, i.e. if F(X)+R: is convex, there exists a map y fromM(X) into X such that the componentsf, of F are y-convex. This allows us to prove inequality (****)when q~ is y,-concave with respect to y (and lower semi-continuous with respect to x ) and inequality (*****) when q~ is also yx-convex with respect to x and y,-concave with respect to y. Actually, we will prove that there exists a vector space W and a map 7t from X into W such that z ( X ) is convex and that f : X K is y-convex $and only i f there exists a convex junction g on z ( X ) such that f = g o z .
-
13.1. Relaxation of compactness assumptions 13.1.1. Existence of a conservative solution
Recall that the two fundamental minimax inequalities (Theorem 7.1.2 for the existence of an optimal decision rule and minisup Theorem 7.1.5) were obtained from Theorem 7.1.1 asserting the existence of X E X satisfying (1)
sup ~ ( 2y ), = vQ = sup inf sup p(x, v)
YEY
K € d X E X Y€K
by estimating this number vQ by infCeecy,x)supycr y ( C ( y ) ,y ) and supyEyinf,,, q ( x , y ) respectively.
398
MINIMAX TYPE INEQUALITIES
[Ch. 13,p 13.1
Therefore, if we want to weaken the assumptions of such results, we need to begin by weakening the assumptions of Theorem 7.1 . l . Actually, we will replace the covering 8 of Y by finite subsets by any covering d of Y satisfying (2)
if K and L belong to d,then KU L belongs to of.
We associate with such a covering the scalar
(3)
vQ(d) = sup inf sup ~ ( xy)., K€d
X € X
YEK
Notice that (4)
VQ
=
VQ(8) 6
v Q ( d ) =s vQ(B)sVQ(GY)
when 8 c ~4 c CB and where 04denotes the covering of Y by all subsets of Y . Theorem 7.1.1 is generalized as follows. Theorem 1. Let X be a subset of a topological spaceaand let 9 :X X Y be a function such that v Q ( d ) isjinite. Suppose thar (5)
+
It
there exists KOE d such that x E+ S U P ~ ~ K p(x, ,, y) is lower semicompact on X
and that, (6)
++ r p ~ ( xy,) = ~ ( xy ), x $ X is lower semi-continuous on U. V y E Y , thefunction x
if x E Y , and
+ if 00
Then there exists X E X such that
(7)
sup p(2,y) =
vQ(d).
YEY
In order to weaken either assumption (5) or assumption (6) (or both), we will replace the proof of Theorem 7.1.1 by a longer one, which use three lemmas. The first states that there exists a generalized sequence x p such that (8)
V y E Y , lim sup p(x,, y )
=s v Q ( d ) .
P
The second lemma shows that the lower semi-compactness of some function x ++ supucKo p(x ,y) implies that there exists a generalized sequence satisfying (8) and (9)
x,, stays in a compact subset and converges to an element X E
U.
Ch. 13, 8 13.11 RELAXATlON
OF COMPACTNESS ASSUMPTIONS
399
-.
The third lemma shows that the lower semicontinuity of the proper functions x px(x, y ) is a sufficient condition ensuring that
(lo)
whatever the generalized sequence { x P Z satisfying the properties (8) and (9), the limit X of { x P bsatisfies supycytp@, y ) 4 &(&).
Remark. Therefore, any assumption implying (9) (called a “compactness assumption”) and any assumptions implying (10) (called a “continuity assumption”) can replace assumptions (5) and (6) of Theorem 1. 13.1.2. Proof of existence of a conservative solution
We begin by proving the first lemma, which does not need any assumption. Lemma 1. There exists a generalized sequence { x ~ of } ~elements xB of X such that (1 1)
V K E d, lim sup sup pl(x,, y ) ] G &(A). P
[VEX
If d!is spanned by a denumerable covering, there exists a {XnlncN
denmerublesequence,
satisfving (11).
Proof. By the very definition of &(d),inequalities (12)
inf sup ~ ( xy, ) X€X
YEK
&(d)
hold for all subsets K belonging to d. Therefore, we can associate With any n E N an element xK, belonging to X such that
On the other hand, the set A! = G ~ X N preordered by the relation (14)
( K I , nl) i (Kz, n2) if and only if K1 c KZand n l s n2
-
is “filtered” or “directed” (any pair {(Ki,nr)}i = 1,2 of& has an upper bound ( K I U K Z ,max(n1, n2))). Therefore, the family (K,n) E A! x ~E X , is ~ a generalized sequence. By taking L c K and m n, we deduce from (13) that
400
MINIMAX TYPE INEQUALITIEs
[Ch. 13, 8 13.1
In other words, if we denote by p = ( K , n) the indices of &! and by xF = xg, the elements of the generalized sequence, inequality (15) can be translated as follows. With any K C a! and n E N, we can associate an index ,uo(n)(namely, po(n) = (K,n)) such that, for any v z= po(n), sup sup +v,
-4n)
YEK
y)
=?=
1 vQ(Oe)+;.
(Take Y = ( L ,m) with L c K and m
==
n.) Therefore,
It is clear that, if A is a denurncrable covering satisfying (2), then the sequence
x/~ is a denumerable sequence since d! = d ? X N is denumerable. 0
Lemma 2. Suppose that the compactness assumption (5) holds. Then there exists a sequence X~ satisfying ( I I ) and (17)
{ x ~stays ~ } in ~ a compact subset and converges to an element 2 C
U.
Proof. Take K = KO,where KO is defined in the compactness assumption (5). By Lemma 1, there exists a generalized sequence {x,,}, of elements xp of X such that (18)
lim sup sup q(x,, y ) ) = inf sup sup q(x,, y ) ) =s wQ(ui?). B
rEJn
GEK.
Thus, we can associate with any n
>0
VBP
(rtKo
an index po(n) such that
In other words, for all v 2 ,uo(n), the element x, belongs to a section of the function x t- supycK o ~ ( xy ,) and so, by the compactness assumption (5), xv lies in a relatively compact subset of U . Therefore, we can extract a generalized subsequence {x,,} which converges to an element X of the closure X of X in U. Furthermore, (20)
cp(xPy,y ) ) =slim sup sup ~ ( x , , ,y ) ) == vQ(Oe). V
P
(rEK
Hence the sequence {x,,}, satisfies properties (11) and (17). 0
Ch. 13,s 13.11
RELAXATION OF COMPACTNESS ASSUMPTIONS
40 1
Lemma 3. Suppose that the continuity assumption (6) holds. Then, for any generalized sequence { x p } of elements xp E X satisfying (11) and (17), its limit i? satisfies
Proof. Consider a generalized sequence { x i } of elements xP E X converging to X E X such that, for any y E Y,
(22)
lim sup p X ( x p y, ) = lim sup p(xp,y ) e ~ Q ( 0 e ) P
P
(since qX(x,y ) = q(x, y ) whenever x E X ) . Since x (23)
++ rpx(x, y ) is lower semi-continuous, we deduce that pX(?, y )
Ir
P
is finite, this implies that Z E X and thus, that ~ ( 2y), = qA2,y ) =s
Since ye(&) =z V Q ( 0 e ) .
lim inf rpx(x,, y ) =slim sup rpx(x,, y ) =s&(of).
0
13.1.3. Existence of optimal decision rules and minisup under weaker compactness assumptions
We give an example of compactness assumptions in the case when (24)
U = F* is the dual of a barreled space F, supplied with the weak topology u(U, F )
or
(25)
U and U* are paired spaces, U is supplied with the weak topology u(U, U*)and U* is supplied with the Mackey topology t ( U * , U ) .
'Then, Propositions 3.1.9 and 3.1.10 state that the condition (26)
3y0 E Y such that p F-+p X p , YO)= S U P ~ C X[(P, x)-rp<x, YO)] is
continuous at 0
implies the compactness assumption (27)
3 y 0 E Y such that x F+ p(x, y o ) is lower semi-compact.
Recall also that condition (26) is equivalent to (28) 28
0 E Int Dom pX*,YO)
402
MINIMAX TYPE INEQUALITIES
[Ch. 13, 0 13.1
We will relax this assumption and prove that conclusions of Theorems 7.1.2 and 7.1.5 remain valid when we replace (28) by the (much) weaker condition (29)
0 E Int
U Dom pt( .,y )
Y €Y
Example. In the case where X is defined by (30)
~ ( xy,) = (L(y),x)+y(y), where L maps
Y into U*,
assumption (28) amounts to writing that
(31)
3yo E Y such that L(y0) E Int (Xb)
(where X b is the positive polar subset of X),because
d X P , v) = o
w ; P-JW)-W(Y).
This assumption requires in particular that the interior of X b is not empty. But the weaker assumption (29) in this case becomes
(32)
0 belongs to Int [L(Y)-Xb],
which can have a meaning even when the interior of X b is empty. We state and prove the following generalization of Theorems 7.1.2 and 7.1.5.
Theorem 2. Suppose that (33)
Uy==o
(i) Y = K,,is a countable union of an (increasing) sequence of compact subsets K,,c Y , (ii) Vx E X , y F--p(x, y ) is upper semi-continuous.
W e make the following compactness assumption (34)
{ (i)
X is a subset of the dual U = F* of a barreled space F, (ii) 0 E Int u Y c y Dom g&(-, y )
the continuity assumption
(35)
V y E Y , x F+ px (x, y) is lower semi-continuous on thespace = F* supplied with the weak * topology
and the concavity assumption
(i) Y is convex, (ii) Q x E X , y I-+
p(x, y ) is concave.
U=
Ch. 13,s 13.11
RELAXATION OF COMPACTNJSS
403
ASSUMPTIONS
Let @(Y,X ) denote the set of continuous decision rules mapping Y into X. Then there exist a solution 2 E X of
Furthermore, i f we make the convexity assumption (i) X is convex, (ii) V y E Y, x t-- ~ ( xy, ) is convex,
then there exists a minisup X E X f ~ q i.e. ~ ,
(39)
sup ~ ( yf ),= sup inf ~ ( xy). , Y€Y
Y€Y X E X
We give examples where Y can be covered by a countable sequence of compact spaces.
UnQ.=]
Example 1. Let V = G*be the dual of the Banach space G. Then V = B,, where the balls B,, of radius n are weakly compact. Therefore, any subsets Y of V can be written PI
Y = (J K,, where K, = YnB,. n=O
Example 2. Let Y c V = G" be a closed convex cone such that
(40)
The cone Y + has a non-empty interior (for the Mackey topology
w,V).
Therefore, if po E Int(Y+), the subsets
(41)
Y, = {y E Y such that (PO,y> =S n }
are compact and form a covering of Y.(See Proposition 3.1.5 and 3.1.9.) Example 3. More generally, let Y c V = G* be a closed convex subset such that 3p0 belonging to the interior of the positive polar subset Yb of Y (42) (for the Mackey topology z(G, V)).
Then the subsets
(43)
Y , = {y E Y such that (PO,y) e n}.
are compact and form a covering of Y. (See Proposition 3.1.5 and 3.1.9.) 281
404
MINIMAX TYPE INEQUALITIES
[Ch. 13, Q 13.1
Theorem 2 is the combination of (a) a proposition stating that compactness and continuity assumptions (34) and (35) imply the existence of .% E X such that (44)
sup ~ ( 2y), = sup inf sup ~ ( xy,) ;
YEY
nEN
%EX YEK.
(b) a proposition stating that continuity and concavity assumptions (35) and (36) imply the following estimate
(c) a proposition stating that concavity and convexity assumptions (36) and (38) imply the following estimate (46)
sup inf sup p(x, y ) = sup inf p(x, y).
nEN x E X Y E K ,
YEY
XEX
We begin by proving (44).
Proposition 1. Suppose that assumplions (33), (34) and (35) of Theorem 5 hold. Then there exists 2 E X such that (44)
sup ~(2, y ) = sup inf sup cp(x, y).
YEY
nEN
xEX YEK.
Proof. We take for a covering of Y the denumerable covering d?= {K,,},. Lemma 1 implies that there exists a denumerable sequence {x,}, such that
Assume for a while that there exists a subsequence {xn,} of {xn}which converges to .% E U. Lemma 3 shows that the lower semi-continuity of the functions x k+ p&, y ) implies that 2 is a solution of (44). Hence it remains to prove that the compactness assumption (34) implies that {x,} lies in a relatively weak compact subset. We shall use the fact that any bounded subset of the dual U = F* of a barreled space F is weakly relatively compact (see Corollary 4 of Appendix A). To prove the latter statement, we choose any p F and check that there exists np such that (48)
sup(p,x)sa(p)
n=n,
<+a.
(Since the sequence is countable, (48) implies
i.e. that the sequence is bounded.)
Ch. 13, 5 13.11
RELAXATION OF COMPACTNESS ASSUMPTlONS
405
By the compactness assumptions (34), we can associate With any p E F an =- 0 such that p / ~ E, Dom &( .,y). Hence there exists yp E Y such that q?Xp/&,,yp) < w . Thus, for any n E N, (P, xn> E~ [v(Xn,Yp)+p>(PIEprY,)I. Since lim SUP,, vn(x,p Y J =sv Q ( d ) , there exists nr, such that p(xn, y,,) =s vb(d?)+e as soon as n a np.Therefore, (48) holds since
UYE
E~
+
This completes the proof. 13 The second estimate (45) follows from the following. Proposition 2. Let Y be a topological space. Suppose that
d is a covering of Y by compact subsets K, (i) X is a topological space,
(ii) b'y E Y, x F+ qj(x, y ) is lower semi-continuous
(i) Y is a convex subset, (ii) V x E X , y t- qj(x, y ) is quasi-concaveon Y . r f @(Y,X ) denotes the set of continuous decision rules mapping Y into X , then
Proof. It is sufficient to prove that for any K € d and C E @(Y,X),there exists 2 E X such that
(53) By replacing K by G ( K )if necessary we can assume that K is a conve-x compact subset of Y . We define the function defined on K X K by
6
(54)
v) = m z ) , Y ) .
@(z,
This function is lower semi-continuous with respect to z (since C is continuous and qj is lower semi-continuous) and quasi-concave with respect to y . Further-
406
MINIMAX TYPE INEQUALITIES
[Ch. 13, 0 13.1
more, K is compact. Thus the Ky-Fan theorem (Theorem' 7.1.3) implies the existence of P E K such that
By setting Z = C ( 3 , we deduce (53). 0 Finally the third estimate (46) follows from Proposition 3, Suppose that (55)
d is a covering of Y by compact subsets o j Y ,
that (56)
Vx
E X, y
F-+ip(x, y ) is
concave and upper semi-continuous
and thar (57)
V y E Y , x F--+ q(x, y) is convex.
Then (46)
sup inf sup ~ ( xy,) = sup inf ( x , y).
KEd
xEX YEK
YEY
XEX
Proof. It is sufficient to prove that for any compact convex subset K of Y, (58)
inf sup &, y) = sup inf cp(x, y).
x € X YEK
Y E K xEX
(If (58) is true, we obtain that
sup inf sup ip(x, y ) = sup sup inf q ( x , y).
KEd
x E X YEK
KEd y€K
xEX
= sup inf ~ ( xy).) , YEY
XEX
To prove (58), we use the fact that (a) K is convex and compact and that, for all x E X,y +-t rp(x,y) is concave and upper semi-continuous. (b) X is convex and V y E Y , x q(x, y) is convex. Then the Nikaido theorem (Theorem 7.1.8) implies the existence of a IIIZU( infJ E K of 9:therefore, (58) holds. 0
++
Ch. 13,s 13.21
RELAXATION OF CONTINUITY ASSUMPTIONS
407
13.2. Relaxation of continuity assmnptions: variational inequalities for monotone correspondences
' (i) 3y0 E X such that x t *)
!-+ q(x, yo) is lower semicompact, (ii) V y EX, x F--q(x, y) is lower semicontinuous for thefinite toPo~ogY (iii) Vx E X,y F.-. Q,(x,y ) is concave, 9
in proving the existence of x' satisfying
We Will prove that assumptions (*) imply the existence of a generalized sequence {xP} satisfying
(***I
vy
E
x,
lim sup q(x/l, y ) es 0 P
and (****)
x,,
stays in a compact set and converges to 3.
Among the functions Q, having the property that (**)holds whenever Z is the limit of a generalized sequence satisfying (* * *) and (** * *), we single out the pseudo-monotone functions, i.e. functions such that (**) holds whenever 3 is the limit of a compact generalized sequence {xP}#satisfying only (* * * *) and (*****)
lim sup q(xP, 2) == 0.
(instead of (****) and (*+*)). Hence there exists a solution 2 of (* *) whenever Q, is a pseudo-monotonefunction satisfying (*). We will apply this result in the cases when dx,
v) = (W),X - Y )
or cp(x, v) =
4w), x-Y)
where L and S are respectively a map and a correspondence from a closed subset X of U into U'.
408
MINIMAX TYPE INEQUALITIES
[Ch. 13, Q 13.2
We say that L and S are pseudo-monotone if the associated function y is d s o pseudo-monotone. We shall prove that completely upper semi-continuous correspondences are pseudo-monotone and that finitely upper semi-continuous monotone correspondences are pseudo-monotone. Moreover, the sum of a pseudo-monotone correspondence and of a finitely upper semi-continuous monotone correspondence remains pseudo-monotone. The introduction of monotone (and pseudo-monotone) correspondences is motivated by the f'dct that, in infinite dimensional spaces, the subdifferential of a convex function is not (generally) completely upper semi-continuous, but is monotone. We then deduce that there e-xists a solution X f X to the variational inequalities ( L ( Z ) , 2 - y ) =s0 for all y E X whenever L is a coercive, bounded, jinitely continuous and pseudo-monotone operator. In particular, such operators are surjective. These results will be extended to the case of correspondences. We shall continue the study of monotone correspondences with a short introduction to cyclically monotone and maximal monotone correspondences. 13.2.1. Variational inequalities
We devote this section to the relaxation of the continuity assumption (1)
V y E Y , x F+ ~ ( xy,) is lower semi-continuous
in the case where q~ is a function defined on the product XXX. The reqmrement (1) is too strong in many instances. We shall illustrate this point with the problem of finding a solution X f X to the variational inequalities.
(2) where (3)
vy E
x,
(L(3,Z-y)
=s 0
X is a closed convex subset of the dual U = F* of a Banach space F, supplied with the weak topology u(F*, F)
:ind where (4)
L is a map from X into U* = F
(we take p(x, y ) = (L(x),x-y)). In most instances, variational inequalities arise from optimization problems or, more generally, from non-cooperative games (see Propositions 4.3.2, 4.3.6 and 9.1.4 for instance). We can extend this problem to the case of correspondences S satisfying (5)
S maps X into U' and takes non-empty compact convex values.
Ch. 13,
13-21
409
RELAXATION OF CONTINUITY ASSUMPTIONS
We shall say that x' is a solution of variational inequalities if there exists p' E U' such that
(6)
{ (ii)(i)
2 E X and p E S(x'), V y E X , ( F , 2-y) =S 0.
We can restate this problem in the following way. (7)
X" E
x is a solution of 0 E S(x')+ N ~ ( x(or ) S(2)n -Nx(x')
#
0).
where N A x ) is the normal cone to X at'x (which is equal to 0 if x E Int (X)) Lemma 1. Suppose that (5) holds. Then 2 e X is a solution of the variational inequalities (6) if and only if
(8)
sup ob(S(2);2 - y )
d
0.
Y€X
Proof. It is clear that if there exists p E S(2) satisfying (F, Z - y )
6
0 for any
y E X , then ob(S(5));2 - y ) s 0 for any y E X.
Conversely,assume that
sup inf (p, . G y ) = sup ob(S(X), 2 - v ) YEX
Y€X
P€S(f)
e 0.
Then, since X is convex, S(2)is convex and compact and (p, X- y ) is separately continuous, we deduce from the max inf theorem (see Theorem 7.1.5) that there exists p < S(%)such that sup
Y€Y
(p,2 - y )
= sup
inf (p, 2 - y ) s 0. 0
Y€Y PESm
Therefore, we have to apply the Ky-Fan theorem (Theorem 7.1.3) to the function p defined by
(9)
d x , v) =
4w;x - Y )
in order to solve the variational inequalities (6). Remark. If X is a cone, any solution 2 of (6) is actually a solution to the problem of finding X e X and p e Xfsuch that p e S(2) and (F, 2) = 0. If X = U, % is a solution of the multivalued equation 0 E S(Z). Definition 1. Let S be a correspondence from a subset X of a locally convex space U into its dual U'. We shall say that
410
[Ch. 13,s 13.2
MINIMAX TYPE INEQUALITIES
(a) S is completely upper semi-continuous if it is upper semi-continuous from X supplied with the weak topology into U*supplied with the strong topology; (b) S is finite4 upper semi-continuous if it is upper semi-continuous from X supplied with the finite topology into U* supplied with the weak topology; (c) S is “coercive” on X
if either X is bounded or
( d ) S is “bounded on X”
if
(11)
lim
IIxII-t-
X€X
IlPll < + sup -
P€S(%)
IIXII
(e) S is “monotone” if for any x, y E X, p E S(x) and q E Sb), then (P-q, x-Y) 0.
Lemma 2. Suppose that U is the dual of a Banach space and that (12)
L and S are coercive and bounded.
Then,for any y E X , the functions x coercive.
-
(L(x),x-y) and x
-.
a*(S(x), x- y ) are
Proof. Since S is bounded on X, there exist constants K and M such that a”(S(x); y ) 4 KI Ix II IIy I I when I Ix II z= M. Therefore, we obtain q(x, y ) = = B(s(x);x-~)- B ( ~ ( x ) ; x ) - a ~ ( ~ ( xB ) ;~y ~) ( ~ ( x ) ; x ) - ~Ilyll.Since lIxll S is coercive, we deduce from these inequalities that q( y ) is also coercive for any y E Y. 0 a,
13.2.2. Existence of a solution to variational inequalities for completely upper semi-continuouscorrespondences
Thanks to Lemma 1 and the Ky-Fan inequality, we obtain the following result.
Proposition 1. Suppose that U is a reflexive Banach space and that the map L and the correspondence Sfrom X into U satisfy (I2) and (13)
L and S are completely upper semi-continuous.
Then there exists a solution x’ E X to variatwnal inequalities (2)and (6).
Ch. 13, p 13.21
41 1
RELAXATION OF CONTIMUITY ASSUMPTIONS
Proof. (a) We will use Theorem 7.1.2 (as in proving the Ky-Fan theorem). By setting p(x, y ) = ob(S(x);x - y ) , we see that p satisfies
(b) Since pl is lower semi-continuous with respect to x for the finite topology (which is stronger than any vector space topology) and since the identity is continuous from X into X supplied with the finite topoIogy (see Proposition 7.1.3), we deduce from the proof of Theorem 7.1.2 that
(c) Furthermore, the coerciveness and boundedness of S implies that (by Lemma 2) V y E Y , x t---
y(x, y ) is coercive.
Therefore, by Proposition 3.1.1, we deduce that (14)
3y0 E
Y such that x t-- rp(x, yo) is lower semi-compact.
At this stage of the proof, Lemmas 1 and 2 and (1.2) and (14) imply that there exists a generalized sequence {xu),, satisfying (15)
x,, stays in a compact subset and converges weakly to x' E
X
and (16)
VJYE X, lim sup y(x,, y ) s tA
0.
=S
(d) It remains to prove the following result, which implies that (17)
VY EX, ~ ( 2y), -Z lim inf q(x,, y ) . P
Proposition 2 . Let S be a completely upper semi-continuous correspondencefrom a closed subset X of a reflexive Banach space U into U'. Then,for any bounded generalized sequence {x,,},, satisfying (15) and (16), its limit x' has the property that
Proof. Let {x,,}, be a generalized sequence satisfying (15) and let ibe its limit. We take p p E S(x,,) such that (p,, x,,-y) = ab(S(x,3; x,-y). Since {xP} is
contained in a weakly compact subset and since S is completely upper semi-
412
MINIMAX TYPE INEQUALITIES
[Ch. 13,$ 13.2
continuous, the subsets S(x,) stay in a (strong) compact subset of U', which is equicontinuous. Hence, we can extract a generalized subsequence (again denoted by) {p,}, which converges strongly to j5 E U*and a generalized subsequence (again denoted by) { x , } ~which converges weakly to f E X . Therefore (p,,, x F - y ) converges to (p,X - y ) and thus, ub(S(Z); 3 - y ) 4 ( j j , Z - y ) = = limp CP,,x,-y,) = lim, a b ( ~ ( x , ) x,-y). ; When U is a finite dimensional space, all vector space topologies coincide and upper semi-continuous correspondences are completely upper semi-continuous. When U is no longer finite-dimensional, the assumption that S is completely upper semi-continuous is much more stringent. For instance, we proved that the subdifferential af of a differentiable convex function is upper semi-continuous from U supplied with the strong topology into F supplied with the weak topology (see Proposition 4.1.8). It is not true in general that it remains upper semi-continuous when U is supplied with the weak topology and U* with the strong topology! Nevertheless, subdifferentials of convex functions are monotone, i.e. (19) (P-4, X-Y) 2 0 V P E v - ( x ) , v q E W Y ) . We obtain (19) by adding the inequalities .f(x)-f(r)
as
(P,x-Y)
S(Y)-f(4=s( 4 , Y -
4
since p E af(x), since 4 E af(v).
Since optimal solutions of convex functions exist, there is some hope of proving the existence of a solution 2 to variational inequalities if we replace the assumption of complete upper semi-continuity by an assumption of monotonicity. Indeed, if we look back at the proof of proposition 2, we notice that the complete upper semi-continuity of L implies much more that we need (eq. (18) instead of inequality (17)). 13.2.3. Pseudo-monotone functions: the Brgzis-Nirenberg-Stampacchia theorem
We are led to introduce the following concept of pseudo-monotone function. Definition 2. We shall say a function tp defined on XX X is "pseudo-monotone" if', for any generalized sequence {x,} satisfying {x,}, stays in a compact set and converges to X (20)
and (21)
lim sup cp(xfl,2) =s 0 C'
Ch. 13,s 13.21
413
REI-AXATION OF CONTINUITY ASSUMPTIONS
its limit X satisfies (22)
Vy E X,
q(3, y ) =S lim inf p(xP,y). P
We shall say that an operator L mapping X into U* is “pseudo-monotone’’ if the function q~ defined by q(x, y) = (L(x),x - y ) is pseudomonotone. Proposition 2 shows that a completely continuous operator is pseudo-monotone. It is also obvious that any function y lower semi-continuous with respect to x is pseudo-monotone. We can extend the Ky-Fan theorem in the following way.
Theorem 1(Brezis-Nirenberg-Stampacchia). Let (23)
us
assume that
X is a closed convex subset,
that (24)
3y0 E X such that x t+ q(x, yo) is lower semi-compact,
and that (25)
, y k-+ p(x, y ) is quasi-concave, { (ii)(i) SUP,,x Xq(y, 0Vx E
Y) =
k t us assume moreover that (26)
I
(i) p is pseudo-monotone, (ii) V y E X , x I- y(x, y ) is lower semi-continuous on X supplied with the finite topology.
Then there exists 2 such that supycx p(2, y ) s 0. Proof. Since p is quasi-concave with respect to y and lower semi-continuous with respect to x for thejinite topology, Theorem 7.1.2 implies that
[The identity is continuous when X is supplied with the finite topology and SUPYEX d Y 7 Y ) 01 On the other hand, we deduce from Lemmas 1.1 and 1.2 that there exists a generalized sequence {x,) satisfying (28)
{x,,} stays in a compact set and converges to x‘ E
X
414
MINIMAX TYPE INEQUALITIES
and (29)
for any finite subset K of X, lim sup sup a,(xP,y P
[YER
[Ch. 13, Q 13.2
’I
-S
vQ e 0.
By taking K = {Z}, we see that the sequence {x;} satisfies properties (20) and (21). Thus, since a, is pseudo-monotone, we deduce that, V y E X, (30)
a,(?, y) s lim inf a,(xP,y ) P
lim sup a,(xP,y) s 0
=G
P
by (29) With K = {y}).
*Remark. By using the pseudo-monotonicity of a,, we do not exploit all the information given by (29). We can introduce a concept of “weak pseudomonotonicity” by replacing the requirement (21) by the weaker one (29) in Definition 2.
Remark. We point out the following result.
-
Proposition 3. If we assume that a, is pseudo-monotone and f :X R is lower semi-continuous, then the function y deJined by y(x, y ) = a,(x, y)+ f (x)- f(y) is also pseudo-monotone.
Proof. Let {x,} be a generalized sequence of xP E X converging to ? and satisfying (31)
Then
Jim SUP (cP(XP9 P
+f(XP)
-fm)4 0.
lim sup a,(xP,x’)+lim inf f ( x P ) - - f ( Z )G P
P
lim sup tcP(xP, x3+f(xP)-f(31
6
P
0.
But, since f is lower semi-continuous, lim inf, f (x,) ==f(Z). Hence lim supr a,(xP, 2) 4 0 and the pseudo-monotonicity of a, imply: (32)
VY
EX,
cP(%
u) lim inf Cp(XP, Y). Q
P
Therefore, VY E
x,
dx’, r>+f(a-f(.Y)
=slim inf a,(xP, y)+lim inff(x,)-f(y) P
ez lim
P
P
inf (a,(XP, y>+f(x,)-f (Y)).
Thus we proved that the function p defined by p(x, y) = p(x, y ) + f ( x ) - f ( y ) is pseudo-monotone. 0
Ch. 13,s 13.21
RELAXATION OF C O N T I ~ T YASSUMPTIONS
415
13.2.4. Existence of a solution to variational inequalities for pseudo-monotone
correspondences
Theorem 2. Let U be a reflexive Banach space and (33)
S be a correspondence from X into U ' with non-empty boundedclosed convex values
where X is a closed convex subset of U.Suppose also that (34)
S is coercive and bounded on X
and that (35)
S is pseudo-monotone andjnitely upper semi-continuous.
Then there exists a solution x' E X to the variational inequalities (6).
Proof. We apply Theorem 1 to the function 9 defined by ~ ( xy), = ab(S(x), x-y). It is coercive with respect to x by Lemma 2. It is pseudo-monotone by Definition 2 and clearly concave with respect t o y . It remains to prove that it is lower semi-continuous with respect to x for the finite topology. This amounts to saying that for any K = {xl, . . .,xn}, the function p defined on Anby (36)
V(A) = ~ ( s ( @/(? A ( ~ ))- ) y ),is lower semi-continuous
where (37)
c n
@(A) =
PXk.
k=l
Let N = .$(K- y)", which is a neighborhood of 0 for the weak topology of U'. Since S is finitely upper semi-continuous, there exists 7 =- 0 such that I[ 1- 2011 6r] implies that (38)
S(B(4)
and that (39)
SUP PE S(SCb,)
= S(B(Ao))+N I (P,m- /?(W I =s P1 &*
We deduce from inequality (38) that (40)
ob(S(B(4), B ( 4 - Y )
ob(s(B(W,B(Ao)-Y) +d(S(/Wo)),/?(4-B(Ao)) + o w B(4-Y)
But, since @(A) E c< K- y , we obtain that (41)
d(N,B(A)-y) a-+c.
416
Also, (39) implies that (42)
[Ch. 13, Q 13.2
MINIMAX TYPE INEQUALITIES
.b(S(B@o,),
+
B(4-B(lo>)
E.
Therefore, inequalities (40),(41) and (42) imply that
~(1) --~(Ao)-E
when
Ill-iloll
eq.0
Taking X = U,we obtain the fallowing surjectivity theorem. Theorem3. Suppose that a correspondence S from a reflexive Banach space U into U*, with non-empty compact convex values, is coercive, bounded, pseudomonotone and finitely continuous. Then there exists ?. € X such that 0 € S ( 3 . 13.2.5. Pseudo-monotonicity of monotone correspondences
Proposition 4. Any monotone and finitely upper semi-continuous correspondence is pseudo-monotone. The sum of a monotone and finitely upper semi-continuous correspondencemd of a pseudo-monotone correspondence is pseudo-monotone. Proof. (a) We begin by proving that a monotone and finitely upper semicontinuous correspondence is pseudo-monotone. Let {x,} be a compact gensatisfy eralized sequence converging to x' and let pP E s(~,) (43)
Iim sup (pP,x,- y ) = lim sup ab(S(x,); x P - y ) c
=s 0.
P
Let ji E S ( 3 . Since S is monotone, then ( P , xP-2)
=zz (PPY
4i-q
implies that 0 = Iim (ji, x,-Z)
s l i m inf ( p P , xP-%)
P
P
and thus, that, by (43), (44)
lim (p,, xP-%) = 0. P
We now choosey E X , z, = Oy+(l-B)T E X(O E]O,l[) andp, E S(Z& We write inequality (pP-pe, x,-zo) a 0 in the following form:
(~&,.x~--+(pe, xP-Z) - 0(pe, ?-Y)+ O ( p P , 9-Y) z- 0
Ch. 13, 0 13.21
417
RELAXATION OF CONTINUITY ASSUMPTIONS
Taking the lirn inf, we deduce from (44) that (45)
(po,3 - y )
e lim inf P
( p P ,2 - y ) .
Since the correspondenceS is upper semi-continuousfor the finite topology, we deduce that pe converges to some p E S(2)when 0 converges to 0. Then d(S(z);Z-y) =s (p, 2-y) =slim inf (pP,2 - y ) = P
lirn inf ( P , ~~, ~ - = y lim ) infd(S(x,); xh-y) P
P
since, by (44), lim inf [(p,,, 2-xP)+(pP, xP-y)l = P
= lim [(pP, 2-xP)]+lim inf [(pP,xP-y)l c
P
= lim inf [(p,, xP -y)]. P
(b) To prove the second part of the proposition, suppose that S = T4-R where T is monotone and finitely upper semicontinuous, R pseudo-monotone. Let {x,} be a compact generalized sequence converging to 2 and satisfying (46)
l h SUP (pP,XP-?)
= lim SUP d ( s ( X P ) ; X P - 2 )
6
0,
P
P
where pP = qp+rPE S(xJ, qP E T(xP)and rP E R(xJ. Since T is monotone, we deduce from (qP-q,xP-Z) a=0 that (rp, xc+
= (P,-q,, xr-2)-s(p,-q,
xu-$
where q is any element of T(2). Therefore lim sup (rP, xP-x') B lim sup (pc, xP-?) P
- lim (q, xc-z) c
P
=s 0.
Since R is pseudo-monotone, we deduce that (47) V y E X, a*(R(Z), 2 - y ) -slim inf d(R(xP), xP-y) c
and, in particular, that limp d(R(xJ, xP- 2) = 0, Since Tis pseudo-monotone (being monotone and finitely upper semicontinuous) we deduce from lim sup d(T(xp),xP-2) Q lim sup (qP, q-3 r
P
4
lim sup (pp,x,- 2) - lim d(R(XP), x,- 2)4 0 P
29
LI
418
MINIMAX TYPE INEQUALITIES
[Ch. 13,O 13.2
that (48)
=slim inf ob(T(x,), x,-.').
d(T(2),.'-y)
P
Let cj E T(Z) satisfy (4,1-y) = d'(T(1),X-y) and let F E R(Z) satisfy (?, 1-y) = d'(R(X), X-y). Let j = q3-F E S(Z). Then d(S(2);1- y ) =s (q+r, 2-y) =slim inf (qp, x,-%)+lim P
=z lim
c
inf [(pP, +-%)I
inf (rP,x p - l ) P
= lim inf ab(S(x,); xc-y). c
We thus have proved that S is pseudo-monotone. 13.2.6. Monotone and cyclically monotone correspondences
Definition3. We shall say that a correspondenceS from U into U* is "cyclically monotone" if (49)
for any finite sequence { { x i , pi}}~risnof the graph G(S), we have 0. ( P I , X I - X ~ ) + ( P ~ , x~-xx~)+ *.. +(pm x n - ~ l )
Of course, any cyclically monotone correspondence is monotone. In fact, the cyclically monotone correspondences characterize the subdifferentials of convex functions.
Theorem 4. (Rockafellar). Let Sbe a correspondence from U into U'. Then S is cyclically monotone i f and onIy if there exists a lower semi-continuous convex properfunctionf such that (50)
for all x E U, S(x) c af(x).
Proof. We begin by proving the suilicient condition. For this purpose, it is sufficient to prove that the correspondence af is cyclically monotone, since dearly this implies that any s c af is also cyclically monotone. Let {{xi,pi}} be a sequence of n elements of the graph of af; then
f
(Xd-f
(xz)
f(x2)-.f(x3)
es -G !
(PI, XI- x2) ( p % x2-x3)
Ch. 13,g 13.21
RELAXATION OF CONTINU~TY ASSUMPTIONS
Adding these inequalities, we obtain that 0 =s(PI,xi-xxa)+(pa, x ~ - x Q ) +
419
-.-+(pn, zn-xi).
Conversely, we now consider a cyclically monotone correspondence S from U into v* and construct an appropriate$ If, for any x, S(x) = 0, the statement is trivial. If not, fix a pair {XO,PO}of the graph of S and definef by (51)
. f ( x ) =sup {(Y P ')}
((p.,x-
€a
X")+
( p 4 x m - x,,-
1)+
..-4- (Po, x1-
xo)).
where S is the family offinite sequences {{xi, pf}}lslsn of G(S). In the first place, sincef is a pointwisesupremum of continuousaBne functionals, it is a lower semi-continuous convexfunction. Its domain is non-empty since (52)
f ( x 0 ) = 0. (Since S is cyclically monotone, ~ ( x ois) the supremum of non-positive scalars (p,, XO- x,,) * +(PO, X I -XO), and thus,f(x0) -S 0. Since 0 = @o, XOXO) +(PO, XO-XO), then { x o , ~ oE} 8 achieves this supremum, which is equal
+-
to 0.) It r e d to check that S(x) c
af(x). For this purpose, we take p E S(x)
and show that (53)
f(x)-f
(r)
(P,x - Y )
for all Y E U.
Let E =- 0. By defhition (51) off, there exists a finite sequence { ~ , , p , } ~ ~ ~ of elements of G(S) such that (54)
f ( x ) .s (p", x-xn)+(p*-l, x.-xn-1)+
Since {x,p} belongs to G(S), we also obtain that
+
***
+(PO, xl-xo)+&-
+- - +
( p , y- x ) (pn, X L X") +(Pa-1, X" -X " 4 ) * (Po, x1- XO) e f(Y). adding these two inequalities, we obtain that f ( x ) - f Q 4 (p, x - ~ ) + E . Letting e tend to 0, we obtain (53). 0 In order to derive the identity S = we are led to introduoe the following definitions. (55)
af,
Definition 4. A correspondence S is said to be maximal cyclically monotone (resp. maximal monotone)if its graph is not strictly contained-inthe graph ofa cyclically monotone (resp. monotone) correspondence.
With such a definition, Theorem 4 implies the following statement. 29.
42 0
MINIMAX TYPE INEQUALITIES
[Ch. 13, 5 13.2
Proposition 5. Let S be a maximal cyclically monotone correspondence. Then there exists a lower semi-continuous convexfunction f such that S = aJ In the next section, we shall characterize maximal monotone correspondences in Hilbert spaces onfy and mention some of their properties. 13.2.7. Maximal monotone correspondences
In this section, U denotes a Hilbert space identified with its dual U*. Recall that a correspondence S from X C U into U is monotone if and only if its graph G(S) c U X U is monotone in the sense that
(56)
a subset G c UX U is monotone if V {x,P), {Y, 4 ) E G,(P-q,x-Y) 0.
We h a l l use the following characterization of a monotone correspondence in Hilbekt space . Proposition 6. Let S be a correspondence from X into U.It is monotone i f and only if (57)
V k > 0, V { x , p ) , {y,q} E G ( 9 , Ilx-y+k(p-q)ll
proaf. Suppose
that S is
-+&2[[p-q112+2k(p-q.
-
Ilx-Yll.
monotone. Then ((x-y+k(p-q)ll2= IIx-y112+ Z- I \ x - y \ \ a since Cp-q.x--Y) 0 for s is
X-y)
monotone. Conversely, assume (57) and take {x,p } and b,q} in G(S). Then (p-q, x- y ) = (1/2Wll x.-Y +k(p- q) 112- II x- y 113- k21 Ip-q 1121 z% -kllp-q I12/2. Letting k converge to 0, we deduce that (p-q, x-y) == 0.Thus 5' is monotone. 0 We also note the following consequence.
Proposition 7. Let S be a monotone correspondence from X it0 U and let Y = S(x> c U. Then (l+S)-l is actually a map satkfying Il(l+S)-l (P)-(l+S)-l(dll =s lIP-dIforal~P,q E y. Proof. Let x E (l+S)-yp) and y E (l+S)-4q).Then p - x E S(x) and q-y E S(y). Inequality (57), with p and q replaced by p - x and q-y, implies that Ilp-all = Iix--~+(p-x-q+y)ll S- Ilx-~lI, i.e. that Ilx-yI1 4 IIp-qII. I f we take p = q, this implies that (1 +S)-l contains a unique point x, i.e. that (l+S)-l is a map. If we take p # q, the .inequality implies that this map is Lipschitz with constant 1.
Ch. 13,§ 13.21
421:
RELAXATION OF CONTINUITY ASSUMPTIONS
It is clear that the union of an increasing family {G,},,, of monotonesubsets of U X U is monotone. Therefore, by Zorn's axiom, each monotone subset is contained in a maximal monotone subset. Thus any monotone correspondence can be extended into a maximal monotone correspondence. In fact, we shall use only the following analytic characterization of maximal monotone correspondences: Proposition 8. Let S be a monotone correspondencefrom X into U.It is maximal monotone if and only if (58)
i f V { y , 4 ) E GQ,
(p-4,
X-V)
z- 0, thenp E S(x).
Proof. This is left as an exercise. 0 We now state the fundamental characterization of maximal m'onotone correspondences. Theorem5 (Minty). Let S bea monotone correspondence from X into U.Then S is maximal monotone k and only if 1+S maps X onto U.In this case, (1 + S F is a map from U into U which is Lipschitz with constant 1.
Proof. (a) Suppose that 1 + S is surjective. We shall prove that (58) holds. For this purpose, take { x , p } to satisfy (p-q, x - y ) a. 0 for all {y, q } E G(5'). Since 1+S is surjective, there exist yo E U and qo E SCyo) such that p + x = = yo+qo. Hencep-qo = -(x--yo). Thus, -Ilx-yol12 0. Therefore x = yo and p = q o E S(y0) = S(x). Proposition 8 implies that S is maximal monotone. (b) Suppose that S is maximal monotone. Let y E U.We shall prove that there exist x E U and p E S(x) such that y = x+p. We write R(x) = S(x)-y, which defines a maximal monotone correspondence from X into U.We have to prove that there exists x E U such that - x E R(x). By Proposition 8, this amounts to finding x E U such that,
Vb, 4) E G(R),
(x+q, X-Y) =s0. Let yo and q o E R(y0) be fixed. If a solution x of (59) exists, it belongs to the
(59)
subset (60)
Xo = { X E X such that (xfqo, X - ~ O ) = = 11xlI2+(x, ~o-yo)-(qo,yo) 0).
This set is obviously bounded. Thus C = &YO) is a weak compact convex subset of U in which we have to find solutions of (59). For this purpose, we introduce the function fdefined on C X G ( R ) by hll
f/y
I,, "I\ - /
Y l "
Y-W\
- Ilrll2r/u
n-*?\-/"
w\
422
MINIMAX TYPE INEQUALITIES
[Ch. 13, 8 13.2
It is lower semi-continuous and convex with respect to x (for the weak topology). Theorem 7.1.1 implies that there exists a solution x E C to the following inequality.
sup Ax, {Y, q } ) CV.4) E G(R)
(62)
SUP
inf
m a f ( x , bi,qr}) = V Q
K E J xEC f = l .
....n
where K = { { y l , ql}, ..., bn, qn}} ranges over the finite subsets of G(R). We define the function f K on & X M n by setting (63)
MA,
p) =
pi(B(A)+qi, p(A)-yi)
i=1
where ~ ( 2 = )
Aiyi. i=1
We have the following estimates. vQ sup inf sup fK(A, p). (64) K E J a c m ac~nr. Now, it is clear that f K is lower semi-continuous and convex with respect to
1 and linear and continuous with respect to p. By the minimax Theorem 7.1.7, there exists a saddle point {&,I PO}. Therefore,
Now, it remains to check that (6s)-
f&,.,u)
6
But, f
.
9
P) =
=
smce (qi-qk,yk-yf)
0 for all p E
i
1-1
k=l
i
i, k - 1
=s 0
A?.
PiPkcs(,) +qi, Yk-Yi)
pipk(qf-qk, Yk-yi)
for all i, k = 1,
0
..., n because R is monotone.
Inequalities (62), (64), (65) and (66) imply that (59) holds. Hence -x f R(x). We state some other properties of maximal monotone correspondences. Propositioa 9. Let S be a maximal monotone correspondencefrom X into U, Then the images S(x) are closed and convex. If a sequence of elements xn converges stongly to x and if a 'sequenceof elements p, E S(xJ converges weakly to p , then p f S(x).
Ch. 13,$ 13.21
RELAXATION OF CONTINUITY ASSUMPTIONS
423
Proof. By Proposition 8, S(x) is the set ofp E U such that (p-q, x-y) 0 for all Cy, q } E G(S). This implies that S(x) is closed and convex. Let x,, converge strongly to x andp, E S(x,) converge weakly top. Hence, for any b,q} E G(S), inequalities (pn-q, xn-y) a 0 hold. In the limit, we obtain that, (p-q, x-y) 0. Then, again by Proposition 8, p E S(x). 0
One can check that a correspondenceS is maximal monotone by proving that 1+ S is surjective thanks to the Minty theorem. Example 1. Let b be a maximal monotone map from R into R. Let 8 be an open subset of R" v d U = LYf2). If x is a function from 8 into R, we set sb(x) :0 bb(x(w)). Let X = ( x E L2(Q)Such that sb(x) E L2(Q)}.Then s b is obviously a monotone correspondence from X into L2(Q).Assume also that 0 = b(0). Then Sb is maximaI monotone. To see this let y E L"(sz)be given. Since b is maximal monotone, for almost all w E a, there exists x(w) E R such that x(w)+b(x(w)) = y(w). Since 0 = = (l+b)-I(O), we deduce that Ix(w)l = l(l+b)-l(y(~))-(l+b)-~(O)I 4 4 ly(o)-Ol. Since y E L2(sa), then x belongs also to L2(sz).Hence 1 +& is surjective. For instance, the map x E Lo(8)I--+ x3 C L2(Q)is maximal monotone. Example 2. Any monotone map S continuous from U supplied with the finite topology into U supplied with the weak topology is maximal monotone.
Proof. Let x and p satisfy (p--S(y), x-y) 0 for all y E U.By Proposition 8, we have to prove that p = S(x). For this purpose, we take any z E U and we set y = Oz+(l-O)x = x-O(x-z) where 8 E 10, 1[. We deduce that {p-S(x-e(x-z)), X - Z ) 60. On letting 8 tend, to 0, the continuity property of S implies that ( p - s ( ~ )x, - z ) == 0 for all z € U.Hence p E s ( ~ ) . Example 3. The most important example of a maximal monotone correspondence is the subdifferential of a lower semicontinuous functi n:
-3 be a proper lower semi-continuous convex function and X = {x E U such that af ( x ) # 0) the domain of the subdgerential Propositim10.Let f : U 1-41 - 00,
o f f . Then af is a maximal monotone operator. Furthtmore, Int(X) = Int(Dom(f)).
x = Dom(f) and
Proof. (a) To prove that af is maximal monotone, we prove that 1 +af is surjective, i.e. if y is given in U,there exists x such that y--x E af(x). In fact, we shall check that this solution is the minimum of thefundion g :z !-+ f (z)+
424
[Ch. 13, 5 13.2
MINIMAX TYPE INEQUALITIES
+ i l l y- z I12. This function is lower semicontinuous and lower semi-compact (since, if p E Dom(f*), the inequality g(z) =S 8 implies that lly-z1I2 4 ~ 4 ( 8 + f * ( p ) + \ I p ( (and ~ ) thus, that z remains in a weak compact subset of v). Therefore, there exists x E U which minimizes g. By Proposition 4.2.6, x is a solution to the variational inequalities f ( x ) - f ( z ) + ( x - y , x - z ) =s 0. It means that y- x E a f (x), i.e., that x E (1 + af)-'(y). Hence af is maximal monotone. (b) Since X c Dom (f),we prove that Dom(f) c X. Since af is maximal monotone, for any k =- 0, there exists a solution x, E X of the equation
(67)
y E
xk
+
af(xk)
where y is given in Dom(f). This implies that for all x E U.
Hence x, converges to y and thus, Dom(f) c X c Dom(f)
.
(c) We prove now that any y E Int(Dom(f)) belongs to X . For this purpose, consider the solutions x, of (67) and set x = y+8z in (68), where 11 zll 1. We obtain that
'Therefore, supk>ol ((y-x,/k), z) I is finite. By the Banach-Steinhauss theorem (see Theorem 10 of Appendix A), we deduce that (y-xk)/k remains if1a weaK compact subset and that a generalized subsequence converges weakly to an element p. We have seen in (b) that xk converges to y . Since f is lower semicontinuous, inequalities (68) imply that (69)
f(y)-f(x)
=G
(p, y - x )
for all x E U.
Hence p E a f ( y ) , i.e. y belongs to X. Since X c Dom(f), this shows that the interiors of X and Dom(f) coincide. 0
Ch. 13, 5 13.21
425
RELAXATION OF CONTINUlTY ASSUMPTIONS
Maximal monotone correspondences have many other properties (which
can be found in the book of Brdzis [3] and the paper of Brtzis and Haraux for instance). We mention some of these without proof. Let S be a maximal monotone correspondence from X into U (a) X and Int(X) are convex. (b) If Int(X) # 0, then, for all x E Int(X), S is bounded on some neighborhood of x. (c) S is surjective if and only if V y E S(x), S-l is bounded on some neighborhood of y. (d) In particular, any coercive maximal monotone correspondence is surjective (compare with Theorem 3). (e) Vk > 0, the Lipschitz map Jk defined by Jk(x) = (1 kS)-l(x) satisfies
+
(70)
J&) converges to the projection of x onto X when k
-.
0.
( f ) Vk > 0, the map S, = (l/k)(I-J,) is maximal monotone and Lipschitz with constant Ilk. It satisfies: Sk(x) E S(J,(x)), Vx E X. Furthermore, Vx E X , Sk(x)converges weakly to the projection of 0 onto S(x)(S, is called the Yosida approximation of S). (g) Let T be a maximal monotone correspondence from Y into U . If either
(71)
Int ( X ) n r f 0
or (72)
V k =- 0, V {x, J’}
E W), (P,sk(x)) Z= 0,
then S+T is also maximal monotone. (h) We shall say that a maximal monotone correspondence T from Y into U satisfies the Brkis-Haraux property if V y E Y,V q E T(Y), 3c = cb,q) such that, V {x, P} E W),( p - q , x - y ) * c. It easy to check that any “tri-monotone correspondence” (satisfying ( p - r , x - y ) 3 (r-q, y-z) when { x , p } , b,q } and {z, r } belong to G(T)) and in particular any cyclically monotone correSpondence satisfies this property. We can prove that if T is a maximal monotone correspondence from U into U satisfying the Brtzis-Haraux property, then
(73)
{
(i) Int ( ( S + T ) ( X ) )= Int ( S ( X ) f T ( U ) ) , (ii) ( S + T ) (X)= S(X)+T(C/).
In particular, S t T is surjective whenever S or T is surjective. Furthermore, if X = U, then Z + S T is surjective (Brtzis and Browder [I, 21) The latter result enables us to solve non-linear Hammerstein integral equations.
426
[Ch. 13,§ 13.3
MINIMAX TYPE INEQUALITIES
(i) Finally, the fundamental property is the existence and uniqueness of the solution of the multi-valued differential equation &/dt E -S(x(t)), x(0) = XO, where xo is given in X . (See Brdzis 131 for instance.)
13.3. Relaxation of convexity assumptions In most cases, we can replace convexity assumptions by assumptions of ])-convexity: If y maps the set M(X)of discrete probabilities into X , we say that f:X R is y-convex if
-
VJm E
4 x > ,fm) -s <wn
In other words, we can usually replace the barycentric operator /3 used in defining ordinary convex functions by any map y. We know that F ( x ) +R: is convex whenever the n components f; of F are convex functions. This result remains true if the functions $i are y-convex. Moreover, the converse statement holds. If F ( X ) +R: is convex, there exists a map y from M(X)into X such that the functions j; are y-convex. This is the reason why the minisup theorem remains true when we assume that the function 9 is yflonvex with respect to x and y e n c a v e with respect to y for any maps yx :M(X) X and yr :J’Z(Y) Y. Also, an optimal decision rule continues to exist when we assume only that 9 is y,-concave with respect to y . We can explain this by the fact that any cone of y-convex functions is the image of a cone of convex functions by a linear operator a* :g at’g = g o at wherez maps X onto a convex subset n ( X ) of a vector space W. In particular, we can construct y-convex functions by taking products f = g o at of convex functions g by a given map a from X onto a convex subset z(X) of a vector space W. The most important example is given by the Lyapunov theorem. We take X to be a a-algebra on a set Q, W = R“ and f i :X R” to be an atomless bounded vector-valued measure. The Lyapunov theorem states that n ( X ) is convex. Hence the set functions A E X g(fi(A))are y-convex whenever g is a convex function defined on R”. This allows the study of games where “nature” is a player choosing an “event” A E X in a set fz of “states of nature”. We also meet “natural”examp1es of maps y fromM(X) into X.For instance, we take X to be the family of the (weakly) compact convex subsets K of a vector space U and the map y to be
-
-
-
-
-.
Ch. 13,s 13.31
RELAXATION OF CONVEXITY ASSUMPTIONS
421
If g is a convex lower semi-continuous function on U (in the usual sense), the function f defined on X by f ( K ) = inf g(x) xPK
is y-convex. 13.3.1. Definition of y-convex functions
-
Definition 1. Let y :&(X) X be a corespondence with non-empty values mapping the set &(X) of discrete probabilities m into subsets y(m) of X . A function fdefined on X i s said to be “y-convex”if and only if V m E M(X), Vx E y(m), f ( x ) &f*(m) = (m, f >.
(1)
A functionf is “y-aflne” y’
12)
V m E W X ) ,V x E y(m), f ( x )
=f’M)
and is “y-concave” if
-f is y-convex. A subset K of X is said to be “y-convex” if
(3)
r(Jm>)= K.
By Proposition 1.3.5, convex functions and convex sets are y-convex functions and convex sets when we take y = /? to be the barycentric operator. We will present two other classes of examples insections 7 and 8 below.
13.3.2. The fundamental characteristic property of families of y-convexfunctions
-.
Let f : X X Y R be a function. It is characterized by the map F from X into the space &(Y) = RY (of functions on Y) defined by (4)
vx E
Let y : A ( X )
-X
x, W ):Y
+
W )( Y ) =f(x,Y).
be a correspondence with non-empty values. If the function
f is y-convex with respect to x, we obtain the following fundamental property. Proposition 1. Suppose that
t5 )
Vy
Y , the function x
-.
f (x, y ) is y-convex.
Then
(6) where R :
F(X)+R; is a convex subset of R Y
denotes the cone of non-negativefunctions on Y.
428
MINIMAX TYPE INEQUALITIES
[Ch. 13,p 13.3
Example. If Y = {I, . . ., n} is a set of n elements, then RY = R", RZ = R" and the map F from X into Rnis defined by (7)
...,S(x, 0, ... A x , n)} E R*.
F(x) = { A x , 11,
Proposition 2. If the n functions x
-
f ( x , i ) are y-convex, then
F + ( X ) = F(X)-tR$ is convex in Rn.
(8)
This generalizes Proposition 1.3.10, which played a fundamental role. Proof of Proposition 1. Let (9)
g=
f: ai(F(xi)+hi) E co (F(X)+RY,)
i=1
where x, E X,hi E R:, ocr 0 (for i = 1, find xo E X and ho E R: such that (1 0)
..., n) and z=l ui = 1. We have to
g = F(xo)+ho.
For this purpose we take
(1 1)
i=1
and
(12)
aiS(xi) E M(X)
xo f y(m) where m =
ho =
C aihi + C aiF(xi)-F(xo). n
n
1-1
151
We have that xo E X since y maps d ( X ) into X . The function -F(x0) is positive since,
gz1 c@(x,)-
by the y-convexity of the function x + f ( x , y). Therefore, the functions h being positive, the function ho is also positive, i.e ho E R:. 0
In fact, this property characterizes sets of y-convex functions on X in the following sense. Proposition 3. Let f :X X Y X into RY satiflying (14)
-
R be a function characterized by a map F from
F(X)+RT is a convex subset o f R y .
Ch. 13, 0 13.31
RELAXATION OF CONVEXITY ASSUMPTIONS
Then there exists a correspondence y :M(X) that the functions x -.f ( x , y ) are y-convex.
-
429
X with non-empty values such
Proof. Let m = CZIa,6(x,) E M ( X ) be a discrete probability on X . Then (14) implies that C;=l a,F(x,) = C~=la,(F(xi)+O)) belongs to F(X)+R: since this set is convex. Thus there exists at least one x E X and h E RI such that CY=la,F(~,)= F(x)+h. In other words, for any y E Y, a i f ( x f ,y ) = = (C;=l a,F(x,)) ( y ) = F(x)(y)+h(y) z = f ( x y, ) . Therefore, if we denote by y(m) the subset of such elements x E X , we see thatfis y-convex with respect to x. 13.3.3. The minisup theorem for yrconvex-y,-concave junctions In the minisup theorem (see Theorem 7.1 S), we can replace the assumptions of convexity of q~ with respect to x and concavity of g~ with respect to y by assumptions of yXconvexity and yr-concavity respectively. Specifically, we consider two correspondences with non-empty values. (15)
yx :M ( X )
-
X
and y y : M(Y)
-t
Y
Theorem 1. Suppose that X is a subset o j a topological space U, that (16)
3 yo E Y such that x
-
q(x, yo) is lower semi-compact
and that
(17)
Vy
Y , x -. px(x, y ) is lower semi-continuous on U .
Suppose also that (18)
V y E Y, x
and
(19)
Vx E X , y
-
~ ( xy ), is yx-convex
p ( x , y ) is yr-concave.
Then there exists a minisup X of q ~ .
Proof. Since assumptions (16) and (17) imply that there exists X such that supucydi?, y ) -s w o , it remains to prove that (20)
V Q = sup inf sup ~ ( xy), s vb = sup inf ~ ( xy). , K€d
X € X
Y€K
Y€Y
X€X
The proof of this inequality is analogous to that of Theorem 7.1.5. We have lo prove that for any
430
K
= {yt
MINIMAX TYPE INEQUALrTIES * * -9
vn}
[Ch. 13,8 13.3
E 8,
since the n functions x +Q(x, yi) are Y-COnvex, then 4+(X) = 4(X)+RT is convex we set 4(x) = {&, YI), . . ., q(x, vJ}. This implies that
(by the separation theorem). Now, since the functions y k-4 q(x, y ) are +,-concave, we deduce that n
(23)
SUP
inf
a w n XEX
C I ' ~ I (yj)x ,=s 9 i=l
for we associate with any A E Mnan element yAE y that
(Eel Ai6(yi)), and deduce
Hence (22) and (23) imply (21). 0
Remark. We can use this theorem to prove that (24)
&(&) = sup inf sup q ( x , y ) = K€d
vb
X€X Y € K
whenever (i) cz! is a covering of Y by compact subsets of Y, (ii) V x E X,y /--+ q(x, y ) is yy-concaveandupper semi-continuous (iii) V y E Y, x I-+ p(x, y ) is yx-convex (see Proposition 1.3). This allows us to replace the assumptions of convexity and concavity of Q in Theorem 1.2 by assumptions of yx-convexityand y,-concavity of Q with respect to x and y respectively.
Ch. 13,$ 13;3]
RELAXATION OF CONVEXITY ASSUMPTIONS
43 1
13.3.4. Existence of optimal decision rules for functions yy-concave with respect to Y
We can also replace the assumption of quasiconcavity by an. assumption of quasi?-concavity in Theorem 7.1.2 concerning the existence of an optimal decision rule. For this purpose, we need to d e h e a “y-finite t ~ p o l ~ g on y ”subsets Y analogous to the finite topology on convex subsets. Dehition 2. Let y be a map from A ( Y ) in to Y. The “y-finite topology” on Y is the strongest topology for which the maps A E A!” y=(A) = y ( z , = l A f N y t ) ) arecontinuous when K = {yl, . . ., yn} ranges over the family of finite subsets.
++
A map C from Y into a topological space X is continuous when Y is supplied with the y-finite topology if and only if (26)
VK
E 8, C ~ isKcontinuous from d” into X.
Also, we define a function cp to be quasi-y-concavewith respect to y if and only if (27)
Vx E X,V a E R, the subsets y-convex.
E Y such that p(x, y ) z- cc} are
We can prove the following theorem.
Theorem 2. Suppose that the compactness and continuity assumptions ( I d ) and (17) hold. Let y be a map from A ( Y )into Y and let @(Y,X ) be the set of continuous maps from Y into X when Y is supplied with the y-fiite topology. If we assume that (28)
Vx E
X,
y
-
p)(x, y ) is quasi-y-concave,
then there exists x’ E X such that
Proof. This is the same as the proof of Theorem 7.1.2. Since assumptions (16) and (17) imply the existence of it such that supuEyy(Z, y ) = vQ, it k sufficient to prove that we can associate with any K = { y ~.,.., yn} and any map C E @(Y, X)an element x E X such that
432
[Ch. 13, 5 13.3
MINXhfAX TYPE INEQUALITIES
For this purpose, we introduce the function cj7 defined on d n X Y by
These subsets are closed since CyKis continuous and pl is lower semi-continuous with respect to x . Since the functions y p(x, y ) are quasi-y-concave, we deduce that V 1E d", 1E Fi where A, = {i such that 1' =- O}. Hence the assumptions of the Knaster-Kuratowski-Mazurkiewicz lemma are satisfied and there exists 2E F,. Thus 1 = Cy&) = Cy(z;=,26(yi)) satisfies inequality (30).
-.
nicA, n;l=l
Remark. This theorem implies that vQ(0e) = sup inf sup p(x, y ) = (33) K € d x € X Y€K
whenever (34)
inf
sup v(C(y),y )
C€Q(Y,X) Y € Y
compact subsets if Y, { (ii)(i) d is a covering of Y isbyquasiy-concave (see Proposition V x E X , y c-t pl(x, y )
1.2).
(see Proposition 1.2). We can replace the assumption of concavity with respect to y by an assumption of y-concavity in Theorem 1.2. 13.3.5. Example: Image of a cone of comex functions by Z*
Let z be a map from a set X into a vector space W such that the image z ( X ) is a convex subset of W. (35) It d e h e s a linear operator Z* from R"(X)into RX by (36)
vxE
x, ( Z ' f )
(XI
=f b ( X ) l .
A question immediately arises. Is the image by a* of the cone of convex functions on z ( X ) a cone of y-convex functions on X? The answer is positive. Let Z' :d ( X ) W be the linear extension of fl defined by
-
(37)
aA(m)=
n
C I'n(xi)
i=1
(see Definition 1.3.4).
wheneverm =
n
C Ai6(xi)
i=l
Ch. 13, 0 13.31
RELAXATION OF CONVEXITY ASSUMPTIONS
433
Since n(X)is convex, d ( m ) belongs to 4X)whenever m E A?@). Therefore, the correspondence yn from M(X) into X defined by (38)
y,
=7c-lnd
has non-empty values. In other words, i f m E M(X), yn(m) = { x E X such that z x = d ( m ) } . Proposition 4. Let z ; X -,W satisfy (35). Then the function f is the image z'g of a convex function g on a ( X ) if and oniy f is a y,,-convex function (where 7, = 7t-st,).
Proof. The first statement is quite obvious. Let f = z'g = g o n where g is convex. Let x E y,(m) = a-Iz''(m) where m = a'd(x,). Then nx = C a h ( x I ) and
Conversely, let f be y,-convex. First, we notice that (39)
if
+I)
= ~ ( x z ) , thenf (XI) = f(x2).
[We have that X I E c 1 n x 2 = p d ( 6 ( x z ) ) and x z E z-lfixl = ylld (&I)) Thus f ( x d = s f ( x ~and ) f(x2) ~ f ( x d . 1 We denote by (z) the equivalence relation defined by (40)
and by (41)
x1(.)xz
0
z ( x d =4 x 2 ) .
the factor space and 8 the canonical { Xl(7t) surjection from onto X / ( z ) . X
Therefore, any map constant on the (It)-equivalence classes can be written y = @ o 8 where @J is defined on X/(n). In particular, 7t = 8 where 2 is a bijective map from X/(n) onto a(X) and f = 30 8 where f maps X/(.) into R. Since % is bijective, we can write 8 =&-lo it and f =fo 8 = P o 2-k =g ci z where g It remains to deduce convexity of g =Po;-' from the y,-convexity of$ 30
434
MINIMAX TYPE INEQUALITIES
[Ch. 13, 9 13.3
Let ElEla'b(ui) be a discrete probability onz(X) and take xi.€c l ( u i ) and
Therefore, g is convex since
=s
n
n
n
i= 1
IE l
i=1
C a'f(x,) = C a'g(7txi) = C aig(ui). 0
13.3.6. Relations between convexity and y-convexity We now answer the fundamental question : is the cone of y-convexfunctions the image by an operator 7t* of a cone of convex functions?
Proposition 5. Let y :M ( X ) - - +X be a correspondence with non-empty values. Then there exist a vector space W and a map iz from X into W such that
{ (ii)(i).c(X)
(42)
is convex,
y =7 t - l ~ ~ .
Proof. Let S * ( X ) be the space of discrete measures on X and A4 be the closed vector space spanned by the set N of measures Gym-m when m ranges over S*(X),i.e. (43)
N = {Gx-m}x~y(rn~,rn~~/n(~.
We introduce (44)
Let 6 : X (45)
factor space { (ii)(i) thethe canonical surjection W
-
=
8*(X)/M, 7tA from S*(X)onto W.
S*(X)be the Dirac operator. We define iz by
iz
= n43 maps X into W via
S*(X)
x - 8 * ( X ) .&8 * ( X ) / M = w. 8
Thus adis a linear extension of Z, i.e. zd( cr'd(x,)) = C a ' d ( S(xi)) = cr,z(x,). On the other hand, since ax-m E A4 when x E y(m) and m E M(X),we obtain that d ( 6 x - m ) = nASx-ZA(m) = zx-7tAm = 0.
Ch. 13,§ 13.31
RELAXATION OF CONVEXITY ASSUMPTIONS
435
Therefore (46)
x E y(m) if and only if x
E a-l aA(m).0
Therefore, Propositions 4 and 5 imply the following theorem.
Theorem 3. Let y be a correspondence with nonwpty values. Then there exist a vector space W and a map a from X into W satisfying (47)
I
(i)
(X) is convex,
(ii) the cone of y-convexfunctions is the image by a* of the cone of convex functions on a(X).
We remark also that (48)
the vector space F of y-a5ne functions equals NL = MI.
Therefore, since 8’(X) is the dual of 8(X)supplied with the topology of pointwise convergence, which coincides with the weak topology o(8(X), &‘(A’)), we deduce that (49)
F is isomorphic with the dual of W
when F is supplied with (the restriction to F of) the topology of pointwise convergence and W is supplied With the factor topology of S*(X) supplied with the weak topology. It is quite obvious that the weakest topoZogy on X for which z is continuous coincides with the weakest topologyfor which all the functions of F are continuous. We shall denote this topology by u,,(X, F),and call it “the weak topology on X associated with y”. It is an Hausdorff topology if and only if eitherfi is injective or, equivalently, if F “separates the points of X” in the following sense (50)
If f ( x ) =f ( y ) for all f E F, then x = y.
Formula (46) shows that fi is injective if and onIy if y is a map. Note also that the weak topology o,(X, F) on X is defined by the semidistances
where K ranges over the family 8 of finite subsets of F. Theorem 3 shows that the properties of convex functions can be “carried” into analogous properties ?C’
436
MINIMAX TYPE INEQUALITIES
[Ch. 13,g 13.3
of y-convex functions. For instance, we can use a vector space of y-affine functions as a "space of perturbations" of minimization problems of functions on X. We define pt" by
and
The function 9); is convex and lower semicontinuous on X arid the function p); is y-convex and lower semi-continuous. As in Theorem 2.4.1, we can characterize yconvex lower semi-continuous functions. Proposition 6. Let F be the vector space of y-aflne functions on X, supplied with the weak topology u,,(X, F). A function q~ is y-convex and lower semi-continzous if and only if (54)
vxE
x, p(x)
= v?(x)-
Proof. If 9, is y-convex, there exists a convex function y defined on I Z ( X )such that vxE
x, p(x) = y ( 4 x ) ) .
The function y is lower semi-continuous since is lower semi-continuouson X supplied with the weakest topology for which rt is continuous. Therefore, if we denote by fb = y ~ ~the ( ~proper ) lower semi-continuous convex function from W into ]- -, - ] , we deduce from Theorem 2.4.1. that
+
where
Therefore
Ch. 13,g 13.31
RELAXATION OF CONVEXITY ASSUMPT~ONS
437
13.3.7.Example: 8-convex set functions
Let U and U*be two paired vector spaces supplied with their weak topologies. Consider the following strategy set : (55)
X is the set of weakly compact convex subsets K of U.
We define the map y = 8 from M(X)into X by (56)
B [ f=1: Ai6(Ki)) = kf:1 AiKi E X
xy=l
(since A' Ki is a weakly compact convex set.) Therefore, the cone of 8-convex .functions is the cone of functions f defined on X and satisfying
For instance, if g is a convex function defined on U,then the function f defined on Xby
is obviously 8-convex.
To prove this we write
c W(Ki). n
=
is1
In particular, for any p E U*, the functions (59)
K
-
a#(K,p) and K F-
n
C Iia#(K,pi)
1-1
are 1-affine. We can give an explicit form for the map n from X into W such that = z-%zA.
B=
438
[Ch. 13, 0 13.3
MINIMAX TYPE INEQUALITIES
We suppose that ‘60)
W is the subspace of 8 ( U * ) spanned by the support functions d ( K ; .) and a*(K; .) when K ranges over X.
a n d n is the map from X into W defined by (61)
VK E X , z(K) = ab(K; *)€ S(V*).
The map z is injective since the lower support functions characterize the closed subsets K. It is clear that = z-17tAsince
,
/ n
n
n
In Section 15.3 we shall study in more detail the weakest topology on X for which a is continuous (when S(U*) is supplied with the topology of pointwise convergence). If M is the symmetric convex hull of a finite subset of U*,we obtain from the minimax theorem that
sup (ub(K1;p)-ab(K2;p)) =
PE M
=
sup inf
sup ( ( p , x1-x2>I
PEM x : € 4 x:€Kz
= sup
inf sup I(p, xl-x2)1
= sup
inf PM(xl-xz).
r t € K z xi€Ki PEM
XLEKZ xiEK1
Hence the topology on X is defined by the semi-distances (62)
d,w(Ki, K2) = p ~ ( z ( K i ) -fl(K2)) = max sup inf PM(xl-xz), sup [X:€KI
xzEKz
inf PM(xl-x2)
x z € K t XIEKI
1
where M ranges over the (symmetric convex hulls of) finite subsets of U*. It is the weak Hausdorff topology we study in Section 15.3. Interpretation. If U describes a commodity space, we can represent X as the family of production sets (which may describe firms, for instance).
Ch. 1 3 , s 13.31
RELAXATION OF C O N V E X I ASSUMPTIONS ~
439
If g denotes a cost function defined on the commodity space, the function f defined on X by (58).associates With each firm K its minimal cost f ( K ) = = inf,,,g(x). Such cost functions are B-convex. 13.3.8. Example: Convex ,functions of atomless vector measures
We mention a very important example of a map from a set X into a vector space W such that z(X) is convex. We takefor the strategy set a u-algebra X on a set 8.
(63)
This is appropriate whenever 8 describes the set of “states of the nature” and a strategy A E X represents an “event”. It is also appropriate when 8 describes “a set of players” and A E X a “coalition of players”. It is quite natural to construct loss functions on X by means of measures p :X R. In particular, we shall devote our attention to loss functions of the following type :
-.
(64)
V A E x , f ( A )= g(pdAh * . *, p n ( 4 ) = g(A4)
where ,u = {pl,
. . ., ,un}is the vector valued measure EX
-
R” and g is a convex function defined on the image p(X) c Rn of the o-algebra (65)
P :A
+
p ( A ) = {pl(A),
* y
pn(A)} E
X by the vector-valued measure p. Such loss functions will be y-convex whenever (66)
p(X) is a convex subset of R”.
The Lyapunov theorem states that such a property is true whenever the measures pi are “atomless”. Definition 3. A set A E X is said to be an “atom” of pi if and only if (67)
(
pi(A) # 0 and for any B c A, B E X , then either
pi(B) = pi(A) or pi(@
= 0.
A measure pj. (resp. a vector valued measure p = {PI, . . ., p,,}) is “atomless” if it has no atoms (resp. if the measures pi are atomless). The explicit statement of Lyapunov’s theorem k given below (see Appendix C).
Theorem 4. (Lyapunov). Let p = { p l y mapping X into R“.
. . .) p,,}
be a vector valued measure
440
MINIMAX TYPE INEQUALITIES
0(68)
p is bounded and'atomless
then (69)
the image p ( X ) is a convex compact subset of R".
[Ch. 13, Q 13.3
CHAPTER 14
INTRODUCTION TO CALCULUS OF VARIATIONS AND OPTIMAL CONTROL
We begin this chapter by extending to infinite dimensional spaces the duality theory for optimization problems presented in Chapter 5 for the case of finite dimensional spaces. Using the minimax inequalities of Section 13.1, we prove the existence of a solution 1 E V* to the dual problem
(*I
F*(-L*F,p) = inf F * ( - ~ * p , p )= - inf F(x, Lx) X € U P E v*
where F is a lower semi-continuous proper convex function and L E &(U, V ) . The second (and independent)section deals with minimization problems with i)n integral criterion and constraints of the form
where A(o) E 2(U,R"). We shall prove the existence of Lagrange multipliers, even when the assumption of the convexity of thefunctions &+q(o,5 ) is dropped. In the third section, we adapt the results of the first section to calculus of variations problems of the form
and optimal control problems
(****)
21
inf
= d
X(O)=€o. X(W)+&)
X(l)==€l
x(4=w4
dd
0
This section has to be regarded as a very short introductionto the field of calculus of variations and optimal control. Finally, we devote the fourth (and independent) section to an introduction to the dynamic programming approach to continuous and impulsive control. problems (which is the subject of a forthcoming book of Bensoussanand Lions). 441
442
CALCULUS OF VARIATION AND OPnMAL CONTROL
[Ch. 14
We now briefly summarize the main results of this chapter. In the general case, we assume that R c U,Y c V and
F is a convex lower semi-continuous function from RXY into R. We associate with F the function H defined on U X V *by W X ,P ) = SUP [(P,Y ) - m Y€V
Yll.
The existence of a Lagrange multiplier will be established either under the Slater condition 3x0 E R such that LOXE Int Y
or under the weaker constraii;t qualification assumption
I
Ubl
(i) R = K, is a countable union of compact subsets of U, (ii) V p E V*,x -t H(x, p ) is upper semi-continuous, (iii) 0 E Int (Y-L(R)).
Ursl
Note that the assumption R = K,,is always satisfied when U is the .dual of a Banach space and in particular, in reflexive Banach spaces and Hilbert spaces. We will point out that the extremality relations
(-L*p, p ) E aF(2, LZ) are equivalent to the relations L*jj E SXH(Z,p )
and ~
E a$(?,$).
In the second section, we prove the existence of a Lagrange multiplier p E Rm*of (**) in a constructive manner, by approximating to it by problems of the form
We study what happens if we drop the assumption of convexity of the functionsf, in the existence theorem for a Lagrange multiplier, and estimate the (positive) length of the consequent duality gap by
Ch. 141
CALCULUS OF VARIATXON AND OPTIMAL CONTROL
443
where we denote by e ( f ) the modulus of non-convexity of a function f.This is defined by
(convex functions are those whose modulus of non-convexity vanishes). This estimate shows that the duality gap shrinks as the number of functions involved in the problem increases and explains why there exists a Lagrange multiplier in tnecaseof aminimizationproblem with integral criterionand constraints. We adapt the first section to the case of problems of calculus of variations whose abstract form is v = inf X€
U(L)
F(x, Lx, yx)
where y maps a subspace U(L)of U onto a space S and L maps U(L) into V. We assume that Uo(L) = Ker y is dense in U.
We construct a dual problem which does not use the transpose L* o f L , but the transpose L*,of its restriction to Uo(L), defined by ( G p , x ) = ( p , Lx) when p E V* = V , x E Uo(L).
We extend this formula when p E V(LE) = { p E V such that L:p E U = U*}. We prove that there exists a unkpe operator B* E B(V(g), E*) such that the abstract Green formula (L*p, x ) = ( p , Lx)+(B.p, 7x1 whenp E V(Lo*),x E W L )
hulds. This leads us to the required dual problem v* = inf PE W
O)
F*(-LGp, p , Pp).
If F is a lower semicontinuous convex function mapping RX Y X C into R (R c U,Y c Y,C c S), the extremality relation becomes the “Euier-Lagrange system”:
{ -L@, p , B*jj} E aF(2, LZ, 7x3. This is equivalent to the Hamiltonian system
444
CALCULUS OF VARIATION A N D OPTIMAL CONTROL
[Ch. 14, $ 14.1
where H is the “Hamiltonian”, defined on UX V X E * by m x , p , 4 = SUP [ ( p ,V)+(% 5)YE v,Z E E
m,Y , 01.
We elaborate an analogous theory in the case of optimal control problems which leads to the famous Pontryagin maximum principle. We emphasize the fact that attention is restricted to the (very particular) “convex” case and to the (much simpler) abstract formulation of the problems of calculus of variations. This avoids the technical difficulties involved in the explicit computation of the conjugates of the functions F(x,y) = f #(o,x(o), 0
Y W )do.
Finally, we devote a short section (Section 14.4) to an introduction to the “dynamic programming” approach to optimal control stopping time and impulsive control problems. This is a borderline topic in the context of the subject matter of this book, since the methods are more those of partial differential equations than of functional analysis which we chose as our main mathematical method. Above all, the rewarding framework within which to use the dynamic programming approach is the “stochastic” one, while this book deals with a deterministic world. The main feature of the so-called dynamic programming approach is to determine the “performance function”
(assumed to be smooth) as the solution of a first order partial differential equation (the Hamilton-Jacobi-Bellman equation). If the performance function is known, we show that we can solve the optimal control problem and even find, at each time 2, the optimal control u(t) as a function of the optimal state ~ ( 2 ) . We shall also use this approach to solve “impulsive control problems”. In this case, the performance function, assumed to be smooth, is a solution to partial differential quasi-variational inequalities (the Bensoussan and Lions quasi-variational inequalities). 14.1. Duality in in6nite dimensional spaces
The existence of Lagrange multipliers of the minimization problem w = inf F(x,Lx) X€X
(where R c U,Y c V,F :RX Y R,L E 2(U,V)) is easily proved when Y is a finite dimensional space (see Chapter 5). -+
Ch. 14,s 14-11
DUALITY IN INFINITE DIMENSIONAL SPACES
445
We shall extend these results in the case when V is an infinite dimensional space and, for this purpose, apply the minimax theorems to the “Lagrangian” defined on UX V* by 4x7 P) = (P,L 4 - W x ,P)
where the “Hamiltonian” H is defined on UX V* by H(x7 P) = SUP YEY
0
7
Y>-
F(x7
u)l.
We check that, when F is convex and lower semi-continuous (with respect to y), the Lagrange multipliers are nothing other than the max-inf of the Lagrangian. We recall that j j is a Lagrange multiplier and f a minimal solution if and only if { -L*p,p } E aF(x’, L2) and we prove that an equivalent assertion is that Lx’ E aPH(T7?,) and L*p E &H(x’,p).
Using the minisup Theorem 7.1.5, we prove that the “Slater condition” 3x0 E R
such that LXOE Int (Y)
implies the existence of a Lagrange multiplier. This is quite a stringent requirement, which is not met for instance when Y is the cone of non-negative functions of a Lebesgue space Lp(s2)whose interior is empty. But when U and V are reflexive Banach spaces we relax this assumption by replacing it with the constraint qualification condition 0 E Int (Y-L(R))
we used when V = R“‘.This is done by applying the minisup Theorem 13.1.2. In the above discussion no use is made of the actual form of the Lagrangian and so we are led to call a “Lagrangian” of a minimization problem
w = inff(x) XEX
wheref’: U --c
1-
00,
+ -1
any function I defined on UX V* satisfying w = inf sup I(x,p). XEU P € Y *
To prove the existence of a Lagrange multiplier (i.e. of a ma-inf of a Lagrangian), we introduce the functions A defined on U X V by 4x9
u) = SUP [(P,r>+I(& PllPEP
446
CALCULUS OF VARIATION A N D OPTIMAL CONTROL
[Ch. 14,$ 14.1
We prove that either the assumption 3 x 0 E U such that 0 E Int Dom 4x0,
-1.
or, in the case of reflexive Banach spaces, the relaxed assumption
u Dom d(x;*),
0 E Int
xE u
implies the existence of such a Lagrange multiplier. What is more, under convexity assumptions, we shall see that defining a duality theory,by means of a Lagrangian is equivalent to the use of a family of perturbed minimization problems w(y) = inf d ( x , y ) X€X
(since w = ~(0)). In this case, vie prove that the set of Lagrange multipliers is the subdifferential av(0) of v at 0. 14.1.1. Lugrangian of a minimization problem under linear constraints
Consider a minimization problem of the type
where
(2)
I
(i) R is a convex subset o f a vector space U, (ii) Y is a closed convex subset of a topological vector space V, (iii) L E l(V,V ) , (iv) X = { x E R such that Lx E Y}, (v) F : R X Y R. -L
We shall not assume any more that V is a finite-dimensional space. Nevertheless, we shall prove the existence of a Lagrange multiplier j5, i.e. of jj E V* satisfying (3)
{ (i) w*
= F*(-L*p,p) = inf,,p F*(-L*p,p),
(ii) w+w* = 0,
where F* is the conjugate function of F. For this purpose, we associate with the function F its “Hamiltonian” and with the minimization problem (1) its “Lagrangian”.
Ch. 14,$ 14-11
Definition 1. Let F be a function from U X V into ]nian” H is the function defined on U X V * by (4)
447
DUALITY IN INFINITE DIMENSIONAL SPACES
vxE
00,
+ -1.
Its “Hmilto-
u, V P E v+, W x , P) = sup [(P,Y ) - m Y)l. Y€Y
The “Lugrangian” of the minimization problem v = inf F(x,Lx) X€X
is the function I defined on UX V* by (5)
y
b,PI
Jw-H(x, PI.
= (P,
In other words, for all x, H(x, is the conjugate function of the functiom We point out the following obvious properties. a)
+-F(x, y).
Proposition 1. The conjugatefunction F* of F can be written
If we assume that (7)
vx
E X, y
I-+ F(x, y ) is convex
and lower semi-continuow
then
Proof. This is left as an exercice. 0 Example. If F(x, y) = f ( x ) + g ( y ) , wheref :R that (9)
+
R and g :Y
-.R,
we deduce-
H(x, P) = g*(P)--f(x)
and that (10)
b P ) = f ( x ) + ( p , Lx)-g+(P).
These definitions are consistent With the definition of the Lagrangkin of a minimization problem v = inf,,, f ( x ) by taking g = y, to be the indicator of Y. The following proposition motivates the introduction of the Lagrangian and relates it to the concept of a dual problem.
448
CALCULUS OF VARIATION AND
OPTIMAL CONTROL
[Ch. 14,§ 14.1
Proposition 2. W e have
sup inf I(x, p ) = - w*.
(1 1)
P € V * xCR
If we assumefurthermore that (7)holdr, then (12)
inf sup I(x,p) = w.
xER P € V*
In this case the Lagrange multipliers are the max-inf of the Lagrangian.
Proof. We have that
inf I(x, p ) = inf [ ( p ,Lx)- H(X, p ) ] xER
xER
=
-sup [(-L*p, x€R
+
x) H(x, p)] =-F*( --L*p, p )
using (6). Hence sup inf l(x, p ) = - inf F*(-L*p, p ) = - v*. PE v* In the same way, we obtain that P E V xER
sup 0,PI = P
SUP P
[(P,L x ) - W x , PI1 = F(x, Lx)
by (8) (since assumption (7) holds). Hence (12) holds. But, to say that p is a Lagrange multiplier amounts to sa*s inf of 1, i.e. inf l(x, p ) = inf sup l(x, p ) = v =- v*. (13)
that p is a max-
x € R PEP
xER
We mention the following property of the Lagrangian.
-
Proposition 3. The function p H(x, p ) is convex and lower semi-continuous. If we assurn8 that I F is a convex function on R X Y , (14) '
then x I--+ H(x, p ) is concave. Proof. This is left as an exercise. 0
We also need the conjugate function of the Lagrangian with respect to p. Proposition 4. Suppose that (7) holds. Then the conjugate function A(x, p * -l(x, p ) is deJind by (15)
A(x, 7 ) = F(x, L x + y ) .
0 )
of
Ch. 14, 0 14.11
DUALITY IN INFINITE DIMENSIONAL SPACES
449
Its domain is equal to (16)
Dom d ( x , -) = Y-Lx
when x E R.
Proof. We check that 4x7
Y ) = SUP [(P, r>+0,PI1 PE v* = SUP [(P,Y PE
v*
+w-H ( x ,PI1
= F(x, Lx+ y )
by (8). Since the domain of F is RXY, it is clear that the domain of d(x, .)is the set of y E V such that Lx+y E Y. 14.1.2. Extremality relations
We have proved (Proposition 5.2.1) that a minimal solution if and only if (17)
p is a Lagrange multiplier and I
{-L*p,p} E aF(x', L2).
Proposition 5. Suppose that (7) holds. Then the extremality relations (17) are equivalent to (18)
L2 E a p H ( 2 , p ) and L*p E SxH(2,p).
Proof. In the first place, (17) amount to (19)
F(1, L2)- F(x, y ) =s (- L*p,I- x)+
(p, LI-
By taking x = 2,this implies that SUP [(P, Y)- F(2, Y)l = (p, JW-F(-% Y
w
= H(F,F).
Hence, the very definition of H implies that (20)
H(I,P)- H ( 2 , p ) 4 (p-p, L I ) .
On the other hand, we can write (19) in the form
(3,r)- F(x, r) =s(L*& x)- F(I, L I ) = (L*p,x- I)+ H(I, p). 31
y).
450
CALCULUS OF VARIATION AND OPTIMAL CONTROL
[Ch. 14,$ 14.1
Taking the supremum over Y , we obtain that H(x, I) H (Z , p ) =s(- L*@,2- x).
This means that L*p E a,H(I, p). Conversely, let LZ E apH(x', p ) and L'p E $#(Z, supp [(p,Y)- H ( x , p)l by (8) and since
p ) . Since F(x, y ) =
(p, L2) = F(2, LI)+ H(I, jj) we deduce that F(Z, LZ)-F(x,y)+(@, Y- LZ)
H(x,~)-H(Z,~).
Since L'p E a,H(x',p), we obtain F ( I ,Lf)-F(x,y )-(j j,L% -y )
=G
(-L*jj, 2-x).
Hence { -L*p, p} E W(3, L$. 14.1.3.Existence of a Lagrange multiplier under the Slater condition
In this section, we shall replace the assumption that V is a finite-dimensional space by the assumption that (21)
V is an infinite dimensional space supplied with the Mackey topol-
ogy Z(V, V*).
Recall that the Mackey topology z(V, V') coincides with the initial topology when V is a barrelled space (see Corollary 3 of Appendix A).
Theorem 1. Suppose that (22)
from R Y into R, { (ii)(i)FV x Ea R,convexfunction F(x, y ) is lower semi-continuous X
is
y
+
and that the following "Slater condition" holds:
(23)
3x0 E
R such that LXOE Int Y (for z(V, V')).
Then there exists a Lagrange multiplier jj f V' Proof. Write P = (p E V* such that, V x E R, l(x, p ) =- - -}. We have to prove that there exists a minisup jj of the function #(p, x) = -t(p, x) defined .on Px R (by Proposition 2). For this purpose, we apply the minisup theorem (Theorem 7.1.5). It is clear that, Vx € X,p t- #(p, x) is convex and lower
Ch. 14,§ 14.11
DUALITY IN INFINITE DIMENSIONAL SPACES
45 1
semi-continuous and that, Vp E P,x t-4(p, x ) is concave. It remains to prove +(p, X O ) is lower semi-compact. that there exists xo such that p By assumption (23) and Proposition 4, we see that 0 E Int Dom ~ ( x o , Since ~ ( x o , is the conjugate function of +(-, XO), we deduce from Proposition 3.1.10 that the function c#J(-,X O ) is lower semi-compact.
+
a).
a)
Example. In the case where F(x, y) =f(x)+g(y), the constraint qualification assumption (23) is implied by (24)
3 x 0 E R such that g is continuous at L x o
(and actually is equivalent to (23) whenever g is convex and lower semicontinuous).
Remark. Assumption (23) is very restrictive. For instance, let us assume that Y is the cone of non-negative functions of a space of functions on Q. If we take either (25)
V = czC(l2) is the space of bounded functions on 52,
(26)
Y = @(Q) is the space of continuous functions on a compact set sd,
or (27)
V = L"(52) is the space of (classes) of bounded measurable functions on 52,
is the space of (classes) of bounded measurable functions on Sa, it is easy to prove that the constant function 1 :a -c l(o) = 1 belongs to the interior of the cone of non-negativefunctions Y = (rep. Y = @+(Q),Y = LT(8))-
a+(@
This property isfalse if we take
(28)
V = Lp(sZ), Y = L$(Q) forp > 1.
We prove this fact in the case when 9 = ]- 1, + 1 J supplied with the Lebesgue measure. To say that LP,(52) has a non-empty interior amounts to saying that its positive polar cone L4,(Q)(I/p+ I/q = 1) is spanned by a non-compact convex subset (for the weak topology) [see Proposition 1.5.51. It is clear that Lt(l2) is spanned by the set
I
x E L'I,(l2)such that ilx(m) dw = 1 -1
452
CALCULUS OF VARIATION AND
OPTIMAL CONTROL
[Ch. 14,s 14.1
To prove that S is not compact, we associate with a function x E S (with compact support) the functions o nx(no) which belong to S (since -t
Ynx(n0)d o = 1). They do not converge in L9(0)(since nx(n, -) converges to
-1
the Dirac measure at O!). Thus, when Y is a cone of non-negative functions, the use of Theorem 1 requires the choice of one of the spaces M(sZ), or L-(0). This is quite a drawback, since these spaces are not reflexive. Furthermore, the characterization of their duals is not always possible. [We know that @*(Q)is the space of Radon measures on the compact set 0.The space Lw(sZ)*is the bidual of U(Q).The characterization of (at least) one supplement of Ll(0) is given by Ioffe-Levin-Valadier : it is the vector subspace of continuous linear functionals p € Lm(sZ)*such that there exists a decreasing sequence of Bore1 subsets A, satisfving A, = 0 and, Vx E Lm(sZ),(p, X ~ - ~ , - X ) = 0 (where xA is the characteristicfunction of A). In other words, this means that p is supported by every A,,.]
@(a)
n;L
14.1.4. Relaxation of the Slater condition
Furthermore, the constraint qualification assumption (23) is much stronger than the “constraint qualification assumption” (29)
0 E Int (Y-L(R))
we used in Theorem 5.3.1 when V is a finite-dimensional space. The question arises as to whether it is possible to replace (23) by (29). We show that the answer is positive by using the minisup Theorem 13.1.2. We assume now that (30)
V = G* is the dual of a barreled space G.
Theorem 2. Suppose that (30)holdr and that (i) F is a convexfunction from R X Y into R, (ii) V xE R,y F(x, y ) is weakly lower semi-contimrous, (iii) V p C P , x H(x, p ) is upper semi-continuous on U,
-.
+
and that
K, is a countable union of compact subsets of U, (i) R = (ii) 0 E Int (Y-L(R)). Then there exists a Lagrange multiplier j j E V*.
Ch. 14, 5 14.11
453
DUALITY IN INFINITE DIMENSIONAL SPACES
Proof. Since a Lagrange multiplier is a minisup p of the function (b defined by 4(p, x ) = -l(x,p), we apply Theorem 13.1.2. The assumptions (31) imply that 4-k lower semi-continuousand convex with respect top, upper semi-continuous and concave with respect to x. It remains to check that the compactness assumption of Theorem 13.1.2 is satisfied. Since d(x, is the conjugate function of 4(-, x ) this compactness assumption is 0)
(33)
U Dom d(x, .).
0 E Int
x€R
Since Dom d(x, -) = Y-Lx, property (33) is nothing other than assumption (32(ii)). Remark. Recall R = uT=K,, l whenever
(34)
Int Rb = 0 for the Mackey topology z(U*, U)
(see Example 3 of Section 13.1.3). This property also holds for any closed subset R of the dual U = F' of a Banach space. We point out the following consequence (among other possible corollaries of Theorem 2).
Theorem 3. Suppose that (35) that
(36)
U and V are reflexive Banach space, f : U - , ] - - , +-I and g : V + ] - - , semi-continuous convex f i c t i o n s
+-I
are fwo lower
and that
(37)
0 E Int (Dom g-L D o m n .
Then there exists a Lagrange multiplier p of the minimization problem
w = inf [ f ( x ) + g ( L x ) ] . X€X
Remark. In the case of reflexive Banach spaces (and, in particular, in the case of Hilbert spaces), we do not have to assume the restrictive constraint qualification assumption (24). In particular, for minimization problem under constraints, Theorem 3 can be restated as follows.
454
CALCULUS OF VARIATION AND OPTIMAL CONTROL
[Ch. 14, 5 14.1
Theorem 4. Suppose that U and V are reflexive Banach spaces, (38) and that
(39)
I
(i) f is a convex lower semi-continuousfunction defined on R c U , (ii) Y is a closed convex subset of V , (iii) L E 2(U,V).
Then, for any w E V satisfying (40) w E Int (Y-L(R)), there exists a Lagrange multiplier p of the minimization problem v = inf f ( x ) . x€R LXEY-w
This result allows us to use Hilbert spaces L2(Q) as constraint spaces. For instance, consider a continuous linear operator L E L?( U,L2(Q))and a cone R of U such that (41)
L maps R onto L"+Q).
Theorem 4 obviously implies the following result [since L(R)-L2,(52) = L$(Q)-L%(Q) = L2(Q)].
Proposition 6. Let U be a Hilbert space, R a closed convex cone of U and L E E
A?( V,Lz(SZ))satisfying (41).
Let X be the subset of R defined by (42)
X = {x E R such that (Lx)( 0 )=s w(o)for almost all o E Q}
where w E L2 is given. Let f be a convex lower semi-continuous convex function defined on R. Then there exists a Lagrange multiplier p E L:(sZ) of the minimization Droblem v = inf f ( x ) . X € X
14.1.5. Generalized Lagrangian of a minimization problem
In the above existence theorems, we used only the fact that we could represent the function f defined byf(x) = F(x, Lx) as the pointwise supremum of the Lagrangian : (43)
f ( x ) = sup I(x,p) whenever x E X P
(see property (8) of Proposition 2).
Ch. 14, 0 14.11
455
DUALITY IN INFINITE DIMENSIONAL SPACES
This representation also holds in the case of the Lagrangian of the optimization problems constructed in Section 5.1 (see Proposition 5.1.1). Therefore, we can construct a “duality theory” whenever such a representation holds.
Definition 2. Letf: U is X . Let
-.1-
-, + -1 be a function defined on U whose domain
We shall say that a function I defined on R X P where R c U and P c V* is a Lugrangian of the minimization problem (44) if
v = inf sup l(x, p).
(45)
x € R PEP
We shall say that p E P is “Lugrange multiplier” ifp is a max-inf of the Lugrangian, i.e. if
u = inf I(x,P).
(46)
x€R
We shall say that the function$’ defined by
f’*(p) =- inf I(x,p)
(47)
x€R
is the loss function of the dual problem. We will set
v” = inf f ‘*(p) = -sup
(48)
PEP
inf I(x,p).
P€P x € R
In this framework, the existence of a Lagrange multiplier can be obtained by applying the minisup Theorems 7.1.5 and 13.1.2to the function defined on P X R by &p, x) = - l(x, p). For this purpose, we introduce the conjugate functions d(x, .) of the functions 4( x ) = - l(x, .) defined on PX R by
+
a,
(49)
4x3
v) = SPU€ PP [(P,r>+4 x , PI].
We mention the following properties of A.
Proposition 7 . f l the Lugrangian satisjies (50)
V p E P, x I-+ I(x, p ) is convex (resp. lower semi-continuous),
456
CALCULUS OF VARIATION AND OPTIMAL CONTROL
then thefunction A satisfes A : R X V -, 1(51)
+ -1
a,
[Ch. 14,s 14.1
is convex (resp. lower semi-continuous)
Proof. This is left as an exercise. 0 Consider the function u defined by v(y) = inf A(x, y). (52) XER
Then properties (46) and (49) show that (53)
v = v(O),
since d ( x , 0) = suppEpl(x*p). We can regard the minimization problem (52) as a “perturbed” problem by elements y E V. Proposition 5.1.4 can be generalized.
Proposition 8. Suppose that there exists a Lagrange multiplier jj. Then jj belongs to the subdi$Terential of v at 0, i.e. (54)
p E av(o).
Proof. For any y E V we have V+(P,Y) =
inf k P ) + ( F , Y ) .
xE R
Since v = v(O), we can write this inequality in the form v(0)- v(y) =s( p , 0 - y ) which means that E av(0).
Remark. We shall prove the converse statement in Theorem 6 below.
Theorem 5. Suppose that the Lagrangian satisfies (i) V x E R, p t--. I(x, p ) is concave and upper semi-continuous, (55) (ii) V p E P , x ++ I(x, p ) is convex.
{
If we assume that either (56) or that
(57)
3 x o E R such that 0 E Int Dom ~ ( x o , a)
{
u:=1-
(i)R = K,,is a countable union of compact subsets K,, of’U, (ii) V p E P,x l(x,p ) is lower semi-continuous, (iii) 0 E Int Dom A(x, .) = Jnt Dom v, (iv) V = G’ is the dual of a barreled space G,
UxEX
then there exists a Lagrange multiplier jj.
Ch. 14, Q 14.11
DUALITY IN INFINITE DIMENSIONAL SPACES
457
Proof. This is a restatement of the minisup Theorems 7.1.5 and 13.1.2 applied to d p , x ) = - 4x7 PI. 0
Example 1. Consider the Lagrangian I(x,p) = f ( x ) + ( p , L(x))-a*(Y; p ) of the minimization problem
Example 2. We can also represent a closed convex subset containing 0 by its gaugea(Y; -)and associate with the minimization problem (58) the Lagranghn defined on R X R , by (60)
RX,PI = f(xI + P ( V ; U X ) ) - 1)
since we clearly have that
+
m
if L(x) E (1 -y)Y, if not.
Hence the perturbed problem is defined by (61)
G(y) =
inf
U)E(1-Y)Y
f(x)
where y E R. Example 3 (Rockafellar). We associated the Lagrangian l(x, p) defined on RxR: by I(x,p) = f ( x ) + ( p , L ( x ) ) with the problem 21 = infLc.+,f(x)(where L maps R into R"). The reason was that w(;y ) = sup 7 ( p , y ) . The question arises as to whether it is possible to represent the indicator y(-R: ; .) of -R: as the pointwise supremum of a function p ( y , p ) , the domain of which is R"XR"* [instead of R"XR",Z. In this case, we can associate With the above problem the new Lagrangian 1 defined on R X R"* (instead of R X RY)by.
(62)
+
e x , P) = f ( x ) P(LX, p ) .
458
CALCULUS OF VARIATION AND OPTIMALCONTROL
[Ch. 14,Q 14.1
The following function p defined by n
(63)
P(Y, P) =
C Pr(Yi,Pi) i=1
where
satisfies the above property. Indeed
[ (ii)(i) ifif
(65)
yi 6 0,SUP~
+
as can be seen by examinine the graDhs of the function p y as 0 and y 2.0.
-
p,.Gy,p) when
I:
Case v
Case v a 0
<0
Fig. 15.
Therefore
Remark. Any kind of function p whose graph has the same shape as the graph of pr satisfies the above property.
Ch. 14, 0 14.11
DUALITY IN INFINITE DIMENSIONAL SPACES
459
14.1.6. Characterization of a Lagrangian by perturbations of the minimization problem
Consider now a family of loss functions x ~ - d ( x y) , where y ranges over a topological vector space V , which are perturbations of the loss function f defined by (67) J'(x) = A(x, 0). Consider the minimal values (68)
w(y) = inf d ( x , y) and w = w(0) .= inf f ( x ) . u
XE
XE u
We shall show that under mild conditions, we can associate with d a Lagrangian IA(x,p ) such that its set of Lagrange multipliers is nothing other than the subdifferential of w at 0. We define I, on UX V* by (69)
Mx, P ) = inf YE
v
Y ) - ( P , v)l.
Proposition 9. If we assume that Q x E (I, y k+ A(x, y ) is convex and lower semi-continuous, (70) then I, is a Lagrangian , i.e.
r f ' we assume also that (72)
.,4 : U X V
-. ] - 00,
+
-
J is a convexfunction,
then 1, satisfies (73)
{ (ii)(i) QQ pxE
E U , p b+ IA(x,p ) is concave and upper semi-continuous, V*, x +-. C(X,p ) is convex.
Proof. Fix x E U and set g ( y ) = d(x,y). Therefore I,(x,p) = -supY [(p,y)-g(y)] = -g*(p) is concave and upper semi-continuous with respect to p. If g is convex and lower semi-continuous, we obtain that
Finally, it is clear that the convexity of d with respect to both x and y implies the convexity of the Lagrangian I with respect to x. 0
460
CALCULUS OF VARIATION AND OPTIMAL CONTROL
[Ch. 14, Q 14.1
The formula (74)
PI1
4 x 3 Y ) = SUP [(P,Y)+ PE v*
shows that the perturbed problems associated with the Lagrangian are nothing other than the initial perturbed problems. In the same way, the Lagrangian associated with perturbed problems is nothing other than the initial LagrangianProposition 10. Suppose that a Lugrangian 1 satisjes (75)
Q' x E R, p
-
I(x, p ) is concave and upper semi-continuous.
Then I = I, and the conjugate function of y
(76)
--*
v ( y ) satisjes
v'(p) = -inf I(x,p). X
Proof. Consider the perturbed problem
(77)
4 x 9 Y ) = SUP [(P,Y)+I(X, P)l* P
Write 8(p) = -l(x,p; Hence 8 is a convex lower semi-continuous function.. Therefore d ( x , y ) = 8 Q and
P) = - e
* w = - sup [(P,Y>- 4 x 9 Y)I Y
= inf [A(%,A- ( p . y)l = L&,P). Y
Now we deduce that v*(p) = -inf [v(y)- ( p , y>] =-inf inf Y
X
I&,
Y)- ( P . u>I =
Y
= -inf I(x,p). 0 X
Theorem 6. Suppose that the Lugrangian satisfies (75). Then mltiplier if and only if
iS a b g m g e
(i) v(0) = w'*(O),
(ii) p E av(o). Proof. We have already proved that any Lagrange multiplier p belongs t o av(0) (Proposition 8). Furthermore, suppinf, I(x,p ) = inf, sup, I(x, p). But (79)
v(0) = inf sup I(x, p ) X
P
Ch. 14,$ 14.21
DUALITY: NON-CONVEX INTEGRAL CRITERION
461
by the very definition of the Lagrangian and ~"(0) = sup [(p, 0)- v'(p)] = sup inf I(x,p) (80) P
P
X
by formula (76). Hence the Lagrangian has a value ifand only i f w(0) = w*'(O). In this case, ifjj E aw(O), jj minimizes v'(p) = - inf, I(x,p), i.e. jj is a Lagrange multiplier [since inf, Z(x, jj) = sup, inf, I(x, p) = inf, sup, I(x, p) = infxf(x)]. 0 Remark. We can also associate with a Lagrangian I(x, p ) a family of perturbed dual problems F(p7 4 )
(81)
= inf X
[4P,x) + ( 4 , x ) l
where
r@, 0) =f'+(P)
(82).
is the loss function of the dual problem. A symmetric study is then possible. 14.2. Duality in the case of non-convex integral criterion and constraints
We shall prove in this section the Aumann-Perles theorem, i.e. there exists a Lagrange multiplier E R"* of the following minimization problem With integral criterion and constraints:
whenever we assume (essentially) that the constraint qualification assumption 0 E Int Dom gA(w) d o U is satisfied, that A(w) E J ( V , R") for almost
(
(1
1)
all o E 2 ! and that g is a lower semi-continuous convex function. But, we do not assume that the functions 5 -, p(w, 5) are convex. We shall prove this theorem in a constructive way, by approximating it by minimization problems of the form
where g is a convex function defined on R", Rt is a convex subset of Ut and where Lt E J(U,, R"). Suppose that the constraint qualification assumption holds, i.e. Dom g-r I C T L,(Rt)). t=l
462
CALCULUS OF VARIATION AND OPTIMAL CONTROL
Ch. 14, $ 14.2
+
If we do not assume that the functions f , are convex, the length w v* of the duality gap is positive, where we set
We will estimate the duality gap by measuring the lack of convexity of a function f in terms of the modulus of non-convexity
(We see that convex functions are those whose modulus of non-convexity e(f) is zero.) Roughly speaking, the less convex the behaviour of a function, the larger is its modulus e ( f ) . We will then prove the fundamental estimate 0 4 w+ w*
4
m+ max p(f,).
T
OrtsT
This estimate shows that the duality gap “shrinks” as the number of functions increases. This explains why the Aumann-Perles holds. The proof uses a fundamental result due to Shapley and Folkman, which states that, in some sense, the sum of a large number of sets is iipproximatively convex. 14.2.1. Modulus of non-convexity of a function Consider
(i) a convex subset X of a vector space U, (ii) the subset M(X)of discrete probabilities m = Eel aid(xi) on X , (iii) the barycentric operator :d ( X ) F+ X associating with any We shall introduce a modulus of non-convexity e ( f ) of a function f defined on X which “measures” the non-convexity off. Definition 1. Let f:X t- R be a function defined on X . We shall say that
is the “modulus of non-convexity” off.
Ch. 14, 0 14.21
463
DUALITY: NON-CONVEX INTEGRAL CRITERION
It is clear that a function f is convex if and only if its modulus of nonconvexity e(f ) is equal to zero. Remark. We can relax the assumption (1) and associate with any
(3)
map y from M(X)into X
the modulus of non-y-convexity
e,, defined by
We notice that a y-convex function is a function such that e , ( f ) = 0. For a given set ,'A we can use y as parameter. For instance, we can associate a map y with a family of functions. Proposition 1. Consider a family (5)
of functions f satisfying
f is lower semi-continuous and lower semi-compact.
Then there exists an optimal map y e : A ( X ) + X associated with @. It is defined by
(6)
yrjF(m) E X miilirnizes on X thefunction, x
F+ sup
Lfc rT
( f (x)-fA(m)).
Proof. The function x i---supfEv( f(x)-fA(m)) is lower semi-continuous and lower semi-compact. Hence, there exists a minimum yrj.(m), i.e.
Therefore,
=
sup inf SUP ((f ( x ) - fd(rn)).0 rn€M(W xEX / ~ r ; t '
For the sake of simplicity, we shall assume that (1) holds. The extensions to the case of y-convex functions are left as exercises. Proposition 2. Let F(x, y ) = f ( x ) + g ( y )be thesum of two functions f : X t-+R andg : Y -, R. Then
(7)
e(F)
e(f)+e(g).
464
CALCULUS OF VARIATION AND OPTIMAL CONTROL
where we set x = '&ofx, and y
19)
'
, (10)
=
[Ch. 14,Q 14.2
E=lu'yi. 0
(ii) Y is a closed convex subset of Y = Rm, (iii) L is a linear operator from U into V, (iv) X = {x E R such that L x E Y } , (v) F : R X Y k + R .
v* = inf F*(-L*p,p). pERm
Recall that -')2
6 pt.
We want to estimate the length v + v* of the duality gap.
Theorem 1. Suppose that (9)holds and that (1 1)
0 E Int (L(R)-Y).
Then there existsp E R" such that (12)
0 =sw+F*(-L*p,P)
=Z
e(F).
Remark. If we also assume that F is convex, then e(F) = 0. We obtain the existence Theorem 5.3.1 for a Lagrange multiplier. Proof. This is analogous to the proof of Theorem 53.1. Consider the operator 4 : R X Y k+RXRm defined by +(X,Y) = (F(X,Y), W ) - Y ) .
Ch. 14, 9 14.21
DUALITY: NON-CONVEX INTEGRAL CRITERION
465
We set (13)
w = infmEJn,(Rxy) 1aiF(xi,yi) where L/nr(RX Y) is the set of discrete probabilities rn = "& aib(xi, yi) such that a'(L(x&-y,) = 0.
Consider (14)
the vector { (ii)(i) the cone Q
6 = (1, 0 ) E RXR"l = 10,
00
[ X {0} c RXRm.
We shall divide the proof into the following three steps Step a : we
6 co ( @ ( R x Y ) ) + Q ) .
Step b: there exists jj such that w s-F*(-L*j,p).
Step c: v
=sw+
e(F).
+
*Step a : If we belongs to co(@(RX Y ) Q), there exists m = -&( R X Y) satisfying
w
zE1 a'&
y,)
E
c a'F(xj,yi) n
2-
i=l
and
The latter equality means that rn E X L ( R X Y). Therefore, we obtain a contradiction of the definition of w. Step b: Since R" is a finite-dimensional space, we can apply the separation theorem to separate we from the convexsubset co(@(RXY ) )+Q. There exists a non-zero continuous linear form {a,p} E (RXRm)*such that (15)
({a,p}, we) = aw
=S
inf [aF(x, y ) + ( p , Lx--y)+ac].
x€R
Y€Y c=-0
We first deduce that inf,,, ac is finite, and thus, that a == 0 and inf,,,ac = 0. Secondly, we deduce from the constraint qualification assumption (1 1) that a =- 0 (see the proof of Theorem 5.3.1). Finally, by setting j = p / a , we deduce from inequality (15) that (16)
w
inf [F(x,y )+(p,Lx)-(p,y )] = -F*(-L*p,p).
x€R
Y€Y
32
466
CALCULUS OF VARIATION AND OPTIMAL CONTROL
[Ch. 14,s 14.2
Step c: By the very definition of w, we can associate with any E =- 0 a discrete probability rn = a'd(x,, yl) satisfying
xy=l
(17)
{ (ii)(i) CLl Cbl
If we set (18)
x=
aiF(xi,yi) 4 W + E, d(L(xl)-yi) = 0.
n
C aixj,
i=1
y
=
n
C aiy;
i=1
we may deduce from (17(ii)) that y = Lx and from (17(i)) that
21
= inf F(x, Lx) =sw f e(F). X€ X
14.2.3. The Shapley-Folkman theorem
We now proceed to state the fundamental result of Shapley and Folkman, which shows that, in some sense, the vector sum of a large number of sets is approximately convex.
Theorem 2 (Shapley-Folkman). Conrider T subsets Kr(lG t =s )'2 of R". I/'
there exists a subset S of at most m indices t such that
Proof. First step :We begin by repiacing the subsets K,by finite subsets K , ( x ) cK,. Since x E C ~ ( C T = ~ there K ~ ) ,exist n real numbers A'€ 10, 1[ such that C1;Ejsnl'= I and points xf(1 e j s rt, I =st G T ) such that
Ch. 14, 5 14.21 DUALITY:
NON-CONVEX INTEGRAL CRITERION
467
We put
and write x, =
c
Ux{E co (K,(X)).
lajan
Second step: Denote by H the set of decompositions x' = { x , } ~ ~ , , ,of the sum x = CT=lx, of elements x, E co K,(x).lf we set ax' = Cb1x,, this subset H can be written (24)
H =
n
co K,(x)rIa-yx).
1,tasT
It is a convex compact subset. Hence there exists at least one extremal point
x' E H. We shall prove that all but in of the components x, of such an extremal point x' belong to K,(x). To see this, assume that the subset S c {I, ..., T } of indices t such that x, does not belong to Kt(x) contains at least (m+1) elements. We shall prove that this implies that x' is not extremal, i.e. a contradiction. Since K,(x) is finite, we can associate with any xt a direction y, € R" and E, =- 0 such that (25)
x,
+ey, E co K,(x)
for any 8 such that 18 I
=S
el.
(See note at the end of the proof if necessary.) Let E = min,,,,, E, Z- 0. SinceScontains at least m+ I elements we can find scalars a, (with a, # 0 for at least one t ) such that
Thus, for any t E S, (27)
{ (ii)(i)
u, = x,+&a,y, belongs to co K,(x),
v, = x,-&a,y, belongs to co K,(x).
We write u, = v, = x, whenever t 6 S. Thus u' = {ur}lstaTand 5 = { v , } , both ~ ~belong ~ ~ to satisfy
nlrraT co K,(x) and
ou' = ox',+& Cafyt = ux' = x,
av'= oxf- e
C a,y, = ox' = x.
In other words, u' and v' belong to H. Since a 32'
z 0 for at least one t, u
# v.
468
CALCULUS OF VARIATION AND OPTIMAL CONTROL
Furthermore,
x' =
[Ch. 14,$ 14.2
+ (u'+ v').
Hence x' is not extremal in H. We thus have proved that the set S has at most m elements. CI Note: Proof of (25). Since x, does belong to co K,(x) and does not belong to K,(x), there exist at least two positive scalars A: E 10, 11 such that xt
==
C Afxf = A f o ~ f+o (1 - A)).: i
where z;=
c--2: 1- y.
xf E co K,(x).
ifio
We set yt = x2- 2: and E, = min(A), 1-A:). Therefore, if 18 I =S E,, then A$'+ 8 E 10, I [ and X,
+8yr = (Afo +8)xp+ ( I - (A? +0 ) ) Z :
E co K,(x). n
14.2.4. ,Sharp estimate of the duality gap
Suppose that Rt where R, is a subset of a vector space U,, (i) R = (ii)f(x) = (1/T) xT=ljXxr)where5 : R, -c R, (iii) L(x) = (1/T) C ~ , L , ( x ,where ) L, E 2 ( U t , Rm) is a linear operator from U,into Rm.
Proposition 2 shows that 1
'T
Using the Shapley-Folkman theorem, we shall prove that we can use in the estimate (12) of the duality gap ( ( m + l ) / T ) maxlsrsTe(f,) ratherthan maxlSrsT e(f;), where m is the dimension of the space R".
Theorem 3. Suppose that (29)
(i) g is a convex function from Y into R, (ii) the subsets R, c U,are convex, (iii) the maps L, are linear.
Ch. 14, 5 14.21
DUALITY: NON-CONVEX INTEGRAL CRITERION
463
r f we suppose that
then there exists p E Rm*such thut
Proof. Consider the proof of Theorem 1 with (32)
w , v) =
r1 r=p f ) + g ( Y ) .
The first and second steps of this proof show that there exists p E Rm* such that
(33)
w
G-
F*(-L*jiyji).
Since
and
we deduce that
The third step of the proof of Theorem 1 is modified as follows. We deduce from (13) (i.e. from the definition of w) that, for any E =- 0,
(35)
(W+&)e
E co ( q R X q + ( t O ,
=[x{o}).
470
CALCULUS OF VARIATION
AND OPTIMAL CONTROL
[Ch. 14,§ 14.2
Since
we can write (36)
co@(RXY)+([O,- [ X { O } ) = G+co
C
1stsT
E,,
where
and where (38)
+
G = k ( Y ) , -V l Y E Y ([O,
-
I x {OH.
Since g is convex, G is a convex subset of RXRm. Therefore, we deduce from (35) and (36) that there exist y E Y and c 3 0 such that (39)
-Y}
(W+E)e-{g(y)+c,
E co
c
IstsT
Et.
But, by the Shapley-Folkman theorem (see Theorem 2), there exists a set S of at most (m + 1) indices such that
Therefore, we can find x, E Rt for t 6 S and m, E &(RJ that
(w+ 9 e - M Y ) +
c,
-v}
=
for t E S such
c
t6.9
Since Lt is linear, Lfm, = L t ~ p tBy . setting x, = & n r whenever t E S, we deduce that
Ch. 14, Q 14.21
DUALITY: NON-CONVEX INTEGRAL CRITERION
47 1
These two statements together imply that 1 W+E==5; j;(xt)+g
c
lstsT
(43) 3V--
m+l max e(ft), T I~IST
Hence, the conclusion of Theorem 3 follows from inequalities (34) and (43). 14.2.5. Applications
-.
Let f:R R be a function defined on a convex subset R of U. Associate with an integer T the function
It is clear that (45)
for any x E R, .f**(x).=sfT(x) ~ f ( x ) .
Proposition 3. Suppose that (i) g is a convex function from Y into R, (ii) R is convex, (46) (iii) L E J(U,Rm).
1
If we suppose that (47)
0 E Int(L(R)-Y),
then there exists@ E Rm*such that VT =
(48)
inf ( f l - ( x ) + g ( L x ) )
x€R
=s-
(f*(-
-c-
v* +
+
+ m+ 1 e ( f 1
L*@) g* (P))
m+l e(f). T ~
472
CALCULUS OF VARIATION AND OPTIMAL CONTROL
[Ch. 14,s 14.2
Proof. This result follows from Theorem 3 with R, = R, f, = f and L,.= L for all t. Therefore, there exists p E R"* such that
Let f : R
-.R be a function defined on a convex subset R c U and L E
J(U,Rm)be a linear operator.
We shall associate with any T the function
It is clear that
Proposition 4. If we suppose that
(51)
y E Int
then there existsp E (52)
L(R),
a"*such that
(v>-(f*L*)* ( Y ) = (-cf)T(Y)-
(Lj>T
(Lf)**( Y ) ==
mfl
e ( f 1.
Proof. We apply Proposition 3 with g = y ( Y ; .). Consider T functions h, : R, R, where R, is a convex subset of the space R". Let [714rrThrbe the inf-convolution of these functions, defined by -+
(53)
0 h,(y) =
1stsT
(see Definition 1-2.2).
inf
%;c;rsTXi=Y
1
l=srsT
hr(xt)
Ch. 14,s 14.21
DUALITY: NON-CONVEX INTEGRAL CRITERION
473
Proposition 5. Suppose that (54)
the subsets R, are convex
and thar
Then
Proof. We apply the above proposition withj; = Th, and L = T I . 0
14.2.6. Extremality relations The extremality relations still hold with an error which can be estimated with the modulus of non-convexity. Proposition 6. Suppose that the assumptions of Theorem 3 are satisfied and that g is lower semi-continuous. Let ji E R"* satisfy inequality (31). If there exists $t E R, satisfying (57)
then
(i) (1/T) xT=ILrgr E ag* (Y), (ii) 2r E af,"(-L:p), (iii) fi(%) = .f:*(%),
Remark. This proposition is analogous to Proposition 5.2.8 concerning the decentralization principle. It shows that this principle still holds with an error which decreases as the number of functions increases. Proof. Condition (57(i)) means that
and (57(ii) and (iii)) mean that Zt minimizes x t-- f (x)+( p , L,x) (see Proposi-
tion 4.1.1), i.e. f;(%)+(p,L,x,) = min [ft(x)+(L:p,x ) ] =-fl(-L?F). X
474
CALCULUS OF VARIATION AND OPTIMAL CONTROL
[Ch. 14,$ 14.2
Therefore,
‘Remark. W e can investigate the properties of a minimum x = x, of the minimization problem. For this purpose, recall that (59)
a , f ( x ) = (p E U* such thatf(x)+f(p)-(p,
x)
a}
is the a-subdverential off at x.
ProPosition 7. Suppose that the assumptions of Theorem 3 are satisjed. Let E Rm*satisfy ineqwlity (31). i f 2 = {Y,}lstsT minimizes
ji
Proof. We can write inequality (31) in the form
m+l
e.
Since all the terms of this sum are non-negative, we deduce that each of the m is less than or equal to ((m+ 1)/T)e, i.e. that statements (60) are satisfied. 0
Ch. 14, Q 14.21
DUALITY: NON-CONVEX INTEGRAL CRITERION
475
14.2.7. The Aumann-Perles duality theorem
Let Q be a bounded open subset 52 of R” and let K be a convex subset of some Banach space F. We introduce the following notation. (62) (63)
Let 164)
U
= L1(sl, F) is the Banach space of (classes o f ) measurable
R
= L1(D,
functions x from D into F with integrable norm I I x(w) I I.
K) is the subset of functions x E L1(D, F)
such that x ( w ) E K for almost all w E Sa. g : R m -c 1- a,+ -1 be a convex lower semi-continuous function whose domain is denoted by Y.
We consider a minimization problem of the form
where the function f and the linear operator L are “integral” in the sense that
We shall assume that tp is a function from S X K into R satisfying (i) for almost every w E D,6 E K -c ~ ( wE,) is continuous, (ii) for every (5 E K,o E Q y(o, E) is measurable, (67) (iii) I v ( w , 5 ) I G C(w)+M 11 5 11 where CEL1(9), M =- 0,
1
+
and that A is a map from Q x F into Rm satisfying (68)
(i) for almost every w E 9,A ( o ) E 2 ( F , Rm), (ii) o A(@)is a measurable function from Q into B(F, Rm), (iii) w 11 A ( o ) 11 belongs to L”(S1).
-+
It follows from the Lebesgue convergence theorem that
(i)f :R = L1(S,K) -c R is continuous, (ii) L E B(L1(Q,F),Rm).
Remark. A function tp satisfying (67(i) and (i) is called a “Carathhodory function”.
476
CALCULUS OF VARIATION AND
OPTIMAL CONTROL
[Ch. 14,s 14.2
In order to define the dual problem of the minimization problem
we introduce the conjugate function f * defined on Lm(1;2,F*) by
and the transpose L* of L defined on R"* by (71)
L*p :0 + A*(o)p.
We define the dual problem (72)
w*
=
inf [S+(--L*p)+g*(p)].
pERm
Remark. Note that the measurable selection theorem implies that (73)
f * ( d= j Q)*(w,4(4)dw R
where, for almost every w E Q, pl*(w, p ) = sup,,,[(p, 5)- p(o, 4 1 (see for instance in the book of Ekeland and Temam [1974, Theorem VIII-1-2, p. 220 and Proposition IX-2-1, p. 251)]. In this case, we can write (74)
v* = inf [ J q * ( w , -A*(w)p) dw+g*(p) pERm*
1.
We will not use this result. 0 Denote by Lo (75)
a ( F , R"), the operator defined by
V t € F, Lo(5) = (J A(w) dw)5.
Theorem 4 (Aumann-Perles). Suppose that hypotheses (64,(67), (68) and the constraint qualiJication assumption (76)
0 E Int (Y-Lo(K))
hold. Suppose also that the moduli of non-convexity are bounded, i.e.
Then there exists a Lagrange multiplier p E R"*, i.e.
(75)
v = - v* = - (f*(-L*p)+g*(p))
Ch. 14, !j 14.21
477
DUALITY: NON-CONVEX INTEGRAL CRITERION
14.2.8. The approximation procedure
Let h be some small parameter, and let CZjh = {Gi},be a partition of 52 into a finite number of subsets Qh, such that 179)
I %h 1
=
-
max meas (Qi) O as h I
-
0.
Denote by Lh (Q, K ) the space of step functions from SZ into K, constant over each member of q h . w e approximate the minimization problem (3) .by
Its dual probIem can be written (81)
vh+ = inf [fh*(-L*p)+g*(p)l p€Rm*
where
Proposition 8. Under the assumptions of Theorem 4, for any h there exists R" such that
Ijh E
(83)
-vh*
=S
vh =S -[fhC(-L*Ph)+g*(ph)]+MICZjhI,
where M is a constant independent of h.
Proof. We apply Theorem 3 with R, = K ,
A([) = T
J y ( w , 5)dw
Q:
and Lf = T f A ( 5 ) d e *fn
(where T denotes the number of elements of the partition of non-convexity off, is estimated by
%h).
The modulus
Finally, the constraint qualification is satisfied. Hence Theorem 3 implies the existence Of& such that (83) is satisfied with M = (m+ l ) @ . 0 Theorem 4 is a consequence of the following more precise proposition.
478
CALCULUS OF VARIATION AND OPTIMAL CONTROL
[Ch. 14,$ 14.2
Proposition 9. Under the assumptions of Theorem 4, there exists a subsequence of the family of jib satisfying (83) which converges to a Lagrange multiplier p E Rm*(i.e. which satisJes (78)).Furthermore, we have the estimate (84)
2)
=
- [f*(-L*p)+g*G)]
- ~ ( - L * ~ h ) - g * ( F h ' h ) + M I1-~ h
Proof. We clearly have the inequalities (85)
-v*
v
4
vh
-[fh*(-L*ph+g*(F)]+MI2hI.
It therefore remains to prove that (a) the family ph lies in a relatively compact subset and thus, that a suitable subsequence converges to p in Rm*. (b) for any (86)
E
=-
0, 3ho such that, V h =s ho,
p ( - L * p ) + g * ( p ) efh+(-L*ph)+g*(Fh)
+
E.
It is clear that (85) and (86) imply -v* e v e vh e - T f ( - L 8 p ) + g * ( ~ ) ] $ M I 2 h I + € .
Letting h and E go to 0, we deduce that (78)
v = -v* = - [ f ( - L ' @ ) f g * ( p ) ]
i.e. p is a Lagrange multiplier. Hence estimate (84) follows from (85) and (78).
Proof of(a). Since 0 E Int(Y-Lo(K)) and the elements 5 can be identified with step functions of Lh(G, K ) for all h, we can write any z E Rmin the form z = e(y-i(A(m)tdo where E =- 0, y E Y and 5 E K. Hence
)
=sg * ( p ) + g ( Y ) + ~ ( - L * P h ) + f ' ( 5 )
s f ( t ) + g ( Y ) - vh < f ( t ) + g ( Y ) - v * Thus the family {&} in bounded, i.e. relatively compact. A suitable subsequence (again denoted by) p h converges to an element F E Rm*.
Proofof (b). Let E be fixed. There exists X E L1 (SLY K) such that
Ch. 14,s 14.31
by the very definition off*. Also, there exist q
- (e
whenever IIP-qIIRm1 7 and
{q,y }
B
479
DUALITY IN CALCULUS OF VARIATIONS
11 %--y Ilv
=s6.
r
0 and 6 r 0 such that
A(w)y(w) dw is a continuous function on Rm*XL1(Q, F), q
)
-
[This follows from the fact that
.
k.lower semi-continuous andf is continuous on On the other hand, we can find hl such that, for any h s hl, lljj-&, ( 1 =s7. Since the space of step functions is dense in L1(Q, K), we can find ht such that, for any h G h2, there exists xh E L,,(a, K) satisfying 11 Z-x,,ll 4 6 (we take x,, = xhJ. Then, for a11 h e ho = min(hl, h2),we deduce from (87) with q = & and y = x,, that I
,a
14.3. Duality in calculus of variations
Consider the following example of calculus of variations (problem of Bolza). 1
+
where 2 denotes the derivative of x and r$ maps ]0,1[ xR"X R" into 1, ] We leave until later a specification of the precise assumptions on 4 and the description of the (infinite-dimensional)space of functions in which we look for a solution of the minimization problem. We shall prove that the dual. problem can be written QO
Q>
480
CALCULUS OF VARIATION AND OPTIMAL CONTROL
[Ch. 14,$ 14.3
We restrict our study to the convex case, where for almost all w E 10, I[, the function {u, v} 4(w, u, w} is convex. We shall prove that j?( .)is a Lagrange multiplier and Z( .) is a solution of the minimization problem if and only if the following extremality relation holds. For almost all w E 10, I[, -+
(*I
{ B ( W > , P<w,}
E
a4(w Z@),
?(o)).
If the function 4 is assumed to be differentiable with respect to {u, v}, this relation becomes p(w) = Du4(w, qw), q4); F(w) = a 4 ( w , ?(w)7
a@)).
We recognize the Euler-Lagrange equation in the classical setting of the calculus of variations. Introduce the conjugate f.ixtion 'I, of the function 4 with respect to v, defined by
(which coincides with the Legendre transform of 4 with respect to v whenever it exists). We shall see that the extremality relations can be rewritten in the form: for almost all w,
(**I
{(ii)( 0
-pL(w) E auy(w,z(w), p(w)), E arv(w, ~ ( c o jj(w)). ),
?(W)
where 8, and 8, denote the super-differential with respect to u and the sub.differential with respect to Y of the function y. If we assume that y is differentiable with respect to {u,r } , this system becoples ( 0 -P(w) = D,y(w, x'(w>,P(w)), (ii) 5(w) = Dry(w,Z(o), p(o)). We recognize the Hamiltonian system in the classical setting of the calculus of variations. Now since we have devised a duality theory in infinite-dimensional spaces and since extremality relations (*), (**) look like the extremality relations (***)
and
{-L*p,p} E
aqx', LZ)
Ch. 14, 5 14.31
DUALITY IN CALCULUS OF VARIATIONS
48 1
it is strongly tempting to deduce the above results from the general theory by setting Y ) = J $4-4 x(4, Y ( W N dw
m,
R
and Lx = Dx = x. This is not as simple as it may appear, since we have to relate the transpose D* of the dflerential operator D with -D in order to link (*) und (** *), (**) and (** * *) respectively. It is time now to become more specific. Under suitable (and natural) assumptions on 4, we can assume that the function F maps UX V into 1- 0 3 , + -1 where U = V = L2 (0,l) is the space of (classes of) square integrable functions from 10, 1[ in R*. Then the minimization problem has a meaning if we take x in the “Sobolev space” H1(O, 1) = {x E L2(0,1) such that i = Dx E L2(0,1)). This definition requires the derivative Dx to be regarded as the derivative in the distributional sense. Supplied with the graph norm, the space H1(O, 1) is a Hilbert space, dense in L2(0,1). The important fact underlying all the theory is the following. The operator y associating With any x E H’(0, 1) it “trace” yx = {x(O), x(1)) E R V R ” satisfies the following properties: maps H 1 (0, 1) onto RnXRR, (*** **) (ii)(i) yKer y is dense in L2(0,l), since one can prove that Ker y = HA(0, 1) is the closure in H’(0, ) of the space @(O, 1) of infinitely differentiable functions with compact support in 10, 1[. The latter statement implies that the transpose (DO)*of the restriction DOto Hi(0, 1) of D is equal to - D by the very definition of the derivative in the distributional sense, i.e.
{
V p E L2(0,I), V x E Hi(0, I), 1
= - J (DP(o>),~(0)) dw.
1
(p(o), Dx(w)) do 0
0
Now the formula for integration by parts relates the transpose D* of D with the transpose Dt = D of DO,i.e. V p E Hl(0, l), V x E H*(O, 1).
-
1
j ( P ( W ) , Dx(w))dw
0
1
= - J ( D P ( 4 , x ( 4 dw+(p(l), x(l))- (P(O), 40))0
33
482
CALCULUS OF VARIATION AND OPTIMAL CONTROL
[Ch. 14,§ 14.3
This formula can be proved directly (approximating by the usual integration by parts formula) or in an abstract way using only the trace property (* * *). We choose the latter approach and prove in fact an abstract “Green formula” which is quite versatile in that it allows various differential operators L to be used. We describe the results briefly. Assume that U(L), U and B are three Hilbert spaces such that
**
[
(i) U(L) c U, (ii) y E L(U(L),%) is surjective, (iii) Uo(L) = Ker y is dense in U.
a(
Let L E U(L), V),where V is another Hilbert space. Assume that U* = U and V* = V for simplicity and that we “know” the transpose Li E A?(V , Uo(L)*) of the restriction LOof L to Uo(L).We define the domain of LE by
V(L8) = {p E V such that G p E U}. We shall prove that there exists a unique operator @* E Pe(V(L:), 8’)such that the abstract Green formula VP
E V ( L 3 , v x E U W , ( G P , X>-(P,LX) = W P , y x )
holds. We then modify the results of Section 14.1 as follows. We associate with a minimization problem w = inf F(x, Lx, y x ) X€
U(L)
the dual problem PI* =
inf
F*(-Gp,p,
P € WG*)
Tp).
We check that the extremality relations can be written
{-UP, F, Pjj} E W(;(n,LZ, yZ) or, equivalently, (i) {Lx’,y q E
w, P, P P ) ,
%.?I
(ii) L;p E axH(Z, 8, pa. where H is the Hamiltonian defined by
Ch. 14,$ 14.31
DUALITY IN CALCULUS OF VARIATIONS
483
We deduce the existence of a Iagrange multiplier under the constraint qualification assumption that 0 E Int Dom w where the perturbation function w is defined by
We end this section with an analogous study in the framework of optimal control problems. 14.3.1. The Greenformula
Consider three Hilbert spaces U,V,stogether with a linear operator L mapping its domain U(L)c U into V and a linear operator y mapping U(L)onto S We require, (1)
and
for the norm llxll = (11~11$+IILxIl~,, U(L) is a Hilbert space (i) y E B(V(L),9)is surjective, (ii) UO(L)= Ker y is dense in U.
Remark. Property (2(ii))implies that U(L)is dense in U.Property (1) amounts to saying that the unbounded operator (U(L),L ) is “closed”. We shall say that assumption (2) is the “traceproperty”. Notation. For any map M defined on U(L),we shall set Mo = MI u 0 ( ~ to ) be the restriction of M to Uo(L). (3) For the sake of simplicity, we shall assume that U and V are identified with their respective duals (4) U = U’ and V = V’. Let i and iobe the canonical injections from U(L)and Uo(L) into U respectively. Since they are injective (resp. have a dense image) their transposes i* E J(U,U(L)*)and :i E a(U,Uo(L)’)have a dense image (resp. are injective). We assume that the transpose ig is identified with the (5) canonical injection from U into UO(L)’ in such a way that (@ 33’
(i) UO(L)c U c Uo(L)*, (ii) each space is dense in the larger spaces.
484
CALCULUS OF VARIATION AND OPTIMAL CONTROL
[Ch. 14,§ 14.3
With these identifications, we have one alternative. If M is any continuous linear operator mapping U(L) into W,we may either “use” the transpose (Mo)* E B( W*, Uo(L)*)of the restriction Mo to Uo(L) or we may “use” the transpose M* E Q( W*, U(L)*)of M . Because of the identijication (6),we will use the transpose (Mo)* of Mo in preference to M . In order to emphasize this point, we introduce the following definition.
Definition 1. Suppose that (l), (2), (4) and (5) hold. We shall say that the transpose (Mo)* of the restriction of M to Uo(L) is the “formal adjoint” of M . .The formal adjoint Lt E A?(V , Uo(L)*)is defined by (7)
vpE
v,v x E
Uo(L), ( L b , x ) = ( p , Lx).
We define the domain of L: by (8)
V(L;) = { p E V such that L$p E U>.
(This has a meaning since U is a dense subspace of Uo(L)*.)We endow VCL:) with the graph norm 1
(9)
llPll = (IIPI12V+ll~Pll~)~
Proposition 1. The domain V(Li)is a HiIbert space.
Proof. If p , is a Cauchy sequence of V(L,*) for the graph norm, then pn is a Cauchy sequence of V and g p n a Cauchy sequence of U.Then p , converges to p in V and g p n to q in U. Since (L:p,,, x ) = (pn,Lx) for any x E Uo(L), we deduce that (q, x) = ( p , L x ) = (L:p, x ) for any x E U&). Thus L& = = q E U.Hence p E V(LE) and pn converges to p. 0 Theorem 1. Suppose that ( I ) , (2), (4) and (5) hold. Then there exists a unkpe operator /I* E &( V(G), 9’)such that (10)
V P E V(Li9,v x E UG), ( G p , x)- ( p , Lx) = (B*P,Y X ) .
Proof. We begin by proving that L*-i*G maps V ( g ) into UO(L)Iwhere i f E &( U,U(L)*)is the transpose of the canonical injection i from U(L)into U. If p E V(L:) and x E Uo(L),we obtain that @*p- i*Gp, x) = (p, Lx)- ( p , iLox) = ( p , Lx-Lx) = 0. Since y is a surjective operator from U(L) onto E, it has a (continuous linear) right inverse cr. Hence cry is a projector whose kernel is Uo(L) and y*a* is a
Ch. 14,$ 14.31
DUALITY I N CALCULUS OF VARIATIONS
projector whose range is UotL)' = (Ker oy)'. (1 1)
485
We write
/P = -a*@* - i*L$ E 2(V(Lz),8.).
Since L*p- i*L& belongs to Uo(L)I, we can write (12)
L*p- i*Gp = y*u*(L*p- i*L$p)) = -y*B+p.
Therefore, by applyingL*p- i * g p +y*p*p = 0 to x E U ( L ) , we obtain formula (10). The operator /?* is unique. Indeed, if formula (10)holds, it can be written (13)
L*p-i*Lg
=
for allp E V(La).
-y*/Pp
If u is (any) right inverse of y, we obtain, by applying cr* to both sides of (13), that B+p = -u*(L*p-i*I$p).
Definition 2. We shall say that formula (12) is an (abstract) Green formula. 14.3.2. Abstract problem of calculus of variations
Now consider a function (14)
F : UXVXE -* 1-
00,
+ -1
and its conjugate function F' defined on U X VXE' by (1 5 )
F*(p, q , 4 = sup [(q,X ) + ( P , X€ u YE v
r>+(..,E>-F(x, Y , 511.
CES
Assume once and for all that (l), (2), (4)and (5) hold. We introduce the minimization problems
and
(17)
v* = inf
PE V ( G )
F*(-I$p,p, F p ) .
These two minimization problems are related by the formula
(18)
0 6 Wf .'12
486
CALCULUS OF VARIATION AND OPTIMAL CONTROL
[Ch. 14,$ 14.3
To see this, observe that, since x E U(L) and p E V(L:), the Green formula implies that
0 = (-GP, X ) + ( P , W + ( g 2 P 9 y:) 4 F(x, k, F ~ + F ( - GP, PPP). ,
Taking the infimum when x ranges over U(L)andp over V(L*), we obtain (18). Definition 3. We shall say that p E V ( g )is a Lagrange multiplier if and only if (i) 0 = w + w*, (ii) w* = F*(-&*p,p,
pp).
Proposition 2. Suppose that F is a lower semi-continuous convex function. Then p is a Lagrange multiplier and x minimizes
x
-
F(x, Lx, y x ) on U(L)
i f and only if (20)
(-L& p , B.p} E a F ( S LZ, yx’).
IfF(x, y , E) =f(x)+g(y)+h(E) where f,g , and h are lower semi-continuous proper convexfunctions defied on U,V and 8 respectively, this relation becomes (21)
(i) (ii) (iii)
- ~ * pE as(%) (0. x’ E ay(-&*F)), p E ag(L2)(or k-E ag*(jj)), B.p E ah(yx) (or y x E ah*(#Yp)).
Proof. We have that {L:p, p , /3*p} belongs to aF(x’,LZ, 2 ) if and only if 0 = (--L& .“)+(p, Lx)+@*p, yx) = [F(?, LZ, YX)+F*(-L@, p , /3*p)]. Extremality relations (21) obviously follow from (20). C l
Remark. Denote by A and A* the “boundary-value” operators (22)
{ (ii) AA* x (i)
: E V(L)+ A x
p E V(U) I-+
E VXE, A*p = {-L$p, B.p) E UX=*. = {Lx, y x }
[Despite the notation used, A* is not the transpose of A!] Since relations (20)imply that
A*jj E a x , @ ( % fi,7x3; p E
a#(%
fi,YZ),
Ch. 14, 0 14.31
DUALITY IN CALCULUS OF VARIATIONS
487
we see that ff satisfies the equation 0 E A*aJ(Z, Lx,y2)-
ax,$Ljj, (72). I,
Remark. When F(x, y , z ) = f(x)+g(y)+h(z), we can write the extremality relations (21) in the form
(i) 2 E a f ( - ~ p ) ) n l i - l [ a g * ~ ) x a h * ( B ' p ) I , (ii) 0 E ag*(jj)xah*(Fjj)-A af'(-Gjj). In particular, if we assume that f is differentiable on
-L: Dom g*
we can write that 2 = Dy(G<-p))
where jj is the solution of the variational inequalities (24)
{ (ii)(i) (LDf*G(-p),
jj-p)+g*(jj)-g*(p) 0,V P E V, (yDf*U(-@),~ p - l t ) + h * ( ~ @ ) - h * ( 3 tG ) 0, Vlt E 8"
Proof. Let ff E U(L) and p E V(L:) satisfy relations (21). Then LIZ = = {L?, y?} E ag*(p)xah*(/3*p)and AZ E &f*(-&). Hence 0 = &-A5 belongs to ag'(jj)x ah*(p*p)-Aaf*(-L:p). Conversely, let p be a solution of (23(ii)). and 5 E a f * ( - G ) such that 7 = fi There exists G, E} E ag*(p)Xah*(@*p) and 5 = 7%.Hence E af(2), LS E ag*@) and y 3 E ah*(b*p3. Finally, variational inequalities (24) make explicit the formulas fi E ag*(jj) and y2 E ah*(/?$) when % = D f * ( - G p ) .
-gjj
14.3.3. The Hamiltonian system
Associate with the minimization problem (10, the Hamiltonkin H mapping U X V X E * into 1- m , -1. This is defined by
+
(25)
w, y, 511.
H(x,p . 4 = sup [(P¶r>+(a,0Y €v bE-P
Then, if the functions {y, 5 ) for every x, we verify that
(26)
w ,Y , Q =
-+
SUP
PE v nES'
F(x, y, 5 ) are convex and lower semi-continuous
[(P,Y>+(fi, E>-H(%, p , 41.
488
CALCULUS OF VARIATION AND OPTIMAL CONTROL
[Ch. 14, Q 14.3
Furthermore, we also have the relation
We now introduce the function I mapping U(L)X V X B * into defined by (28)
4%p , 4
1-
a,
+ -1
= ( p , W+(% yx)- H(x, p , 4
Proposition 3. The function I defined on U(L)X V X E " is a Lagrangian of the minimization problem v = in€ F(x, Lx, yx), i.e. .=€ W L )
(29)
F(x, Lx, yx) = sup I(x, p , z).
y,
PE n€3
Proof. This follows immediately from relations (26). 0 Proposition 4. Let F be a lower semi-continuous convex proper function and H its Hamiltonian. Then the extremality conditions (20)
{-&*F, p , p'p} E aF(% LX, y2)
are equivalent to the relations (30)
A2 {Lx', E aP, { (ii)(i) GP E Z x ~ ( ip, ' ,/~*p). =
yfi}
,JI(Z, p , p'p),
Proof. This is analogous to the proof of Proposition 14.1.3. Suppose that {-Lip, 3, /I*p} E aF(% LX, yZ}. Then, F(%L%yZ)-F(x,y,E) G(-JZ,~~,X'-X)+(P,L 2 - y ) ~ -t-
@*P, yZ- E).
Taking x = i , we deduce that AX = {LX, yX} belongs to a,,H(Z,p,@*p), since N(2, -,.) is the conjugate function of F(2, -). This amounts to writing that (p, LX)+ (fp, yZ) = F(2, LX, yX)+H(x, p , /I*p). Hence we can write a,
t (P, y)+(B *-P , 4)-F(x,
Y, E)
p*p). Taking the supremum over {y, t},we deduce that Gjj E a,H(2, p,p*p). ==
(L@,x-?)+H(x', j%
Conversely, suppose that (30) holds. Since A3 E a,. ,H(2, p , /I*p),we have that
(31)
(P,L$+(B+p, ~ 2=) F(2, LZ, yZ)+H(?,p, 87).
Ch. 14,$ 14.31
DUALITY IN CALCULUS OF VARIATIONS
489
By definition of H(x, F, f p ) , we have that
(p,y)+(PF, E ) == F(x, Y , E ) + W , F,B*F). Since L*p E 8,H(x, F, p*F), it follows that
(32)
(33)
H ( x , p , P*F)- H(2, p, B*P)
(Lip, x-
q.
We deduce from the three inequalities (31), (32) and (33) that F(X, LZ, yz)- F(x, y , E )
+(PP,72- E),
=s(-
GF,2-
x)
+(F, LX- y) +
which means that {--I-$, j?,y*F} E aF(2, L2, g?).
Remark. In the case when (34)
w ,Y , E ) = Fo(x, y)+h(E),
the Harniltonian can be written
(35)
H ( x , p , n) = ffo(x,p)-h*(5).
The system (20) can be written
(36)
{ G p ,p} E WO(% { (ii)(i) pp ah(@.
Lz).
-
E
The conditions (20(i)) are called the Euler-Lugrange conditions and conditions (20(ii)) the transversaliry conditions. The system (30) can be written
(37)
{
(i) LZ E apH0(x,p ) and Gp E %Ho(2, (ii) yz E ag*(,Q).
PI,
Conditions (37(i)) are called Humilronian conditions. 14.3.4. Lagrangian of a problem ojcalmlus of variations
We have to make explicit the dual problem associated with this Lagrangian. and check that it is the same as (17). Proposition 5 . Suppose that ( I ) , (2), (4, (5) and
(38)
vq, P , % F*(q, p , Jc) = sup [(q,x) %€ U(L)
hold. Suppose also that (39)
35,y, E such ihat x
-
F(?,Y,
+ H(x,p , 741
g) is continuous at 2 (on v).
490
[Ch. 14,§ 14.3
CALCULUS OF VARIATION AND OPTIMAL CONTROL
Then
Proof. We can write
We show that this quantity is infirite ifp 4 V(LE),i.e. if - g p written (40)
I I -Gpllu
= sup I(-Up, x)l X€
UOW)
fJ
U.This can be
=+ -.
IlXIIu-1
- +-
Therefore, there exists a sequence x,, E UO(L),such that Ilx,,llv = 1 and - (L*p, x,,) . Even if it means replacing F(x, y, 5) by F(x- I,y, E) and x,, by Axn, we can assume that there exists {y, such that F( y, 5) is bounded above on the unit ball of U. Hence, since xn E Uo(L),
Z-
.,
s}
SUP n
2 SUP
n
3 U+
tH(xm P,~ ) - ( G P ,xn)l
+ F)-
t(p, 9) (G SUP n
F(xn, 39 $1-
[(-Up, xn)]
=+
(GP;P,
xn)~
03
where a = (pyY)+(fiy ~)-supIIxllsl~(x,Yy 5). Suppose that p E V(L:) and that P*p # s. Sincep E V(L;), we can apply the Green formula. Hence - I(x, p, a) = H ( x , p , n)- (Lip, x ) +(/3*p-z, y c ) . Since y maps U(L) onto c", we deduce that whenever s*p # z. sup - I(x, p, s) = + t~
X€
U(L)
Suppose that p E V(LE) and P*p = az. Then
+
Y PP)1 sup - I(x, p , n) = sup [(--LZPP,x ) H(XY P
X€
U(L)
X € U(L)
= F*( -L;p, p,
pp)
14.3.5.Existence of a Lagrange multiplier
Consider the function v defined on V X c" by (41)
v(y, 5 ) = inf F(x, Lx+y, yx+E). x € U(L)
by assumption (38). 0
Ch. 14, Q 14.31
DUALITY IN CALCULUS OF VARIATIONS
491
We assume that (42)
the domain of F is R X Z where R c U,Z c V X 8 .
If A is the operator defined by Ax = {Lx,yx}, then 143)
Z-A(R) is the domain of o.
It is clear that
Proposition 6. Suppose that (38), (39) and (42) hold. Then, i f p is any Lagrange
multiplier, we have (45)
P E a,vcv, 0, B*P E
WY,
6).
Proof. By Theorem 1.6, we know that av(y, E) is the set of Lagrange multipliers. By Proposition 5 we know that if {p, a} is a Lagrange multiplier, then z = B'p. Hence (45) holds.
Theorem 2. Suppose that (46)
F is a convex lower semi-continuous function on U X V X B , whose domain is R X Z where R c U and Z c V X B
and that (47)
V { p , n}, x
-
H(x, p , n) is upper semi-continuous on U.
I$ (48)
0 E Int ( Z - d ( R ) ) ,
then there exists a Lagrange multiplier. Proof. This is analogous to the proof of Theorem 1.2. We apply the minisup theorem Theorem 13.1.2 to the function $(p, a; x ) = - Z(x, p, a), whose conjugate function with respect to {p,n} is {y, 5 ) F(x,Lxfy,yx+5). 0
-
14.3.6. Exdmple : The Dirichlet variational problem
We now consider the most famous classical example of the calculus of variations. This is the same as Bolza's problem but set in n dimensions. Let C? be a smooth bounded open subset of R" with boundary r.
492
CALCULUS OF VARIATION AND OPTIMAL CONTROL
[Ch. 14,§ 14.3
(a) The operator L = grad and L: = -div The space (49)
P(Q) = {xEL2(SZ)such that Dix E L2(@ for i = 1, . . ., n}
is called the Sobolev space. The derivative D,x = ax(o)/aw,is to be understood as a derivative in the distributional sense, i.e. it is defined by (50)
J ( D ~ x(0) ) ~ ( 0dm)
= - J X ( W ) D i y ( ~dw )
R
R
for all infinitely differentiable functions y with compact support in SZ. With the graph norm
Hl(S2) is a Hilbert space. If we set (51)
u =LyQ,
v = L2(SZ)”,
we see that (52)
W ( 0 )is the domain of the operator L = grad
defined by
Lx = grad x = { D g , . . ., Dnx}. One can prove the following fundamental trace theorem. If SZ is smooth enough, there exists a Hilbert space E of functions 5 defined 1
on I’(calledHZ (0) such that the trace operator y defined by yx = xlr satisfies
(53) and (54)
{
(i) y E ~(zP(sz),~ + (isrsurjective, )) (ii) H:(Q) = Ker y is dense in L2(Q)
H:(SZ) is the closure in W(SZ)of the space of infinitely differentiable functions with support in SZ.
Hence, we deduce from (50) that the transpose to Hi(SZ) is defined by (55)
Gp =-divp
=-
n
C Dipi. t=1
of the restriction of grad
Ch. 14,§ 14.31
493
DUALITY IN CALCULUS OF VARIATIONS
Let (56)
D’(n) = {p E L2(G). such that div p E Lz(SZ)}
be the domain of div, supplied with the graph norm.
1
Theorem 1implies the existence of a unique operator B*E .l?(Dl(Q),H-T(r)) 1
1
,
where we set HZ(r)* = H-” (0,such that (57)
V p E D1(Q), Vx
EH W ,
J (-divp)x- J ( p , grad x) n
R
=
r
$P.
Comparing this with the classical formula, we can set (58)
B+p =-(p,
6) where TI denotes the normal to I‘
[For n = I and 52 = ]0,1[, we have D1@) = W(Q), (59)
3 = R2, 3’ = R2,
.YX
B+P
= {~(O),x(1)}~
=
{~(o),- ~ ( l ) }
and (57) is nothing other than the formula for integration by parts.] (b) The Dirichlet problem
Let suppose that (60)
-.1- -, + -1
FO :L2(sl)xL2(sz)” convex function
whose conjugate function is denoted by F l . If 5 Dirichlet variational problem t o be (61 1
2)
=
is a lower semicontinuous,
inf Fo(x, grad x ) = inf [Fo(x, grad x)+y(5; yx)] X€H1(0)
yx=E
X € H W )
and its dual problem to be
which is a Neumann variational problem. The Euler-Lagrange system can be written (63)
1
E HT(r)is given, we define the
{div p , p } E aFo(Z, grad 3; yx = 5.
Example. In the case where
494
CALCULUS OF VARIATION A N D OPTIMAL CONTROL
[Ch. 14,s 14.3
this system becomes divp = 1 and
p
= grad Z,
1.e. 3 = div grad Z = Ax.
The Euler-Lagrange system is then
The minimum X is the solution of the Dirichlet problem for the LaplacianLet HObe the Hamiltonian defined by
Then the Hamiltonian system can be written
(i) grad Z E apH0(?, p), (ii) - div p E i m l ( Z , jj), (iii) p = 6. Remark. Assumptions (38) and (39) are, of course, satisfied when the functions x --c Fo(x,y) are continuous with respect to x. We refer to the book of Ekeland and Temam and to a note of Berliocchi and Lasry for minimal assumptions. In the case when n = 1, the comprehensive study of the Bolza problem is due to Rockafellar.
Remark. Note that we can deduce from a measurable selection theorem that if
Ch. 14,$ 14.31
DUALITY IN CALCULUS OF VARIATIONS
495
where, for almost all w E B,
4*(w, r, s) = sup [r-u+(s, v)-(p(w, u€R o€IW
and ytw; x , s) = sup [(s, w€R-
u, v)l
4-4 h u, a.
[See the book of Ekeland and Temam (1974, Theorems VIII-2, p. 220 and Proposition IX-2-1, p. 251).] Since (q, X ) + ( P , u)-Fo(x, Y)-Fo(P, d= = J [(do), x < 4 >+ ( P ( W ) , u ( 4 )- 4(w x ( 4 Y ( 4 ) II
-4+(o, dw),P ( 0 ) ) I dw = 0 if and only if the integrand is equal to 0 (since the integrand is always nonpositive), we deduce that ( q , p } E F ( x , y ) if and only if, for almost all w, {q(w),p(w)} E &j(w, x(o), ~ ( 0 ) This ). allows us to write the global relations (63) and (67) locally, i.e. for almost w E Q, (63 bis)
{div p(w), p(w)} E a4(wy Z(o), grad Z(w))
and
{
(67bis)
(i) grad :(a) E asw(a,34, PW), (ii) - div p ( o ) E &p(w, Z(w), p(w)).
14.3.7. The maximum principle for optimal control problems It is possible to regard an optimal control problem as a particular case of a problem in the calculus of variations. However, we prefer to outline an analogous study in the framework of Control Theory. Consider (71)
{
a Hilbert space W (of “controls”), a continuous linear operator M f &(W,V )
and the constraint
Lx = Mu where x E U(L), u E W relating a “control” u to a “state” x of the system. Let (72)
-
]+ -1 be a lower s emi-continuous, convex function.
F : UX W X S
00,
496
CALCULUS OF VARIATIONAND OPTIMAL CONTROL
[Ch. 14,$ 14.3
We consider the “optimal control” problem
(73)
v = inf F(x, u, yx). Lx-MU
In order to construct the dual problem, we introduce the “Hamiltonian” H defined by
Then, we can write
We construct the Lagrangian of the problem defined by (76)
I(x, u ; p , r , 4 = ( p , Lx)+(r-M*p, ii)+(z,yx)-H(x, r, 4.
It is clearly a Lagrangian since
(77)
SUP
P. r, n
F(x, u, yx) if Lx = Mu, ifLx z Mu.
4 x , u, P,r, 4 =
Proposition 7. Suppose that (75) holds, that
and (79)
3% ii,
F
such that x
-.
F(x, ii, i) is continuous on U at 2.
Then the dual problem is defined by (80)
M*p, Yp)i f p E V(Lo‘), inf I(x, u, p , r, z) = F*(-Gp, r = M’p, 72 = Bfp, UEW + otherwise.
-
x E U(L)
Proof. This is analogous to the proof of Proposition 5 and is left as an exercise. 0
The above Lagrangian is associated with the following perturbations : (81)
u(y,
k, E )
=
inf
L*=Mu-y
F(x, u+k, yx+E).
Proposition 8. A Lagrange multiplier p exists whenever 0 E Int Dom v and x H(x, r,)I. is upper semi-continuous on U.
-.
Ch. 14,g 14.31
DUALITY IN CALCULUS OF VARIATIONS
497
Let p be a Lagrange multiplier. It is easy to check that {Z, ii} is an optimal solution if and only if (82)
Mii, { (ii)(i) Li? {-Gp, M*p, pp} E W(z, =
ii,
p).
By a proof analogous to the proof of Proposition 4, we deduce that conditions (%!@)‘are equivalent to (83)
PA { (ii) G{n,FYX) E aPmhH(z, M*A no. (i)
E &H(Z, M*p,
In the case where (84)
F(x, u, E ) = Fo(x, u)+h(E)
and where Ho(x,r ) = sup,, [(r, u)-Fo(x, u)], these relations can be written (85)
{
(i) Gp E % H ~ ( z ,M*F), (ii) ii E i3,Ho(S, W p 3 , (iii) B+F E ah(y2).
Relation (85(i)) is called the “adjoint equation”, relation (85(ii)) is the famous “maximum principle” and relations (85(iii))are called “transversality conditions”. Notice that (85(ii)) can be reformulated as follows: (86)
E maximizes (p, Mu)-Fo(Z, u).
Example. Consider the case when B = It, TI, (87)
and (88)
U = L2(B,R”), W = L2(9,Rp),
H = R”XR”
U(L)= W(B), Lx(w) = Dx(w)-A(w)x(o), (Mu) ( w ) = M(w)u(w), Yx = ( 4 t h X(T)),
B*P = {-P(O, P(T)}
where (89)
-
A ( * ) E Lw(B;L?(R”,R”)); M ( - )E L-(B; 2(RJ’, R“)).
Let Fo:L2(Q,R”)XL (9,Rp)-1 - ,+ -3 be a convex lower semi-continuous proper function and C be a subset of R”X R”. We consider the optimal control problem w = inf Fo(x, u) under the constraint Dx(m)- A(@)x(w) = M(o)u(w)almost everywhere and (90) {XW, E c.
xm}
34
498
CALCULUS OF VARIATION AND OPTIMAL CONTROL
[Ch. 14,$ 14.3
Under the assumptions (78) and (79),the dual problem is (91)
TI* =
inf
PEWQ
[F*(Dp-A*p; M*p)+a+(C;{p(t), -p(T)}].
The Hamiltonian system becomes :
(i) -Dp- A*p E &JTo(X, M*F), (ii) ii E a,Ho(z, M*p), (iii) {p(t), -F(T)} is normal to C at
{x'(t), 28).
The perturbation function is equal to (93)
vol, k, 5 t , ET)
inf
=
Dx(o)--A(w)
XCW)
Fo(x, 4.
= M a )Nu)+ W w ) k(w)+y(ru) x(w)€C-{€4. Pr)
Hence, we deduce that (94)
1
(i) iX.1 E a,.v(O, O,O, 0,) (ii) M * ( - ) p ( . )E &(O, 0, 0,O) (iii) p(t1 f aElv(O,0, 0, 0,01, (iv) -F(T) E a ~ p ( 00,, 0,O).
In particular, if we associate with any 6 the optimal control problem (95)
v(t, E ) =
the solutions p(E, (96)
a)
inf
Fo(x, u)
D%(o)--A(w) x(m)=M(m) ~ ( w ) , W E It. T [ a ,e. (NO. x(n1 € C - K 0 )
of its dual problem satisfy the condition
p(5, t ) E -a&
E).
-.
Consider the case where there is no state constraint, i.e. where (i) FO : U X K R, where K c W,K is convex, f 0 (97) (ii) C = CoXRn where COc R", COis convex, # 0
{
Then, we assume (89), (97) and (72), (78),and (79) (with Fo replacing F ) we may deduce the existence of afunction p satisfving (92).[If we set R = U ( L ) x X K , Y = { O X C o } and A(x, u) = {Dx- Ax-Mu, x(O)}, the constraint qualification 0 f Int(Y-A(R)) is satisfied since clearly, A maps U ( L ) X K onto L2(0,T; R")X R".]
Example (the quadratic case). Suppose that FOand h are quadratic functionals:
+
(i) PO(% u) = j?' [ j ( s , x(s)-~O(s))~+k(s, u(s)- uO(s))*] d ~ , (ii) h(5) = : g ( 5 - 5 T ) 2
Ch. 14,$ 14.31
DUALITY IN CALCULUS OF VARIATIONS
499
where we set (i)j(s, El2 = (J(sK, E), dEY = (GE, 5)(ii) 4 8 , r1I2 = (W)% r1) where ('O0)
J(s) E A?(R", R"), G E &(Re, R") and K(s) E ,P(Rp, Rp) are selftranspose, positive semidefinite matrices such that the functions s + J(s) and s -c K(s) belong to Lw(sd, A?(R*, R")) and L-(S2, L!(RP, Rp)) respectively.
Consider the differential equations describing the state of the system :
The adjoint system (92(i) and (iii)) becomes (102)
(0 -K4 { (ii)p(T)
= A*(s)p(s)- 4 s )
= - G(x(T)-
ET),
(m-xo<s)),
since the Hamiltonian can be written T
H ( x , r,n) = - 1 J j ( s , x(s)r
++
T t
XO(S))~
ds
[k,(s,r(4)'+(r@h
~ o ( ~ ) )ds+:g,(ZYi-(n, l
ET)
where we set (103)
kJs, r)2 = (K(s)-lr, r), g,(fi)2 = (G-ln, n).
The maximum principle states that u maximizes the function u k - ( p , Mu)-
- Fo(x,u) over K,i.e. (104)
u(s) = uo(s)+K(s)-l M*(s)p(s).
On writing u(s) in the system (IOI), the control probIem reduces to solving the two-p 3int boundary value problem for the system (9 W = 4 s ) x(s)+N(s)p(s)+M(s) UO(S), = A*(s)p(s)-JJ(s) (x(s)--o(s)), (iii) x(r) = E,p(T) = - G(x(T)- 5 ~ )
(3-d(s)
(105)
where we set (106) 34.
N(s) = M(s)K(s)-1 M*(s).
500
CALCULUS OF VARIATION AND OPTIMAL CONTROL
[Ch. 14, fj 14.4
14.4. Optimal control and impulsive control problems
Consider the optimal control problem v(t, t ) =
inf U(.)
4r)=ds,
-w.Y(J))
x(t)=€
[jP(& x(4, .(4) &+h(x(l?)
1-
{u@))EK
The approach of dynamic programming is to take the initial time t and the initial state 5 of the system as parameters and to show that the “optimal performance function” {t, t }-,v(t, t )is a solution of a first order partial differential equation
by assuming a lot of smoothness. This equation is the Hamilton-JacobiBellman equation. We shall prove that knowledge of the performance function allows one to “solve” the optimal control problem in the following way. We shall assume that for any fixed { t , 5}, there is a unique control u = D*(t, 5) achieving the maximum of the function
over K. This decision rule D” is called a “feed-back” map, which yields the “optimal closed loop control” U(t) at each time t via the formula C(t) = D”(t, X(t))
where is the optimal state. We shall also prove the following fundamental principle of optimality. An optimal pair {Z(-), is(.)} has the property that, whatever the initial state E at time t is, the remaining states X(s) must constitute an optimal state when s ;a z with regard to the state ?(z) resulting from the ‘firstinitial state. We will give the Hamilton--Jacobi-Bellman equation explicitly in the case of quadratic control problems. In this case, the performance function w(t, 5) is quadratic with respect to 5 :v(t, 5) = #‘(t)5,E)+(r(t)y E)+o(t). We show that P(-), I(.) and a(.) are solutions of a non-linear system of differential equations called the “Riccati system”. Knowledge of P( -)and r( .) enables one to construct the “closed loop control”. ?(a)
Ch. 14,s 14.41 OPTIMAL
CONTROL AND IMPULSIVE. CONTROL PROBLEMS
501
In summary, we have two methods for solving optimal control problems. The maximum principle is particularly well-suited when the performance function is not smooth (which can happen) and leads to a two-point boundaryvalue problem for a system of two differential equations. The principle of optimality assumes some smoothness and leads to a non-linear partial differential equation. Next, we use the dynamic programming approach to solve optimal stopping time problems of the type
We shall prove that the performance function v, assumed to be smooth, is a solution to the Bensoussan-Lions variational inequalities
{ and
(i) w t , oiat+(av(t, OPE,y(t, o ) f V ( t ,E ) 0, (ii) ... d t , El-h(t, E) -S 0, (110 [ ~ tE)-h(t, , 511 [av(t,t)iat+(av(t, w(t,E))+
om,
E)] = o
v(T, 5) = h(T, E )
We use the performance function v to construct the optimal stopping time 3 as follows. We define the "stopping set" 8 by
8 = {{t, E} E [0,TIXR" such that ~ ( t5), = h(t, 5)). Then 3 is the smallest of the elements z E It, T [ such that {z,E} E S. We disregard in this section all technical difficultieswhich arise from the lack of smooth-
nss of &? f ~ C ! U f l J&yu!yeed & L,$&y?ml&rn, A bvpk Qf Bensoussan and
Lions (1978) will Study thoroughly such problems from both the optimal
c o n t d and the partial differentialequations view pojjjts,
14.4.1. The Hamilton-Jacobi-Bellman equation for a control problem
Consider an optimal control problem denoted by
under the constraint (*)
{
n(s) = y(s, x(s), u(s)) x(t) = E
for s E It, T[, R = dx/ds,
502
CALCULUS OF VARIATION AND OPTIMAL CONTkOL
[Ch. 14,s 14.4
and
(3)
u(s) E K
for s E It, T [ .
We assume that (4)
K c RP is a convex compact subset
and that (5)
~ I : ] O , T [ X R ~ X KR- + { y:]O,T[XR"XKR"
are a t least twice differentiable on a neighborhood of 10, T[XR" X K . (We do not attempt a treatment in terms of minimal assumptions in this introductory presentation.) The main assumption we make is that (6)
the performance function v defined on [0, TI X R" by (1) is continuously differentiable.
This assumption is by no means necessary, but allows a quick develclpment of the theory. We shall prove that the performance function is the solution of the Hamilton-Jacobi-Bellman equation. fioposition 1. Suppose that (6)holds. Then the performance function is a solution of the Jirst order partial differential equation (7)
{ (ii)( 9 v(T, 5 )
aNt, Ellat = maXUEK[(- avo, WaE, y ( t , 5, u))- v ( t , 6,u]), = h(5).
Remark. Consider the case when (8)
Y(t, E , u) = M(t)u+A(t)E.
Then
(9)
The last term of the right-hand side inequality is just the Hamiltonian of y.
Ch. 14,§ 14.41 OPTIMAL CONTROL AND
IMPULSIVE CONTROL PROBLEMS
503
Proof. We begin by proving the terminal condition (7(ii)). If we take t = T , then
We now prove that v satisfies the partial differential equation. Let t < T. We introduce At (which will tend to 0) and split the control function u into the form (10)
d s ) = u1(s)+uz(s)
where u(s) if s E It, t + A t [ ,
(1 1)
O
ifs 4 [t+dt, T[,
0
if s E It, t+At[,
u(s)
if s
[t+At, T[.
Hence the solution x of the differential eq. (2) a n be split in the following way : = xul( is solution of
x(
0 )
a)
and x(-) = xa1(.;u2)is solution of (13)
{W
= ~ ( 8 x(s), ,
ds))
x(t+dt) = x,,(t+At).
Consider
Then we can write
E It+&
T[,
504
CALCULUS OF VARIATION AND OPTIMAL CONTROL
[Ch. 14,$ 14.4
In computing the infimum over u = u l f u 2 , we take first the infimum over uz when u1 is fixed and then the infimum over ul.] We now use the Taylor series expansion formula
where we set
1
+at( t , Q +o(dt). av
Therefore, eq. (15) becomes
+-Odt( A 0
*
Hence, by letting dt tend to 0, we can check that v is a solution of the partial differential eq. (7).
Ch. 14,s 14.41 OPTIMAL CONTROL
AND IMPULSIVE CONTROL PROBLEMS
505
14.4.2. Construction of the closed loop control
Assume now that we know the performance function w(t, 5). We shall assume that there exists a unique control P ( t , E ) E K which achieves the maximum over K of the function (19) u I--[( - aw/a5(t,5), y(t,t,u))- q(t, 5, u)],which depends smoothly
upon t and 5.
This decision rule is called a “eed back map” or a “closed loop control”. This gives the optimal control at each t as shown in the following proposition.
Proposition 2. Suppose that (6) and (19)hold. Then,for each t, the optimal con-trol ii(t) and the optimal state Z(t) are related by (20)
ii(t) = P ( 2 , x’(t)).
Proof. Consider the solution z( -) of the differential equation (21)
{ .w
= Y(S,
z(t) = 5.
z(s),w s ,
We differentiate the function s -I
m)),
v(s, z(s)) on It, T [ and obtain
506
CALCULUS OF VARIATION AND OPTIMAL CONTROL
[Ch.14,$ 14.4
Hence, by integrating from t to T,
-This means that the state z( .) which is the solution of the differential equation for the control u( defined byu(t) = D x ( t, z(t)),achieves the minimum. Hence It is the optimal state and u is the optimal control. a)
14.4.3. The principle of optimality
We now proceed to prove the principle of optimality. Proposition 3. Let x' be an optimal state associated with an optimal control u' of the minimization problem v(t, 6). Then the restrictions Z, and ii, to ]z, T [ (where t -= t < T ) are respectively an optimal state and an optimar control of the problem w(Z(z), z).
Proof.W e have, by definition, that
If the restriction {Zs,fiZ} is not an optimal pair of the problem u(Z(z), z), there exists a pair (2, a} such that
We shall exhibit a contradiction by proving that the pair
{%,fi}
defined by
Ch. 14,s 14.41
OPTIMAL CONTROL AND IMPULSIVE CONTROL PROBLEMS
507
yields a loss smaller than v(t, 5). Indeed, we have that
W, El =S
T
7
q(s, ZM, t
W) dr+ J q(s, W, ti@))
ds+h(R(T))
t
= v(t, 5). 0
Proposition 4. Suppose also that v is twice dgerentiable. Suppose that (6) and (19) hold. Then the function - av( ,S)/a5 is the solution of the adjoint system (28)
{ (ii>(0j ( T )
m,
* l d t = - (Dy(t, m),P(t))+@(t, = - Dh(x'(T)),
m,W ) ,
when D = a185 denotes the gradient with respect to 5.
Proof. Let Z(.) be the optimal state and let ii(t) = D + ( t , X ( t ) ) be the optimal control. Write p(t) = - av( f, x(t))/a5. It is clear that P Q = Dh(Z(7'))since v(T,3(7')) = = -h(Z(T)).
We now compute the derivative of the ith coniponent pi(-) of$(-). Then
Now compute a2v(t, E)/aEia5, by using the Hamilton-Jacobi-Bellman equation. SinceD"(t, E) is the unique solution which achieves the maximum in (19), we deduce that
On setting 5 = Z(f) and U ( t ) = D*(t,Z(t)), eqs. (29) and (30)yield
which is the adjoint equation.
508
[Ch. 14,s 14.4
CALCULUS OF VARIATION AND OPTIMAL CONTROL
14.4.4. The quadratic case: Riccati equations
We continue the study of the quadratic control problem initiated in Section 14.2.7. Consider the following norms (32)
1
1
j ( s , 0 = ( 4 4 5 , Q2, g ( 0 = (G5, t)%,W ,7) = (WS)V,% ),
1
associated with the duality operators (33)
J(s) E L(R", R"*),
G E 4 R " , R"*),
K(s) E
J?(Rp, Rp').
We introduce linear operators (34)
A(s) E d ( R " ,R"),
M(s) E J?(Rp,
a")
and set
(35)
N(s) = M(s)K(s)-l M+(s).
We shall assume that the functions J( .), K( smooth. We set (36)
1
a),
A( .),M(
a),
N( are su5ciently a)
(i) q(s, x , u) = +j(s, x - x & ) ) ~ + + k(s, u - u ~ s ) ) ~ ,
(ii) h(E) = +&E-ET)~,
(iii) y(s, x , u) = A(s)2+M(s)u
and study the quadratic control problem
I
%(s) = A(s) x(s)+M(s)u(s),
x(t) = E
without constraints on the control (i.e. u ( - ) ranges over Lz(t, T; R")). It is clear that, for each t, 5 t-+ v(t,5) is a quadratic function, which can be written (38)
v(t, 5) = +(WP,E)+(r(t),
E)+m
where, for each t, (39)
(i) P(t) E A?(R",R"*) is a symmetric matrix, (ii) r(t) E Rn*, (ui) a(t) f R.
Ch. 14,$ 14.41
OPTIMAL CONTROL AND IMPULSIVE CONTROL PROBLEMS
PropoSition 5. The Hamilton-Jacobi-Bellman
509
equation can be written
with ‘theterminal condition (41) v(T, E ) = ig(5-ET)2. The performancefunction v can be written in the form (38) where P( .),r( .) and are solutions of the “Riccati system of equations”
I
(a)
(42)
(i) P ( t ) = P(t)~ ( tP(t)) A*(t)~ ( t ) - ~ (A(t)-J(t), t)
(ii) i ( t )
= P ( t ) . N ( t )r ( t ) i - J ( t ) xo(t)-A*(t) r(t)-P(t) M ( t ) uo(t),
(iii) a(t) = ( ~ ( tr(t), ) r(t))- +j(t, X O ( ~ ) ) ~(r(t), M ( t )uo(t)) with the terminal conditions (43) P ( T ) = G, r(T) = -GET, a(T) = +g(5T)2. Then, i f P ( r( .) are solved, the optimal co-statefunction p( .) is given by a),
+
-F
(44)
I
Proof. In the case of quadratic control problems, we have that
1
- -j(4
2
E- xo(t))?
510
CALCULUS OF VARIATION AND OPTIMAL CONTROL
[Ch. 14,$ 14.4
But
Thus we have proved (40). Let US write v(t, 5 ) = i ( p ( t ) ,6, t.)+(r(t), 5)+a(t) where, for each t, P ( t ) E -@(R",R"*)is a symmetric operator. Since (47)
av
1
.
at ( t , 4 ) = y(WE,E)+(W,E)+a(t)
and since
This implies that "Riccati" eq. (42) hold.
Ch. 14, p 14.41
OPTIMAL CONTROL AND IMPULSIVE CONTROL PROBLEMS
51 1’
Since for all 5 E R”the terminal condition can be written
+
f ( P ( T ) E , E)+(r(T), 5 ) U ( T ) = v(T,5) = +(G& E)-(GET, l ) + : ( G h , b), we obtain the terminal conditions (43). Let P(.) and r(.) be solutions of the ficcati equation. It is clear that P(t) is symmetric for all t. Let p( .),Z( -) be defined by (44)and (46). Then
-at aF (ti = p ( t )q t ) + i ( t ) + P ( t ) n(r) = -P(t)N(t)p(t)+A*(r)F(t)-P(t) A(t) x ( t ) + P ( t ) 9(t) - J ( t ) (x’(t)-xo(t))-P(t) M(t)uo(t) = A*(f)p(f)-J(t)(Z(t)-xo(t))
+PO)[w-( ~ ( tx’(t)+M(t) ) uo(t)+~Ct)~(t))]
= A*(r)p(r)- J(t) (~(2)+P(t)
xo(t))
[W- [A(t)z(t)+M(t) uo(t)-W ( P ( 0 x(0
+ r(t))I.
Hence, since z( is the solution of (46), we see that p( -)is the solution of the adjoint system a)
-2 at 0) = A*(t)p(r)-J(t) (z(t)-xo(t)), This proves that %(-) is the optimal solution and that the feed-back map is given by
a(r) = u0(t)- K(t)-l M*(r)[P(r)z(r)-t r(t)], since the function u +( - av/a5, M(r)u)- $k( t, u- uo(t))2achievesits maximum at (50)
av P ( t , 5) = uO(t)-K(r)-lM*(i)- ( t , 5).
at
14.4.5. The Bensoussan-Lions variational inequalitiesof a stopping rime problem
We consider an optimal “stopping time” problem
512 where x(
CALCULUS OF VARIATION AND OPTIMAL CONTROL a)
[Ch. 14,g 14.4
is the solution of the differential equation
(i) k(s) = y(s, x(s)) for s E It, Ti, I (ii) x(t) = t. where g~ :10,T[X R" R and y :10,T]X R" R"are smooth functions which (52)
-
-
satisfy the requirements implying existence and uniqueness of a smooth solution. We shall say that the function v defined by (51) is the performance f i c t i o n of the optimal stopping time problem. We shall assume that the performance function v is continuously differentiable. (53) [Actually, it is sufficient to assume that w is Lipschitz].
-
Remark. Since the function t j: ~ ( sx(s)) , ds+ h(z,x(z))is continuous and z ranges over a compact set, there exists an optimal stopping time 2. Interpretation. We can use such a model to determine the optimal time to sell a commodity 5 available at time t, whose state evolves according to (2), when y(s, x(s)) is the maintenance cost at times and h(s, 5) is the value of E at times. Proposition 6 (Bensoussan-Lions). Suppose that (53) holdr. Then the performance function v is a solution to the variational inequalities: for any {t, 5 ) € (0, TI _-
and the terminal condition (55)
v(T, 5) = hG", 5).
Proof. The terminal condition v(T, f ) = h(T, 5 ) follows from the definition by taking t = z = T.Now, by taking z = t, we deduce that (56)
r
v(t, 5 )
f v ( t , x ( t ) ) dt+h(t, ~ ( t )=) h(t, x(r)).
t
Hence (54(ii)) is satisfied. We now take At (which will converge to 0) and z a t+ At. We write
i
5 ) =s 94%x ( 4 ) ds+h(t, X W ) t"
Ch. 14,$ 14.41
OPTIMAL CONTROL AND IMPULSIVE CONTROL PROBLEMS
513
Taking the infimum over z E [t+dt, TI, we deduce that (58)
t+dt
j &,x(s)) &+v(r+dt,
v(t, 5)
x(t+dt)).
t
Using the Taylor formula, we can write x(t+dt) = 5+05 where
A5
=
= dt.ax/at+o(dt).
Then
Dividing,byAt and letting At tend to 0, we deduce that (Mi))holds. It remains to check that (54(iii)) holds. This amounts to proving that
whenever (60)
v(t,
5) -=
w,5).
Let t be a point satisfying (60) and T be the optimal stopping strategy. Then 3 =- t (If not, we would have v(Z,4) = v(t, 5) = h(3, t), which is impossible). Let dt be so small that t+At < 3. Since
I+At
f
~ ( s ~, ( s ) )&+v(t+dt,
x(t+dz)),
t
we deduce from the Taylor formula applied to x(t+d) and v(t+Az, x(t+Ar)) that
v(r, 5)
[
v(t, E)+dt cp(t, E)+
(2,
E ) +\z / (t,5), y(t, E))]
as in the proof of Proposition 1. Dividing by dt and letting A t converge to 0, we deduce that
Hence (59) follows from (61) and (54(i)). 35
+o(dt)
514
CALCULUS OF VARIATION AND OPTIMAL CONTROL
[Ch. 14,§ 14.4
14.4.6. Construction of the optimal stopping time
Knowledge of the performance function w allows us to find the optimal stopping time. We introduce the following “continuation subset @” of [O,T]X X Rndefined by (62)
@ = { {t, 5 )
E [0,TI X Rnsuch that w(t, 5 ) -Z h(t, E)}
and the “stopping subset J” defined by (63)
s = { {t, 5 ) E [O,T]X R such that v(t, 5) = h(t, t)}.
Proposition 7 . Suppose that (56) holds. VE E R“ is given, the optimal stopping time Z is the smallest of the elements z E [t, TI such that {t,5 ) belongs to the stopping set 8. Proof. Let 5 E R” be fixed and let Z be the smallest of the elements z E [t, T ] such that {z,E} belongs to the stopping set a. This set is non-empty since {T,E } belongs to 8 by (52). Consider the function s F-+w(s, x(s)). Its derivative is equal to
= -y(s, 4 s ) ) whenever s E It, 5[
[since in this case, {s, 5) belongs to @, (i.e. v(s, x(s)) < h(s, x(s)) and thus, equality (59) holds]. Hence, by integrating this equality from t to 5, we deduce that
h(z, x(Z))-w(t, E ) = “(2,x(Z))-w(t, f
=-
y(s,
XW)
dr;
I
i.e. 3 achieves the minimum. 0 14.4.7. The Bensoursan-Lions quasi-variational inequalities of an impulsive control problem
We study an “impulsive control problem” (65)
q(s, x(s)) ds+Nt(u)
1
Ch 14,g 14.41 OPTIMAL. CONTROL AND IMPULSIVE CONTROL PROBLEMS
515
where u ranges over the set Atof discrete vector measures u =
(66)
2 Ei6(8,)
I-1
where 8, E [t, TI, 5, E R’f, N,(u) = m
on the interval [t, T [and where x(.) is the solution of the following systems of differential equations:
(67)
I
if s E [ I , el[, i f s E I&, U, ifs E [el, 6i+l[, i f s E [em,T[,
ko(s) = YJ(S, X O ( S ) ) ; x ~ ( t = ) 5. &(s) = ~ ( 8 ,xl(s)); =~~(8$+5~, Ai(s) = y(s, xds)); km(s) = y(s, xm(s));
xi(&) = xi-l(&)+E,, xm(~m)=xm-l(em-l)+Em
We assume that y :10, T [X R“ -* R and y :10, T [XR” -* R” are smooth functions which satisfy the requirements for the existence and uniqueness of a smooth solution. Interpretation. The typical example of such an impulsive control problem arises in inventory management. We regard R” as the space of “inventories” 6. We denote by x(t) the state of the inventory at time t. An “impulsive control” u= Ei6(8,) denotes a “supplying policy”. At time Or, the firm orders E, E R:. The evolution of the state x( .)is described by the system of differential ~ ( s x(s))ds , as the maintenance cost of the inventory eqs. (67). We regard %(-)from z1 to t2. We add to the maintenance cost q(s, x(s))& the sum N,(u) = m of the fixed costs (assumed to be equal to 1) arising ’at each order of a commodity. The aim of the problem is to find the optimal times 8, at which to place the orders as well as the optimal quantities E, to order. Many other management problems (production maintenance, marketing, finance, etc.), can be “modeled” by impulsive control problems.
c;nEl
I::
IT
We shall require that the performance function v(-, smooth. For instance, we shall assume that (68)
0 )
defined by (65) is
the performance function is continuously differentiable.
We shall prove that is a solution of quasi-variational inequalities. For this purpose, we define the operator R associating with a function v(., -) the function Rv(-,-)defined by (69)
R d f ,5 ) = 1
+ inf
v(t, 51t.q).
We introduce the correspondence S defined by (‘70) 35.
Sv = {w(
., .)such that w -s Rv}.
516
CALCULUS OF VARIATION
AND
OPTIMAL CONTROL
[Ch. 14,§ 14.4
Proposition 8 (Bensoussan-Lions). Suppose that the regularity assumption (68)
hl&. Then the performane function v(.,
a)
is a solution to the following quasi-
variatwnal inequalities
(71)
v f S(v)
and
cind the terminal condition (73)
w(T, E ) = 0.
The quasi-variational inequalities (7I), (72) are called the “BensoussanLions” quasi-variational inequalities. Proof. The terminal condition o(T, -5) = 0 is obvious since NAu) = 0 and JFg(s, x(s)) ds = 0. Consider now t -= T.We use controls of the form
Therefore, since N,(u) = 1+N,(w),we deduce that
= l+u(t,E+q).
Taking the infimum as q ranges over R?, we obtain that
(3
v(t, E )
i.e. that v E S(v).
-G
1 + inf v(t, t + q ) = Rv(t, 0, 9ER%
We now check that (72(i)) holds. For this purpose, we take controls u = -= 8e.o < 6, < T and At such that t + d t -= el. = ~ - l E , S ( O , ) where t -=
Ch. 14, $ 14.41
OPTIMAL. CONTROL A N D IMPULSIVE CONTROL PROBLEMS
517
Then we can write
Taking the infimum over the discrete measures u = we deduce that (79)
v(t, 0
f+dI 4
Ip(s,4 s ) ) dr+ v(t+dt,
xtl5,8(8,) E
%(t+dt)).
t
On using the Taylor expansion formula, this implies, as in the proof of Propositions 1 and 6, that
Dividing by dt and letting dt converge to 0, we obtain (72(i)). Jt remains to prove (72(ii)), i.e. that
whenever
Let i2 = xTl&d(8,)be an optimal control. If (81) holds, them t < 81 necessarily. [If not, we would have ii = $(t)+G and v(t, 5) = l+v(t, € + f ) = = Rv(t, E), which is Impossible]. We choose dt such that i+dt -G 81. Then
The Taylor expansion formula implies that
518
[Ch. 14,o 14.4
CALCULUS OF VARIATION AND OPTIMAL CONTROL
Dividing by At and letting At tend to 0, we deduce that
Hence (82) and (72(ii)) imply (81). 0 14.4.8. Construction of the optimal impulsive control
We now show how to use the performance function v to construct the optimal control. We define the "continuation set" @ c [0,TIXR" by
@ = {{t, 5 ) E [0,TIXR" such that v(t, E )
(83)
<
Rw(t, E)}
and its complement 8, the "stopping set"
8
(841
=
{ {t, E } E
[0,TI XR" such that v(t, E) = Rv(t, €I}.
Theorem 1 (Bensoussan-Lions). Suppose that the performance function satisfies (68). Then the optimal impulsive control 11 = C E,S(B,) is obtained by induction as follows: 81 is the first time t E It, T] such that Lz,5 ) belongs to the stopping set 8, v(B1, x(B1)+5) on RT. (ii) E R: minimizes 5
(i)
-.
(85)
If 186)
{%t,
&} are known for k
=si, we construct x,(
-) as the solution of
i(~ = Y)( S , x ~ ( s ) )for s > 8,, x,(Bi) = xi-i(Bi)+
(i)
F,
B,+I is thefirst time z E ]Bi, TI such that the st0ppin.q set 8. E R; minimizes 5 +-- v(8t+1,X j ( B i + l ) + E ) ) on RT.
{t,x,-l(Bi)+Ei)} belongs to
(87) (ii)
Ei+l
Interpretation. Let 5 be the initial inventory at time t. If {t, 5 ) E 8,we order a stock fj which minimizes q F+ v(t,t+q). I f not, we let the state x( .)of the stock evolve until a time 81 at which {z,t } reaches the stopping set 8. At such a time B1, we order a new commodity 5 1 and let the state x ( - ) of the inventory evolves from the position x(B1)+El at time B1 and so on.
Ch. 14,$ 14.41 OPTIMAL CONTROL AND
IMPULSIVE CONTROL PROBLEMS
519
7 Stoppin!
Fig. 16.
Proof. Consider the function s .-t
v(s, ?(s)). Its derivative is equal to
Suppose that { t , C} belongs to @. Let 81 E It, T[be the smallest of the elements z such that {z,5 ) belongs to the stopping set &. Then, for any s E It, el[, we have ~ ( s 5) , -= Rv(s, 5) and thus, by (88) and (go), d - v(s, qs)) = -q(s, qs)). (89) ds 'Integrating this equation from t to 81,we obtain e; Rv(B1, x'(81))- v(t, 5) = - q(s, x'(s))ds. t
If we write Rv(&, ~ ( 8 ~=) l+v(&, ) (90)
;e
?(&)+El),
+
we obtain that
v(t, E) = J q(s, x(s)) ds+ 1 v(&, x(el)+51). 1
520
CALCULUS OF VARIATION AND OPTIMAL CONTROL
[Ch. 14, Q 14.4
Let Z1(s) be the solution of the differential equation nl(s) = y(s, 21(s)) if s =- 81, ~l(81)= Z ( B ~ ) + L
(91)
Let 8, c 18, T ] be the smallest of the elements z E 181, T] that {z,?(el)+ 51) belongs to the stopping set 8. Then, for any s E ]81,82[, we have the rela tion (89), which, integrated from 81 to 02,yields
+
Writing Rv(Bz, xl(Bz))= 1 ~(82,xl(8z)+ F2), we have that 4
v(8t 2(81)+ E l ) = J
(92)
e;
+ ~ ( 8Zi(Bz)+ ~ 3 Ez).
~p(s,~ I ( S ) )ds+ 1
Define by induction a sequence of times 8,and of vectors Zi such that
where Z,(
(94)
a)
I
is the solution of the differential equation
(i) ?,(s) = p(s, xi($)) if s Bi, ~ , ( 8 i = ) %i-,(Bi)+ F i , (ii) Bi+, is the smallest of the z E ]Bi, T ] such that {t,Z i - l ( B i ) + F i } belongs to the stopping set 8. (iii) Ei+, E R$ achieves the minimum of 5 I-+ w(8i+1,
Adding eqs. (93), we obtain that (95)
T
v(t, E ) = J ~ ( s2(s)) , ds+Nt(u3, t
Z(Bi)+E)
CHAPTER I5
FIXED POINT THEOREMS, QUASI-VARIATIONAL INEQUALITIES AND CORRESPONDENCES In this chapter we prove some more fixed point theorems and some existence theorems for the critical points x of a correspondence S (i.e. the solutions of 0 E S(x)). We also study quasi-variationalinequalities(introduced in Chapter 9) and complete our account of the properties of correspondences.
15.1. Fixed point and snrjectivity theorems for correspondences This section is devoted to the proof of several existence theorems for the critical points of a correspondence and their application to fixed point and surjectivity theorems. We obtain another proof of the Kakutani theorem (Theorem 9.2.3). Tie results we present here are very useful in Mathematical Economics. We choose to deduce all results from an adaptation (due to B. Cornet) of a theorem of Browder and Ky-Fan which states that, i f X is a convex compact subset of a locally convex space U and if S is an upper hemi-continuous correspondence with non-empty closed convex images satisfying the Ky-Fan boundary condition: V x E X, S(x)nT,(x) f 0 (where T,(x) = N,(x)- is the “tangent cone” to X at x), then there exists a critical point x of the correspondence S. The proof of this theorem is based on the Ky-Fan inequality (Theorem 7.1.3). In particular, we are led to introduce “inward” and “outward” correspondences s. If S has compact, convex images, these terms mean respectively that Vx E X , S ( x ) n ( x + T ~ ( x )f) 0 and S ( x ) n ( x - T ~ ( x )f )
0.
Let X be a compact, convex subset of U and let S be an upper hemi-continuous correspondence frorn X into U with non-empty closed convex images. If S is either inward or outward, then S has a fixed point. If S is outward, then for all y E X there exists x E X such that y E S(x). We then consider a family of correspondencesS( .,A) depending on 1 E [0,1] 521
522
FIXED POINT THEOREMS
[Ch. 15,s 15.1
.,
We shall prove that if S( 0) satisfies the assumptions of the Browder-Ky-Fan A ) (if any) do not belong to the boundtheorem and if the critical points of S(-, ary of X,then s(-,1) does have critical points (Granas’s theorem). Finally, we mention several generalizations of the Browder-Ky-Fan theorem which may be useful.
15.1-1. The Browder-Ky-Fan existence theorem for critical points Let U be a locally convex Hausdorff vector space and let U’ be its dual. We assume once and for all that (i) Xis a convex compact subset of U, (ii) S is an upper hemi-continuous correspondence from X into U with non-empty closed convex images (see Definition 2.5.2). Recall that upper semi-continuous correspondences from X into U supplied with the weak topology are upper hemi-continuous (see Proposition 2.5.1).
Definition 1. Let S be a correspondence from X into U.We shall say that x is a “critical point” of S if 0 E S(x). To prove the existence of a critical point of S,we have to add to assumption (1) a consistency assumption relating S and X. Definition 2. Let X be a closed convex subset of U . We shall set (2)
1
(i) Nx(x) = { p E U*such that (p, x ) = maxy,x ( p , y ) ) , (ii) a ( X ; p ) = { x E X such that ( p , x ) = minyEx(p, y ) } , (iii) Tx(x) = Nx(x)- = {x E U such that ( p ,x) e 0, V p E Nx(x)}
and say that N,(x)is the “normal cone” to X at x, a ( X , p ) is a “supporting set” of X and that Tx(x) is the “tangent cone” to X at x. Weshall say that S satisfies the “Ky-Fan boundary-condition” if (3)
vpE
u*,v x E
a ( x ; p ) , u#(s(x),P) 3 0.
We begin by proving a characterization of the Ky-Fan boundary condition. Proposition 1. Sup?ose that S satisfies (4)
vx E
x, s ( x ) ~ T ~ ( x#) 0.
Then S satisfies by Ky-Fan boundary condition. The converse is true if the images S(x) are compact.
Ch. 15,$ 15.11
523
FIXED POINT AND SURJECTIVITY THEOREMS
Proof. We begin by noticing that (5)
x E a ( X ; p ) if and only if - p E Nx(x).
Let p E U ' and x E a(= p). Since there exist y f S(x) nTx(x)and - p E E N,(x) = Tx(x)-, we deduce that a # ( S ( x ) , p ) ( p , y ) = 0. For the converse, we assume (3). Since Nx(x) is convex and S(x) is convex and compact, the minisup Theorem 7.1.5 implies that there exists 7 E S(x) such that O =s
inf
-P€Nx(x)
o+(S(x),p) =
inf
-PEN&)
sup ( p , y ) =
Y€SW
inf
(p,jj)
-PENx(X)
The latter inequality implies that 7 E T,(x), and thus, that (4) is satisfied.
[7
Remark. Notice that, if x E Int(X) (for the initial topology of V ) ,then N d x ) = = 0, T,(x) = u. Remark. If S is a (single valued) map from X into U,then the Ky-Fan boundary condition can be written :
either S(x) E Tx(x) for all x f X or V p E U*, Q x E a ( X ; p), (P, S ( 4 )
0.
Theorem 1. Suppose that (1) holds. If S is upper hemi-continuous and satisjies the Ky-Fan boundary condition (4), then there exists X E X such that 0 E S(2).
Proof. Let us assume that the conclusion is false: we will contradict the Ky-Fan boundary condition (3). Since for all x E ,'A we assume that 0 B S(x), we deduce that there exists p E U * such that a # ( S ( x ) ; p ) -= 0 = @,O) (since S(x) is closed and convex). If we set
(6)
V(p) = {x E
X such that o*(S(x), p ) < 0 } ,
the contradiction of the conclusion amounts to Writing (7)
Since the correspondence S is upper hemi-continuous, the subsets V(p) are open. Since X is compact, this implies that
524
and that there exists a continuous partition of unity {@I, this covering of X. We introduce the function a :XX X R defined by
-.
(9)
[Ch. 15,
FIXED POINT THEOREMS
a(x,Y ) =
...,8),
8
15.1
subordinate to
It
C P d X ) (Pi, x--Y).
f=1
It obviously satisfies the assumptions of the Ky-Fan Theorem 7.1.3. Hence there exists 3 E X such that sup,,,,a(Z, y ) =s0, i.e. (10)
(p, 2 ) 6 inf ( p , y ) in which p = Y EX
C Pi(2)pi,
iEI
..
where I = {i = 1, . ,n such that Bi(Z) =- 0).This set is non-empty. Inequality (10) amounts to sayingthat Z E a(X, F). It therefore implies that a*(S(Z), p) z-0. But, since Pf(Z) =- 0 and x , E I / l i ( Z ) = 1, we obtain that
.++(S(Z),F )
C Pi(?) o+(s@),pi) < 0,
fEI
because if i E I, then Pi(?) 'r0 and thus, Z E V(pi), i.e. a*(S(Z), p,) Therefore, we have obtained a contradiction. 0
-= 0-
Corollary 1. Suppose that U is a Hilbert space (supplied with the weak topology) and that X is the unit ball of U . US is an upper hemi-continuous correspondence from X into U with non-empty closed convex images satisfVing (11)
Vx
E Xsuch that llxll = 1, then a*(S(x),
x) z= 0,
then S has a critical point 2. Proof. Identify U with U*.Then Xis weakly compact and, V p E U*,a ( X ; p ) = 0 if p = 0 and a ( X ; p ) = {-p/llpll} if p # 0. Therefore (11) obviously implies the Ky-Fan boundary condition (3) for -S. Hence - S has a critical point (which is also a critical point of S)!0 Corollary 2. (Altman).Let U be a Hilbert space (supplied with the weak topology) and let X be the unit ball of U. Let S be an uppet hemi-continuouscorrespondence from X into U with non-empty closed convex values satisfying (12)
Vx E Xsuch that
Then S has a critical point.
llxll = 1,
3 u E S(x), . llu[12z=Ilu-x[12-[lxllz
Ch. 15,s 15.11 FIXED
525
POINT AND SURJECTIVITY THEOREMS
Proof. Since (u, x ) = ~(Il~11~+11~112-lI~--x112)
it is clear that condition (12) implies condition (11) of Corollary 1. Thus there exists a critical point. 0
Definition 3. We shall say that a correspondence S from X into U is 'semicoercive" if
It is clear that any coercive correspondence is semi-coercive (see Definition 7.2.1).
CoroHary 3. Let U be a Hilbert space supplied with the weak topology. Let S be a correspondence from U into U with non-empty closed convex images satisfying
(14)
S is semi-coercive and upper hemi-continuous.
Then S has a critical point. M f . L e t a = limx-tao*(S(x), x)/llxll =- 0. We take E = +a. There exists A =- 0 such that o"(S(x), x ) 2 ful Ix I I whenever 11 x 11 =- A. In particular, if 1 Ix I I = A, un(S(x),x ) a f aA. Hence the restriction of S to the ball of radius A satisfies the boundary condition (11) of Corollary 1, and thus, has a critical point in this ball. 0 Corollary 4 (Comet). Let X = [- 1, + l]" be the unit cube of Rnand I: =fi, . . .,A) be a continuous map from X into R" satisfying the boundary condition (15)
V i = 1, ..., n, f;(x) 4 0 whenever xi .f;:(x) z- 0 whenever xi = - 1.
=+ 1
and
Then there exists Z E [ - 1, + 11" such that F(2) = 0. Proof. Let p Rn*.We set I,' = {isuch that pi =- 0) and pi < O}. Then a(X, p ) = { x E [- 1, +I]" such that x, = xi = - 1 if i E I;'}. Hence, if x E a ( X ; p ) ,
(p;F(x)) =
cPm)+ c
i€I;
i€I;
P'f;:(X)
0
ZF
= {i such that i E I; and
+ 1 if
526
[Ch. 15,§ 15.1
FIXED POINT THEOREMS
when (15) holds. Thus (15) is nothing other than the Ky-Fan boundary condition. We shall consider now the case when X is a simplex. Lemtm 1. Let x = J"' = { x C : R vector (1, . . ., I}, we have
such that
c;=l.'= 1). If 1 denotes the
(i) a(x,11) = -43% (16)
(ii) V p # 11, a(X, p )
= a(X, q) = { x E M nsuch that (q, x) = 0 )
where q = p - (minlsjsn pj)l.
.
Proof. Let p = {pl, . .,p"} E R"'. We set Z(p) = {i such that pi = minlsjs,,pj}. Hence if X = Mn, we obtain that
(17)
a(X,p) = {x E
M nsuch that xi
>q =
= 0 for all i E f ( p ) } .
To see this, let 2 C _/nn minimize (p, x ) over Mn. Suppose that i belongs to Z(p) and j 6 Z(p). Take y E M"such tht yi+ y' = 1j2 = a and 9 = 2 for k f i, j . Hence
'&,,
for any y i E [0,1 1. Since p i - q =- 0, this implies that 3'= 0. Now,
since Z(p+Al) = Z(p). We can therefore consider only the linear forms p such that mhlsj,,pj= 0. Hence, since Z(p) = {isuch that pi =- 0}, we deduce that x E JY" satisfies (p, x ) = 0 if and only if x' = 0 for all i E I(p). Thus (17) and (18) imply (16).
zsl
Corollary 5 (Cornet). Let X = A"", let H = { x E R" such' that x, = 0 ) and let F be a continuous map from M" into H . If F satisjies the boundary condition (19)
f;:(x) 2 0 whenever xi = 0,
then there exists 1 E M s u c h that F ( 2 ) = 0.
Proof. We apply Theorem 1 and Lemma 1. If p = 11, then a ( X ; p ) = M* and (p, F(x)) = L c;;=lfl(x)= 0. If p # ill and x E a(X, p), then x' = 0
Ch. 15,
15.11
whenever i E Z(p)
527
FIXED POINT AND SURJECTIVITY THEOREMS
=
{ j such that p'
(P, W)) =
=
c
=- q
c h(4
P'f;:(X)+Cl
EI@)
= mink p'}. Hence
i w )
c (p'-q)f;:(x)
2s
0
€I@)
by (19).
Thus the Ky-Fan boundary condition is satisfied. 0 Corollary 6. Let X = &''and let F be a continuous map from Mninto R satisfying the "Walras law", i.e.
(20)
v x E A",( x , F(x)) =
Then there exists X E A(2) = 0 or both.
c x&) n
i=l
= 0.
such that, for any i = 1, . . ., n, either X = 0 or
Proof. We apply Corollary 5 to the map G defined by
(21)
Vi = 1, . . ., n, gj(x) = zjh(x).
Since x, = 0 obviously implies that g,(x) = 0, there exists x E
(22)
V j = 1,
. . .,n,
such that
gj(Z) = 0. 0
k, then we deduce that v j = I, . . .,n, f i ( X ) = 0.
Remark. If 2 E (23)
We obtain the same conclusion under other assumptions below.
Propodtion 2 (Cornet-Lasry). Let F be a continuous map from A"' into R" satisfying (24)
1
(i) Vx E An, (x, F(x)) = 0, (ii) F(&) is bounded below, (iii) 3a =- o s.t., ~x 6 lim infy+, ~ l f = ~ f i (z=x a. )
Then there exists IE &'such
An,
that F(I) = 0.
Reirawk. This result is the connter part of the Debreu Theorem 8.2.2.
528
[Ch. 15,
FIXED POINT THEOREMS
0 15.1
Remark. Notice that the more familiar condition
implies the “boundary condition’’ (4l(iii)). Proof. By (24(ii)) there exists a scalar a -= 0 such that infxEA,,fi(x) =- a for all i. We introduce the correspondence T mapping Aninto A!’defined ’ by
where E
= -2-
(I.
n2a
=-
0 (since a
>
0 and a
<
0).
We shall prove that the correspondence S defined by 1 S(X) = T(x)-- 1
n
is upper semi-continuous (i.e. is closed), satisfies the Ky-Fan boundary condition (4) and that any X satisfying 0 E S(X) belongs to &’and is a solution of F(X) = 0.
(a) The correspondence T is closed. Let {y,, x,} be a sequence of points of the graph of T converging to b,x } . If x E &’,it is clear that y = T(x). If x 6 &’, we have to check that (y, x ) =z (l/n)-e. If there exists an infinity of integers p such that x, 4 4,then inequalities (y, x,) 4 (l/n)- E imply that (y, x ) 4 =s (l/n)- E, i.e. that y E T(x). If there exists an infinity of integers p such that xP E &’, then n
(YP.
x,)
=
n
C xi,pfi(xpip)- a1=1 C xip
i=1
ifi(xp)- na
i=1
by the Walras law (24(i)).
-
c n
i=l
-a ft(XP)
-
Ch. 15,s 15.11
By assumption (24(iii)), we have, since a n
C
529
FIXED POINT AND SURJECTIVITY THEOREMS
m P ) -
<
0,
-
na =-u-na
i=1
when p is large enough. Hence,
-a
(YP, X P )
=s-
1
1 a .z-+2n n2q
=
1 n
--&
since we can choose a such that a
O<--=l. nu
Thus, ( y , x ) continuous.
=s
(l/n)-
E.
Hence the graph is closed and T and S are upper semi-
(b) S satisfies the Ky-Fan boundary condition. This is clear if p = 11, since = A[l- 11 = 0 for x E Let p E R"' satisfy m i n p j = 0 and let Z(p) = {i such that pi =- O}. Let x E a(&'; p ) . Then if i E Z(p), ei = (0, . . ., 1, 0, . . ., 0} belongs to T(x) since (ei, x) = xi = = 0 =s (l/n)- E because xi = 0 when x E a(J&!"; p ) . Since T ( x ) is convex,
a(M";1) = A"and 1 ( 1 , S ( x ) )
y=-
k.
1 ei belongs to T ( x ) and z = y - - 1 n I I ( P )I i E I ( P ) 1
C
belongs to S ( x ) and satisfies
since IZ(p)I
=sn.
(c) By Theorem 1, there exists f such that 0 E S ( f ) . Such an X belongs to zixi =s (l/n)- 8, which is impossible. (1 /n)l with Hence X E &. This implies that, for any i,
2. Otherwise 0 = z-
Zi= b The Walras law (41(i)) implies that Cy=lA(.I?) f;:(X) = b = 0 for all i = 1, . . ., n. 0 36
Cy=lX, = b = 0. Hence
530
[Ch. 15, 0 15.1
FIXED POINT THEOREMS
Remark. The above Theorem 1 is equivalent to the'.following result due to Ky-Fan. Theorem 2 (Ky-Fan). Let S and T be two upper hemi-continuous correspondences s a t i ~ing y (26)
and
(27)
V x E X , S(x)-T(x) is non-empty, convex, and closed
v p E u*,v x E a(x; 3 u E s(X) and v E ~ ( x ) such that ( p , u ) a ( p , v).
Hence there exists X E X such that S(2)nT(2) # 0 Proof. Theorem 2 follows from Theorem 1 on replacing S by S-T. Theorem 1 follows from Theorem 2 on taking T = 0. IJ.I.2. Properties of inward and outward correspondences
It will be convenient to use the following concepts introduced by Ky-Fan. Definition 3. We shall say that a correspondence S is "outward" (resp. "inwurd") if (28)
w;
v p E u+,v x E PI. ( P , x ) (rev. o+(s(x), P ) 2 ( p , x))
O*(S(X),
P)
"strongly inward" if (29)
vp E
u*,vx E qx;p )
a*(S(x), p ) z= ."(X, p ) .
It is clear that any strongly inward correspondence is inward and that any correspondence mapping X into itself is inward. We also notice that S is outward (resp. inward) if and only if I-S (resp. S-I) satisfies Ky-Fan the boundary-condition(3). If a correspondence S satisfies (30)
v x E X , S ( X ) n ( X + W x ) ) f 0,
then S is inward. The converse is true if the imagesS(x)are convex and compact. In the same way, any correspondence S satisfying 13 I )
vx E
x,
S(x)n(X-Tx(x)) # 0
Ch. 15,p 15.11
FIXED POINT AND SURJECTIVITY THEOREMS
531
is outward, the converse being true if the images S(x) of S are convex and compact (see Proposition 1). We point out the following examples of outward and strongly inward correspondences.
Proposition 3. The condition
implies that S is outward and the condition
implies that S is stongly inward. Proof. We prove the first statement. Condition (32) implies that there exists a(% p), i.e. such that (p, y ) = (p, x) = minZEx(p,2). Hence ab(S(x),p ) =s( p , y ) = (p, x), and (28) implies that S is outward. The proof of the second statement is similar. 0
y E S ( x ) such that y E
By applying the Browder-Ky-Fan theorem to the correspondences Z-S and S-I, we obtain the following fixed point theorem.
Theorem 3 (Ky-Fan). Suppose that holds ( I ) . If S is either inward or outward then there exists a fixzd point of S. In particular, Theorem 3 implies the Kskutani fixed point Theorem 9.3.4. We now consider any fixed y in X . Applying the Browder-Ky-Fan theorem to the correspondences y-S and S-y, we obtain the following result. Theorem 4 (Rogalski-Cornet). Suppose that ( I ) holds. If S is either outward or strongly inward, then,for any y E X , there exists a solution I E X to the multivalued equation y E S(.Y). Proof. Notice that, if S is outward, the correspondence y-S satisfies the Ky-Fan boundary condition since
Thus there exists a solution ito the equation 0 E y-S(Z). l(i*
532
FIXED POINT THEOREMS
[Ch. 15, 0 15.1
Consider now the case when S is strongly inward. Then the wrrespondence S-y satisfies the Ky-Fan boundary condition since
(35)
v p E u*,vx E a(x; PI, U+(SO-Y, P) = q s e ) , PI- (P,Y) 3 u"(S(x),p)-u*(X,p) 2% 0.
Hence there exists a solution T to the equation 0 E S ( 3 - y. 0 In particular, we deduce the following statement from Proposition 3. Theorems. Suppose that (1) holds and that S satis-s either condition (32) or condition (33).Then,for m y y E X,there exists 2 such that y E S(2). Proof. Obviously, condition (32) implies that S is outward and condition (33) rmplies that S is strongly inward. Hence Theorem 5 follows from Theorem 4. 0
We point out the following particular cases.
Proposition 4. Suppose that (1) holds. US is upper herni-continuousand maps X into itself, then either the condition (36)
v p E U',
W ( x ;Pll c qx;PI
or the condition
(37)
VP E
u*, s[a(x;P)1 c a
m -PI
implies that S maps X onto X . Proposition 5. Suppose that X is the unit ball of a Hilbert space U and that I; is a weakly continuous map from X into X satisfying
for every x such that llxll = 1, F(x) = fx. (38) Then F ( X ) = X. Proof. We apply the above Proposition 4 since in this case,
Example. We take
(39)
u = R"*, x = M , ax = A~\J&.
Ch. 15, 0 15.11
FIXED POINT AND SURJECTIVITY 'THEOREMS
533
Then, for any p E R" such that maxJp' w min, d = 0, a ( x , p ) = { X E ax such that (p, X ) = O} by Lemma 1. We obtain the following result.
Prop~~ition 6 (Cornet). Let S be an upper hemi-continuous correspondencefrom into h e y , with non-empty convex compact values. I f S satisjks the boundary condition V X E ax, S(X) = {y E Mi such that (y, X) = 0}, (40) then S maps M"onto itserf. 15.1.3. Critical p i n t s of homotopic correspondences
From Theorem 1 we can deduce other existence theorems using the PoincarC method. We adapt a result due to Granas. Suppose that (41)
{
(i) U is a finite dimensional space, (ii) X is a convex compact subset of U whose interior is nonempty.
Then the boundary
ax = 4 I n t ( X ) is distinct from X.
Theorem 6 (Granas). Suppose that (41)holdr.Let S be an upper hemi-continuous correspondence from X X [ O , 11 into U with non-empty closed convex imagesW e assume that,for t = 0, the correspondence x t-+S(x, 0) satis&s the Ky-Fan boundary condition (6). If we assumefurthermore that (42)
V t E [0, 1[, all the critical points of S( -,t ) (if any) belong to WX),
then the correspondence x
F+ S(x, 1)
has critical points.
.,
Remark. The correspondencesS( 0) and S( -,1) are said to be "homotopic". Theorem 6 states that under assumption (42), a correspondence homotopic to a correspondence satisfying the Ky-Fan boundary condition also has Critical points. Proof. We shall assume that S(., 1) does not have critical points and obtain a contradiction. We write A = ax which is a closed subset of X,and (43)
B = { x E X such that 3 t E [0, I] satisfying 0 E S(x, t)}.
534
FIXED POINT THEOREMS
[Ch. 15, 0 15.1
The set B is non-empty since it contains the critical points of S( .,0), which exist by the Browder-Ky-Fan Theorem 1. It is closed, since S is an upper hemi-continuous correspondence from X X [O, 11 into U . Moreover, A n B = 0 because, if x E A and if t E [0, 1[, assumption (42) implies that x 4 B. If x E A and t = 1, x B since we assumed that S( 1) has no critical points. If d(x, A) = inf,,, d(x, y ) denotes the distance from x to A, then d(x, A) +d(x, B) =- 0 for all x E X . We introduce the continuous function f defined on x by
a
(44)
a,
+
d(x, E [O, 11. f(x) = d(x, A)+d(x, B )
It is equal to 0 on A and to 1 on B. We define the correspondence T o n X by
(45) T(x) = S ( x , f ( x ) ) , which is obviously upper semi-continuous with non-empty closed convex images. If x E A, T(x) = S(x, 0). Therefore, T satisfies the Ky-Fan boundary condition (3) and thus, has a critical point X by the Browder-Ky-Fan Theorem 1. Therefore 0 E T(Z) = S(i?,f(X)).By the very definition of B, this implies that 2 E B and thus, thatf(X) = 1. Hence 0 E S(Z, l), which is a contradiction. In particular, we obtain the following result.
Theorem 7. Suppose that (41) holds. Let R and S be two upper hemi-continuous correspondences from X into U with non-empty closed convex images. Suppose that (i) R satisfies the Ky-Fan boundary condition (3), (46) (ii) VS a 0, V x E ax, 0 4 R(x)+sS(x). Then the correspondence S has critical points.
{
Proof. Theorem 7 follows from Theorem 6 by setting S(x, t ) = (1 - t)R(x)+ t S(x). Assumption (46(i)) obviously implies assumption (42).
+
As a useful particular case, we mention the following result.
Tbeorem 8. Suppose that (41) holds and let xo belong to the interior of X . Let S be an ,upper hemi-continuous correspondence from X into U with non-empty closed convex images. If (47) Vs > 0, v x E then S has critical points.
ax,
XO
4 x+sS(x),
Proof. Theorem 8 follows from Theorem 7 by taking R(x) = x-XO. 0
Ch. 15, 5 15.11
FIXED POINT AND SURJECTIVITY THEOREMS
535
151.4. Other existence theorems for critical points
The Ky-Fan boundary condition is only one of the possible sufficient conditions for the existence of a critical point of a correspondenceS.It is a particular case of a family of boundary conditions which we are about to describe. Consider two locally Hausdorff vector spaces U and V, a subset X of U and a correspondence S from X into V. The boundary conditions depend upon (48)
a continuous linear operator L E
a(U, V)
and a function f from XX X into R satisfying (49)
I
(i) Vx E X, y t--+f(x,y ) is concave, (ii) V y E X,x +-f(x, y ) is lower semi-continuous, (iii) V y E X,f ( y , y ) =-s0.
We assume that (50)
X is a convex compact subset of U.
The Ky-Fan Theorem 7.1.3 implies that the subsets (51)
a,,,-(X; p ) = { X E X such that supyExIf(x, y)+(L*p, x-y)l
O}
are non-empty. Notice that, when U = V, L = Z and f = 0, a,,,(X; p ) = a ( X ; p). The Browder-Ky-Fan Theorem 1 can be extended as follows. Theorem 9. Suppose that assumptions (48), (49) and (50) holds. Let S be a COTrespondence from X into V which is upper hemi-continuous with non-empty closed convex images. Suppose moreover that (52)
vpc
u*,vx E a L , A x ; p ) , a*(s(x),p)
0.
Then there exists a critical point of S. Proof. This is the same as that of Theorem 1 except that the function a defined on X X X by (9) is replaced by the function
(53)
4x3 V ) = f ( x , Y ) -
n
C Bt(x) (L*Pi,x-Y).
is1
Property (10) becomes I E a,,,(X; p). Then o"(S(Z),p) a= 0 by assumption (52) and a*(S(Z),ji) < 0 by construction. Thus the non-existence of a critical point of S implies a coptradiction. c]
536
[Ch. 15, 5 15.1
FIXED POINT THEOREMS
Remark. An important example of functions f satisfying (49) is the function fdefined by f ( x , y ) = g(x)-g(y) where g is a lower semi-continuous convex function. We shall now relax the assumption that X is convex compact.
Theorem10. Let S be an upper hemi-continuouscorrespondencefrom a topological space X into a H a u s h t f locally convex vector space V , with non-empty closed convex images. Suppose that (54)
1
(i) either X is compact, (ii) or, more generally, that there exist P I , . .,pn E V* such that { x E X such that ... ,,d ( S ( x ) ) ,pi) e 0 } is compact.
.
Finally, we assume that (55)
there exists a continuous map C ,from V* supplied with { the Jinite topology into X such that sup,, ab(S(Ce(p), p) Qs
> 0,
s a.
V*
Then there exists a critical point of S.
then any critical point X of S is a solution to the variational inequalities suppc f (Z, p ) 0. But such a solution exists by Theorem 7.1.2. To see this, observe thatf is lower semi-continuous with respect to x(for Sis upper hemi-conmaxi,l,. , ,,"f ( x , tinuous),fis concave with respect to p and the function x pi) is lower semi-compact by assumption (54). Thus there exists a solution I to 1-4
~ i n fsup u b ( ~ ( ~ , ( p ) ) s, pi n) f e=-0
(by assumption (55)).
pEV*
E
= 0.
s-0
[7
In fact, when X is a subset of a reflexive Banach space, we can relax the compactness assumption (54).
Theorem 11. LetU and V be two reflexive Banach spaces, let X be a closed subset of U (supplied with the weak topology) andlet S be an upper hemi-continuouscorrespondence from X into V with non-empty closed convex images. Suppose that (57)
there exists M E &(V, U)such that u,,x{x-
MS(x)} is bounded
Ch 15, 0 15-21 and that (58)
QUASI-VARIATIONAL INEQUALITIES
537
V I Ew 0, there exists a continuous map C,from the unit ball B' ofV' into Xsuch that s u p p E ~ob(S(CE(p)), + p ) s E.
Then there exists a critical point of S.
Proof. Consider the function f defined on X X B * by (56). Any solution 2 E X to supPEv.f(~,p)e 0 is a critical point ,of S. To prove that such a solution exists, we can apply Theorem 13.1.2, which is a generalization of Theorem 7.1.2. However, we give a direct proof. The functions.Ldefined on B' X B* by &(q, p) =f(C;(q),p) satisfy the assumptions of the Ky-Fan Theorem 7.1.3. Hence there exists 4 E B* such that &(g, p) supqcB.&(q, q) G &(by (58)). Thus v X = infxCxsuppcB.f(x, p) 4 ~ ~ p , , ~ ~ . f p) s ( p=s, E for all E =- 0. Thus v+ = 0. Now, there exists a minimizing sequence of elements x, E X such that suppEB+ f ( x , p ) e l / n . By the Fenchel inequality, we obtain V q E B',
( 4 , xn) < f ( x , , M*qjfsup [(q,r)-f(y, M*q)!
; + sup 1
VEX
Y€X
sup (q, .y-Mv)
U€S(.Y)
(hy assumption (57)).
Hence the sequence of elements x,, is bounded, i.e. relatively compact in r/ for the weak topology. A generalized subsequence converges to an element Z E X which satisfies f ( Z , p) e lim infx,,f(xn, p) 0 since f is lower senii-continuous with respect to x. Therefore 5 is a critical point of S. 0
Remark. Taking M
= 0, assumptien (57) rcquises that Xis bounded, and thus
relatively compact for the weak topology.
Remark. In Theorem 1 1 ,we need assume only that 7 i is the dual of a barreled space. We can also extend Theorem 11 to the case when Y is a FrPchet space. 15.2. Qpasi-variational inequalities
We shall extend existence Theorem 9.3.1 for a solution 2 to.the quasi-variational ineqdities
538
FIXED POINT THEOREMS
[Ch. 15, 0 15.2
by replacing the assumption of lower semi-continuity of pl with respect to x by the assumption of pseudo-monotonicity introduced in Section 13.2. In particular, we can prove the existence of a solution 3 to (i) x' E S ( 3 , (ii) V y E S(f), (L(z),F - y )
4
0
when Xis a convex compact subset, L is monotone, bounded and finitely continuous map from Xinto U fand S a continuous correspondence from X into X. We can relax the latter requirement by using another fixed point theory. Assume for instance that L is linear and U-elliptic. Let T be the map associating with any x E U the solution T(x) E S(x) to the variational inequalities (***)
(L(T(x)),T ( x ) - y )
==
0 for any y E S(x).
Then any solution x to(**)is a fixed point of T.In most instances, U is a dense subset of a Hilbert space V , ordered by a closed convex cone P of positive elements satisfying P = { x € Vsuchthat(x,y)~OVVyEP}.
(For instance, V = L2(Q),P = I,:(@.) Define a sub-solution xb and a super-solution xx of T by
X6eT(Xb) and T ( x + ) s x * respectively. We shall prove that, when T is increasing, the existence of a sub-solution 2 and a super-solution xx satisfying xb == x* imply the existence of a fixed point 2 satisfying xb 4 2 =sxx. We next prove that the map T defined by(***) is increasing whenever L satisfies. @(SUP (x, O)), SUP ( -x, 0))) 0 and S satisfies the property If XI =S x2, Y I E S W , Y Z E S ( d then inf (YI,Y Z ) E S(xd and sup (YI,Y Z )E ~ ( x z ) . 15.2.1. Selection of fixed point by pseudo-monotonefunctions
We shall extend Theorem 9.3.1 by replacing the continuity assumption by .an assumption of pseudo-monotonicity (see Section 13.2).
Ch. 15, 0 15.21
QUASI-VARIATIONAL INEQUALITIES
539
Theorem 1. Suppose that (1)
X is a convex compact subset
that a function q~ : X X X
-+
R satisfies
(i) v y , x +-+q(x,y ) is lower semi-continuousfor the finite topqlogy and pseudo-monotone, (ii) V x ,y t--+ q(x, y ) is concave (iii) sup,,x q(y, Y ) 4 0 and that a correspondence S from X into X satisfies
(3)
S is upper semi-continuous with non-empty closed convex images.
W e assume also that thefunction q~ and the correspondence S are consistent in the sense that .4)
{ x E X such that
E(X)
= S U P ~ ~ S q(x, ( ~ ) y ) e 0) is closed.
Then there exists X E X Fch that
(i) 2 E S(2), (ii) S~PY€S(~, 94% Y ) == 0. Proof. This is analogous to the proof of Theorem 9.3.1. By assuming the conclusion false, we deduced that the compact set Xis contained in VOU U;,,V(pi), where Vo = { x such that a(x) =- 0} and V(p) = { x E X such that ( p , x)- o*(S(x),p ) 0). We introduced a continuous partition of unity @o, . . .,/?,} subordinate to this covering and the function a defined on XX X by
-
(6)
a(x ,Y) = Bo(x) ~ ( x Y)+ 7
n
C / ? i ( X ) (Pi, X T Y ) .
i=l
We continue by applying the Brkzis-Nirenberg-Stampacchia Theorem 13.2.1 to deduce the existence of X such that (7)
[The function a is pseudo-monqtone. Let a (compact) generalized sequence {x,} converge to ? and satisfy lim suppa (x,, Z) == 0. Since the functions Jo, . . . B, are continuous, we deduce that lim sup a(x,, P
x) = Bo(?)-lim sup q(xp,2)=s 0. /I
540
FIXED POINT T€BOREMS
[Ch. 15,§ 15.2
Hence either e0(X) = 0 or Po(X) > 0 and lim sup, cp(x,, 3)=s0. In this w e , the pseudomonotonicity of implies that V y E X, ~ ( 2y , ) 4 lim inf p(x,, 1)).
(8)
Id
In both cases, we deduce.that G@, y ) lini inf, u(x,,y) PO(^) lim inf,q(xp,y)+ -f p;(x,,> (P, x , A J We end the proof by contradicting (7)as in Theorem 9.3.1. 0
Z=l
W e now prove a result analogous to Theorem 9.3.2. TheoremZ. (Joly-lvlosco). Let X be cf>iivexand compuct. Suppose'that a map L. from x inio U ' satafgs
L is monotone,finitely continuous and bourided,
(9)
rmd that
S is cc cbntinuous correspondence from X into X with non-empty closed convex images.
(lei
Then there exxisrs a solutior: X of the quasi-vm'ational inequalities
(0 x' E S(3, (ii)
Proof. We apply Theorem I to the case when rp is defined by q ( x , y )=(L(x),x- y). By Proposition 13.2.5, y is pseudomonotone and finitely continuous with respect to x. Hence assumptions (1),(2)and(3) are satisfied. In order to apply Theorem 1 it remains to prove that property (4) holds. Let {x,) be a generaked sequence satisfying a(x,,),cg 0 and converging to 2. w e have to prove that a ( 3 6 0. we fix any y E S(rl). Since S is lower semi-continuous, there exists B generalized sequence CV,} of elements y, E S(x,) which converges to y E S(3. Since L is monotone, we deduce from the definition of a and the boundedness of L that I
(J%J, 4 i - Y ) 6( U X A s - u J + W ,), Y,,-Y> =s4 X J + IIL(-G)11, I I Y,-Y I I =G MIIYP-P 11.
(L(Y),X,-Y>
Letting x,, and y,, converge to x and y strongly, we deduce that (12)
vy
E S ( 3 , (L(y),2 - v )
6
0.
Ch. 15, Q 15.21
QUASI-VARIATIONAL INEQUALmES
541
Taking y = Z+e(z-X) E S(Z), we obtain that (13)
vz E s(q,
a),Z-
(L(x’+qZ-
.)0.
Hence, by letting 8’ go to 0, we infer that (14)
a(2) =
sup (L(?), Z - Z )
Z€S(x3
-s
0.
15.2.2. Fixed point theoremfor increasing maps
Let V be a Hilbert space (identified with its dual V’) andP a closed convex cone of V satisfying (15)
P = { x E V such that (x, y ) Z. 0 tiy E P}.
We shall use P as the cone of non negative eiements defining on‘ F a vmtcr ocdering (16)
x
a=y
if and only ifx-y F P.
Example. The main example is the case where V = Lys;):,, P = L;(Q). Property (15) is trivially satisfied. We can apply the Zorn lemma in V thanks to the following proposition. Proposition 1. Let V and P satisfy property (15). Any nonempry “cotnplete” subset M boundedfrom above (for the ordering) has an upper bound. Proof. Since M is completely ordered, we can write M in the f m n A4 = {.yl}icl where I is a completely ordered set and where i QI j implies that xj- xi E P. Let u E V be an upper hound: M c a-P. Hence the family of real nuisibers (xi, a-xJ is increasing and bounded. Hence, it converges: if j 3 i a ic, we have
= (tz-xJ)-t(xf-xfo) E P+P = P. This implies that sinre a-xi.-(xj-x,) I xj-xi l2 converges to 0, i.e., that {x,} W e Cauchy generalized sequence of Y. Since V is a Hilbert space, x, converges to 2. For any y t P, we hake {x, y) = lim, (x!, y) (xi, y ) fcr every j . Hence 2 B x, for any j E I. On the other hand, if z E Y bounds N above, i.e. if z Z- xj for al! j , then (2,7 ) lim, (xi,y ) =: (X,y ) for all y E P. Hence z a x’. We have proved that X is the upper bound of M. z
542
FIXED POINT THEOREMS
[Ch. 15,
0 15.2
Proposition 2 (Birkhoff-Tartar). Let V and P satisfy (Is) and let T be an increasing map from V into itself. Suppose that there exists a sub-solution xb and a super-solution x* of the equation T ( x ) = x defined by (18) xb=sx*, XbeT(Xb) and T ( x # ) e x * . Them the subset o f f x e d points 5 of T satisfving xb =s X G x* is non-empty and has a smallest and a largest element (for the ordering).
Proof. We write
X
= { x f V such that xb G x
4
x# and x e T(x)},
Y = { y e Vsuchthatxb=sy<x# and y Z= T ( y ) } , 2 = { x E Xsuch that x = s y for ally E Y}. Then 2 is non-empty (since xb E 2).By the Zorn lemma, there exists a maximal element X* E 2. By the very definition of 2,x* E X. Hznce 2 =sx* Q x+ and x* =s T(x*). Since T is increasing, xb s T(xb)B T(x*)4 T(x*) < X I and T(x*)=sT(T(x*)).This implies that T(x*)also belongs to.'A On the other hand, when y E Y, it satisfies X* s y . Thus T(x*) -s T ( y )4 y Hence T(x*)E 2. Since X* s.T(x*) and x* is maximal, this implies that x* = T(x*)is a fixed point of T. If f is any other fixed point between xb and x", it belongs to Y and thus, x* =G X by the definition of 2.Hence x* is the smallest fixed point of S. We can prove the existence of the largest fixed point of S by replacing 2 by 2 = { y E Y such that x =sy for all x E X}.0
15.2..9. Quasi-variational inequalitiesfor increasing correspondences Consider a Hilbert space V and a cone P c V satisfying (15) and another Hilbert space U contained and dense in V , supplied with a stronger topology. Denote by x !--- x+ the unique ortogonal projection of x onto P (see Section 2.2.1). Hence the map x F---(x)- = x - x + is the orthogonal projection onto -P for which any x E V can be split into the unique decomposition (19)
x = x + -x-
E P - P such that ( x + , x - ) = 0
We assume that there exists c =- 0 such that if x E U, x+ E U and (20)
Ilx+ llu
c IIx"(I.
We introduce a continuous linear operator L E -e(U, U*)which is U-elliptic, 1.e. (21)
there exists C =- 0 such that V x E U, (Lx, X) a-C I I x 11%.
Ch. 15,
15.21
543
QUASI-VARIATIONAL INEQUALITIES
We assume furthermore that L satisfies the property vx E
(22)
u,
(Lx+,x - )
0.
For simplicity, we use the notation (i) inf ( X I ,x2) = X I - (XI- x2)+ = x2(ii) sup (XI, x2) = x2+ (XI - X Z ) + = XI
(XZ-XI)+,
-+
(x2-
XI)+.
Finally, we introduce a correspondence S with non-empty closed convex values which satisfies the following monotonicity property. (24)
If XI =S X Z , y l E S(XI), Y Z E S(XZ),then inf (YI, Y Z )E S(xd and SUP ( Y I ,YZ)E S(x2).
We consider the following problem. Find % E V satisfying (25)
{ (ii)(i) VyE Z
S(Z),
E S ( 3 (Y%)-.f,2- y ) =G 0,
wheref is given in U*. By Theorem 9.2.2, there exists a unique solution x = T(z)of the variational inequalities (26)
{ (ii)(9
x E S(Z),
(L(x)--f, x- y)=s 0 for any y E S(z)
thanks to the assumption of Gellipticity of L. Thus, it is clear that a solution i of (25) is nothing other than ajixed point X = T(F)of the map T defined above. We shall say that A? is a sub-solutiov and xx a super-solution if we have (27)
Xb
respectively. "beorem 3 (Tartar). Let V , P and U satisfy (15) and (20). Suppose that L E E 2 ( U , U*) is U-elliptic and satisfies (22) and that the correspondence S from V into U with non-empty closed convex values satisjes property (24). v g and x x are respectively a sub-solution and a super-solution with xb =s x x , then the subset of solutions of (25)sathfying A? -s x 4 xx is non-empty and has a smallest and a largest element.
Proof. We apply the Birkhoff-Tartar Theorem 2 to the map T defined by (26). We have to check that T is increasing.
544
FIXED POINT THEOREMS
[Ch. 15, Q 15.2
Let X I x2 and let T( x d E S(x1) and T(x2) E S(x2)be the solutions of the variational inequalities
T(xd-yl) { (ii)( 9 ((L(T(xi))-J, L(TW)--f,
0 for any y1 E ~ ( x l ) , T ( x z ) - - Ys ~0) for any y 2 E s(x2).
(28)
Assumption (24)on S implies that Y1
= T(x1)- (T(x1)-T(x2))+ f S(Xl),
Y2
= T(X2)+(T(xl)-T(x2))+ E S(x2).
Taking such elements y l and y 2 in inequalities (28) and adding, we obtain that
(29)
(L[T(Xl)--W 2 ) 3 , (T(Xl)--T(X!d)+) 4 0.
Since
(T(xd--TT(xz))= (T(X1)- T(x2)),.- (T(x+-T(xz))we deduce that ( w U 1 ) -
Wz)).,,
(Wd- T(.Q))+j
=3s(L(T(x1)-T(xd)),(T(Xl)--T(Xd)+) .s 0
by assumption (22).
Hence, the U-ellipticity condi?ion (21) implies that
(T(xl)--T(xd)+ = 0,
i.e. that T(x1)-T(.x2)E -P.
Thus the map T is increasing. Example. Let I2 be a smooth open bounded subset of RR.We take
Assumptions (15) and (20) are satisfied. We take for the operator L the operator defined by "
c;
where uu(o) t,tja c II E j i2 and ao(w) c t,6 E R"and for almost all E 0. Hence .Lis U-elliptic. Furthermore, property (22) is satisfied since, in this w e
w
x+ = sup (x, O),
x- = sup ( - x , 0).
Ch. 15,s 15.31 UPPER AND
LOWER SEMI-CONTINUOUS CORRESPONDENCES
545
Finally, consider the case of two increasing functions MI and M2 mapping WQ). Then the correspondence S defined by
LZ(Q) into
(32)
S(X)= (y E H’(SZ) Such that MI(x) =Sy
Mz(x))
=S
satisfies the monotony conditions (24). 15.3. Other properties and examples of upper and lower semiantinoow c o r r e spondences
We have proved that the subdifferential af of a continuous convex function is upper semi-continuous. Some other examples of semi-continuous correspondences are given in this section. For instance, we have often used the correspondence S defined by S(y) = { x E
R such that L x E Y-y}
a(
where R and Y are closed subsets cones of Frgchet spaces U and V and Lf U,V). We shall prove that, if the constraint qualification assumption holds, i.e.
0 E Int (L(R)-Y), then the correspondence S is lower semi-continuomand closed (With non-empty closed convex values). We have also met correspondences S defined by S(y) = = {x E R such that V p E 9, y(x, p , y) 4 0) where y maps RXPX Y into R. We shall prove that such a correspondence S Iower semi-continuous and closed if, for instance,
1
(i) y is continuous, (ii) V y E Y , Vp E P , x t-+ y(x, y, p ) is convex, (iii) P is compact, (iv) V y E Y,3xy E R such that, Vp E P,y(xy,p,y) < 0.
We also prove an important theorem due to Michael: if S is a lower semicontinuous correspondence with non-empty closed convex images mapping a compact set Y into a Frechet space U,there exist a continuous selection s of S, i.e. a continuous map s such that V x E X,s(x) f S(x). Next we show how to supply the family @(U) of non-empty (weakly)closed convex subsets of C’ with a topology (the weak Hausdorff topology) in such a way that continuous correspondences from Y into U (supplied with the weak topology) are the continuous maps from Y into @(v). 37
546
FIXED POINT THEOREMS
[Ch. 15,
15.3
Finally, we prove that upper hemi-continuous correspondences with compact convex images are upper semi-continuous and similarly for lower semicontinuous correspondences. 15.3.1.Lower semi-continuity of preimages of linear operators
in most examples, we used in optimization and game theory the subset X of the form. (1)
X = { x E R such that Lx E Y }
where (i) U and V are Banach spaces, (ii) R c U and Y c V are closed convex subsets, (iii) L E &(U, V ) . This motivates the study of continuity properties of the correspondence S mapping V into U defined by
(3)
S ( y ) = { x E R such that Lx E Y - y } .
Theorem 1 (Robinson). Suppose that (2) holds. If the constraint qualifcation assumption (4)
0 E Int (L(R)-Y)
(i.e. 3 6 =- 0 such that S ( y ) # 0 when II y II =s 6 ) holds, then the correspondenceS has non-empty closed convex images and is closed. Furthermore, there exist y =-0, c z- 0 and d 0 such that if y E B,(y) and if x E R, there exists xu E S(y) such that
Hence S is lower semi-continuous at 0. Remark. In the case where R = U and Y = {0}, the constraint qualification assumption amounts to saying that L is surjective. Then Theorem 1 implies the famous open mapping theorem of Banach. Theorem 2 (Banach). Let L €2(U,V)be a surjective operator mapping a Frkchet space U onto a FrPchet space V . Then L is open (i.e. maps open subsets onto open subsets). If we assume furthermore that L is injective, then L-l E 2(V,U) is continuous.
Ch. 15, 5 15.31 UPPER A N D
LOWER SEMI-CONTINUOUS CORRESPONDENCES
547
Proof of Thmrem 2. In this case, Sol) = L-*). It is clear that the lower semicontinuity of S amounts to saying that L maps open subsets onto open subsets. Proof of Theorem 1. The proof is quite involved and so we introduce an intermediary lemma. We begin by indicating how the theorem follows from the lemma.
Lemma 1. Suppose that (2) and (4) hold. Then there exist a ball B J a ) of U and a ball B,(q) of V such that
(6)
~ ~ (c7~) r ~ ~nRI ( a y. )
First step: Deduction of Theorem 1 from Lemma 1. (a) We begin by observing that the constraint qualification assumption (4) implies that the images S(y )are non-empty when 11 y 11 -= 8. They are obviously convex and closed. (b) We prove that S is closed. Let {u,,x,} be a generalized sequence of point of the graph of S (i.e. such that Lx,+y, E Y) which converges to {y,x}. Since L is continuous and since R and Yare closed, we deduce that Lx+y E Y . (c) Let 8 =- 0 be such that B,(6) c L(R)-Y. We take y = min(6, $7). We fix x E R. If Lx E Y -y, we take xu = x. Otherwise, we choose z E Y-y such that I I Lx- z [ I = d,-,(Lx) = d,(Lx+y). We deduce from (6) that there exist u E B,(a)nRand e E Y - y such that -y(Lx+z)/(lILx+zII) = Lu-v because B,(y) c L(B& nR ) - Y+B,(y). We set 2 = (I lk - z II)/(r+ II Lx- ZI I) E lo, 11. Multiplying the latter equation by 1,we obtain.
L ( l u + ( l - 1 ) x ) = Lv+(l-;l)z E Y - y ,
since v and z belong to Y-y. We set xu = Au+(l-A)x, which belongs to R since both u and x belong to R. Hence xy E S(y).Finally, Ilx-xyll = IlW-u)ll .=s ~(ll~ll+llull) = = (dr(Lx+r))l(r+du(Lx+y)) (I1 xll+ II ull) z s (l/y)du(Lx+y)(IIxII+a)
since I I u II BE a. Hence (5) is proved. (d) We prove that S is lower semi-continuous at 0. If x E S(0) and E > 0 are fixed, we choose e = min(y, e/c(d+ 11 XI[). I f y E B,(e) c B,(y), we deduce from ( 5 ) that there existsx,, E S(y)such that IIx-x,II =scdY(Lx+y)(d+IlxII) ~ c e ( d + I I x l l ).S E. 37f
548
[Ch. 15,§ 15.3
FIXED POINT THEOREMS
Second step Proof of Lemma I .
(a) We be&
by proving that there exist q
=- 0 and u > 0 such that
mV(7) c L[B~(u) nRI- Y.
(7)
Denote by F the convex pet (8)
u
Fn = L[Ba(m)n R]- Y .
Because T = l B U ( n ~n)R = R, it follows that u;=lFn 2 L(R)- Y has a nonempty interior thanks to assumption (4). We deduce from the Baire theorem (1) that one of the sets' F has a non-empty interior. Let u E Int F,. Since 0 E Int(L(R)- Y),there exist a =- 0, x E R and y E Y such that -au = Lx-y E L(BA1I xll)nR)-Y. We set A = l/(a+l). So, 0 = (l-A)u+il(-au) belongs to the interior of Fa,where a = (1 -A)n+ +ill1 x 11. Hence there exists q =- 0 such that (7) holds. (b) We now prove that the conclusion (5) of Lemma 1 holds. We deduce from (7) that, for any integer n, (9)
md~n) c 2-nFa,
where x), = 2 3 . Let y be any element of Bv(q). Since 2 y f 2 B,(q) c Fa,we can find zo E Fa such that 2y- zo E BV(q).Assume that we have constructed a sequence z k ( 0 -z -G K 6n- 1) such that
Since zk E Fa = L[B,(u) nR]- Y , there exist that z k = J!&)-yk. Consider the sequence (13)
un =
f
xk
E B,(u)
nR
and yk E Y such
2-&Xk.
k-0
The Bake theorem states that in a complete metric space V,if a denumerable union of closed subsets F. has a non-empty interior, then one of the F,'s has a non-empty interior.
Ch. 1 5 , § 15.31 UPPER
AND LQWER SEMI-CONTINUOUS CORRESPONDENCBS
549
It is clear that {u,} is a Cauchy sequence of U since
1
c m
1/
~ ~ u n -=u ~ ~ 2-kxk ~ e+ k=n
rn
2-kIIXkl( 6 :
k=n
c 2-&& k=n m
Therefore, since U is complete, this Cauchy sequence {un} converges to u, in
U.
Since B,(a) c R is closed and convex, then
and thus,
Therefore, because
( it follows that lim + (16)
lim y - k n + +
n---
n-c-
c 2-kyk 1 = 0, n
k-0
2-kvk = ~ ( u , ) - y . k-0
Since Y is closed and convex, we deduce in the same way that v, = lim n+O0
+
n
2-kyk
k-0
belongs to Y and that y = L(u,)-
V,
E L[Bu(u)nRl- Y.
Thus (6) holds. 0 Remark. In the case when R and Yare cones, we can restate the theorem in the following way. Theorem 3. Suppose that U and V are both Banuch spaces and that R and Y cue closed convex cones sati+fying L(R)- Y = V. Then there exists a constunt M such that
(17)
Vy E
V,3 x E S(y ) such that
and thus, S is lower semi-continuous on V .
llxll
4
MllylI
550
[Ch. 15, $ 15.3
FIXED POINT THEOREMS
Proof. By Lemma 1, we know that B,(q) c L[B,(u)n R ] - Y . Let y E V . Then -q(y)/lly 11 belongs to B,(q) and thus, there exists P E: Bu(a)n R such that -3 6 , l l ~ l l ) E L2-Y. Then x = (Ily]l/q)Z belongs to S(y) and l l x l l = = (llYll/V) llnll == (l[yll/q)a.Therefore, we have proved(17)withM=a/q.O 15.3.2. Lower semi-continuity of correspondences defined by constraints
We consider now another correspondence S we used in optimization theory (see Section 5.1), defined by (18) where
S(y) = { x E R such that V p E P , y(x, p , y )
0)
(i) R is a closed convex subset of a topological vector space U, (ii) P is a closed convex subset of a topological vector space V*, (iii) Y is a'topological space, (iv) y maps R X P X Y into R.
(19)
Proposition 1. Let assume that V p E P, that function { x , y } F+ y(x, p , y ) is lower semi-continuous. (20) Then the correspondence 5' defined by (19)is closed. Proof. The graph of S is the set of pairs { x , y } such that y(x, p, y ) == 0 for all p E P. It is obviously closed when (20) holds. 0
Theorem 4. Suppose that (i) V p E P,y E Y , x y(x, p , v) is convex, (21) (ii) V x E R, ( p ,y} t-+ y(x, p , y ) is upper semi-continuous and thar (i) P is compact, (22) (ii) V y E Y, we can associate x, E R such that Y(Xy,P,Y) -=z 0 V P E p .
{
1-4
I
Then the correspondence S defined by (19) is lower semi-continuous.
Proof. The proof follows from the following two lemmas.
Lemma 2. Given the assumptions of Theorem 4, VXO E S(y0) and VO E 10, 11, there exists a neighborhood &(yo) (23) such that y ( O x y o + ( l - ~ ) x o , p ,Y) < 0 V p E P,VY E Ne(yo).
Ch. 15,s 15-31
UPPER AND LOWER SEMI-CONTINUOUS CORRESPONDENCES
551
Proof. Write (24)
- co = SUP Y(XY0, P, Yo). PEP
The number co is strictly positive by assumption (22). Let x = 8 xu,+(' -8)xo where xo E Solo). Since y is convex with respect to x, we deduce that (25)
Y(X,P , Yo) 4 eY(X.v0, P, YO)+ (1 - 8 ) Y(X0,P, Yo) -ec0 E P.
vp
Since y is upper semi-continuous With respect to (p, y } , we can associate with of p and yo such that any p E P open neighborhoods N(p) and N,(~o) (26)
Y(X, 4, Y )
Y(X,
P,Y O ) +
E =G-
8co+
E <
0 vq f N P ) , Y E NP(Y0)
(where we take E -= 8co). Now, since P is compact, it can be covered by a finite union Uy.=lN(p,) of neighborhoodsN(p,). If we set
nN,,(Y~), n
wY0)=
1-1
we deduce from (26) that y(x, q, y ) over P. 0
-= 0 whenever y
E ZVBcyo) and q ranges
Lemma 3. Suppose that (25) and property (23) of Lemma 5 hold. Then the correspondence S defined by (18) is lower semi-continuous.
Proof. Let xo E S(y0) and let N ( x 0 ) be a neighborhood of XO. Let xu, be the element associated with yo by (22(ii)). Then x = 8x,,o+(1-8)xo = xo+ + O(xyo-xo) belongs to N(xo)when 8 =- 0 is small enough. Thus, assumption (23) implies the existence of a neighborhood N,(yo) of yo such that y(x, p , y ) = = y(8x,,,+(l-f3)xo, p , y ) 0 for all p E P and y E No(y). This implies that x E S(y)n N ( x o )whenever y E N & ~ o )0 . 15.3.3. Continuous selection theorem
Let S be a correspondence mapping a topological space Y into a topological space U.A selection s of S is a map from Y into U satisfying (27)
VY E y ,
S(Y)
E S(Y).
The problem we shall solve is the following: Is there a continuous selection of a lower semi-continuous correspondence? Under convenient hsumptions, the answer is positive.
552
FIXED POINT THEOREMS
[Ch. 15, 4 15.3
Theorem 5 (Michael). Suppose that (28)
space { (ii)(i) YU isis aa compact Frkchet space
and that (29)
S is a lower semi-continuow correspondence with non-empty closed convex images S(y).
Then there exists a continuous selection s of S . Proof. Consider an increasing sequence P1 =s Pz e . . . =z P, e . . . of seminorms P, defining the topology of the Frechet space U.We use the decreasing subsequence of neighborhoods B, of 0 defined by B, = { x E U such that P,(x) -c 2-Cn+l)1. We construct by induction a sequence of continuous functions s, : Y satisfying
++ U
(Hl) S 1 W E S(Y)+Bl and, for n
2,
(Hn)
{ (0
sn(y) E S(y)+Bn,
(ii) s n b ) E
sn-
&)+ 2Bn-
1.
Once these functions are constructed, we shall prove that they converge to a continuous function s satisfying
(27)
4 Y ) E S(Y) V Y E‘Y.
First step: construction of s1. Let m :Y ++ U be any (not necessarily continuous) selection of the correspondence S. Since S is lower semi-continuous, there exists an open neighborhood Nl(y) of y such that vz E NlW,
(al(Y)+Bl)rlS(z)f 0.
This can be rewritten as (30)
v z E N1(y),
4 9 E S(z)+B1.
Since Y is compact there exists a continuous “partitition of unity” 48: dinate to a covering { N l M ) } by kl open neighborhoods Nl(yf).
subor-
Ch. 15,§ 15.31
UPPER AND LOWER SEMI-CONTINUOUS CORRESPONDENCES
We set s1(4 =
553
c B:(4 4 Y I ) . ki
i=1
This is a continuousfunction. Let z be fixed. By (30), s1(z)
=
c ki
B X Z ) Ol(YI)
i=l
E
5
i=l
Bf(z) (S(z)+B1)
c S(z)+B1 since S(z)+B1 is a convex subset and B;(z) tion 31 satisfies property (HI).
0,
cfxl
@;(z)= 1. Thus the func-
Second step: construction of Suppose that the continuous functions s, satisfying (Hk) are constructed for 2 =sk s n. We construct s,,+~ satisfying (H,+&. For this purpose, we choose u,(y) satisfying u I t wE M Y )
+~~1n S(Y),
this set being non-empty by (H,,(i)). Since S is lower semi-continuous, there exists an open neighborhood NLyl(y) of y such that v z E NATl(Y), an(y)
E S(z)+B,+1.
Since s, is continuous, there exists an open neighborhood N2:l(y) of y such that v z E NAy1(y), s”z) E sn(y)+Bn. Therefore,>f N,,+,(y) = NAy,(y) nNFjl(y), we obtain that
Let &’+’ be a partitition of unity subordinate to a finite covering {Nn+l(y;+l)}of the compact set Y . Write
Then s,,+~is a continuous function. Let z E Y be fixed. We obtain that
+
c S(z)+Bn+1
since S(z) B,+ is . convex.
554
[Ch. 15, 0 15.3
FIXED POINT THEOREMS
On the other hand, by (31) and (Hn(ii)), we have that sn+ 1 ( z )
E k ~ ’ ~ : + ~( s( nz( y)T + l ) + B n ) i=l
Therefore, we have proved that (H,,,) tive construction of the sequence {s,}.
is satisfied. This completes the induc-
Third step: construction of s. The sequence {sn} is a Cauchy sequence in the pasce Q(Y,v) of continuous functions, supplied with the topology of uniform convergence, because, for every n and every z E Y , Pn(Sn+k+ l(z)- s n ( z ) )
c
=
k
= Pn(
c
j= 0
k
s
c k
sn+j+l(z)-Sn+j(z))
pn+j(Sn+j+l(z)-sn+j(z))
j=o
=s
Pn(Sn+j+l(Z)-Sn+j(z))
j=O
G2
2 k
2-(n+i+l).
j= 0
Since Uis complete, the space @(Y,v) is also complete. Thus the sequence sn converges to a continuous function s E @(Y, U).Since sn(z)E S(z)+& we deduce that s(z) E S(z) = S(z) for any z (since S(z) is closed by assumption). Remark. The theorem remains true if we replace the assumption “Y is compact” by the weaker assumption “Y is paracompact”. We used the compactness assumption only to construct continuous partition of unity associated with a covering {Nn+l(y)}of Y.But “paracompact” spaces are (by definition)the topological spaces in which we can associate a continuous partitition of unity with such coverings. Metric spaces are paracompact. Application: Surjectivity of lower semi-continuous correspondences. As an application, we deduce the following surjectivity theorem from the above selection theorem (and Proposition 1.4).
Theorem 6. Suppose X is a convex compact subset of U.If U is a Fr6chet space and S is lower semi-continuousfrom X into itself and satisjes either the condition
V P E U*¶
W ( x ;PI1 = a(x; PI
or the condition
VP E
u*, s[a(x;P)I c q x ; -P)¶
then S maps X onto itself.
Ch. 15,s 15.31
UPPER AND LOWER SEMI-CONTINUOUS CORRESPONDENCES
555
15.3.4. Weak Hausdorfl topology on thefamily of closed subsets of topological vector spaces Since a correspondence S with non-empty closed values from Y into a locally convex Hausdofl space can be regarded as a map from Y into the family
(32)
q(v)of non-empty closed subsets of U,
the question arises as to whether it is possible to endow @(v)with a topology for which continuous maps are nothing other than continuous correspondences. We have already met such a topology in Section 13.3.7.
Suppose that U is supplied with the weak topology a(U, U’),defined by the seminorms
when K ranges over the family 8 of finite subsets f U*(where cos (K)denotes the symmetric convex hull of K). We thus define on q ( U ) the “semi-distances” (34)
dir(A, B) = max ( ~ K B), (A
4)
where we set
(35)
&(A, B) = sup inf Y€A
x€B
PK(~-X).
Proposition 2. The functions d,(A, B) are semi-distances. The family @(v> of closed subsets of U supplied with the topology defined by the semi-distances &(A, B) when K ranges over the finite subsets of U* is a Hausdorff space. Definition 1. We shall say that this topology is the “weak-Hausdorfl topology”. Proof. By the very definition, the functions d,(A, B) are symmetric. If A , B , C are three subsets, 6,(A, B) =z 8,(A,C)+ 6,(C, B). To see this, let y belong to A , x to B and z to C . Sincep,(y-x) s pK(y-z)+pK(z-x), we deduce that
Taking the infimum over C,we deduce that
556
FIXED POINT THEOREMS
[Ch. 15,s 15.3
Finally, taking the supremum over A, we deduce that &(A ,B) e 8,(A, C)+ +aK(C,B). To provethat(7(U)is a Hausdorff space, we have to prove that, if d,(A, B) = 0 for all K’s, then A = B. Let y E A. Since 6g(A, B) = 0 for any K,
then inf,,, pK(y-x) = 0 for any K. This implies that y belongs to the closure of B. Then A c fi = B, since B is assumed to be closed. We show in the same w a y t h a t B c A =A.ThusA =B.O
Remark. It is clear that the two following statements are equivalent: (a) A c B+B&), (b) VY E A, ( Y + B K ( E ) ) ~#B 0, where we set B&) (37)
= {x
E U such that p&)
.sE).
We deduce the following result.
Proposition 3. I f A c B+B,(E), then 6,(A,B) then A c B +Bx(2E).
E.
Conversely, ifd,(A,B)
-s
E,
ProOz. This is left as an exercise. 0 Proposition 4. Let S be a correspondence with non-empty closed values mapping a topological space Y into a locally convex Hausdorf space U.If S is upper semi-continuous at yo, then
(38)
For any j n i t e set K , for arly E =- 0, there exists a neighborhood N(yo) of yo such that d,(S(y), S(y0)) =z E whenever y E NCyo).
The converse is true ifS(y0) is compact.
Proof. If S is upper semi-continuous at yo, then, for any finite subset K and any E =- 0, there exists a neighborhood N(y0) of y o such that
(39)
VY E NYO),
m9 c S(YO)+BK(E).
This implies that 6,(S(y), S(y0)) 4 E whenever y E N(y0). Conversely, if 6,(S(y), S(y0))G $ E whenever y E Ndyo), we deduce from Proposition 3 that (39) holds. Thus, S is upper semi-continuous at yo, since any neighborhood N of the compact set S(y0) contains a subset S(yo)+BK(c).0
Ch. 15, 9 15.31 UPPER
A N D LOWER SEMI-CONTINUOUS CORRESPONDENCES
557
Proposition 5. Let S be a correspondence with non-empty closed images mapping a topological space Y into a locally convex space U.
If
(40)
For any finite subset K, V E =- 0, the& exists a neighborhood N(y0) of yo such that &(S(yo), S(y)) Q E when y E N(Yo),
then
(41)
S is lower semi-continuous at yo.
The converse is true if we also assume that (42)
S(y0) is compact.
Proof. If &(S(YO), S(y)) =S +E when y E N(yo), then S(y0) c S(y)+BK(c) (by Proposition 4). Thus, V x E S(yo),(x+BK(&))r l S(y) = 0 when y E NCyo). This implies that S is lower semi-continous at yo. Conversely, suppose that S is lower semi-continuous at yo and that S(y0) is compact. Then we can cover &'(yo) with a finite number of open neighborhoods xi+BK(&)where xi E S(yo), i.e.
Since S is lower semi-continuous, there exist n neighborhoods N,(yo) of yo such that (44)
y E N,,(yo) implies that S(y) n(xi+ BK(E))f 0.
Let N(y0) = n ; = l N q b o ) . Then any x E S(y0) belongs to a set xI+BK(e). Furthermore, for any y E N(yo), we know that xi E S(y)+B&). Thus x E E x,+B&) c S ( ~ ) + ~ B , (when E ) y E N(y0). Therefore,S(y~)c S ( ~ ) + ~ B , ( E ) , i.e. 8,(S(y0), Sb))4 2~ when y EN(yo). 0 15.3.5. Relations between hemi-continuity and semi-continuity
Let X ( U ) denote the family of non-empty closed convex subsets. In this case, we have a characterization of the Hausdorff semidistances. Proposition 6. The following inequality is always true:
558
[Ch. 15, Q 15.3
FIXED POINT THEOREMS
Proof. We can write
= sup sup inf ( p , y-x). P€X
v€A x€B
Since (47)
sup inf ( p , y- x ) G inf sup ( p , y-x), P€K
x€B P€X
XEB
we deduce that a =ssup inf sup ( p , y- x) = sup inf p&YEA
x€B p€K
Y€A
xEB
x) = &-(A, B).
If we assume that B is convex, then inequality (47) is actually an identity by the minisup Theorem 7.1.5, since we can replace K by its convex hull, which is compact. In this case, we have
This characterization of the semi-distances implies a Characterization of upper and lower semi-continuous correspondences.
Theorem 7 (Castaing). Suppose that S has non-empty closed convex images. It is upper hemi-continuous whenever it is upper semi-continuous. The converse is true whenever (48)
the images S(y) are compact.
Proof. The first statement has already been proved (Proposition 2.5.1). W e prove the converse in the case of finite dimensional space only. Associate with any pair {p, E } E U*X 10, [, the subset (49)
Fp. e
= {Y E
U such that ( p , y ) =G~'(S(XO),p)+ &}
and with any finite sequence J of pairs { p , E } the subset (50)
FJ=
n F~,~.
Iple)€J
Ch. 15, 9 15.31
UPPER AND LOWER SEMI-CONTINUOUS CORRESPONDENCES
559
To say that S is upper hemi-continuous at xo amounts to saying that we can associate with any finite sequence J a neighborhood Nj(x0) such that (51)
~(NJ(xo c) F) J .
Now, let N be any fixed open neighborhood of S(x0). By (51), it is enough to prove that for some J, F, c N. Let G = 5 be the complement of N and G, = G n F p Suppose that V J, F, is not contained in N,i.e. that 'd J, G , # 0. The subsets G, have the finite intersection property (since G, n G , f, G,",). Furthermore, there exists a finite sequence JOfor which FJ, is compact. Therefore, there exists y € G,. Since y € G and since S(XO)c N,then y 6 S(XO),i.e. there exists p E U*such that (p, y ) =- a*(S(xo), p ) . This is a contradiction since y E F, where J = JoU {p, E } with E -= ( p , y)-a#(S(x~,),p).. Hence
nc,cc,o
(52)
there exists J such that S(N,(xo)) c F j c N. 0
Theorem 8 (Valadier). Let S be a correspondence with non-empty convex corn-pact imagesfrom Y into a locally convex space U. If (53)
S is lower semi-continuous at yo
then (54)
V p E V*, thefunction y ~-+a*(S(y),p ) islowersemi-continuousatyo.
The converse is true if we also assume that
(55)
the images S(y) stay in a compact subset C .
Proof. If S is lower semi-continuous at yo into U supplied with the weak topology, we deduce from Propositions 6 and 7 that for any finite subset K c U and any E =- 0,
whenever y ranges over a neighborhood N(y0) of yo. Conversely, Suppose that 5' is not lower semi-continuous at yo, i.e:that there exist E =- 0 and a neighborhood xo+K* (where K is the convex symmetric hull of a finite subset) of xo E S(yo) such that for any neighborhood N(y0) o f yo, there exists yN < N(y0) such that E
(57)
GK(S(YO), S(YN))= max Ea*(S(uo);p)-a#(s(vlv); P)I PEK
= aY(S(y0);P N ) - ~ # ( W N ) ; P N )
560
[Ch. 15,§ 15.3
FIXED POINT THEOREMS
the maximum is achieved at pN E K since K is compact and the support functions u"(S(y), p) are continuous with respect to p). Since K is compact, a subsequence (also denoted by) pN converges to p' E K We shall prove that y u"(S(y); p') is not lower semicontinuous, i.e. that ~4
&
(58)
- e u*(S(yo);p')--*(SCyN);
2
p')
for N large enough.
We have that, U'(s(y0); pN)
U'(S(y0); pN-p')fU'(SbO);
=S
p').
Let C denote the compact set containing images S(y). Then ux(S(yN);p')
=S C * ( s ( y N ) ;
p8-pN)40"(S(yN); pN)
=so.*(C;~ * - ~ N ) + u ' ( ~ ( Y N ) ; P N ) .
Therefore, (59)
E
"U*(S(yO); p N ) - U x ( S ( y N ) ; pN) =S U*(S(yO);P')-ur(S(yN); p*)
U*(S(yO),pN-p*)
+U*(C, p*-pN)-
Now, since K is an equicontinuous subset of U*,the weak topology coincides on K with the topology of uniform convergence over the compact subsets. Therefore, since S(y0) and C are compact, (60)
a
~ # ( s ( y o ) , p ~ - p * ) + ~ * ( C , p * - pe~ ) E for p~ close enough to p'.
Hence (58) holds and the proposition is proved. 0
APPENDIX A
SUMMARY OF LINEAR FUNCTIONAL ANALYSIS
In this appendix we summarize the main results of linear functional analysis which are used in this book. We recall (without proof) the statements of the results known as the HahnBanach theorems. We proceed by proving that if two spaces U and F a r e paired, then F is the dual of U and U the dual of F when they are supplied with the weak topologies .r(U, F) and a(F, U). We use this result to prove that the space of discrete measures is the dual of the space of all functions. We then study the topologies of uniform convergence over subsets of a covering defined on spaces of functions and give sufficient conditions implying that the duals coincide. We employ these results to construct stronger topologies on U and F, using the fact that U and F are spaces of continuous linear forms on F and U respectively. In particular, we prove the Mackey theorem stating that the duals of U supplied with the Mackey topology and the weak topology coincide. Finally, we show that when U is a Hausdorff locally convex vector space and F = U*,then U sis also the dual of U supplied with the weak and the Mackey topology. Furthermore, we prove the Banach-Steinhaus theorem and deduce that the initial and the strong topologies coincide when U is barreled. Recall that we have proved the open mapping principle in Section 15.3 (see Theorem 15.3.2) and that other results of functional analysis are scattered throughout the book (the study of gauges and support functions in Section 1.4, polarity in Section 1.4, Hilbert spaces in Sections 2.2 and 2.3 and the continuity of convex functions in Sections 3.3 and 3.4).
1. Hahn-Banach theorems
Several connected results are known under the name “Hahn-Banach theorem”. We recall several of these statements. (a) If U is a Hilbert space, it is easy to deduce from the projection theorem (see Proposition 2.2.1) the following result. 38
561
562
APPENDIX A
Theorem 1. If K is a non-empty closed convex subset of a Hilbert space and i f x 4 K, then there existsp E U*,p # 0, such that 8 ( K , p ) -= ( p , x). (b) If U = Rn is a finite dimensional space. Theorem 1 implies the following separation theorem. Theorem 2. If K is a non-empty convex subset of a finite-dimensional space and i f x 6 K, then there exists p C Rn*,p # 0 such that o'(K, p ) =G ( p , x). (The proof of Theorem 2 uses the fact that the unit sphere is compact.) Note that the minisup theorems of Chapters 7 and 13 used only Theorem 2 and not the more sophisticated results we are about to state. (c) Case of Hausdorf locally convex spaces. The separation theorems follow from the following extension theorems.
-
Theorem 3. Let U be a vector space and let p be a positively homogeneous convex function defined on U.Let M be a vector subspace of U and f :M R be a linear form satisfying f ( x ) p(x) for all x E M . Then there exists a linear form f : U R defined on U which extends f and which satisfies f ( x ) =sp(x) for ail x E U . --t
This result depends upon Zorn's lemma. Notice that we did not assume that U is endowed with a topology. When U is a Hausdorff locally convex space, Theorem 3 implies the following.
Theorem 4. Let M be a vector subspace of a Hausdog locally convex space U . Let f be a continuous linear form dejked on M. Then there exists a continuous linear form defined on U which extendsf. Theorem 3 also implies the following separation theorem.
Theorem 5. Suppose that K is a non-empty open convex subset of a Hausdorfs locally convex space U. If x 6 K, then there exists p E U*, p f 0, such that a'(K; PI =S ( p , x). Finally, Theorem 5 implies the following generalization of Theorem 1. Theorem 1 bis. Let K be a non-empty closed convex subset of a Hausdorf locally ' such that o'(K; p ) < ( p , x). convex space U. Zfx 6 K, there exists p C U 2. Paired spaces
We begin by showing that the dual U* is not (0).
LINEAR FUNCTIONAL ANALYSIS
563
Proposition 1. Let U be an Hausdorfl locally convex vector space. Then there exist non-zero continuous linear fonns. Furthermore, the bilinear form { p , x ) E U*X u p(x) IS not degenerate.
-
-.
Proof. Let xo # 0. Let M = Rxo be the one-dimensional vector space spanned by xo and let f : M R be the continuous linear form defined byf(Ax0) = 1. Then Theorem 4 implies the existence of continuous linear form p on U which extendsf and is not equal to 0 (since ~ ( x o=f ) ( x o ) = 1). Now,if p(x) = 0 for all x , then p = 0 by the very definition of a zero form. On the other hand, the equations p(x0) = 0 for all p E U* imply that xo = 0. (If not, if xo # 0, there would exist p E U* such that ~ ( x o=) 1 # 0.) 0 Now, let F be a vector space isomorphic to V'. Then there exists a bijective linear operator j from (I*onto F. Write (p, x ) =j - l ( p ) (x). Proposition I implies that (1)
the bilinear form {p, x } E F X U
-
( p , x ) is not degenerate.
It is remarkable that (1) is also a sufficient condition for F to be isomorphic to U*.To see this, associate with a bilinear form (., -) on F X U the weak topoIogies a(F, U)and a(U, F) on F and U defined by the families of seminorms (2)
PM(P>
= SUP I(P, x>l, x€M
=u
and (3) obtained when Mand Krange over the families S of finitesubsets of U and F. Denote by Fa and V, the vector spaces F and U supplied with the weak topologies a(F, U) and a(U, F). Theorem 6. Let { p ,x } ( p , x ) be a non-degenerate bilinearform on theproducz FX U of Zwo vecrorxpacees.Thenthere exhzx a b&kc/ivehew operarorjmqpzhg F onto the dual (Val*of U, and a bgective linear operator k mapping U onio the dual (F,)* of F,. +
Proof. It suffices to prove the first statement. We define j from F into (UJ* by (4)
V p E F, j p :x * j p ( x ) = ( p , x).
It is clear that j is linear. It is injective because, if if j ( p ) = 0, then jp(x) = = ( p , x) = 0 for all x E U.Thus, p = Osince the duality pairing is non-degefi38.
564
APPENDIX A
erate. It remains to prove that j is surjective. Let f : x f( x ) be a continuous linear form on V . We prove that it can be written f ( x ) = (p, x ) wherep E F. Sincef is continuous, there exists a semi-norm pK associated with a finite subset K = {pl, . ..,p,} c F such that -t
If(x)l
=z c p ~ ( x )where c
=- 0.
Therefore the equalities ( p i , x ) = 0 imply thatf ( x ) = 0. Introduce the linear ~ , and its image operator L from U onto R" defined by L x = {(pi,x ) } ~ =..,,,, T = L ( U ) c R". We define the linear form g on T by g(t) =f ( x j whenever t = Lx. It is clear that g(t) does not depend upon the choice of the solution x of L x = t, since f ( x ) = 0 whenever Lx = 0. Since T c R",g can be writAiti. Therefore, we have proved that ten g(t) =
Eel
f ( x ) = g(Lx) = wherep = C;=l A%i.
n
C Ai(Pi, X )
1x1
= ( P , X)
This result allows us to prove the following theorem.
Theorem 7 . The space S*(X)of discrete measures on a set X is (isomorphic to) the dual of the space S ( X )= Rx of functions definedon X , supplied with the topology of pointwise convergence, defined by thefamily of semi-norms (5)
P L V ) = SUP I f ( x ) l X € L
obtained when L ranges over the family 8 ofJnite subsets of X .
cy=l
Proof. It is clear that the bilinear form defined by (m, f) = a'f(xi) is non-degenerate. On the other hand, it is easy to check that the Lemi-norms pK (see (5)) of the topology of pointwise convergence and the semi-norms p K (see (3)) of the weak topology a(8(X), a * ( X ) ) define the same topology. Hence S * ( X ) is isomorphic to S ( X ) * by Theorem 6. 0 3. Topologies of uniform convergence More generally, let d! be a covering of the set X satisfying (6) if K and L belong to of, then K U L belongs to d. We denote by d ( X ) the vector space of functions defined on X which are bounded on every subset L in the covering d.We can then define the seminorms pL( .) by (5) as L ranges over &.We shall supply & ( X ) with the family
LINEAR FUNCTIONAL ANALYSIS
565
of semi-norms pL with L ranging over d.This topology is called the "topoiogy of uniform convergence over the subsets of d". The space d(X) is complete. Let (24 be the family of all subsets of X. Then a ( X ) is the Banach space of bounded functions on A'. Clearly, we have M ( X ) c d ( X ) c S(X), the canonical injections being continuous. Since U ( X ) is dense in S(X),any space &(X) is dense in S(X). Since the canonical injection j from d(X)into S(X)has a dense image and is injective, its transpose j* is injective and has a dense image. W e will identry j * with the canonical injection from S*(X)into d * ( X ) . Now, let %(X) c d(X)be a subspace of functions. We shall denote by Z a ( X )this space supplied with the topology induced by d,called the topology of uniformconvergence over the subsets of of.In particular, any space % ( X ) of fundions on X Can be supplied with the topology of pointwise convergence. It will be denoted by %,,(A'). The transpose j * E &(%,'(A'), %,(X)) of the identity map j (which is continuous from %;,(X) into %,,(x>), is injective with a dense image. Therefore, we will ia'entifv %:(X) with a dense subspace of %:(X). The question arises as to whether the duaIs Z z ( X ) and %:(A') coincide. For this purpose, we denote by Y :X %:(A') the map defined by
-.
(6)
Wx)
: f E W X ) * (w4,f) =m>.
[If %(X) = d ( X ) , Y is the Dirac operator. Otherwise, Y(x) is the restriction of the Dirac measure 8(x) to Z ( x ) . ] Theorem 8. Let %(X) be a subspace of d ( X ) . Suppose that
(7)
VL E d, Y ( L )is a compact subset of %:(X)
when %:(X) is supplied with the weak-topology u(%z(X), %,(X)). Then
%,'(x> = %:(x).
Proof. We have to prove that any POE %z(X) belongs to %:(XI. Since is continuous on %=(A/), there exists a semi-norm pL such that (8)
PO
I PO,^) I PLU)for anyf E %a(x).
But we can write, by (6),
where COs Y ( L )is the closed convex symmetric hull of Y ( L )in %t(X). Since Y(L)is compact by assumption, c&% ( Y ( L ) is ) compact on %z(X) and thus, is also a compact subset of %:(A').
566
APPENDIX A
1 (i.e. if Now, inequalities (8) and (9) show that if sup#Ec3 yp(L)I (p,f)l f E [ z s Y ( L ) ] * ) ,then (PO,f) G 1. Therefore po E [cos Y(L)]*#.Since S s Y ( L )contains 0 and is convex and closed in %I(X) (because Gs Y(x) is compact in %;,*(X)), we deduce that [Zs Y(L)]** = Gs !P(L). Hence p o E cos Y ( L ) c aeZ(X).
Corollary 1. Let X be a topologicalspace. Let @,(X) and @,(X) be the spaces oJ continuousfunctions on X supplied with the topology of uniform convergence over compact subsets and the topology of pointwise convergence respectively. Then
q ( x )= qx).
Proof. It is clear that the operator Y from X into @,*(X)is continuous (since, if x p converges to x, then, for any f E @(A'), (!P(x#), f ) =f(x,) converges to ( Y ( x ) , f ) =f ( x ) , for f is continuous). Then Y(L) is compact for any compact subset L of X.Assumption ( 5 ) is satisfied when we take the covering of X by all the compact subsets. 0 Corollary 2. Let X be a compact space. Let @,(A') and @,(X) be the spaces of' continuousfunctions supplied with the topology of uniform convergence and the topology of pointwise convergence respectively. Then @,(' X ) = @;(A'). 4. Topologies associated with a duality pairing
Consider two paired spaces U and F. By Theorem 6, we can identify U with the space &(F) of continuous linear forms on F supplied with the weak topology u(F, U).Therefore, since any continuous linear form is bounded over bounded subsets, U = B(F)c &(F) whenever 04 is a covering formed by bounded subsets of F,. We shall denote by u(U, F) the topology induced by cd(F). It is given by the family of semi-norms pR defined by (3) as K ranges over G+?. We distinguish in particular the following two cases. (a) Case cd = @ is the family of all bounded subsets of Fa. The topology j3(U, F ) is called the strong topology on U. (b) Case ~4 = % is the family of compact subsets of Fa. The topology t(U, F) is called the Mackey topology on U. We will denote by U, = B&F) and U, = &,(F) the space U supplied with the topologies &U, F) and z(U,F) respectively.Theorem 8 implies the following.
Theorem 9 (Mackey). Let U and F be two paired spaces. Then F = (UJ*is also equal to (U,)', i.e. the dual of U supplied with the M a c k y topology.
LINEAR FUNCTIONAL ANALYSIS
567
Proof. We take X = F and we consider the spaces U,= OP,(F) and U, = J,
If we denote by r3 the covering of the dual U' of U by the subsets ABF when p ranges over the family of semi-norms and A E R,, we see that the initial topology is nothing other than the topology E(U,U*).We leave the proof of the following lemma as an exercise. Lemma 1. A closed convex symmetric subset of U*is equicontinuous if and only i f it is equal to AB: for some semi-norm and some 1 =- 0. Then the initial topology is the topology E(U,U*)associated with the covering & of U*by the equicontinuous subsets of linear forms on U.We check that equiconfinuoussubsets are relatively weakly compact. Indeed, if K c U*is an equicontinuous subset of linear forms, its closure K in S ( U ) is also an equicontinuous set of linear forms (and thus, K c U*). Furthermore, R is bounded. Since 8(U) = RU,the Tychonov compactness theorem shows that R is compact. We have the inclusions S c 8 c Zj c B. This means that the injections U,:-r U,-c U = U,-,U,are continuous. The Mackey Theorem 9 implies that ua= U* = u,*c us'. Remark. Since U is a topological vector subspace of &(U*),which is complete, the closure of U in &(U*) suppIies a completion of U.0 The following question remains to be answered. Under what conditions does the initial topology E(U,U*)coincide with the strong topology?
568
APPENDIX A
5. The Banach-Steinhauss theorem
This amounts to giving sufficient conditions for a bounded subset of continuous linear forms to be equicontinuous. The following result (sometimes called the Bmach-Steinhaus theorem) answers this question. Theorem 10. Let U be a barreled space and let V be a Hausdoif locally convex space. Then any pointwise bounded set of continuous linear operators.fromU into V is equicontiizuous.
Proof. Let 25 c B(U,V) be a pointwise bounded set of linear operators B. This implies that, for any semi-norm q defining the topology of V,we have that z(x) = sup,,,q(Bx) -= + a.Therefore, the function x n(x) is clearly a lower semi-continuous semi-norm. Since U is barreled, this implies that 7t is actually continuous (by definition of a barreled space). Then there exists a semi-norm p on U w d c =- 0 such that
-
This means that % is equicontinuous. 0 Corollary 3. Let U be a barreled space. Then the initial topology, the strong topology and the Mackey topology coincide, i.e. Up= U, = U,.
Remark. Recall that any Frechet space is barreled, because by Proposition 3.3.4, any lower semi-continuous semi-norm is continuous. In particular, we mention the following consequence which we have often used. Corollary 4. Let U be a barreled space. Any (weakly) bounded subset o j U* is weakly compact.
A bounded subset of a locally convex space U is not necessarily weakly compact. If U = L1(- 1, +I), we saw that the bounded set of functions x nJ(nx) (wheref E L1(- 1, 1)) is bounded but that it is not weakly compact. The result is true if and only .if U is reflexive (i.e., barreled and satisfying CJ = (U*)*). -t
+
APPENDIX B
THE KNASTER-KURATOWSKI-MAZURKIEWICZLEMMA
We devote this appendix to the proof of the Knaster-Kuratowski-Mazurkiewicz lemma, which we used to prove the Ky-Fan inequality (see Theorems 7.1.2 and 7.1.3) and, consequently, the Brouwer and Kakutani fixed-point theorems (see Theorems 9.3.5 and 9.3.4). This lemma Will be deduced from the Sperner lemma, which is of interest in itself. For this purpose, we break up a simplex into a sequence of “barycentric subdivisions” which yield coverings of the initial simplex by simplexes whose diameters converge to 0. By studying the structure of these interlocking simplexes, we will prove the Sperner lemma. For the sake of completeness, we shall deduce the Brouwer fixed point theorem from the Knaster-Kuratowski-Mazurkiewicz lemma. Recall that the Brouwer theorem implies the Ky-Fan inequality (using the Lasry Theorem 7.1.4; see remark ending Section 9.3). 1. Buycentric subdivision of simplexes
...
Consider (n+ l)-a$inely independent points xo, xl, ,xn of a vector space. We shall say that co(X0, xl, . .., x“)is a n-simplex and that co(+, 21, . ., x’k) is a k-dimensionalface of the simplex. Notice that there are n+ 1 faces of dimension n- 1 of a n-simplex. We shall construct the barycentric subdivision of an n-simplex into (n+ l)! smaller n-simplexes by induction. The construction involves taking the barycenter 1 r” = -((xo+xl+ n+ 1
*’*
.
+P)
of the simplex as the common vertex for all new smaller n-simplexes of thesubdivision. Thus, barycentric subdivision is carried out by induction from simplexes of lower dimensions to those of higher dimensions. For n = 1, the 1-simplex co{xO, x’} is subdivided into the two l-simplexes co{xO, 9) and cob1, xl} where yl = -&?+xl) is its barycenter. 569
570
APPENDIX B
Suppose that we can perform the barycentric subdivision of a k-simplex into (k+ l)! simplexes of dimension k for any k < n and let us prove that this property holds for k = n. Consider the n+ 1 faces Si(n-l) of dimension n- 1 of the n-simplex co{x0,xl, .., x"}. They can be subdivided in n! = (n- 1 l)!simplexes Tj,n;l)of dimension n- 1 by assumption. There are (n+ I)! = (n+ l)n! such (n- I)-simplexes 7,/,;-1). Put T g . = co (T;fll y'"') (1)
.
+
u
where
1 v" = -((Xo+xl+ *.- +%*) n+ 1 is the barycenter. It is clear that it is a n-simplex. Since any point of the n-simplex which is neither the barytenter y" nor lies in any proper faces belongs to a segment joining the barycentcr y" to some simplex T$-'), we have covered co(xo, .,2')by (n+ l)! simplexes of dimension n having the barycenter f' US .common vertex. For n = 2, we obtain the 6 = (2+ I)! following simplexes.
..
.o
Fig. 17.
By iteration of formula (l), we see that any n-simplex T(")of the barycentric subdivision can be written in the form T(")= co (yo, yl, . .,y") "(2)
+
.
where, yo = xl", y1 = a(xl" dl)7 1 yk = -(x'o+x'1+ (3) k+ 1
. - . 3
**.
+X'k),
...
1 y" = ($0 + xh + n+l €or a convenient permutation {io, it
...,
+Xi")
...,in} of the (n+ 1)-indexes.
KNASTER-KURATOWSKI-MAZURKIEWICZ LEMMA
57 1
In other words, any n-simplex of the barycentric subdivision is the convex hull of the barycenters of a strictly increasing sequence {+}, {+, x"}, . . ., {+, 21, . . xi", . . , {fi,. . . x'n} of subsets of {xo, . , x"}. Now, let T(")= co(yo,y l , . . .,y") be any n-simplex of the barycentric subdivision and
.
.
..
be any (n- ])-face of Ten). Then we can write either
or
with 0 e k n-1. In the first case, we see that T("-') is the face of the only simplex T'")= = co (yo,y l , . . .,f - 1 , y"). In the second case, we have the following situation : yk-l is the barycenter of {xo, . . .,x'k-'} and yk+l is the barycenter of {xo, . . ., X ' k - l , d k 7 &+l}. Let j( be the barycenter of (9,.. . 7 dk-',dk}and be the, Then baryanter of {A?, ...,x i k - I , X'~+I}. (7)
3") = co (yi, . . .,yk-1, jF, . . .,y")
and
are the only two n-simplexes of the subdivision having T("-l)as a face. Now, if we notice that case (5) happens if and only if T("-l) lies in some (n- 1)-face of the initial n-simplex co(x0, . . .,xy, we have proved the following result. Proposition 1. Any (n- l)-face T("-l)of any n-simplex of the barycentric subdivision of co(xO,b. . .,x"> is
(i) either a face of exactly one n-simplex of the subdivision VT("-" lies on a ( n - I)-face of co(9, xl, . . .,x"), (3) or a common face of exactly two n-simplex of the subdivision otherwise.
572
APPENDIX B
Exrrmple. Ifn = 2, first case
---
is the face of
...
Secondcase
ion face of
-.
..
.~-.
Fig. 18.
2. Seqnence of barycentric subdivisions
By performing the barycentric subdivision for each of the (n+ l)! simplexes of dimension n of a n-simplex co(X0, . . ., x"), we obtain a covering by smaller n-simplexes. By iterating this process v-times, we obtain the so-called Y&barycentric subdivision of the simplex. Let us state and prove the counterpart of Proposition 1. Proposition 2. Let F-l)be any (n- l)-&ace of any n-simplex of ttie v&-baryThen it is centric subdivision of co(9, xl, . . ., 2).
(i) either a face of exactly one n-simplex of the v*-subdivision ifl-%es (n- l)-faceofco(X0, xl, . . ., x"),
in a
(ii) Or a common face oj exactry two n-simplexes of the v'h-subdivision otherwise.
Proof. We proceed by induction of v.'Proposition 1 proves the case where v = 1. Suppose that Proposition 2 holds for v - 1. (a) First case. The simplex TP-l) has been obtained by subdividing a (n- 1)simplex Ty'i') of the (vsubdivision. Since TP-* lies in a (n- 1)-face of co (9, ...,x"),p-7')also lies in such a face. Then, by the induction assumption TP-7') is the face of a unique n-simplex T F p Moreover, 7'f-l) is a (n-1)simplex of the barycentric subdivision ofTE,, and it lies on its (n- I)-face p-7').
KNASTER-KURATOWSKI-MAZURKIEWICZ LEMMA
573
xo First care al
Fig. 19.
e-')
Then Proposition I implies that is a face of exactly one n-simplex of the barycentric subdivision of P J p O n the other hand, if the subdivision of any n-simplex Splentails TF-l), then SFl must have T?") as a face and thus, it equals TEp Therefore, TF-') is a face of exactly
el.
(b) Second case. (b-I) Firstly, assume that has been obtained by subdividing Tp-i') wich does not lie on a face of co(9, ...,3).Then can only be a face of a n-simplex Ty)'of the barycentric subdivision of a n-simplex having TG') as a face. By the induction assumption, there are exactly two such n-simplexes pzland p2pThe first part of Proposition 1 implies that each of these simplexes, when subdivided, entails exactly one n-simplex (and p:'')) having as a face. (b-2) Assume that TF-') cannot be obtained by subdividing any (n- 1)-simplex of the (v- I)* subdivision. Let TEIl be the unique n-simplex whose subdivision entails T"-l. Then e-')cannot be on any (n- 1) face of Tpp Proposition 1 implies that T?'')is a common face of exactly two n-simplexes of the subdivision of T:Zl(which then belong to the vth subdivision). Any other n-simplex of the (u- 1)lhsubdivision, apart from TZl, does not include TF-') and thus, never yields a n-simplex having as a face in the subdivision. 0
v-')
c-')
PJl
e-')
c-')
3. The Sperner lemma Theorem 1 (Sperner's lemma). Let Svbe the uth barycentric subdivision of a n-simplex S = co(X0, xl, . . .,fl.Suppose that u is an application mapping each
574
APPENDIX D
vertex y , = xbo2%' of a n-simplex Tr' of %, onto a verlex x i = uy, such that (9)
the index i satisfies A: =- 0.
Then there exists a n-simplex TF) = co (y:, . . ., y:) oj'TiT,such that
(10)
Y i = 1 , ..., n, a y ' , = x i .
Proof. Let 55,be the family of the n-simplexes of the d" barycentric subdivision and 8,be the sub-family of simplexes TP) = co (y:, y:, . ., y 3 satisfying
.
(1 1)
u{y!, y;, . . ., y:} = {XO, XI, . . ., x"}.
We shall prove by induction that the number qnof elements of 8,is add (and thus, that there exists at least one simplex p )satisfying (11)). For n = 0, this is obviously true. Suppose that the result holds for n-1. We prove it true for n. For this purpose, we shall say that an (n- 1)-face of a simplex P ' o f 55"is regular y its vertices have xo,XI, . ,2-las images. Let us denote by T I ,Tz, . ..,T , the n-simplexes of the v&-subdivision 'Ti, of co(x", xl, . . ., Y') and Z, the number of regular faces of Ti.We shall determine the sum Z I + Z Z + +z, in two different ways.
..
--
- -.
) (a) the sum zl+z2+ +z, is odd. Indeed let q,,-l be the number of ( n - 1)simplexes T y - l ) = co(y& ...,y=-l) contained in co(X0, xl, ..., 2-l) and , . . ., X-'} = {x", xl, . ., 2-l}.By the induction hywhich satisfy ~ { y ;y:, pothesis, this number qn-l is odd. Now, we compute the number of (n- 1)-simplexes which are regular faces T2-l) of n-simplexes of 55".The following possibilities can occur.
.
.
Tf-') is contained in co(9, xl, ..., 2-l).If a{y:,yi, . .,fl-'} = ., ."-I}, it is the regular face of exactly one n-simplex TT) by the first statement of Proposition 2. Then, there are qn-l such simplexes, by the very definition of qnW1. If u{y:, yi, .,fl-l} # {x", xl, ...,A?-'} it is not a regular face. (x)
= {xo,XI, .
.
..
(/I TY-* ) is contained in a (n- 1)-face of co(P, xl, ..., x3, but not in the In this case, T,("-l)cannot be the regular face of a n face co(xo,xl, . ..,?-I). simplex T?) because of assumption (9). ( y ) T,("-')is not contained in any ( n - 1)-faceof co(xo, xl,
. . ., 9).In
this case, the second statement of Proposition 2 implies that TF-l) is the regular face of exactly two n-simplexes of Zl,(which have TP-l) as a common face).
575
KNASTER-KURATOWSKI-MAZURKIEWICZ LEMMA
Therefore,
(b) Computation of q,,. We prove now that
(13)
tl+t2+
.*'
+z, = qn+21.
If Ti belongs to 8,,it obviously has only one regular face (zj = 1). There are qn such simplexes. If Ti = co(y:, y:, . . .,y:) does not belong to 8,, then either be, ...,g}does not contain {xo, XI, . . .,2-l)and thus, has no regular face (zi = 0) or b :, . .., yy"} does contain {xo, .xl, . . ., Y - l } .In this case, zf = 2. To see this, suppose that TIdoes not belong to 8, and that Ti = co{yf, ...,fl} is such that ay; = 9, uyt = xl, . . .,a<-l = Y-l. Then we have afl # x"' (because T, 4 8,).If, for example, cry: = xo, then Ti has two regular faces: . . . y:-') and co(<:, yt, . . .,A'-'>. co (y:, dY It follows from these considerations that (13) holds. Hence, we deduce from. (12) and (13) that (14)
qn = qn-l+2(k-I)
is odd.
4. The Knaster-Kuratowski-Mazurkiewicz lemma We state this result.
+
Theorem 2 (Knaster-Kuratowski-Mazurkiewicz lemma). Consider n 1 closed subsets Ff of an-simplex co {xo,xl, . ,x"} such that any face co {xt,,, xil, ,.'*} is contained in the union FI,UFt,U UFf,.Then the intersection FI # 0.
..
--
...
nf;'
To prove this, we use the Sperner lemma to construct n simplexes TT) = co (yp, . . ., y:) of the v"-barycentric subdivision such that y: E Fi for all i. Intuition suggests as v increases, the n-simplexes of the vth-subdivision will decrease in such a way that the sequences y; will all converge to the same point ye which will necessarily belong to the intersection n;==,FI.We begin by proving the following proposition.
Proposition 3. Let TP)be any n-simplex of the vth subdivision of S = co (xo, xl,, ., x").Then
..
(
)'
diameter (Tp)) -c nJ diameter (S).
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APPENDIX B
Proof. This inequality follows by iteration from the case when v = 1. This case will be proved by induction on n. The case n = 1 is obvious. Suppose that it is true for n- 1. We deduce that it holds for n. Let T = Tp)be the convex hull ofyO,yl, . . .,y" whereyo = xo,yl = +(x"+xl), . . .,y(") = (l/(k+l)) (x"+. . . . ..+2), . ... We must prove that (16)
n diameter ( T) =s-diameter (S). n+ 1
We have that
1 n+l
=s-
Therefore, for any x =
C .1I xi-
izj
n
x / II 6 -~
n+ 1
diameter (9.
c;-oA i d E S,
in particular, taking x = fl, we deduce that
Oh the other hand, by the induction assumption,
rnax
(18)
Od,/sn-l
4 -
n- 1 diameter (R)=s n
l ~ y ( f ) - y (e ~-) ~ ~
n- 1 n diameter (S) =s-diameter (S) n n+ 1
..
where R = co (2, ., 2-l).Then (17) and (18) imply that (19)
max
Ori,jsn-1
]Iy(')-y(j)II 4-
It remains to prove that
n diameter (9. n+l
KNASTER-KURATOWSKI-MAZURKIEWICZ LEMMA
(Since(16)follows from (19) and (20).) For this purpose, take a = and b = C;sofijyjbelonging t o T.We have
577
zs0a,f
Proof of Theorem 2. Let 53, 6e the vth-barycentricsubdivision of the simplex S = co (xo, xl, . . .,xy. We construct the map u as follows. If y, is the vertex of a n-simplex TP)fo Zj,, we have (21)
y, =
C AiJi n
I-0
=
C A$* s
k=O
where {io, .. .,i,} is the set of indexes j such that A{ =- 0. Then co (x’”, dl, . . . ...,2*) is contained in F,o u UFI, by assumption. Thus y, belongs to at least one closed subset F1where i E {io, ..., is}. Write (22)
ay, = xi
for such an index i.
Therefore, the assumption of the Sperner lemma is satisfied. Hence there exists at least a simplex TF)= co{yf, . . .,y,”} such that 07: = xi. By the very construction of a, this means that y: e Fi for all i = 0, . . .,n. Now, by Proposition 3, we have that ’
Since S is compact, we can extract subsequences (again denoted by) y: which converge to the same element y . (by (23)). Since y. = Iim yt and F, being closed, y+ E 4 for all i. 0 We state the following corollary. 39
Fi = {x E S such that xi 2 h ( x ) )
+xi, = 1. If x were not contained in any of Fie, Fil, ..., Fit, we would have w - f k ( x ) for all ik. Adding these ‘t inequalities, we would obtain the contradiction 1=
c
k=O
Xfk c
c fi,(x)
k=O
=z C J ; : ( X ) i =O
= 1.
nbo
Thus the assumptions of the Knaster-Kuratowski-Mazurkiewicz lemma are F,, i.e. Z satisfying A(%)=G X for all i. satisfied and so there exists 2 E We cafinot have Zi =-A@) for at least one.i.sincesuch an inequality yields the contradiction 1 i1. Hence 5ti =fi(Z) for all i, i.e. Iis a .fixed point off. Corollary 2. If we assume that
X is homeomorphic to an n-simplex S, (25) any continuouS map f mapping X inro itself har afixed point.
KNASTER-KURATOWSKI-MAZURKIEWlCZ LEMMA
579
Proof. If q :X -c S is the homeomorphism, the map g = qfq-' is a continuous application from S into itself, which has a fixed point J. Since J = yfq-47), we see that x = is a fixed point off: 0
9-v)
In particular, one can prove that convex compact subsets of finitedimensional vector spaces are homeomorphic to simplexes. This implies the existence of a fixed point of continuous maps from a convex compact subset of R" into itself.
APPENDIX C
LYAPUNOV'S THEOREM ON THE RANGE OF A VECTOR
VALUED MEASURE
Recall the statement of Lyapunov's theorem. Tbeorem. Let oi? be a a-algebra on a set 52 and p, (i = 1, measures. Define
...,n) be n non-atomic
Then p(&) is a convex compact subset of R".
Proof. We associate with
p the continuous linear operator L E B(Lm(12),R")
defined by LV = { J v ~ P *~*, * , J v ~ P ~ }
+
where we set Lw(Q) = L"(s2, d,da) with o = 1p11+ Ip,l. Each pi is absolutely continuous with respect to a. The Radon-Nikodym theorem implies that there exist n functions fr E L'(s2) such that dpi =fi do. Then L can be written in the form
Consider the set 'zj =
{q E Lm(12)such that cp(o) f [0, 11 a.e.}.
(This is the set of fuzzy coalitions: see Section 10.4.4.) We have seen that 2, is convex and weakly compact. Thus L(%) is a convex compact subset of R" (since L is obviously continuous on L" supplied with the weak topology u(LoD, Ll)). We shall conclude by proving that p(&) = I,(%). Since p(&) is obviously contained in L('si),it remains to prove that L ( Z ) p(ci4). Let x E L ( T ) and define %, = {t E 'si such that Lt = X} 580
LYAPUNOV’S THEOREM
58 1
which is a weakly closed convex subset of ‘7i. Hence ‘7ixis weakly compact. By the Krein-Milman theorem (see Proposition 3.4.9) there exists an extremal point zo.If we prove that any extremal point zoof “6, is a characteristic function, we thus deduce that any u E L(’7i)is the image p(&) = Lz, of a characteristic function. Therefore, suppose that zo belongs to 5,and is not a characteristic function. Then there exists 8 E 10, *[ and A E uf (such that u(A) >.O) satisfying 8 =z zo(w) 1- 8 for almost all w E A. Consider the subspace
Since a(A) > 0 and u is non-atomic, the dimension of V is greater than n. Therefore, the kernel of the restriction LI, of L to V in non-zero. Thus there exists p E V such that Lp = 0. We can choose such a v satisfying -0 =sy(o)=z 8 for almost all w. We see that zo = $(zo-q)+$(to+y) and that both z0-y and zo+p belong to “6 since 0 -G z o ( o ) f q ( ~ )G 1 a.e.) and satisfy L(zo-q) = This is a contra= L(zo+p) = x. Hence ZO- q~ and zofy both belong to Zx. diction. 0
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Chapter 1 The results of this chapter must be regarded as classical. Material on convexity in finite or infinite dimensional vector spaces can be found in almost any text on functional analysis (e.g. Treves, Dunford and Schwartz, Schwartz and Yosida). The foundations of the general theory of convex sets and functions is mainly due to Minkowski, who introduced the idea of characterizing closed convex sets by gauge functions and support functions. In the case of infinite dimensional spaces, see Hormander. For other results on the separation of convex sets, we refer to the papers of Klee. The introduction of indicators is due to Fenchel.
Chapter 2 The results of Sections 2.1 and 2.2 are classical. The general conjugacy correspondence defined in Section 2.4 was discovered by Fenchel. A comprehensive study of the consequences of the conjugacy theorem can be found in Rockafellar [a] in finite dimensional spaces and in Moreau [I] in infinite dimensional spaces. We refer to these books for further comments. Finally, the results of Section 2.5 are taken from the book of Berge [I].
Chapter 3 The results of Section 3.1 are classical, as well as the definition of proper maps (see Bourbaki [2]). The results of Section 3.2.2 are due to Arrow and Debreu. The results on continuity of convex functions must be considered as classical. The fact that a compact convex subset is the convex hull of its extremal points was first proved by Minkowski and generalized by Krein and Milman for infinite dimensional spaces. See also the papers of Aggeri [l] and Bronstedt [2]. 582
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583
Chapter 4
The explicit development of the theory of subdifferentiability is relatively recent. The reader should refer to Moreau [I] and Rockafellar [6] for a general review of the literature. For further generalizations, see the papers of Clarke who defines generalized gradients for locally Lipschitz functions. Fox a review, see Aubin [13]. The results on differentiability and variational inequalities are classical. The relationship between the Legendre transform and the conjugate function was noted by Fenchel. See also the papers of Ioffe, and Ekeland [7]. The relationship between subdifferentiability and differentiability from the right is due to Moreau. The differentiability properties of a pointwise supremum are due to Valadier. Section 4.4 is essentially the papers of Ekeland [3] and Ekeland and Lebourg. The order relation used in the proof is a device due to Bishop and PheIps and used by Bronstedt [3] and Browder in different settings. See the papers of Edelstein, Asplund, Asplund and Rockafellar, and Lebourg for further applications. A review of these applications appears in Ekeland [8].
Chapter 5
The duality theory for constrained minimization problems is historically based on the famous paper of Kuhn and Tucker, who used the differential calculus approach. Slater seems to have been the first to use a “constraint qualification assumption” of the type 0 € Int(L(R)-Y). Theorem 5.3.1 is due to Fenchel and Proposition 5.3.4 to Uzawa (see Arrow et a].). The marginal properties of Lagrange multipliers (Proposition 5 . I .4)was first discovered by Gale in the context of economics. See also the book of Karlin. An extensive literature has been devoted to duality theory. The most general results seem to be the theorems of Clarke [8] and Aubin-Clarke [2].
Chapter 6
We note only that the main concepts of game theory we present in this chapter were defined early by economists. The model of duopoly was introduced in 1838 by Cournot. Edgerworth, in 1881, considered the two-person game defined
584
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in Section 6.4. He assumed that the players could make any trade they wished. He defined the core (called the contract curve) and showed that it was nonempty. Earlier, in 1874, Walras defined the general equilibrium concept of an economic system made up of households and firms:Edgeworth recognized that the Walras equilibrium did belong to the core. In 1909, Pareto defined an “optimal resource allocation”. For the study of stability of two-person games, see Rockafellar [6], and McLinden [l]. Chapter 7
The first fundamental theorem of game theory was the minimax theorem of Von Neumann in 1928. He deduced it from the Brouwer theorem. Later, in 1941, Kakutani presented a simplified version of Von Neumann’s theorem and proved his famous fixed point theorem. In 1958, Sion [I], [2] proved Theorem 7.1.7. Actually, only separation tbeorems are needed to prove the existence of the minisup, as was recognized by Nikaido [2], [3], Moreau [I] and Rockafellar [6]. Theorem 7.1.4 is due to Lasry (unpublished) and is used in Lasry-Robert for the study of topological degree for correspondences. It is also used to prove that the Brouwer theorem implies the Ky-Fan inequality, proved in Ky-Fan [ 121. We deduce it from Theorem 7.1.2, due to Aubin [8]. See also the papers of Koenig, and Simons. The next two sections on extensions of a game are a summary of the thesis of Moulin [3], to which we refer for further comments. The theory of repeated games, introduced by Aumann and Mashler, was continued by Mertens, and Zamir. For games with imperfect information, see for instance, Ponsard, and Ponsard and Zamir. We refer to a forth-coming book of Mertens and Zamir. Chapter 8
The study of the Walras model can now be considered as classical. See the fundamental book of Debreu [l], in 1959. The first proof of the existence of an equilibrium in a series of alternative models are due to Wald (1933) and later, by Arrow-Debreu (1954) and McKenzie [l] (1954), Nikaido [5]. The most general version is in Debreu [I].
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585
We also refer to the books of Nikaido [5], Ekeland [4], [9] and Arrow and Hahn and, in particular, to Chapter 1 of the latter. Notice that we have left aside the problem of utility theory. We havechosen to use loss functions (minus utility functions) instead of preference orderings. For the relationship between these two concepts, we refer to Debreu [l]. See also Kannai [6] for the characterization of convex loss functions. A comprehensive study in these models in the framework of quadratic optimization with explicit computation of solutions is made in Aubin [16]. We did not develop in this book further extensions o f the Walrasian model such as finiteness properties Debreu [l] and uniqueness properties Balasko [ 11. Stability is studied in Arrow and Hahn, for further developments see Balasko [31. The global requirement of uniqueness has revealed itself to be excessively strong and was replaced by that of local uniqueness and the stability of the set of equilibria. For such an approach, using differential topology, we refer to the papers of Balasko [2], [3], Dierker, and Smale [ 11. See the papers of Fuchs for the study of structuraI stability. Broader interpretations and further extensions of the Walras model have been proposed. They include infinite-dimensional commodity spaces (T. Bewley [l], [2]), public goods (Milleron, Champsaur [2], [2], [4], Foley [l], [2], Rosenthal), fiscality (Guesnerie [ l]), uncertainty (Arrow [4], Debreu [ 11, Radner, Dreze [l], [5], Guesnerie and De Montbrial, Malinvaud 131, Stigum, Starett, etc.), transaction costs (Hahn [3], Kurz, Starr [I], [2]), money (Grandmont [l], [2], Grandmont-Laroque [ 11, Starr, [2]). Time dependent economies are studied in Grandmont [l], 121, Green [I], De Montbrial [ 11. Recently, the study of micro-economic foundations of macroeconomies is initiated’ by Malinvaud and Younes. For labour-managed and participatory economies, see Dreze [6]. “Barter curves” of exchanges economies and planification models can be studied in the framework of differential inclusions. See Antosiewicz and Cellina, Aubin [ll], [14], Aubin, Cellina and Nohel, Aubin and Clarke [l], Aubin and Day, Champsaur, Dreze and Henry, Cornet, Dreze and de la Vallee and Poussin, Malinvaud [4], Smale [3], [4], etc. Chapter 9 Theorem 9.1.1 on the existence of a non-cooperative equilibrium is due to Nash [ 11. Another proof using the associated variational inequalities was given by Rosen. The existence of solutions of variational inequalities for quadratic functionals was proved by Stampacchia. The proof given here is due to Lions
586
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and Stampacchia. The concept of Walras-Cournot .equilibriumwas introduced by Gabsewicz and Vial. Theorem 9.3.3 can be found in Arrow and Debreu. The fixed point theorems are due to Brouwer and Kakutani. Section 9.4 is the first part of the fundamental paper of Arrow and Debreu. Chapter 10 For the extension of the results of Section 10.1 to infinite dimensional spaces, we refer to Kurz Majumbar [l], [2], Radner [5]. See also Peleg [2], [3], [4] for properties of Pareto optima, Guesnerie [2] for the non-convex case. The selection procedure of Section 10.2.7 is due to Nash [2], it was improved by Harsanyi [l], [2]. For the concept of “second best”, we refer to Guesnerie 111. The concept of the core was introduced by Gillies [l], [2]. We refer to Aumann [6] for a survey. See also the papers of Billera [l], [2], Scarf [5], Shapley [9], [lo], [ll], 1121, [13], [14] etc. In the case of indivisibility, we refer to Henry [l] and Shapley and Scarf. The core of an economy with public goods is studied in Champsaur et al. The concept of fuzzy set is due to Zadeh. Other concepts of solutions of games have been devised. For instance, the concept of “Von Neumann-Morgenstern solution” of a game with sidepayments (see Von Neumann-Morgenstern) and the concept of bargaining set, introduced in Aumann-Maschler [ 11 and extended to games without sidepayments by Assher [l], [ 2 ] . The use of Debreu-Scarf coalitions, i.e., replicated economies or games, was introduced by Debreu and Scarf in the case of economies. Proposition 10.4.1 is due to Castaing [l]. The idea of using a continuum of players is due to Aumann [2]. An extensive study of economies and games with continuum of players lies in the books of Hildenbrand [6] and of Aumann and Shapley. Another approach uses non-standard analysis (see Brown and Robinson [ 11, [2] and Ali-Kahn). The idea of canonical cooperative equilibrium can be found in Shapley [4] (A-transfer-value) and Baudier. The concept of nucleolus was introduced by Schmeidler [l]. Fuzzy games were introduced by Aubin [9], [lo], but the idea was dormant in former papers (Ekeland [4], for instance). Further generalizations of fuzzy games can be applied in sociology (see Aubin, Louis-Gudrin and Zavalloni): each player is described by a vector of Rk (beliavioral profile) and fuzzy coalitions are replaced by “groups of participation”, described by matrices whose entries belong to [O, I].
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Chapter 11
An extensive literature is devoted to games with side-payments. We refer to the book of Owen [3] for general comments. For market games and relationships between economies and games with side-payments, see Shapley and Shubik [l], [4], [5] and Billera [3]. The concept of balanced games was introduced by Bondareva [ 13, [2] for games with side-payments and by Scarf [2] for games without side-payments. Theorem 11.2.2 is due to Scarf IS]. The concept of Shapley-value was introduced in Shapley [2]. The results of Section 11.3 are analogous to the results of the book of Aumann and Shapley dealing with non-atomic games. The idea of fuzzy-game with side-payments also lies in Owen [5], who introduced the linear extension operator (and in E. Baudier, personnal communication). See Cornet [3] for the study of his extension operator. Chapter 12 The equivalence Theorem 12.3.1 between the fuzzy core and the set of equilibria of an economy is analogous to the theorems of Debreu and Scarf, Aumann [2], and Hildenbrand [6]. We refer to the book of Hildenbrand [6]. The proof of non-emptiness of the fuzzy core (see Aubin [5]) implies the theorem of Scarf [2]. Analogous ideas can be found in Shapley [3] and Baudier.
Chapter 13 Theorem 13.1.1 is related to the minimax theorems of Rockafellar (see [6]). Theorem 13.2.1 was stated in a slightly more general form in Brezis et al. The concept of monotonicity was introduced by Minty [ 11 and Zarantonello. The concept of pseudo-monotonicity is due to Brezk [I]. Several versions of Theorem 13.2.2 are due to Browder, Lions, Stampacchia, etc. (See the books of Lions [3], BrBzis [3], Browder [3], Pascali for further comments.) The paper of Brondstedt [4] is closely related to the use of y-convex functions, introduced in Aubin [3]. See also Aggeri [2]. Other generalization of convexity can be found in the papers of Avriel, and Berge [l].
Chapter 14 Other proofs of Theorems 14.7.1 and 14.1.2 can be found in Rockafellar {ll]. See also the book of Laurent and the papers of Joly and Laurent. The
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characterization of the dual of L” is due to Ioffe and Levin. The example 10.1.3. of “augmented Lagrangian” is due to Rockafellar [13]. See also the paper of Goldstein [I 1, [2] and Goldstein and Tretiakov [l], 121, Berbekas. The presentation of duality theory via the perturbations of the minimization problem is due to Rockafellar (see his book [6] and [111 for instance). The Shapley-Folkman theorem appeared first in a paper of Starr [l]. See also Appendix B of the book of Arrow and Hahn. The proof we give in Section 14.2.3 is due to Ekeland. See also another proof in Arsttein [S]. The ShapleyFolkman theorem was used by Hildebrand et al. to prove the convergence of the core to the set of equilibria. The results of Section 14.2 can be found in Ekeland [6] and Aubin-Ekeland. Theorem 14.2.4 implies the results of Aumann and Perles (see also Arkin and Levin). An extensive literature deals with the calculus of variations and control theory. We mention in particular the books of Ekeland and Temam, Lions [4] ioffe and Tikomirov [4], Young [4]. For the maximum principle in control theory, we refer to the book of Pontryagin. For applications of control theory to management problems, see the book of Bensoussan et al. For control problems with infinite horizon, see Halkin [3] and Aubin and Clarke [3]. The latter paper deals also with constraints on the state and locally Lipschitz cost functionals. Problems of calculus of variations involving convex Hamiltonians (instead of concave-convex Hamiltonians as in this book) are studied in Aubin and Ekeland [2]. We chose an approach which follows closely the papers of Rockafellar 181, [9], [I41 and Berliocchi and Lasry [l], [2], [3] using the abstract Green formula proved by Aubin [l] (see [2]). For the non-convex case, see the papers of Clarke [ 11, [3], [5],[6], 171 and Aubin and Clarke [2]. The dynamic programming approach was introduced by Bellman in the framework of control problems. i t was used by Bensoussan and Lions for stopping time problems and impulsive controls. We refer to forthcoming books of Bensoussan and Lions for further comments.
Chapter 15
The redaction of Section 15.1 is due to Cornet, who used a version of a theorem of Browder [2] and Ky Fan [124 to prove several surjectivity and fixed point theorems. Corollaries 15.1.3 and 15.1.4 are due to Cornet, Proposition 15.1.2 to
COMMENTS
589
Cornet and Lasry. It generalizes a classical existence theorem in Mathematical Economics, see McKenzie [l] and Debreu [l]. The concepts of inward and outward correspondences were introduced by Browder [ 11, [2] the version used here is due to Ky-Fan [I21;the concept of strongly inward correspondence is due to Cornet. Theorem 15.1.4 is due to Rogalski for outward correspondences, to Cornet /4] for strongly inward correspondences, Theorem 15.1.3 is due to Browder [2] and Ky-Fan [2]. Theorem 15.1.6 is a version of a theorem due to Granas : it extends the Leray-Schauder theorem to correspondences. Other fixed point theorems for dissipative maps are found'in Aubin and Siege1 and Ekeland [8]. The selection theorem 15.3.5 was proved by Michael [I], [2], [3]. See other selection theorems in Cellina and Antosiewicz related to the study of multivalued differential equations. We refer to Lasry and Robert for the study of the topological degree for correspondences. For other references on non-linear functional analysis we refer to the books of Cronin, Lions [3], Krasnoselski [l], [2], Pascali, J.T. Schwartz [ 11, Vainberg. Theorem 15.2.1 is analogous to the results of Joly and Mosco. The results of Sections 15.2.2 and 15.2.3 are due to Tartar [l], [3]. The results of Section 15 3.1 are due to S. Robinson [3] and others.
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SUBJECT INDEX
Adjoint equation (optimal control) 497 Altman theorem 524 Appropriation of the economy 243 Arrow-Debreu-Nash theorem 283 Asplund theorem 127 Aumann-Perles theorem 476 Balanced game 346, 382 Banach theorem 546 Banach-Steinhauss theorem 568 Barrier cone 34 Barycentric operator 24 Bensoussan-Lions variational inequalities 512 Bensoussan4ions quasi-variational inequalities 516 Best compromise 307 Bipolar theorem 32 Birkhoff-Tartar theorem 542 Bishop-Phelps-Browder theorem 121 Bounded correspondences 4 10 Brezis-Haraux property 425 Brezis-Nirenberg-Stampacchia theorem 413 Brouwer theorem 284, 578 Browder-Ky Fan theorem 522 Canonical fuzzy core 326 Castaing theorem 558 Clarke’s generalized gradient 583 Closed loop control 505 Closed correspondence 70 Coercive correspondences 4 10 Coercive functions 63 Commodity space 5 Concave functions 25 y-concave functions 427
Conjugate functions 60 Conservative solution 68, 208, 266, 397 Constrained noncooperative equilibria 282 Constraint qualification hypothesis 79, 88, 156, 550 Continuation subset 514-518 Convex cover 345, 376 Convex functions 24, 61, 90, ))-convex functions 427 Cooperative equilibrium 329, 375 Cooperative games characteristic form 314 strategic form 312 with side payments 338 Core 171, 312, 315, 320, 322, 339, 342, 378, 379, 391 Cornet extension operator 361, 367 Cornet-hsry theorem 527 Cournot decision rule 187 Cranas theorem 533 Critical point 521 Debreu-Gale-Nikaido theorem 250 Debreu-Scarf coalitions 3 18 Decentralization principle 151 Decision rules 152, 164, 21 1, 227, 289-307 405-43 1 Decomposition principle 11, 58, 150 Demand correspondences 176, 178,182,244, 246-253 Differentiability Frichet 124 from the right 115 GLteaux 111 local E 124 of a pointwise supremum 118 Dual problem 137, 145 617
618
SUBJECT INDEX
Duality gap 195, 468 Duality map 52 Duopoly 183
Ky Fan theorem 530-53 1
Edelstein theorem 129 Edgeworth box 192 Ekeland-Lebourg theorem 126 Ekeland variational principle 123 Epigraph 10 Euler-Lagrange equations 489 Extension of games 220 with exchange of information 223 without exchange of information 227 Extremal points 100 Extremality relations 145, 449, 473, 486
Lagrange multipliers 137, 145, 450,496 Lagrangian 136, 447, 459, 489, 496 Lasry theorem 214 Lax-Milgram theorem 273 Least-core 331 Legendre transform 113 Linearized extension of maps 24 Lions-Stampacchia 272 Lower semi-compact functions 43, 76 Lower semi-continuous functions 43, 61 Lower semi-continuous correspondences 68, 545, 550, 551 Lyapunov theorem 439, 580
Fenchel theorem I56 Finite topology 21 1 Formal adjoint 484 Fuzzy coalitions 316 Fuzzy core of a representation 378 Fuzzy games 320 fuzzy economic games 321,391 fuzzy market games 343 Gauges 38,97 Gradient 111 Green’s formula 485 Hahn-Banach theorem 31, 561 Hamiltonian 446-487 Hamilton-Jacobi-Belman equation 501 Impulsive control problem 515 Imputation 309 lndicator 27, 62 Inf-convolution 12, 153, 159 Inward correspondences 530 Ioffe-Levin-Valadier theorem 452 Iterated games 230 Joly-Mosco theorem 540 Kakutani theorem 284
Knaster-Kuratowski-Mazurkiewicz theorem 213, 575 Krejn-Milman theorem 101 Ky Fan boundary condition 522
Ky Fan inequality 21 3
Mackey’s topology 566 Marginal profit 114 Maximal monotone correspondences 4 B Maximum principle 497 Max-inf. 197 Michael theorem 552 Minisup theorem 197, 216, 429 Minty theorem 421 Mixed extension 222 Mixed strategy 19, 82 Modulus of non convexity 462 Monotone correspondences 410418 Moreau 582 Moulin theorem 233 Nash bargaining solution 309 Nash theorem 268 Nikaido theorem 217 Non cooperative equilibrium 169, 266 Non cooperative Walras equilibria 285 Non satiation property 48 Normal cones 107, 522 Normalized games 394 Nucleolus 333-369 Orthogonal complement 30 Orthogonal left and right inverses 56, 57 Outward correspondences 530 Owen’s extension operator 366 Pareto minimum 170,295
SUBJECT INDEX
Pareto multiplier 297 Polar subset and cone 30, 78 Price 6 Principle of optimality 506 Prisoner’s dilemma 178 Product of a function by an operator 11,94, 152, 158 Production function 40 Production set 35 Profit function 35 Projection of best approximation 51 Proper function 9 Proper map 83 Pseudo-equilibrium 302 Pseudo-monotone function 412 Pseudo-monotone map 412 Pseudo-monotone correspondence 415, 416 Quasi-convex 44 Rates of transfer 297 Recession cone 34 Resource operator 6 Representation of a game 373 Riccati equation 508 Robinson theorem 388-546 Rockafellar theorem 41 8 Rogalski-Cornet theorem 531 Saddle point 197 Scarf theorem 348, 383 Section of a function 10 Selection function 305 Semi-coercive functions 76 Sequential extension 226 Shadow minimum 167, 265, 306 Shapley-Folkman theorem 466 Shapley value 360 Simple games 369 Sion theorem 218
619
Slater condition 160 Sperner lemma 573 Stackelberg equilibrium 187 Stackelberg disequilibrium 187 Stopping time problem 51 1 Strong topology 566 Subdifferential 105 Subgradient 105 Superdifferential 105 Support function 27, 62, 99 Supporting set 522 Targent cone 522 Tartar theorem 543 Threat functionals 309 Topologies uniform convergence over subsets 564 Mackey 566 strong 566 weak 561 Upper semi-continuous68,109 completely 292, 293 finitely 410 Upper semi-continuous correspondences 66 Upper semi-continuous functions 44 Valadier theorem 559 Value of a two-person game 196 Values of a fuzzy game 353 Variational inequalities 114, 269, 408, 410 Von Neumann theorem 222 Walras-Cournot equilibrium 277 Walras equilibria 194, 245, 248, 252, 256, 259, 391 Weak Hausdorff topologies 438, 555 Weak topology 563 Weierstrass theorem 45
