Language, Games, and Evolution: Trends in Current Research on Language and Game Theory

Lecture Notes in Artificial Intelligence

6207

Edited by R. Goebel, J. Siekmann, and W. Wahlster

Subseries of Lecture Notes in Computer Science FoLLI Publications on Logic, Language and Information Editors-in-Chief Luigia Carlucci Aiello, University of Rome "La Sapienza", Italy Michael Moortgat, University of Utrecht, The Netherlands Maarten de Rijke, University of Amsterdam, The Netherlands

Editorial Board Carlos Areces, INRIA Lorraine, France Nicholas Asher, University of Texas at Austin, TX, USA Johan van Benthem, University of Amsterdam, The Netherlands Raffaella Bernardi, Free University of Bozen-Bolzano, Italy Antal van den Bosch, Tilburg University, The Netherlands Paul Buitelaar, DFKI, Saarbrücken, Germany Diego Calvanese, Free University of Bozen-Bolzano, Italy Ann Copestake, University of Cambridge, United Kingdom Robert Dale, Macquarie University, Sydney, Australia Luis Fariñas, IRIT, Toulouse, France Claire Gardent, INRIA Lorraine, France Rajeev Goré, Australian National University, Canberra, Australia Reiner Hähnle, Chalmers University of Technology, Göteborg, Sweden Wilfrid Hodges, Queen Mary, University of London, United Kingdom Carsten Lutz, Dresden University of Technology, Germany Christopher Manning, Stanford University, CA, USA Valeria de Paiva, Palo Alto Research Center, CA, USA Martha Palmer, University of Pennsylvania, PA, USA Alberto Policriti, University of Udine, Italy James Rogers, Earlham College, Richmond, IN, USA Francesca Rossi, University of Padua, Italy Yde Venema, University of Amsterdam, The Netherlands Bonnie Webber, University of Edinburgh, Scotland, United Kingdom Ian H. Witten, University of Waikato, New Zealand

T· L· G

Texts in Logic and Games

Texts in Logic and Games (TLG) was a book series created in 2007 and published by Amsterdam University Press from 2007 to 2009. The Editor-in-Chief of the series was Johan van Benthem (Amsterdam & Stanford), the Managing Editors were Wiebe van der Hoek (Liverpool), Benedikt Löwe (Amsterdam), Robert van Rooij (Amsterdam), and Bernhard von Stengel (London). The following volumes were published by Amsterdam University Press: Volume 1 Johan van Benthem, Benedikt Löwe, Dov Gabbay (eds.) Interactive Logic Selected Papers from the 7th Augustus de Morgan Workshop, London, December 2007 Volume 2 Jörg Flum, Erich Grädel, Thomas Wilke (eds.) Logic and Automata: History and Perspectives, December 2007 Volume 3 Giacomo Bonanno, Wiebe van der Hoek, Michael Wooldridge (eds.) Logic and the Foundations of Game and Decision Theory (LOFT 7), June 2008 Volume 4 Krzysztof R. Apt, Robert van Rooij (eds.) New Perspectives on Games and Interaction, December 2008 Volume 5 Jan van Eijck, Rineke Verbrugge (eds.) Discourses on Social Software, March 2009 From 2010 onwards, TLG will become part of the Springer FoLLI series and continue to publish books at the interface between logic, computer science, and game theory in the area of “intelligent interaction” linking into economics, linguistics, mathematics, philosophy, social sciences, and other congenial areas.

Anton Benz Christian Ebert Gerhard Jäger Robert van Rooij (Eds.)

Language, Games, and Evolution Trends in Current Research on Language and Game Theory

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Series Editors Randy Goebel, University of Alberta, Edmonton, Canada Jörg Siekmann, University of Saarland, Saarbrücken, Germany Wolfgang Wahlster, DFKI and University of Saarland, Saarbrücken, Germany Volume Editors Anton Benz Zentrum für Allgemeine Sprachwissenschaft 10117 Berlin E-mail: [email protected] Christian Ebert University of Tuebingen, Department of Linguistics 72074 Tuebingen, Germany E-mail: [email protected] Gerhard Jäger University of Tuebingen, Department of Linguistics 72074 Tuebingen, Germany E-mail: [email protected] Robert van Rooij University of Amsterdam, ILLC 1090 GE Amsterdam, The Netherlands E-mail: [email protected]

Library of Congress Control Number: 2010941374 CR Subject Classification (1998): I.2, F.2, F.1, F.4.1, H.3 LNCS Sublibrary: SL 7 – Artificial Intelligence ISSN

0302-9743

ISBN-10 ISBN-13

3-642-18005-1 Springer Berlin Heidelberg New York 978-3-642-18005-7 Springer Berlin Heidelberg New York

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. springer.com © Springer-Verlag Berlin Heidelberg 2011 Printed in Germany Typesetting: Camera-ready by author, data conversion by Scientific Publishing Services, Chennai, India Printed on acid-free paper 06/3180 543210

Foreword

Recent years have witnessed an increased interest in formal pragmatics and especially in the establishment of game theory as a new research methodology for the study of language use. Game theory and decision theory (GDT) are natural candidates if we look for a theoretical foundation of linguistic pragmatics. Over the last decade, a firm research community has emerged with a strong interdisciplinary character, where economists, philosophers, and social scientists meet with linguists. Within this field of research, three major currents can be distinguished: one is closely related to the Gricean paradigm and aims at a precise foundation of pragmatic reasoning, the second originates in the economic literature and is concerned with the role of game theory in the context of language use, and the third aims at language evolution seen either from a biological or from a cultural perspective. This volume grew out of two conferences, one organized at ESSLLI in 2007 on language, games, and evolution, and the other organized at the ZAS in Berlin on games and decisions in pragmatics in 2008. Both were funded by the ZAS, Berlin. Based on a selection of contributions to these conferences, we invited additional papers which together provide a state-of-the-art survey of current research on language evolution and game theoretic approaches to pragmatics. Special thanks go to Tikitu de Jager, who helped us greatly with the preparation of the latex manuscript of this volume!

Table of Contents

Language, Games, and Evolution Language, Games, and Evolution: An Introduction . . . . . . . . . . . . . . . . . . . Anton Benz, Christian Ebert, Gerhard J¨ ager, and Robert van Rooij

1

Part I: Non-evolutionary Approaches: Synchronic Phenomena How to Set Up Normal Optimal Answer Models . . . . . . . . . . . . . . . . . . . . . Anton Benz

14

Strategic Vagueness, and Appropriate Contexts . . . . . . . . . . . . . . . . . . . . . . Kris de Jaegher and Robert van Rooij

40

Now That You Mention It: Awareness Dynamics in Discourse and Decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michael Franke and Tikitu de Jager The Role of Speaker Beliefs in Determining Accent Placement . . . . . . . . . James German, Eyal Sagi, Stefan Kaufmann, and Brady Clark

60 92

Part II: Evolutionary Approaches: Diachronic Phenomena Evolutionarily Stable Communication and Pragmatics . . . . . . . . . . . . . . . . Thomas C. Scott-Phillips

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Simulating Grice: Emergent Pragmatics in Spatialized Game Theory . . . Patrick Grim

134

Signaling Games: Dynamics of Evolution and Learning . . . . . . . . . . . . . . . Simon M. Huttegger and Kevin J.S. Zollman

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Pragmatics, Logic and Information Processing . . . . . . . . . . . . . . . . . . . . . . . Brian Skyrms

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Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Language, Games, and Evolution: An Introduction Anton Benz1 , Christian Ebert2 , Gerhard J¨ager2 , and Robert van Rooij3 1

1

Zentrum f¨ ur Allgemeine Sprachwissenschaft 2 Universit¨ at Tuebingen 3 Universiteit van Amsterdam

Introduction

Recent years have witnessed an increased interest in formal pragmatics and especially the establishment of game theory as a new research methodology for the study of language use. Within this field of research, three major currents can be distinguished: one is closely related to the Gricean paradigm and aims at a precise foundation of pragmatic reasoning, the second originates in the economic literature and is concerned with the role of game theory in the context of language use, and the third aims at language evolution seen either from a biological or from a cultural perspective. It has often been stated that language is used for doing something1 . But it is not used as a hammer is used to drive a nail into wood. The utterance of a sentence has no immediate causal effect on the world except that it modulates the density of air according to a certain pattern. In order to achieve something with language, the speaker or author has to get an addressee involved. Even if his aim is only to inform the addressee about some fact, he can only achieve his aim if the addressee arrives at the intended interpretation and believes the speaker to be trustworthy. In short, the success of a linguistic act depends on the behaviour of both the producer and the recipient. But this means that language use satisfies the abstract characteristics of a game in the sense of game theory. In a very general sense we can say that two agents play a game together whenever they each have to choose between several actions such that their preferences over their own actions depend on which action the other agent chooses. Obvious examples are card playing, chess, or soccer. In pragmatics, the two agents are commonly called speaker and hearer. Game theoretic approaches to linguistics have divided into several lines of research, addressing questions of language evolution, typology, Gricean pragmatics, and the credibility of market signals. There are several, and as it seems, independent roots from which these lines originated. There has to be mentioned at least Michael Spence’s (1973) work on job market signals, John Maynard Smith’s (1982) work on evolutionary biology, Ariel Rubinstein’s (2000) book an economics and language, and early work on game theoretic pragmatics by e.g. Arthur Merin (1999) and Prashant Parikh (1990; 2001). If we had to name the 1

E.g. (Searle, 1969; Clark, 1996).

A. Benz et al. (Eds.): Language, Games, and Evolution, LNAI 6207, pp. 1–13, 2010. c Springer-Verlag Berlin Heidelberg 2010 

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single most influential work on game theory and language, then this would have to be David Lewis’s monograph on conventions (1969). Lewis’s aim was to answer a challenge posed by Quine (1936) who claimed that “in order to apply any explicit conventions that were general enough to afford all logical truth, we would already have to use logic in reasoning from the general conventions to the individual applications” (Lewis, 1969, Foreword by Quine). Hence, what counts as logical truth cannot be settled by convention. A similar argument can be brought forward against the idea that the meaning of language is conventional. Every convention seems to presuppose an agreement. How else could a convention be adopted by a population? But in order to come to an agreement the members of the population have to communicate with each other, hence, they have to use some sort of prior language in order to reach their agreement. It seems to follow that the idea of the conventionality of linguistic meaning leads into a vicious circle, and, hence, that linguistic meaning cannot be conventional. Against this background, Lewis set out to show that conventions can be established without conscious agreement simply by coordination of behaviour. Lewis defines conventions to be certain kinds of regularities in the behaviour of a population. The following is an example of a non-linguistic convention, which originated in Rousseau (1755, Part II): (1) Rousseau’s stag hunters. There is a party of hunters. They have the possibility to hunt stag together or hunt rabbit individually. If they hunt stag together, they are provided with meat for several days. If they hunt individually, then they can only hunt rabbit which provides them with meat for only one day. They have only success hunting stag if everybody joins in. If one hunter drops out, then all others who still go for stag will achieve nothing. Obviously, if they establish a convention to hunt together in their community, then this will be most profitable for all of them. The convention to hunt together in their community is a regularity in the behaviour of the hunters which answers to a coordination problem. For two hunters, the following table shows a possible representation as a strategic game: stag rabbit

stag rabbit 2, 2 0, 1 1, 0 1, 1

This table shows the possible choices for the two hunters. Both can choose between hunting stag and hunting rabbits. The first hunter chooses between the row actions, the second hunter between the column actions. If they both choose stag, they both achieve a payoff of 2; if both choose rabbit, they both are guaranteed a payoff of 1; if one of them chooses rabbit, the other stag, then the hunter choosing rabbit still receives a payoff of 1, but the one choosing stag only achieves payoff 0. Clearly, they are better off if both decide to hunt stag. The behaviour of the hunters can be described by their strategies. A strategy of a player tells for each of his information states which action he will choose.

Language, Games, and Evolution: An Introduction

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In the model of the stag hunt game, there is only one information state, hence, a player’s strategy will just choose an action. The strategies of a set of players are in equilibrium if no player can improve by changing his strategy given that the other players keep to their strategies. In the stag hunt game, the strategies of all hunting stag and of all hunting rabbit are both in equilibrium. According to Lewis, conventions answer to recurrent coordination problems. For example, the hunter community has to decide again and again which type of animal they are going to hunt. Common wisdom has it that a true convention has to be an arbitrary solution to the coordination problem. If there is some physical or other necessity that enforces certain behaviour, then this behaviour cannot be called conventional. As the conventional behaviour answers to a coordination problem, it follows that one of the main reasons for adhering to it is the expectation that other members of the community also follow the convention. Furthermore, it must be in everybody’s interest to follow the convention. Lewis even adds the condition that everyone has to have an interest that everyone else follows the convention. He calls a strategy pair which satisfies all these conditions a coordination equilibrium. The conventions which determine linguistic meaning are a special sort of conventions. They can be described as equilibria in a signaling game. The simplest examples of signaling games consist of two players, the speaker and the hearer. The speaker wants to communicate a specific meaning M ∈ M. For this purpose he chooses some form F ∈ F. The hearer has to interpret F and choose some M  ∈ M as interpretation. Speaker and hearer have a common interest to communicate the meaning which the speaker had in mind. Hence, the coordination problem is solved successfully if M = M  . The signaling and interpretation strategies of speaker and hearer are represented by functions S : M → F and H : F → M. The coordination problem is recurrent, and ideally, it should be solved for every meaning M the speaker may want to communicate. If the latter condition holds, then the composition H ◦ S is the identity map on M. Lewis calls this a signaling convention. Clearly, a signaling convention is an equilibrium which solves the coordination problem. Lewis’s goal was to explain the conventionality of language meaning. He achieves this goal by defining the meaning of forms relative to a signaling convention: If S, H is given, then the meaning of a form F is identified with S −1 (F ). Actual languages are instances of signaling conventions. This ultimately leads to a reduction of linguistic meaning to a property of behaviour. It is instructive to look at Lewis’s (1969, Sec. IV.1, p. 122) classical Paul Revere examples: (2) A scene from the American War of independence. The sexton of the Old North Church informs Paul Revere about the movements of the British regular troops, the redcoats. The only possibility to communicate with each other is by use of lanterns. A possible signaling strategy of the sexton may look as follows:

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a) if the redcoats are observed staying home, hang no lantern in the belfry; b) if the redcoats are observed setting out by land, hang one lantern in the belfry; c) if the redcoats are observed setting out by sea, hang two lanterns in the belfry. The common goal of Paul Revere and the sexton is to warn the countryside about the movements of the British regulars. To achieve this, Paul Revere has to interpret the sexton’s signals correctly. The successful interpretation strategy looks as follows: (3) a) if no lantern is observed hanging in the belfry, go home; b) if one lantern is observed hanging in the belfry, warn the countryside that the redcoats are coming by land; c) if two lanterns are observed hanging in the belfry, warn the countryside that the redcoats are coming by sea. The strategies described in (2) and (3) can be represented as follows: S

stay land sea 0 1 2

speaker’s selection strategy

0 1 H stay land

2 sea

hearer’s interpretation strategy

The top row lists the possible information states of the agent: the set of states M = {stay, land, sea} for the sexton, and the set F = {0, 1, 2} with the number of lanterns for Paul Revere. The pair S, H is clearly a signaling convention. It is arbitrary, as there are other strategy pairs which would be equally successful, both have an interest in following it, and they are both interested in the other one following it. Lanterns in a belfry have no meaning in themselves. Only by the fact that the sexton and Paul Revere have agreed on a certain strategy pair do they get meaning. In general, a form F means M in a language community because the community has adopted a signaling convention where speakers choose F if they want to express M , and hearers choose M if F is signaled to them. Once the signaling convention described in (2) and (3) is established between the sexton and Paul Revere, we can define the meaning of one lantern in the belfry as the British troops are setting out by land. In this sense, signaling behaviour comes prior to the signal’s meaning. Hence, in order to explain the emergence of conventional meaning and to answer Quine’s challenge, it suffices to explain the emergence of the underlying signaling convention. A signaling convention may come into existence by explicit agreement. But it may also develop from precedents set by previous behaviour. Another possibility which was not yet seen by David Lewis is the slow evolution of signaling conventions driven by mechanisms similar to biological replication. The latter approach is part of a large body of research on the evolution of cultural phenomena. It derives part of its attractiveness from the lack of any assumptions about the rationality of agents, which sharply differentiates it from classical game theory.

Language, Games, and Evolution: An Introduction

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The research paradigm of evolutionary game theory dates back to the work of the evolutionary biologists George Price and John Maynard Smith in the nineteen seventies (Maynard Smith and Price 1973; Maynard Smith 1982). They noticed that Darwinian evolution has a strategic aspect which can aptly by modeled with game theoretic techniques. One of their standard examples is the Hawk-Dove game. Suppose two animals of the same species are competing for a resource, for instance a territory in a favourable habitat. Each contestant may adopt one of two strategies: – Hawk: Escalate the conflict until you are injured or the opponent retreats. – Dove: Display but retreat as soon as the opponent attacks. The payoff of the interaction is measured in terms of differential fitness. This is the change in the expected number of offspring that the contestants will have. Winning the territory increases fitness, but getting injured decreases it. The following payoff matrix captures these ideas: hawk dove hawk 1, 1 7, 2 dove 2, 7 3, 3 This is a strategic situation because the gain in fitness of an agent depends both on its own action and on the action of the other player. The crucial ideas of evolutionary game theory are (a) that games are played repeatedly between the members of a population, and (b) that strategies are assumed to be heritable dispositions, not the outcomes of rational decisions. Suppose a population consists almost exclusively of hawk players. Then dove players have, everything else being equal, on average a higher fitness than the population average. As a consequence, the proportion of dove players within the populuation will increase from one generation to the next. If, however, the proportion of dove players exceeds a certain threshold (80% for the current example), hawk players will have a higher fitness, and thus the hawk sub-population will grow faster then the dove sub-population. In the long run, the population will always converge to a state with 80% hawk players and 20% dove players. A population state that is an attractor for the evolutionary dynamics, like the 80/20 mixture in the example, was dubbed Evolutionarily Stable Strategy (ESS) by Maynard Smith and Price (1973). It can be characterized in game theoretic notions that differ only slightly from the concepts of rationalistic game theory, even though the underlying interpretation is quite different. The logic of evolutionary game theory does not depend on the causal mechanisms of biological replication. It is applicable in all scenarios where a certain strategic behavior is being replicated, and the likelihood of replication is correlated to the outcome of the strategic interaction. This also applies to cultural transmission of behavior via imitation, including communicative behavior. Let us return to Lewis style signaling games. Under an evolutionary interpretation, the chance that a certain communication strategy (of the sender or the receiver) is replicated is positively correlated with the average communicative success of this strategy within a population of communicating agents. If the

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number of meanings and forms coincide, the ESSs of a signaling game turn out to be exactly the signaling conventions (i.e. the pairs of strategies where the sender plays a bijection from meanings to forms and the receiver plays the inverse bijection). So even if a population starts in a state where there is little population wide correlation between meanings and forms in the signaling behaviour of the agents, the evolutionary dynamics may lead to a state where all agents use the same convention that leads to reliable communication. Linguistic semantics is concerned with the conventionalized aspects of interpretation, while pragmatics relates interpretation to language use. While there is thus a division of labor between these two sub-disciplines, there is no clear boundary. Many aspects of conventionalized semantics are not arbitrary but have a foundation in the demands of language use. Evolutionary game theory has been used to model how aspects of language use (like biased frequency distributions over meanings) shape aspects of conventional semantics. The emergence of linguistic meaning and pragmatic cooperation is addressed in the chapters of Thomas C. Scott-Philipps, Patrick Grim, Brian Skyrms, and Simon Huttegger & Kevin J. S. Zollman. All three major currents of research mentioned in the beginning can be developed on the basis of Lewis’s signaling games. In this volume, signaling games are applied in models of language evolution, to Gricean pragmatics, and in investigations into the reliability of linguistic signals, which is a common topic in the economic literature on language use. Lewis (1975) characterizes conventions pre-theoretically as “[. . . ] regularities in action, or in action and belief, which are arbitrary but perpetuate themselves because they serve some sort of common interest” (Lewis, 1975, p. 4). The common interest is to solve a coordination problem. The most natural examples are those where we have aligned preferences, and we use language to solve our problem how to act accordingly. But how can we account for communication if the preferences of participants in a discourse are not completely aligned? In an important article, the economists Crawford and Sobel (1982) show that in a Lewisian cheap talk game communication is possible only if the preferences of speaker and hearer are aligned. To establish this fact, no mention was made of any externally given meaning associated with the messages. What happens if we assume that these messages do have an externally given meaning, taken to be sets of situations? The interesting question is now not whether the game has equilibria in which we can associate meanings with the messages, but rather whether there exist equilibria where the messages are sent in a credible way. As it turns out, the old question concerning informative equilibria in signaling games without pre-existing meaning and the new one concerning credible equilibria in signaling games with messages that have a pre-existing meaning are closely related. Farrell (1988, 1993), Rabin (1990) and Stalnaker (2005) show that costless messages with a pre-existing meaning can be used to credibly transmit information if it is known by the receiver that it is in the sender’s interest to speak the truth. An important topic in game theoretic analyses of communication is how to account for the possibility of communication if the preferences are less harmonically

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aligned. Perhaps the first answer that comes to mind involves reputation and an element of reciprocity. These notions are standardly captured in terms of the theory of repeated games (e.g. Axelrod and Hamilton, 1981). The standard answer to our problem how communication can take place if the preferences are not perfectly aligned both in economics (starting with Spence, 1973) and in biology (Zahavi, 1975; Grafen, 1990; Hurd, 1995) doesn’t make use of such repeated games. Instead, it is assumed that reliable communication is also possible in these circumstances, if we assume that signals can be too costly to fake. Costs can come about in two ways: production costs and social costs. In animal communication it is normally production costs that count. Zahavi (1975) proposed an appealing explanation for why peacocks have such very long tails, and stags such enormous antlers. Individuals who can afford themselves such exaggerated traits must be strong individuals that can readily tolerate the survival costs of the trait. Consequently, Zahavi argued, females should pick males with these ‘handicaps’ because they have made it through a survival filter. So, by showing one’s handicap, an agent can communicate his true quality/ability in an honest way. Handicaps make honest communication possible, even if the preferences of the individuals involved are not fully aligned2 . Over the years, the use of this handicap principle has been extended from sexual selection to a number of other animal communicative behaviors. Noteworthy are the analyses of begging baby birds, alarm calls, and threat behavior of animals when contesting resources3. These analyses assume that there exists a variety of ways in which communicative displays can handicap the signaler. That is, in different signaling systems, the costs of messages are determined in different ways. In the original examples of sexual signaling by peacocks and others, the costs were merely costs of production. For threat display — when signals are sent for a strategic reason — however, the costs are not incurred by the producer (these are negligible), but by the receiver. The receiver can verify the message (by ignoring the threat, i.e. attacking), and the cost depends on the type of the sender: high quality senders do better (when attacked) than low quality senders. In these cases, the costs might be called social costs. Note that if costs play a role for natural language it is most naturally via these social costs. One of the central tasks of linguistic pragmatics is to explain how speakers can communicate more by their utterances than they literally say. For example, why does the speaker of the following sentence normally communicate that some, but not all, of the boys came to the party? (4) Some of the boys came to the party. Or, why does B in the following exchange communicate that the garage is open, if B is known to have expert knowledge on issues at hand (Grice, 1989, p. 31)? 2

3

The economist and sociologist Thorstein Veblen (1899) already suggested a similar explanation for the seemingly ridiculous squandering of resources by the wealthy classes he observed: The wealthy engage in conspicuous consumption in order to advertise, i.e. signal, their wealth. See Bergstrom’s http://octavia.zoology.washington.edu/handicap/

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(5) A: I am out of petrol. B: There is a garage round the corner. There exist a variety of game and decision theoretic explanations of these phenomena. Although these explanations share some basic assumptions about decision making, the whole field is less homogenous than one might expect. On the one side, we can distinguish approaches which are based on classical and evolutionary game theory, as well as approaches based on decision theory. On the other side, there are also considerable differences concerning the empirical assumptions which determine the choice and the setup of the models. Among the approaches based on classical game theory we can distinguish between, for example, the framework of Prashant Parikh (2001), the optimal answer model (Benz and van Rooij, 2007), and the recently developed iterated best response model (Franke, 2009; J¨ager and Ebert, 2009). Parikh assumes that semantic meaning is highly underspecified. As a consequence, the hearer can choose between the literal interpretation and any enrichment of it. The payoffs of speaker and hearer are not only determined by communicative success and the production costs of linguistic expressions, but any contextually relevant aspect may have an effect on payoffs. Parikh then assumes that speaker and hearer adopt the Pareto Nash equilibrium of the resulting signaling game. A Nash equilibrium is (weakly) Pareto optimal if there is no other Nash equilibrium that yields strictly better results for both the sender and receiver. In the optimal answer approach, underspecification of linguistic meaning plays no systematic role. It is assumed that the hearer chooses an action or a state of the world as interpretation based on the semantic meaning of an utterance. The implicature of uttering F is then explained by the indicated meaning of the speaker’s signaling act, i.e. it is identified with S −1 (F ). It thus spells out, in a game theoretic framework, Grice’s heuristic that implicatures are propositions which the speaker must have believed in if one assumes that he follows the cooperative and other pragmatic principles when making his statement. In Example (5), the implicature that the garage is open is inferred from the fact that learning that there is a garage around the corner will induce the hearer to go to this garage. If the speaker is fully cooperative and if it is assumed that he knows whether the garage is open or not, it follows that he must have known that it is open when making his utterance because otherwise his answer would not have been optimal. The iterated best response (IBR) model takes its starting point from considerations about the limited reasoning capabilities of speaker and hearer. As in the optimal answer model, the underspecification of linguistic meaning plays no significant role. Given signaling and interpretation behaviour determined by pure semantics, the IBR model takes into account the effect of each interlocutor’s reasoning about the other interlocutor’s strategy. As an example, let us consider 4. If the hearer starts out with the literal interpretation of the forms some and all, then she can choose between the states of the world all coming (θ∀ ) and some but not all coming (θ∃¬∀ ) when receiving the message some, and she must choose the state all coming (θ∀ ) when receiving the message all. This leads to the following strategy H0 :

Language, Games, and Evolution: An Introduction

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 some → θ∃¬∀ , θ∀ (1.1) all → θ∀ If the speaker assumes that the hearer chooses θ∃¬∀ and θ∀ with probability 12 when receiving message some, then his best response to this strategy is to send some if some but not all came, and all if all came. This leads to strategy S1 :   θ∃¬∀ → some S1 = (1.2) θ∀ → all H0 =

If the hearer knows that the speaker will reason like this and adopt S1 , then she can infer that the following strategy H2 is her best response to the speaker’s signaling behaviour:   some → θ∃¬∀ H2 = (1.3) all → θ∀ Now, it is again the speaker’s turn. But he will notice that his best response to H2 equals his best response to H0 , hence S3 = S1 . Correspondingly, the hearer will notice that her best response to S3 is the same as her best response to S1 , hence H4 = H2 . This ends their reasoning about each other as the strategy pair (S1 , H2 ) is known to be stable in the sense that no further reasoning about each other can lead to a different equilibrium. But (S1 , H2 ) is such that the speaker signals some if some but not all came, and the hearer interprets it as such, hence, the implicature in (4) is explained. One of the first papers applying evolutionary game theory to pragmatics was (van Rooij, 2004a). This approach was further developed by Gerhard J¨ager who worked out detailed evolutionary models of a variety of linguistic phenomena (J¨ ager, 2007a,b). Purely decision theoretic models where proposed by Arthur Merin (1999) and e.g. in (van Rooij, 2004b; Schulz and van Rooij, 2006).

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Overview

This volume collects eight articles addressing various topics on game theoretic approaches to language. They divide into two groups. The first group, consisting of the four papers by Thomas C. Scott-Phillips, Patrick Grim, Brian Skyrms, and Simon M. Huttegger & Kevin J. S. Zollman, address questions of language evolution. The second group, consisting of the papers of Anton Benz, Kris De Jaegher & Robert van Rooij, Michael Franke & Tikitu de Jager, and James German, Eyal Sagi, Stefan Kaufmann, and Brady Clark, address synchronic phenomena. As the latter group uses the more familiar frameworks of classical game theory, we start with this group. The first chapter ‘How to set up normal optimal answer models’ by Anton Benz investigates the rules for constructing a game theoretic model for a given discourse. He is especially interested in examples involving preference statements as in: (6) Common context: Peter can only afford cheap wine. Peter: Where can I buy red wine? Bob: At the Wine Centre. +> Peter can buy red wine at a low price at the Wine Centre.

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He addresses the question of how to determine the set of possible worlds Ω and their probabilities, and how to define the action sets and their utilities. Only part of the answers to these questions is explicitly stated in (6). The construction rules are based on multi-attribute utility analysis and a number of normality assumptions. For example, the set Ω will be constructed as a set of attributevalue functions, and the normality assumptions include e.g. assumptions about the independence of elementary events and the even distribution of probabilities. Vagueness is a pervasive feature of natural language. This has often been regarded as a deficiency, especially by scientists, logicians, and philosophers. However, it seems that the use of vague expressions is not such a bad thing, and might even be beneficial compared to their precise counterparts. Kris De Jaegher & Robert van Rooij in the chapter on ‘Strategic vagueness, and appropriate contexts’ discuss game-theoretic rationales for vagueness, and for the related concepts of ambiguity and generality. As it turns out, ‘standard’ game theoretical models can only predict vagueness to be beneficial in case they assume a conflict of interest. But vagueness is not limited to these contexts. In order to explain vagueness in general, the authors propose that various forms of ‘bounded rationality’ should play a role. They discuss the costs of using expressions, and a limitation of the contexts in which a vague word can be used appropriately. Finally, they relate vagueness to Prospect Theory, one of the best-known theories of bounded rationality. In standard semantics, a question like ‘Did you leave the keys in the car?’ is modeled as a request for information, and a statement like ‘You might have left your keys in the car’ is counted as true iff it is compatible with the state of affairs that the keys were left in the car. Michael Franke & Tikitu de Jager in the chapter ‘Now that you mention it: awareness dynamics in discourse and decisions’ notice that this semantics doesn’t account for the use we make of such sentences. What such sentences do, intuitively, is to make the addressee aware of possibilities they were unaware of before. The authors propose a pragmatic model for the effects of becoming aware of possibilities in pragmatics. According to Franke and de Jager, unawareness is different from uncertainty; it is not introspective, i.e. if you are unaware of where the keys are, it doesn’t mean that you hold the explicit belief that the keys are nowhere to be found, and it can be easily overridden. The authors provide a model of the dynamic update effects of being made aware of something. They notice that typically, we don’t just make the addressee aware of new possibilities for their own sake. Utterances that do so typically constitute advice as well, in the above case the advice to search the car. To account for this extra effect, they propose that the dynamics changes the decision problem of the addressee such that searching the car becomes the best action to perform after the awareness update. ‘The Role of Speaker Beliefs in Determining Accent Placement’ by James German, Eyal Sagi, Stefan Kaufmann, and Brady Clark adds a gametheoretical approach to the classic discussions on the relationship of focus marking and nuclear accent placement. Based on previous empirical work, the authors consider cases of mismatches between the actual placement of nuclear accents

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by speaker and the predictions of contemporary theories. As they argue, such deviations from grammar are expected under a game-theoretic view of utility maximization that takes into account prior beliefs as well as utterance and processing costs of the speaker and hearer. They devise a game-theoretic model of accent placement with a Pareto Nash Equilibrium solution concept and they give a precise formal characterization of the dominance relation over the strategies in terms of conditions on beliefs and costs. The second group of papers starts with the chapter on ‘Evolutionarily stable communication and pragmatics’ by Thomas C. Scott-Phillips. He explores possible answers to the question of how signaling can remain evolutionarily stable given that individuals have incentives to be dishonest. He distinguishes between three aspects of being cooperative: (1) material cooperation which means that interlocutors pursue the same extra-linguistic goals; (2) the communicative cooperation which means that interlocutors must agree on the same meaning of signals; and (3) the informative cooperation which means that the interlocutors are honest in their use of linguistic signals. Communicative cooperation is a prerequisite for conventional meaning, and informative cooperation for mutual trust. He discusses three factors which can ensure the evolutionary stability of communicative and informative cooperation: (1) causal relationships between signals and meanings, which he calls indexes; (2) causal relationships between the costs of signals and their meanings, which he calls handicaps; and (3) costs associated with dishonesty, which he calls deterrents. Patrick Grim in the chapter ‘Simulating Grice: emergent pragmatics in spatialized game theory’ investigates the possibilities for emerging conversational conventions in spatialized game theory. In Grim’s spatial models, agents are arranged on a two-dimensional lattice. They do not interact randomly with all other agents in the population but only with their immediate neighbours. Strategies spread in the population by imitation of the most successful neighbour. In his study of the emergence of Gricean maxims, Grim assumes an environment of random events of local significance, which agents can communicate to each other, and in response to which they have to act appropriately. The meaning of signals is pre-defined. The maxims itself are encoded by probabilistic constraints on the sender’s signaling behaviour; e.g. the maxim of quality is encoded by a probability with which the sender will choose a true signal. Grim presents simulations in which the population converges on truth telling and the maximisation of relevance. Hence, the dynamics of local interaction, self-advantage, and imitation of successful neighbours is sufficient to produce conventions of quality, quantity, and relevance without presupposing the highly reflective agents which standard Gricean pragmatics assumes. Grim shows that even primitive forms of implicatures can emerge in his models. Simon Huttegger & Kevin J. S. Zollman in the Chapter ‘Signaling games: dynamics of evolution and learning’ discuss the evolutionary dynamics of signaling games. The authors start with a closer investigation of the dynamics as given by the replicator equations and a perturbation thereof, the selectionmutation dynamics. Taking these two models as baseline cases for comparison,

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they extend their discussion to more sophisticated dynamic models, in particular models of learning in signaling games. While there are some close similarities between learning and evolutionary models, Huttegger & Zollman conclude that there are many cases of learning rules that always converge to a non-pertubed signaling system while there is no version of the replicator dynamics that could model this result. Brian Skyrms in the chapter ‘Pragmatics, logic and information processing’ takes up the discussion on the conventionality of logic going back to Quine (1936). Skyrms approaches the issue by asking whether some rudimentary form of logic can evolve out of information transfer in sender-receiver games. He presents a slightly modified version of Lewis’s (1969) signaling games in which the sender may not observe the state of the world exactly. In a population of monkeys, for example, this may lead to the evolution of signals of the form ‘leopard or snake’ which may be an optimal warning signal when a member of the population discovers the danger but cannot see what threatens them exactly. If a second monkey then signals ‘No snake’, and the whole troop runs, then obviously the members of troop performed a simple form of logical conclusion. Skirms calls signals of the form ‘leopard or snake’ proto truth-functional. He then addresses the question whether proto truth-functions can co-evolve with logical inference, or whether they have already to exist.

References Axelrod, R., Hamilton, W.: The evolution of cooperation. Science 211(4489), 1390 (1981) Benz, A., van Rooij, R.: Optimal assertions and what they implicate: a uniform game theoretic approach. Topoi - an International Review of Philosophy 27(1), 63–78 (2007) Clark, H.H.: Using Language. Cambridge University Press, Cambridge (1996) Crawford, V., Sobel, J.: Strategic Information Transmission. Econometrica: Journal of the Econometric Society 50(6), 1431–1451 (1982) Farrell, J.: Communication, Coordination and Nash Equilibrium. Economic Letters 27(3), 209–214 (1988) Farrell, J.: Meaning and credibility in cheap-talk games. Games and Economic Behavior 5, 514–531 (1993) Franke, M.: Signal to Act: Game Theory in Pragmatics. PhD thesis, Universiteit van Amsterdam. ILLC Dissertation Series DS-2009-11 (2009) Grafen, A.: Biological signals as handicaps. Journal of theoretical biology 144(4), 517– 546 (1990) Grice, H.P.: Studies in the Way of Words. Harvard University Press, Cambridge (1989) Hurd, P.: Communication in discrete action-response games. Journal of Theoretical Biology 174(2), 217–222 (1995) J¨ ager, G.: Evolutionary Game Theory and Typology. A Case Study. Language 83(1), 74–109 (2007a) J¨ ager, G.: The evolution of convex categories. Linguistics and Philosophy 30(5), 551– 564 (2007b) J¨ ager, G., Ebert, C.: Pragmatic Rationalizability. In: Riester, A., Solstad, T. (eds.) Proceedings of Sinn und Bedeutung, vol. 13 (2009)

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Lewis, D.: Convention. Harvard University Press, Cambridge (1969) Lewis, D.: Languages and Language. In: Gunderstone, K. (ed.) Minnesota studies in the philosophy of science, VII, vol. 1, pp. 163–188. University of Minnesota Press (1975); reprinted in his Philosophical Papers I Maynard Smith, J.: Evolution and the Theory of Games. Cambridge University Press, Cambridge (1982) Maynard Smith, J., Price, G.R.: The logic of animal conflict. Nature 246(5427), 15–18 (1973) Merin, A.: Information, Relevance, and Social Decisionmaking: Some Principles and Results of Decision-Theoretic Semantics. In: Moss, L., Ginzburg, J., de Rijke, M. (eds.) Logic, Language, and Information, vol. 2, pp. 179–221. CSLI Publications, Stanford (1999) Parikh, P.: Situations, Games, and Ambiguity. In: Cooper, R., Mukai, K., Perry, J. (eds.) Situation Theory and its Applications, vol. 1, pp. 449–469. CSLI Lecture Notes, Stanford (1990) Parikh, P.: The Use of Language. CSLI Publications, Stanford (2001) van Quine, W.O.: Truth by Convention. In: Lee, O.H. (ed.) Philosophical Essays for Alfred North Whitehead, pp. 90–124. Longmans, Green and Co., New York (1936) Rabin, M.: Communication between Rational Agents. Journal of Economic Theory 51, 144–170 (1990) Rousseau, J.-J.: Discours sur l’origine et les fondements de l’in´egalit´e parmi les hommes. Marc-Michel Rey, Amsterdam (1755) Rubinstein, A.: Economics and Language. Cambridge University Press, Cambridge (2000) Schulz, K., van Rooij, R.: Pragmatic Meaning and Non-Monotonic Reasoning: The Case of Exhaustive interpretation. Linguistics and Philosophy 29, 205–250 (2006) Searle, J.R.: Speech Acts. Cambridge University Press, Cambridge (1969) Spence, A.M.: Job market signaling. The quarterly journal of Economics 87(3), 355–374 (1973) Stalnaker, R.: Saying and meaning, cheap talk and credibility. In: Benz, A., J¨ ager, G., Van Rooij, R. (eds.) Game Theory and Pragmatics, pp. 82–101. Palgrave Macmillan, China (2005) van Rooij, R.: Signalling Games select Horn Strategies. Linguistics and Philosophy 27, 493–527 (2004a) van Rooij, R.: Utility of Mention-Some Questions. Research on Language and Computation 2, 401–416 (2004b) Veblen, T.: The theory of the leisure class: An economic theory of institutions. Macmillan, New York (1899) Zahavi, A.: Mate selection–a selection for a handicap. Journal of theoretical Biology 53(1), 205–214 (1975)

How to Set Up Normal Optimal Answer Models Anton Benz Zentrum f¨ ur Allgemeine Sprachwissenschaft, Berlin

Abstract. We investigate the role of multi-attribute utility analyses in game theoretic models of Gricean pragmatics, i.e. for finding a model of the linguistic context of an utterance and for the calculation of implicatures. We investigate especially relevance implicatures of direct answers. The work is based on the optimal answer model (Benz, 2006; Benz & v. Rooij, 2007). We argue that multi-attribute utility functions play an essential role in finding the appropriate models. We concentrate especially on default assumptions which are necessary in order to calculate the correct implicatures. As normality assumptions play a central role in the construction of these models, we call them normal optimal answer models. We introduce rules which provide guidelines for setting up these models.

1

Introduction

This paper can be characterised as a preparatory study for a theory of relevance implicatures in discourse. In order to set up such a theory, we need to know how to construct a game theoretic model of an utterance situation from a given discourse. We argue that multi-attribute utility functions play an essential role in this process. The core examples are of the following form: (1) Peter: I have to buy wine for our dinner banquet. I will get into trouble with our secretary if I spend too much money on it. We still have some white wine. Where can I buy red wine? Bob: At the Wine Centre. +> Peter can buy red wine at a low price at the Wine Centre. The root question in (1) makes no reference to the price of wine. Nevertheless, the contextually given objective of buying wine at a low price has an impact on the implicature of the answer. These objectives are provided by the linguistic context which stands in a background relation to the question. In (1), we find a domain object d, the wine shop, and two relevant attributes, S( . ) for selling wine, and L(.) for selling at low prices. In addition, the inquirer can just perform a random search with expected utility 0 < ε < 1. The essential parameters of the answering situation are shown in Table 1. Going to shop d is an optimal choice in world w1 only. Hence, the inquirer can infer from the recommendation to go to d that the actual world is w1 . An obvious question is, how do we arrive at the game theoretic model starting from the discourse in (1)? This question involves first of all the question how to A. Benz et al. (Eds.): Language, Games, and Evolution, LNAI 6207, pp. 14–39, 2010. c Springer-Verlag Berlin Heidelberg 2010 

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Table 1. The table for Example (1) Ω S(d) L(d) search Good(d) w1 1 1 ε yes w2 1 0 ε no w3 0 1 ε no w4 0 0 ε no

determine the set of possible worlds Ω. Then, we have to ask, how do we know the speakers and hearers knowledge about Ω? How do we know their utilities over outcomes and the actions between which the hearer can choose? Only part of the answers to these questions is explicitly stated in (1). In this paper, we propose a procedure for constructing a game theoretic model from a discourse which provides answers to these questions. This procedure is based on a multiattribute utility analysis and some default rules. For example, the set Ω will be constructed as the set of all attribute-value assignments, and the default rules introduce assumptions about the independence of elementary events and the even distribution of probabilities. The paper divides into three parts. First, we will introduce the game theoretic framework on which we base our analysis. This is the Optimal Answer (OA) model. The version presented here is slightly more general than (Benz, 2006; Benz & v. Rooij, 2007). The second part, Section 3, contains the main body of results. We introduce the multi-attribute utility theory and use it for analysing an extended example. Based on this analysis, we generalise the procedure used there and introduce a general prescript for constructing optimal answer models. Finally, in the last section, we show how to use normal optimal answer models for deriving systematic classifications of dialogue situations and calculating the associated lists of optimal answers. In particular, we provide a complete list of all situations in which the addressee has to choose a domain object based on preferences defined by two attributes.

2

The Optimal-Answer Model

Grice (1989, p. 26) characterised conversation as a cooperative effort. The contributions of the interlocutors are not isolated sentences but normally subordinated to a joint purpose. In this paper, we will always assume that questioning and answering is embedded in a decision problem in which the inquirer has to make a choice between a given set of actions. His choice of action depends on his preferences regarding their outcomes and his knowledge about the world. The answer helps the inquirer in making his choice. The quality of an answer depends on the action to which it will lead. The answer is optimal if it induces the inquirer to choose an optimal action. We model answering situations as two-player games. We call the player who answers the expert E, and the player who receives the answer the inquirer I. For Grice, the information communicated by an answer divides into two parts, the semantic meaning of the answer and its implicated meaning. In our definition

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of implicature, which we provide later, we closely follow Grice’s original idea that implicatures arise from the additional information that an utterance provides about the state of the speaker: “. . . what is implicated is what is required that one assume a speaker to think in order to preserve the assumption that he is observing the Cooperative Principle (and perhaps some conversational maxims as well), . . . ” (Grice, 1989, p. 86) In a game theoretic model, what the speaker utters is determined by his strategy s, i.e. a function that selects a sentence for each of his possible information states. When the inquirer receives answer F , then he knows that the expert must have been in a state K which is an element of s−1 (F ) = {K | s(K) = F }, i.e. the set of all states which are mapped to F by s. Lewis (2002, p. 144) calls this the indicated meaning of a signal F . We identify the implicature of an utterance with this indicated information. This identification implies that, once we know s, the implicatures can be calculated. Hence, all depends on how we can know the speaker’s strategy s. This knowledge will be provided by the Optimal-Answer (OA) Model and its later modifications. 2.1

Optimal Answers

The OA model tells us which answer a rational language user will choose given the inquirer’s decision problem and his own knowledge about the world. Instead of introducing full signalling games (Lewis, 2002), we reduce our models to the cognitively relevant parameters of an answering situation. We call these simplified models support problems. They consist of the inquirer’s decision problem and the answering expert’s expectations about the world. They incorporate the Cooperative Principle, the maxim of Quality, and a method for finding optimal strategies which replaces the maxims of Quantity and Relevance. In this section, we ignore the maxim of Manner. A decision problem consists of a set Ω of the possible states of the world, the decision maker’s expectations about the world, a set of actions A he can choose from, and his preferences regarding their outcomes. We always assume that Ω is finite. We represent an agent’s expectations about the world by a probability distribution over Ω, i.e. a real valued function  P : Ω → R with the following properties: (1) P (v) ≥ 0 for all v ∈ Ω and (2) v∈Ω P (v) = 1. For sets A ⊆ Ω it is P (A) = v∈A P (v). The pair (Ω, P ) is called a finite probability space. An agent’s preferences regarding outcomes of actions are represented by a real valued function over action-world pairs. We collect these elements in the following structure: Definition 1. A decision problem is a triple (Ω, P ), A, u such that (Ω, P ) is a finite probability space, A a finite, non-empty set and u : Ω × A → R a function. A is called the action set, and its elements actions; u is called a payoff or utility function.

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In the following, a decision problem (Ω, P ), A, u represents the inquirer’s situation before receiving information from an answering expert. We will assume that this problem is common knowledge. How do we find a solution to a decision problem? It is standard to assume that rational agents try to maximise their expected utilities. The expected utility of an action a is defined by:  P (v) × u(v, a). (1) EU (a) = v∈Ω

The expected utility of actions may change if the decision maker learns new information. To determine this change of expected utility, we first have to know how learning new information affects the inquirer’s beliefs. In probability theory the result of learning a proposition A is modelled by conditional probabilities. Let H be any proposition and A the newly learned proposition. Then, the probability of H given A, written P (H|A), is defined as P (H|A) := P (H ∩ A)/P (A) for P (A) = 0.

(2)

In terms of this conditional probability function, the expected utility after learning A is defined as  EU (a|A) = P (v|A) × u(v, a). (3) v∈Ω

I will choose the action which maximises his expected utilities after learning A, i.e. he will only choose actions a where EU (a|A) is maximal. We assume that I’s decision does not depend on what he believes that the answering expert believes. We denote the set of actions with maximal expected utility by B(A), i.e. B(A) := {a ∈ A | ∀b ∈ A EUI (b|A) ≤ EUI (a|A)}.

(4)

The decision problem represents the inquirer’s situation. In order to get a model of the questioning and answering situation, we have to add a representation of the answering expert’s information state. We identify it with a probability distribution PE that represents his expectations about the world: Definition 2. A five-tuple σ = Ω, PE , PI , A, u is a support problem if (Ω, PE ) is a finite probability space and Dσ = (Ω, PI ), A, u a decision problem such that there exists a probability distribution P on Ω, and sets KE ⊆ KI ⊆ Ω for which PE (X) = P (X|KE ) and PI (X) = P (X|KI ). The last condition says that PE and PI are derived from a common prior P by Bayesian update. It entails: ∀X ⊆ Ω PE (X) = PI (X|KE ).

(5)

This condition allows us to identify the common ground in conversation with the addressee’s expectations about the domain Ω, i.e. with PI . The speaker knows the addressee’s information state and is at least as well informed about Ω. Hence, the assumption is a probabilistic equivalent to the assumption about common

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ground that implicitly underlies dynamic semantics (Groenendijk & Stockhof, 1991). Furthermore, condition (5) implies that the expert’s beliefs cannot contradict the inquirer’s expectations, i.e. for A, B ⊆ Ω: PE (A) = 1 ⇒ PI (A) > 0. The expert E’s task is to provide information that is optimally suited to support I in his decision problem. Hence, we find two successive decision problems, in which the first problem is E’s problem to choose an answers. The utility of the answer depends on how it influences I’s final choice: Inquirer I Asks ↓ • ↑ expectations of I Ω, PI 

Expert E answers ↓ Q? −→

• ↑ expectations of E Ω, PE 

I decides for action ↓ A −→

• ↑ expectations of I Ω, PI ( . |A)

Evaluation ↓ a −→

• ↑ utility measure u(v, a)

We assume that E is fully cooperative and wants to maximise I’s final success; i.e. E’s payoff, is identical with I’s. This is our representation of Grice’s Cooperative Principle. E has to choose an answer that induces I to choose an action that maximises their common payoff. In general, there may exist several equally optimal actions a ∈ B(A) which I may choose. Hence, the expected utility of an answer depends on the probability with which I will choose the different actions. We can assume that this probability is given by a probability measure h(.|A) on A. Then, the expected utility of an answer A is defined by:  EUE (A) := h(a|A) × EUE (a). (6) a∈B(A)

We add here a further Gricean maxim, the Maxim of Quality. We call an answer A admissible if PE (A) = 1. The Maxim of Quality is represented by the assumption that the expert E does only give admissible answers. This means that he believes them to be true. For a support problem σ = Ω, PE , PI , A, u we set: Admσ := {A ⊆ Ω | PE (A) = 1}

(7)

Hence, the set of optimal answers in σ is given by: Opσ := {A ∈ Admσ | ∀B ∈ Admσ EUE (B) ≤ EUE (A)}.

(8)

We write Ophσ if we want to make the dependency of Op on h explicit. Opσ is the set of optimal answers for the support problem σ. As answers are propositions in our model, i.e. sets A ⊆ Ω, it trivially follows that all propositions can be expressed. The behaviour of interlocutors can be modelled by strategies. A strategy is a function which tells us for each information state of an agent which actions he may choose. It is not necessary that a strategy picks out a unique action for each information state. A mixed strategy is a strategy which chooses actions

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with certain probabilities. The hearer strategy h(.|A) is an example of a mixed strategy. We define a (mixed) strategy pair for a support problem σ to be a pair (s, h) such that s is a probability distribution over P(Ω) and h(.|A) a probability distribution over A. We may call a strategy pair (s, h) a solution to σ iff h(.|A) is a probability distribution over B(A), and s a probability distribution over Ophσ . In general, the solution to a support problem is not uniquely defined. Therefore, we introduce the notion of the canonical solution. Definition 3. Let σ = Ω, PE , PI , A, u be a support problem. The canonical solution to σ is a pair (S, H) of mixed strategies which satisfy:   |Opσ |−1 , A ∈ Opσ |B(A)|−1 , a ∈ B(A) S(A) = . (9) , H(a|A) = 0 otherwise 0 otherwise We write S( . |σ) if S is a function that maps each σ ∈ S to the speaker’s part of the canonical solution, and H( . |Dσ ) if H is a function that maps the associated decision problem Dσ to the hearer’s part of the canonical solution. From now on, we will always assume that speaker and hearer follow the canonical strategies S(.|σ) and H(.|Dσ ). We make this assumption because it is convenient to have a unique solution to a support problem; the only property that we really need in the following proofs is that H(a|A) > 0 ⇔ a ∈ B(A) and S(A|σ) > 0 ⇔ A ∈ Opσ . The expert may always answer everything he knows, i.e. he may answer KE := {v ∈ Ω | PE (v) > 0}. From condition (5) it trivially follows that B(KE ) = {a ∈ A | ∀b ∈ A EUE (b) ≤ EUE (a)}. If expert and inquirer follow the canonical solution, then: Opσ = {A ∈ Adm | B(A) ⊆ B(KE )}. (10) In order to show (10), let A ∈ Adm and α := max{EUE (a) | a ∈ A}. For a ∈ B(A) \ B(KE ) it holds by definition that EUE (a) < α and H(a|A) > 0. EUE (A) is the sum of all H(a|A) × EUE (a). If B(A) ⊆ B(KE ), then this sum divides into the sum over all a ∈ B(A)\B(KE ) and all a ∈ B(A)∩B(KE ). Hence, EUE (A) < α, and therefore A ∈ Opσ . If B(A) ⊆ B(KE ), then the answering expert knows that answering A would induce the addressee to choose a sub-optimal action with positive probability. Hence, we can call an answer A misleading if B(A) ⊆ B(KE ); then, (10) implies that Opσ is the set of all non-misleading answers. 2.2

Implicatures of Optimal Answers

An implicature of an utterance is a proposition which is implied by the assumption that the speaker is cooperative and observes the conversational maxims. More precisely, Grice linked implicatures to what the hearer learns from the utterance about the speaker’s knowledge. The speaker’s canonical solution maps his possible information states to utterances. Hence, the hearer can use this strategy to calculate what the speaker must have known when making his utterance.

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As the canonical solution is a solution, it also incorporates the information that the speaker is cooperative and follows the maxims. We treat all implicatures as particularised implicatures, i.e. as implicatures that follow immediately from the maxims and the particular circumstances of the utterance context. The answering expert knows a proposition H in a situation σ iff PEσ (H) = 1. Hence, if the inquirer wants to know what the speaker knew when answering that A, he can check all his epistemically possible support problems for what the speaker believes in them. If σ is the support problem which represents the actual answering situation, then all support problems σ ˆ with the same decision problem Dσ are indiscernible for the inquirer. Hence, the inquirer knows that the speaker believed that H when making his utterance A, iff the speaker believes that H in all indiscernible support problems in which A is an optimal answer. This leads to the following definition: Definition 4 (Implicature). Let S be a given set of support problems with joint decision problem (Ω, PI ), A, u. Let σ ∈ S, A, H ⊆ Ω be two propositions with A ∈ Opσ . Then we set: A +>σ H ⇔ ∀ˆ σ ∈ S (A ∈ Opσˆ → PEσˆ (H) = 1),

(11)

If A +>σ H, we say that the utterance of A implicates that H in σ. The definition entails: If A ⊆ H, then A +>σ H.

(12)

Hence, our use of ‘implicate’ deviates from the common usage in which implicated information is something extra in addition to entailed information. In our usage, entailed information is part of the implicated information. This is just a matter of convenience. If no confusion can arise, we simply drop the subscript σ in +>σ . As the hearer has to check all support problems in S, it follows that we arrive at the more implicatures the smaller S becomes. If S = {σ} and A ∈ Opσ , then A will implicate everything the speaker knows. The other extreme is the case in which answers implicate only what they entail. We show in Proposition 7 that this case can occur. We are interested in cases in which the speaker is a real expert. If he is an expert, then we can show that there is a very simple criterion for calculating implicatures. We can call the speaker an expert if he knows the actual world; but we will see that a weaker condition is sufficient for our purposes. To make precise what we mean by expert, we introduce another important notion, the set O(a) of all worlds in which an action a is optimal: O(a) := {w ∈ Ω | ∀b ∈ A u(w, a) ≥ u(w, b)}.

(13)

We say that the answering person is an expert for a decision problem if there is an action which is an optimal action in all his epistemically possible worlds. We represent this information in S: Definition 5 (Expert). Let S be a set of support problems with joint decision problem (Ω, PI ), A, u. Then we call E an expert in a support problem σ if ∃a ∈ A PEσ (O(a)) = 1. He is an expert in S, if he is an expert in every σ ∈ S.

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This leads us to the following criterion for implicatures: Lemma 6. Let S be a set of support problems with joint decision problem (Ω, PI ), A, u. Assume furthermore that E is an expert for every σ ∈ S and that ∀v ∈ Ω ∃σ ∈ S PEσ (v) = 1. Let σ ∈ S and A, H ⊆ Ω be two propositions with A ∈ Opσ . Then, with A∗ := {v ∈ Ω | PI (v) > 0}, it holds that:  A +> H iff A∗ ∩ O(a) ⊆ H. (14) a∈B(A)

Proof. We first show that (∃a ∈ A PEσ (O(a)) = 1 & A ∈ Opσ ) ⇒ ∀a ∈ B(A) : PEσ (O(a)) = 1.

(15)

= 1 and PEσ (O(b)) <  1. Then EUEσ (b) = Let a, b be such that PEσ (O(a))   σ σ σ P (v) · u(b, v) < P (v) · u(a, v) + v∈O(a) E v∈O(a)∩O(b) E v∈O(a)\O(b) PE (v) · σ σ u(a, v) = EUE (a). With KE := {v ∈ Ω | PE (v) > 0} it follows that b ∈ B(KE ), and by (10) that PEσ (O(b)) = 1.  b ∈ B(A). Hence, b ∈ B(A) implies + ∗ Let A := a∈B(A) O(a). We first show that A ∩ A+ ⊆ H implies A +> H. Let σ ˆ ∈ S be such that PEσˆ (H) = 1. By (15)  that A ∈ Opσˆ . We have toσˆshow σ ˆ + σ ˆ ∗ PE (A ) = PE ( a∈B(A) O(a)) = 1 and by (5) PE (A ) = 1; hence PEσˆ (A+ ∩A∗ ) = 1, and it follows that PEσˆ (H) = 1. Next, we show A +> H implies A∗ ∩ A+ ⊆ H. Suppose that A∗ ∩ A+ ⊆ H. Let w ∈ A∗ ∩ A+ \ H. From condition ∀v ∈ Ω ∃ˆ σ ∈ S PEσˆ (v) = 1 it follows that σ ˆ there is a support problem σ ˆ such that PE (w) = 1. As w ∈ A+ , it follows by (10) that A ∈ Opσˆ . Due to A +> H, it follows that PEσˆ (H) = 1, in contradiction to w ∈ H. A∗ is the equivalent to the common ground updated with A. In the context of a support problem, we can interpret an answer A as a recommendation to choose one of the actions in B(A). We may say that the recommendation is felicitous only if all recommended actions are optimal. Hence, A+ represents the information that follows from the felicity of the speech act of recommendation ∗ which is associated to the  answer. It should  also be mentioned that B(A) = B(A ) by Definition 4; hence a∈B(A) O(a) = a∈B(A∗ ) O(a) It is not uninteresting to see that the expert assumption on its own does not guarantee that an utterance has non-trivial implicatures. There are sets S in which the conditions of Lemma 6 hold but in which answers only implicate what they entail: Proposition 7. Let S be a set of support problems with joint decision problem σ (Ω, PI ), A, u. Assume that for all X ⊆ Ω, X = ∅ : ∃σ ∈ S KE = X and ∃a ∈ A O(a) = X. Then, for all σ ∈ S with A ∈ Opσ it holds ∀H ⊆ Ω : A +>σ H ⇔ A∗ ⊆ H. Proof. Condition ∀X = ∅ ∃a ∈ A O(a) = X trivially entails that E is an expert σ for all σ ∈ S. Condition ∀X = ∅ ∃σ ∈ S KE = X entails the second condition of

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Lem. 6: ∀v ∈ Ω ∃σ ∈ S PEσ (v) = 1. Then, let A ∈ Opσ and let a∗ be  such that ∗ ∗ ∗ O(a ) = A ; as B(A) = B(A ), it follows that {O(a)|a ∈ B(A)} = {O(a)|a ∈ B(A∗ )} = O(a∗ ) = A∗ . Hence, by Lem. 6, A +>σ H iff A∗ ⊆ H. This proposition also shows that the conditions of Lemma 6 are less restrictive than they might seem to be. 2.3

Examples

In this section we consider two examples. In both, the answering expert knows the actual state of affairs. This means that we can use Lemma 6 for calculating implicatures. For more examples, we refer to (Benz & v. Rooij, 2007). We start with the classical Out-of-Petrol example (Grice, 1989, p. 31): (2) I : I am out of petrol. E : There is a garage round the corner. (G) +> The garage is open. (H) We can assume that B’s assertion is an answer to the question “Where can I buy petrol for my car?” We distinguish four worlds {w1 , w2 , w3 , w4 } and two actions {go-to-g, search}. Let d be the place of the garage. Let G(d) mean that d is a petrol station, and H(d) that the place is open. Let the worlds and utilities be defined as shown in the following table: Ω w1 w2 w3 w4

G(d) H(d) go-to-d search + + 1 ε + − 0 ε − + 0 ε − − 0 ε

The answering expert knows that he is in w1 . We assume that PI and ε are such that EUI (go-to-d|G(d)) > ε, i.e. the inquirer thinks that the expected utility of going to that garage is higher than doing a random search in the town. Hence B(G(d)) = {go-to-d}. We see that O(go-to-d) = {w1 } ⊆ H(d). Hence, by Lem. 6, it follows that G(d) +> H(d). As second example, we choose an example with several equally useful answers which make the speaker choose different actions: (3) Somewhere in the streets of Amsterdam... a) I: Where can I buy an Italian newspaper? b) E: At the station and at the Palace but nowhere else. (S) c) E: At the station. (IN(s)) / At the Palace. (IN(p)) IN(d) stand for d has Italian newspapers. The answers IN(s), IN(p) and S are equally useful with respect to conveyed information and the inquirer’s goals. The answer in b) is called strongly exhaustive; it tells us for every location whether we can buy there an Italian newspaper or not. The answers IN(s) and IN(p) are called mention-some answers. All answers are optimal, and neither IN(s) implicates that ¬IN(p), nor IN(p) that ¬IN(s). There are two relevant domain objects, the palace p and the station s. We arrive at a model with four worlds and the utilities shown in the following table:

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Ω IN(p) IN(s) go-to-p go-to-s search w1 + + 1 1 ε w2 + − 1 0 ε w3 − + 0 1 ε w4 − − 0 0 ε We assume that the probabilities of IN(p) and IN(s) are equal, and that 0 < ε < 1. For d = p, s, it is EUI (go-to-d|IN(d)) = 1 > ε and B(IN(d)) = {go-to-d}. We find O(go-to-p) = {w1 , w2 } = [[IN(p)]] and O(go-to-s) = {w1 , w3 } = [[IN(s)]]. As [[IN(p)]] ⊆ [[¬IN(s)]] and [[IN(s)]] ⊆ [[¬IN(p)]], it follows by Lemma 6 that neither IN(p) implicates ¬IN(s), nor IN(s) implicates ¬IN(p).

3

Normal Optimal Answer Models

In this section, we address the question of how to find an OA model given the information we find in a discourse like (1). We will see that normality assumptions play a crucial role in constructing these models. We therefore call these OA models normal OA models. In our introductory example (1), the implicature depends on contextually stated preferences for certain attributes of domain objects. In this section we introduce the multi-attribute utility theory1 (MAUT), a sub-field of applied decision theory which studies decision problems with multiple objectives. We apply it to modelling desires as e.g. expressed by ‘I want to buy cheap red wine.’ We will see how the utility function of the OA model can be replaced by a Boolean combination of certain elementary events which describe the success conditions of actions in terms of basic attributes. This allows us to succinctly represent payoff functions by predicates Good which take domain objects as their arguments. For example, in case of ‘I want to buy cheap red wine’ this predicate is defined by Good (d) ⇔ Italian-wine(d) ∧ cheap(d). After having justified this representation, we study an extended example and show how MAUT and a number of normality assumptions enter into the construction of OA models. On the basis of this example, we then describe a general procedure for constructing normal OA models. 3.1

Multiple Attributes and Utilities

In a game or decision theoretic model preferences are represented by a utilityor payoff function. It assigns to every outcome of the game or decision problem a utility value which represents the overall preferences of a player. In practice, these preferences are often the result of balancing different aspects of the outcome against each other. When buying a new car, the customer may compare different offers according to e.g. price, fuel consumption, insurance group rating, noise level, equipment, warranties, and resell price. Each of these aspects can be evaluated in 1

See (Keeney & Raiffa, 1993), (French, 1988, Ch. 4).

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isolation. They define different attributes of the cars. In order to come to a decision, the customer may first evaluate each of the offers according to each attribute, weight the attributes against each other, and then choose according to the result. In practical applications of decision theory, e.g. for modelling the consumer behaviour in retail shopping, the main task is to find a list of relevant attributes a1 , . . . , an and a function U , such that U (a1 , . . . , an ) represents the overall payoff function u of an agent. This means, if the agent has to decide between products d1 , . . . , dm , then U (a1 (d1 ), . . . , an (d1 )), . . . , U (a1 (dm ), . . . , an (dm )) represent his preferences over the outcomes of buying d1 , . . . , dm . We call the function U a multiattribute utility function. For a sequence of attribute values a1 (dj ), . . . , an (dj ) of a domain object dj , we write a(dj ), and for the sequence a1 , . . . , an of attributevalue functions, we write a. Multi-attribute utility theory investigates the properties of multi-attribute utility functions and their applications; especially it is concerned with the constraints which are imposed on the utility functions by the fact that they depend on an array of attribute values. In general, the values ai (dj ) can be assumed to be real numbers, but they are not utilities themselves. A higher number does not indicate greater desirability. If ai measures the number of seats in a car, then an intermediate number like 5 may be optimal. If the number of seats is higher or lower, then the utility diminishes. Attributes may represent (measurable) properties like fuel consumption per 100km or (categorical) properties like ‘has air conditioning’. In the latter case, the attributes have Boolean values and we can set ai (dj ) = 1 if dj has property ai , and ai (dj ) = 0 if it doesn’t. In the following, we consider only cases in which the attributes represent categorical properties. We use MAUT in particular for modelling attitude reports expressing desires, as e.g.: (4) a) John likes vanilla ice. b) Peter wants to buy a bottle of cheap red wine. The sentence (4a) says that John likes ice cream which has the property to be vanilla ice. We cannot directly transform statements in (4) into statements about preferences. Statements about desires have an absolute form. In decision theory, we have to translate such absolute statements into statements about preferences over outcomes of actions. This is our next goal. Representing the property of having vanilla taste by an attribute a with values 0, 1, and quantifying over the objects d between which J has to choose, we arrive at a formula of the form ∀d (J desires d iff a(d) = 1). Doing this also for (4b), we arrive at the following logical forms: (5) a) ∀d (J desires d iff a(d) = 1). b) ∀d (J desires to verb d iff a(d) = 1). The formulas in (5) state conditions for when J desires something. They are not yet statements about the preferences expressed by desires. If (4a) is true, then the attribute ‘having vanilla taste’ is a relevant attribute of ice cream in situations in which John has to choose between different instances of ice cream.

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Hence, we explicate the predicate desire in (5) as a short form for a statement about preferences over attribute combinations. These preferences over attribute combinations can be represented by a multi-attribute utility function U . In a case like (5a), in which there is only one relevant attribute a, we replace our first approximation ∀d (J desires d iff a(d) = 1) by the bi-conditional: ∀d, d [U (a(d)) > U (a(d )) ↔ a(d) = 1 ∧ a(d ) = 0].

(16)

Now we have translated the absolute statement ‘John desires d iff d has property a’ into a statement about preferences. If there is only one attribute that influences decisions, then this representation seems to be appropriate. But we may find more complex descriptions of preferences: (6) a) I like vanilla ice and I like strawberry ice. b) I like vanilla ice but only together with strawberry ice. c) I like vanilla ice and I like strawberry ice, but I don’t like them together. Every statement describes a complex condition that depends on the attributes of ice cream. These conditions naturally translate into a condition Good on the combination of attribute values a(d) of domain objects d. In general, a multiattribute utility function U depends on the whole sequence a of attribute-value functions. Hence, we replace the simple condition a(d) = 1 ∧ a(d ) = 0 in (16) by a condition involving a predicate Good . This leads to a more general logical form for J likes Good (a(d)): ∀d, d [U (a(d)) > U (a(d )) ↔ Good (a(d)) ∧ ¬Good (a(d ))].

(17)

In (4b) and (5b), there is an action type act which is applied to different domain objects d. As there is only one action type, we can describe the preferences over action-argument pairs act -d by a condition Good over attributes of domain objects. In support problems, the utility measures u are defined as functions which map world-action pairs to real numbers. If the action set is generated by a single action type act and a set of domain objects d, then (17) leads to the condition: u(v, act -d) > u(v, act -d ) iff v |= Good (a(d)) ∧ ¬Good (a(d )).

(18)

As payoff functions are unique only up to linear rescaling, we can set without loss of generality:  1 if v |= Good (a(d)) (19) u(v, act -d) = 0 if v |= ¬Good (a(d)) It is this special form which we use when constructing normal OA models in Sec. 3.3. It allows us to concentrate on the predicate Good when setting up the model. Finally, we want to line out how our analysis is related to so-called ceteris paribus interpretations of preference statements, e.g. (Hanson, 2001). Let’s turn

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to the simple case in which the utilities only depend on a single attribute as e.g. in (16). It may be argued that, in fact, in a statement like (4a), it is meant that John prefers vanilla ice over other flavours of ice cream all else being equal (ceteris paribus). Let us assume that U depends on a sequence a0 , . . . , an of attributevalue functions one of which represents the attribute vanilla taste. Let’s say this is attribute a0 . Then the logical form of a preference statement like (4a) under a ceteris paribus interpretation is: ∀d, d [(a0 (d) = 1 ∧ a0 (d ) = 0 ∧ ∀i = 0 ai(d) = ai (d )) → U (a(d)) > U (a(d ))]. (20) In this case, the value of attribute vanilla flavour a0 is only decisive if all other attribute values of the two domain objects are equal. If we only compare two domain object for which all attributes a1 , . . . , an have the same value, then we can disregard them and (20) is equivalent to (16). But, in general, (20) and (16) are not equivalent. Often, background assumptions which influence an agent’s preferences are not explicitly stated. For example, in the situation where a customer has to choose between different cars, he may state that he prefers cars with low fuel consumption. But probably he would be very surprised if the sales attendant thereupon showed him their offer of solar cars. The stated preferences refer to a salient set of relevant alternatives. This may seem to favour the ceteris paribus condition as it asks all alternatives to be equal with respect to all other attributes. But we think this is too strong a condition. For example, if the customer states that he prefers cars with low fuel consumption, then this renders the colour of the car irrelevant. It is possible to decide on the basis of the customer’s preference statement that he prefers a green car with low fuel consumption over a blue car with high fuel consumption. This would not be possible under a ceteris paribus interpretation. Hence, we prefer an interpretation of preference statements which assumes that utilities depend on the explicitly stated attributes only as long as everything else is normal. The normality condition is not made explicit in (16). In what follows, we treat it as an implicit assumption. Which condition is appropriate, the ceteris paribus condition (20) or (16), is an empirical question which only the analysis of examples can answer. The aim of the next section is to uncover a number of implicit assumptions which we make when interpreting examples similar to (1). We may distinguish two types of normality assumptions. Some explicitly enter into the construction of OA models, some remain implicit in the background. We have just mentioned the assumption that attributes for which no preferences are explicitly stated are disregarded, and it is assumed that the domain objects are normal instances with respect to them. This normality assumption is an assumption which we make when setting up a normal OA model, but it is not one that is made explicit in the model itself but remains implicit in the background; i.e. we don’t find an explicit representation of these disregarded attributes in the model, nor a condition which tell what counts as being normal with respect to them. What this section aims at are the normality assumptions which must be explicitly represented in the OA model in order to account for a) the optimality of answers and b) their

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implicatures. Finding these normality assumptions is a theory-driven process. Hence, we have to check at the end that the existence of these assumptions is also supported by examples. 3.2

The Analysis of an Example

Preferences may depend on attributes in different ways, as indicated in (17). In the following example, the decision depends on several attributes which must all be satisfied to make the outcome optimal: (7) J: I am looking for a house to rent in a quiet and safe neighbourhood, close to the city centre, with a small garden and a garage. E: I have an offer for you in Dorothy Avenue. The answer (7) clearly implicates that the offered house is located in a quiet and safe neighbourhood close to the city centre and has a small garden and a garage. The same pattern occurs if only one but not all of the attributes must be satisfied: (8) a) A: I want to see a classical Beijing opera tonight or Chinese acrobatics, but I don’t want to go to one of these modern tea houses which mix both things. What can I do tonight? B: You can go to the Lantern Tea House! b) John loves to dance to Salsa music and he loves to dance to Hip Hop, but he can’t stand it if a club mixes both styles. J: I want to dance tonight. Is the Music in Roter Salon ok? E: Tonight they play Hip Hop at the Roter Salon. In (8a), the answer implicates that the Lantern Tea House shows either a classical Beijing opera or a Chinese acrobatics performance but not both, and in (8b) that there is only Hip Hop at the Roter Salon. We consider the last example (8b) in more detail. In accordance with our assumptions in Section 2 we assume that the answering expert is fully cooperative and knows all about the inquirers expectations. In addition we assume that he knows that there plays Hip Hop at the Roter Salon and not Salsa. The first two lines of (8b) introduce the relevant attributes and the properties of the inquirer’s utility function. There are two attributes. Let d range over dance locations; we write: H(d): There is Hip Hop at d;

S(d): There is Salsa at d.

What has to be explained by an OA model is: a) why is H(d) an optimal answer, and b) why does H(d) implicate that not S(d)? What we now want to find out is which assumptions we have to make in addition to what is explicitly stated in (8b) in order to answer these questions. We assume that these additional assumptions are automatically accommodated. The first task in the application of the OA approach is the construction of the support problem Ω, PE , PI , A, u which models the answering situation after

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J’s question ‘Is the Music in Roter Salon ok? ’ From the first sentence of J’s utterance, we know that this question serves to solve his decision problem of where to go for dancing. Hence, we can conclude that the action set A consists of all acts going-to-d in which d ranges over dance locations. In addition, we can assume that there is an alternative act stay-home. The background provided in the first sentence tells us how the utility function u is defined. We can translate the sentence into a property of dance locations: Good(d) iff (H(d) ∨ S(d)) ∧ ¬(H(d) ∧ S(d)). The arguments of u are world-action pairs (v, a). Obviously, the outcome of performing act going-to-d in world v is desired iff v |= Good(d), hence, we can assume that:  1 ⇔ v |= Good(d); (21) u(v, going-to-d ) = 0 ⇔ v |= ¬Good(d). There are three assumptions that enter here. First, we have to assume that H(d) and S(d) are the only attributes that count, i.e. there are no hidden objectives which are relevant for finding an optimal answer. Second, the objectives H(d) and S(d) are of equal weight, i.e. J does not prefer e.g. H(d) ∧ ¬S(d) over S(d) ∧ ¬H(d). Third, (21) also supposes that H(d) ∧ S(d) is equally dis-preferred to ¬H(d) ∧ ¬S(d). This leaves us with the problem of finding Ω, PE , PI , which is neither given by the background, nor by the question itself. We first consider the probability distributions PE and PI . We first turn to PE . It is clear that E’s answer can only implicate that there is no Salsa playing at Roter Salon if it is common knowledge that E knows which music they are playing at Roter Salon. Hence, if r is the Roter Salon, then we must assume that PE (Good(r)) = 1 or PE (Good(r)) = 0. We now turn to PI . In (8b), nothing is stated about the inquirer’s prior expectations about the music playing at different locations. Hence, we may assume that the inquirer has no reason to expect that Salsa playing at a location d is more probable than Hip Hop playing at a location e. This leads to the assumption that for X ∈ {H, S}: ∃α ∈ (0, 1) ∀d PI (X(d)) = α. We can also see that in (8b) nothing is said about the dependencies between events. Hence, we can assume that Salsa playing at one place is independent of Hip Hop playing at another place. This leads to the assumption that for all places d, e, d = e, the events H(d), S(d), H(e) and S(e) are probabilistically independent, means that for all non-empty subsets E ⊆ {H(d), S(d), H(e), S(e)}:  which PI ( E) = X∈E PI (X). Definition (21) entails that EUI (going-to-d ) = PI (Good (d)). If α is the probability of the elementary events H(d) and S(d), then, by independence, it follows that PI (Good (d)) = 2α(1 − α). Furthermore, for d = e and X ∈ {H, S, Good } it is PI (Good (d) ∧ X(e)) = PI (Good (d)) × PI (X(e)). Furthermore, we have to make an assumption about the relation between the expected utilities of stay-home and going-to-d before and after learning that S(d) or H(d). Let X ∈ {H, S}, then we assume: EUI (going-to-d ) < EUI (stay-home) < EUI (going-to-d |X(d)).

(22)

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This means, the inquirer believes that it is more probable that a place plays only Salsa or only Hip Hop. If we do not make this assumption, then the addressee might still decide to stay at home after learning that H(r), in which case H(r) would not be an optimal answer as it fails to induce the addressee to choose an optimal action. Hence, the assumption is necessary for explaining the optimality of H(r). We will say more about neutral acts like stay-home later on. We now show that these assumptions indeed entail the implicatures observed in Example (8b). Let r denote the Roter Salon. Then, after learning that H(r), we find for all other places d: PI (Good(d)|H(r)) = PI (Good(d)) < PI (Good(r)|H(r)).

(23)

Assumption (22) together with proposition (23) imply that learning H(r) will induce the hearer to go to the Roter Salon. As EUE (H(r)) = 1 and as there is no possible answer A such that EUE (A) > 1, it follows that H(r) is optimal. The following table shows a model for (8b). As we assumed that the speaker knows that only Hip Hop is played at Roter Salon, he knows that the actual world is w2 : (9) Ω H(r) S(r) stay-home Good(r) H(r) ∈ Opwj w1 1 1 ε 0 no w2 1 0 ε 1 yes w3 0 1 ε 1 no w4 0 0 ε 0 no Let, as usual, [[ϕ]] = {w ∈ Ω | w |= ϕ}. Then O(going-to-r) = [[Good(r)]] = ∗ ∗ {w2 , w3 }. As [[H(r)]] = [[H(r)]] = {w1 , w2 }, we find [[H(r)]] ∩ O(going-to-r) = {w2 } = H(r) \ S(r). From Lemma 6, it follows that H(r) implicates that only Hip Hop is played at Roter Salon. This proves the claim. Finally, this leads to the last parameter which was left unexplained, the set of possible worlds Ω. What counts for I’s decision are only the truth values of H(r) and S(r). Hence, we can identify the set of possible worlds Ω with the set of all truth value assignments to the formulas H(r) and S(r). There are two obvious questions to which this example gives rise: First, are the assumptions also necessary for deriving the implicature, and second, is there a general structure which we can extract from this example and apply to the interpretation of similar examples? We start with the second question. 3.3

The Construction of Normal Models

In this section we present a procedure for constructing a normal OA model. The construction starts with a given set of domain objects and a sequence of attributes. In (8b), the first sentence introduces a set of two attributes which we represented before by the formulas H(x) and S(x). John’s question introduced the domain object Roter Salon. With these two parameters, we can construct a suitable set of possible worlds. Let D be a set of domain objects. We denote

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attribute-value functions for a fixed sequence of n attributes by (ai )i=1,...,n . We assume that all attributes range over values in {0, 1}. Then we set: Ω := {(ai )i=1,...,n | ai : D → {0, 1}}.

(24)

This means, Ω is the set of all attribute-value functions which map the objects in D to their respective values in {0, 1}. In this section, we write a for the elements of Ω. If |D| = m, then |Ω| = 2mn. In (8b), it is D = {r} and n = 2, hence, |Ω| = 4. The assumptions that enter into this construction can be summarised as follows: I Assumptions about Objectives: 1. Only attributes of elements of D count. 2. Completeness of objectives: (ai )i=1,...,n represents all objectives. We add a strong default assumption about the speaker’s knowledge: II Expert assumption: ∃a ∈ Ω PE (a) = 1. In analogy to the construction of possible worlds from attribute-value functions in (24), we define the probabilities of these worlds in terms of the probabilities of elementary events which are constructed from the same attribute-value functions. For each attribute ai , we can introduce a prime formula Ai such that a |= Ai (d) iff ai (d) = 1. We then define [[ϕ]] = {a ∈ Ω | a |= ϕ}, and set E(n, D) := {[[Ai (d)]] | d ∈ D, 1 ≤ i ≤ n}. E(n, D) is a set of elementary events. For example, in (8b), it is the set {H(r), S(r)}. If the reader of an example has no further information about the elementary events, he has no reason to expect that one event is more probable than the other. Hence, by the principle of insufficient reason, he should treat them as equally probable and independent of each other. This is captured by the following conditions: III Laplacian Assumptions: 1. Equal undecidedness: ∀X, Y ∈ E(n, D) : PI (X) = PI (Y ).

(25)

2. Probabilistic independence: ∀E ⊆ E(n, D) : PI



 E = PI (X).

(26)

X∈E

If PI (Ai (d)) is known for all i, d, then the Laplacian assumptions uniquely define PI . To see this, let α := PI (Ai (d)) and nv := |{(i, d) | v |= Ai (d)}|. As before, let m = |D| and let n be the number of attributes. Then, PI (v) = αnv × (1 − α)mn−nv . In particular, if α = 12 , i.e. if the addressee is equally undecided with regard to the truth of Ai (d) and ¬Ai (d), we find PI (v) = 2−mn . In (8b), the first sentence not only introduced the relevant attributes but also stated a constraint on the multi-attribute utility function U . The sentence

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stated that dancing at a place d is desirable iff H(d) ∨ S(d) but not H(d) ∧ S(d). Let us abbreviate this condition by Good (d). This condition leaves the multiattribute utility function U highly underspecified, but it is natural to interpret the statement such that the utility of dancing at d is equal in all worlds in which Good (d) holds, and equal in all worlds in which ¬Good (d) holds. This motivates the following normality assumption about utilities: IV Equal weight of objectives: All values which can be treated as equal are treated as equal. The last parameter that is needed to set up a normal decision problem for an example is the set of actions. This set is closely connected to the set of domain objects D. Looking again at (8b), we see that the domain objects, the clubs that play dance music, are an argument, in this case the goal, of the actions between which the inquirer has to choose. If there is only one type of action, then we can identify the set of hearer’s actions with a set {act -d | d ∈ D}. A multi-attribute utility function U represents preferences over states of the world. These states are the outcomes of actions between which an agent can choose. With U as before, the preferences over world-action pairs must satisfy the following condition: u(a, act -d) = U (a1 (d), . . . , an (d))

(27)

As argued for in Section 3.1, this leads to a representation of desirability by a payoff function u which is defined over world-action pairs as in (19) using the predicate Good . In (8b), we also introduced a neutral action, stay-home, or doing nothing. In the Out-of-Petrol example, (2), the neutral act was a random search in town. That we need such assumptions can be best seen from (8b). If the action set only consisted of the acts of going to places d, then, if there is only one place, the inquirer has no choice and any answer would be optimal. Hence, the assumption of a default action is partly a technical device that allows us to reduce the set of possible worlds and the set of hearer’s actions to those explicitly stated in the examples. It is clearly desirable to have a uniform characterisation of the default act. In the Out-of-Petrol and the Hip-Hop examples, we could unify their characterisations by assuming that they consist of a random sequence of acts act -d which are followed by the act of doing nothing if none of them is successful. Alternatively, we could think of the default act as a lottery over the acts act -d followed by the act of doing nothing if act -d fails. This depends on whether or not it is reasonable to assume that the acts act -d can be performed in sequence. Hence, in general, we can assume that the neutral act consists of a random sequence of acts from {act -d | d ∈ D} followed by an act of doing nothing, including the special case in which the random sequence has length zero. In addition, we have to assume that the length of the sequence causes some costs which are implicitly represented in the payoff function u. The HipHop example as represented in (9) is an example in which the sequence has length zero. This assumption was necessary as D contained only one element r; hence, if the random sequence had length longer than zero, then the first act of

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the random sequence could only be the act of going to r, which means that there is only one act to choose for the hearer, namely going to r. What we need, in any case, is an action set A which consists of at least two actions with distinct outcomes for which a property similar to (22) can be proven. Furthermore, as it is a default act, it must be the alternative which the hearer would choose if he had to choose before learning anything from the speaker. Although we can’t claim to fully understand the nature of the normality assumption which is involved here, we make the following tentative assumptions about the existence of a neutral act l: V Neutral alternative: i) l ∈ B(Ω) and ii) B(KE ) = {l} ∨ l ∈ B(KE ) with KE = {v ∈ Ω | PE (v) > 0}. The first condition says that l is a best choice if the hearer doesn’t learn any new information. It entails EUI (l) > maxd EUI (act -d). The second condition means that if the answering expert can recommend any act different from l, then the neutral act l is no longer an optimal choice. Together, these default rules define a unique support problem σ for each sequence (ai )ni=1 , D, U, α, l in which (ai )ni=1 is a sequence of attributes, D a set of domain objects, U a multi-attribute utility function, α the probability of the elementary events Ai (d), and l the neutral act. We can even simplify this description as only the numbers n of attributes and m of domain objects count. Furthermore, in all those cases in which U only ranges over values in {0, 1}, we can characterise U by a formula Good(d) which only contains formulas of the form Ai (d) with d = 1, . . . , m. Finally, if there is no reason to assume otherwise, α = 12 . Then, σ only depends on a quadruple n, m, Good( . ), l. A full justification of these default rules can only be given by examples. Table 2 shows the combined table for examples (8a) and (8b). In both examples, there is one domain object d and two attributes. The speaker recommends an action, going-to-d . This recommendation is optimal in world wi if wi |= Good (d). This is the case in w2 and w3 . Hence, this recommendation implicates that the domain object d has either attribute A1 or A2 , but not both. From this, we can see that the assertion of A1 (d) or A2 (d) implicates that the domain object has only the asserted attribute. Table 2. Joint table for examples (8a) and (8b) Ω A1 (d) A2 (d) Good(d) A1 (d) ∈ Opwi w1 1 1 0 no w2 1 0 1 yes w3 0 1 1 no w4 0 0 0 no

In Table 3, we show the model of (7). Again, there is only one domain object but six attributes. By the rule about equal weights of objectives, Good(d) only holds if it has all attributes A1 , . . . , A6 . The speaker recommends an action act-d, hence, we have to find out which worlds are elements of O(act -d). This

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can only be the world in which d satisfies all attributes. Hence, the recommendation implicates that d has all of the desired attributes. A possible objection may arise with respect to the number of domain objects. Whereas in (8b) the inquirer explicitly asks about one object, no such restriction exists in (8a). Hence, one would expect an underspecified number of domain objects. But the representation of these objects is unnecessary due to the existence of the neural act l. Table 3. The table for Examples (7) Ω A1 (d) . . . A5 (d) A6 (d) Good(d) w1 1 . . . 1 1 1 w2 1 . . . 1 0 0 ... ... ... ... ...

3.4

Violations of the Normality Assumptions

In this section we look at the effects of violations of the normality assumptions. We show that their violations also lead to different implicatures. We present a series of variations of (8b) which violate one of the rules. As we assume that the rules hold by default, only an explicit statement of their invalidity or general world knowledge can violate them. We start with the expert assumption II. If it is known that the answering person has only partial knowledge about the music playing at Roter salon, the implicature from H(r) to ¬S(r) may be suspended: (10) E: I know that Hip Hop is played at the Roter Salon tonight. But I don’t know whether they mix it with other music. Still, this answer may induce the inquirer to go to the Roter Salon, but the implicature that there is no Salsa is clearly cancelled. The normality assumptions I and IV are a kind of exhaustification principles. Hence, it is not possible to violate them directly by stating new objectives or more details of the utility function. Hence, we assume that violating facts belong to unstated background knowledge. For example, assume that in (8a) it is common knowledge that the inquirer only wants to go out if he finds a place close by. Then, this additional objective has the effect that B’s answer “You can go to the Lantern Tea House” implicates that the Lantern Tea House is close by. Let us assume that, in Example (8b), it is common knowledge that the inquirer’s preferences for Salsa far exceed that for Hip Hop. If he asks “I want to dance tonight. Where can I go to?”, then the answers “Tonight they play Hip Hop at the Roter Salon” has the additional implicature that there is no place where they only play Salsa. Violations of III are probably most interesting. In the following example, we find an explicitly stated dependency between playing Hip Hop and playing Salsa:

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(11) The Roter Salon and the Gr¨ uner Salon share two DJs. On of them only plays Salsa, the other one mainly plays Hip Hop but mixes into it some Salsa. There are only these two Djs, and if one of them is at the Roter Salon, then the other one is at the Gr¨ uner Salon. John loves to dance to Salsa music and he loves to dance to Hip Hop but he can’t stand it if a club mixes both styles. J: I want to dance tonight. Is the Music in Roter Salon ok? E: Tonight they play Hip Hop at the Roter Salon. Let gr denote the Gr¨ uner Salon. The dependencies imply that PI (Good(r)|H(r)) = 0 and PI (Good(gr)|H(r)) = 1, Hence, after hearing that H(r), the inquirer can infer that going to the Gr¨ uner Salon is the optimal choice. This implies that the answer H(r) is also optimal with EUE (H(r)) = 1. Indeed, in this context H(r) implies that the music at Roter Salon will be bad. The last normality assumption, V, is probably the most problematic one. It is also difficult to test. That is mainly because it serves a technical purpose. Besides being a means for simplifying the models, it likewise prevents some unwanted predictions of optimality of answers. Assume that in the Out-of-Petrol example (2), there is a single road which the inquirer can either follow to the left or to the right. To the left, there are fields, bushes, and trees. To the right, there is the beginning of a town. In this scenario, turning to the right is probably more promising than turning to the left. But this entails that saying nothing is already an optimal answer if the answering expert knows that to the right there is a petrol station. This possibility is ruled out when assuming that there exists a neutral act with the properties stated in V.

4

A Classification of Support Problems and Implicatures of Simple Answers

The aim of this section is to show how to use the normal optimal answer approach for deriving systematic classifications of support problems representing dialogue situations and calculating the associated lists of implicatures for simple answers. In Section 3.3, we saw that the default rules define a unique support problem for each quintuple (ai )ni=1 , D, U, α, l in which (ai )ni=1 is a sequence of attributes, D a set of domain objects, U a multi-attribute utility function, α the probability of the elementary events Ai (d), and l the neutral act. In this section, we only consider cases in which U can be characterise by a formula Good(d) which is in turn a Boolean combination of formulas Ai (d) representing the multiple basic attributes. In addition, we assume that the probability of all elementary events equals 12 . We have seen that these restrictions imply that the normal support problem describing the dialogue situation is uniquely characterised by a quadruple (ai )ni=1 , D, Good( . ), l. In this section, we consider two types of dialogue situations. First, the type of situations with one domain object

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and two attributes; and second, the type of situations with two objects and one attribute. The first type of dialogue situations includes the Out-of-Petrol example (2), the second the Italian newspaper example (3). It should become clear how to extend these classifications to the general case, i.e. to the cases with an arbitry number of domain objects and attributes. In examples (8a) and (8b), we find one relevant domain object and two attributes which have an influence on the decision of the addressee. The model was shown in Table 2. From this table, we can derive all normal models for situations with one domain object and two attributes if we systematically list all possible utility functions which can be described by a Boolean predicate Good . The following table shows the definition of the predicate Good in examples (8a) and (8b): (12) Ω w1 w2 w3 w4

A B Good 1 1 0 1 0 1 0 1 1 0 0 0

In case of example (8b), attribute A denotes the playing of Hip Hop, and B the playing of Salsa at the Roter Salon. going-to-r is good just in case the Roter Salon has one but not both attributes. The first column of Table 4 shows all possible predicates Good which can be defined from two attributes A and B. The four numbers listed in each row tell us whether Good is true in the worlds Table 4. A complete list of all Good predicates for the Out-of-Petrol examples 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0

Good 1 1 1 1 1 0 1 0 0 1 0 1 0 0 0 0 1 1 1 1 1 0 1 0 0 1 0 1 0 0 0 0

1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0

    − − −     −    

A        − − − −     

· · · · · · · · · · · · · · · ·

· · · · · · · · · · · · · · · ·

  − −   −       −  

B · · · · · · · · · · · · · · · ·

       − − −   −   

· · · · · · · · · · · · · · · ·

· · · · · · · · · · · · · · · ·

¬A ·  · − ·  ·  ·  · − ·  ·  ·  · − ·  ·  ·  ·  · − · 

  −    −    −   −  

· · · · · · · · · · · · · · · ·

¬B  · − ·  · − ·  ·  ·  ·  ·  · − ·  ·  ·  ·  · − ·  ·

    −  −     − −   

w1 , . . . , w4 which are defined in (12). Hence, for example, in the fifth row, the entry 1 0 1 1 means that choosing the only domain object is good in worlds w1 , w3 , w4 and inferior to the neutral act in world w2 . The other columns in

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Table 4 show entries for the four answers A, B, ¬A, and ¬B. Each row has four entries for each answer. They again relate to the four worlds w1 , . . . , w4 . A ‘’ means that the answer makes the hearer choose an act which is optimal in the world to which the  entry belongs; a ‘−’ means that the answer will lead the hearer to choose a sub-optimal act; and a ‘·’ means that the answer is not admissible. Hence, in the fifth row, the entry −, , ·, · for answer A says that A induces the hearer to choose the inferior act in w1 , it induces him to choose the optimal act in world w2 , and it is not admissible in w3 and w4 . In this row Good is defined such that the neutral act is preferred in w2 , and dispreferred in w1 ; hence, the ‘−’ and ‘’ entries in the first and second position for answer A both say that A induces the hearer to choose the neutral act. In order to see that A indeed recommends the neutral act, we must compare the expected utilities of the different acts. For each row, it is assumed that the expected utility of the neutral act is minimally higher than the expected utility of choosing the domain object. Hence, if learning A increases the probability of Good , then this is interpreted as a recommendation to choose the domain object, otherwise it is interpreted as a recommendation to choose the neutral act. In the fifth row, learning A decreases the probability of A, hence it recommends the neutral act. As second example for the systematic application of normal models, we consider the case of two domain objects with one attribute. This class of examples includes the Italian newspaper example (3). The table in (13) shows the four possible worlds which can be defined by two objects a, b and one attribute A: (13) Ω A(a) A(b) w1 1 1 w2 1 0 w3 0 1 w4 0 0 As the predicate Good (d) must be a Boolean combination of the prime formula A(d), there are four possible definitions. They are listed in the first column of the next table: (14)

Good (d) Good (a) Good (b) A(d) ∨ ¬A(d) 1 1 1 1 1 1 1 1 A(d) 110 0 101 0 ¬A(d) 001 1 010 1 A(d) ∧ ¬A(d) 0 0 0 0 0 0 0 0

The four entries for the formulas Good (a) and Good (b) show whether the respective formulas are true or not in the worlds w1 , . . . , w4 defined in (13). We again assume that the expected utility of the neutral act is minimally higher than the expected utility of choosing one of the domain objects a, b. Hence, if learning an answer increases the probability of Good (d), then this is interpreted as a recommendation to choose a domain object. The table in (15) shows for each definition of Good (d) and each of the worlds w1 , . . . , w4 whether the simple answers A(a), A(b), ¬A(a), and ¬A(b) are optimal:

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A(a) A(b) ¬A(a) ¬A(b) ········ (15)   · ·  ·  · · · −  · − ·  −···−····· ········ Each row corresponds to one of the definitions of Good (d) shown in (14), and the four entries for each answer to the worlds w1 , . . . , w4 . Hence, the entry , −, ·, · in the third row below A(a) says that if Good (d) is equivalent to ¬A(d), then answer A(a) induces the hearer to choose the optimal action in world w1 but misleads him in w2 . This can be calculated as follows: as learning A(a) does not increase the probability of Good (b), the neutral act still has the highest expected utility; hence, answer A(a) must be interpreted as a recommendation to choose the neutral act, which is optimal in w1 but not in w2 . Among other reasons, the neutral act was introduced for representing the effect of unmentioned alternative domain objects which are in the choice set of the hearer. If the neutral act consists in first trying two objects a and b in a random order, and then, if unsuccessful, doing nothing, then an answer which discourages the choice of one domain object d will be equivalent to a recommendation of the other object. This changes the table in (15). The result is shown in (16): A(a) A(b) ¬A(a) ¬A(b) ········ (16)   · ·  ·  · · ·  − ·  · − −··−······ ········ As an example, we consider the following variation of the Italian newspaper example (3): (17)

I: I heard that I can buy Italian newspapers at the station or at the Palace. Do you know where I should go to? E: The Palace doesn’t sell Italian newspapers.

If we identify the Palace with domain object a, then looking into the second row under ¬A(a) in (16) shows us that answer ¬A(a) can only be successful in world w3 . Hence, the hearer can infer that A(b), i.e., that he can buy an Italian newspaper at the station. These two examples, the situations with one domain object and two attributes, and the situations with two objects and one attribute, show the potential of normal optimal answer models for categorising dialogue situations and systematically deriving the associated lists of optimal answers. At the same time, these examples also point to some limitations of the model. Let us consider Example (17) again. The negated answer ¬A(a) is generally classified as a partial answer which implicates that the speaker does not have full domain knowledge. Hence, the answer leads to a suspension of the normality assumption that the speaker is a domain expert. In order to arrive at the predicted implicature, it

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has to be common knowledge that there can be no doubt that the speaker is an expert. A similar problem arises with the Out-of-Petrol examples in Table 4. We consider the following example: (18) a) I: I want to have a house with either both a garden and a balcony, or with neither a garden nor a balcony. b) I: I want to have a house with both a garden (A) and a balcony (B). c) E: The house in Shakespeare Avenue has a balcony (A). For this example, the entries in the seventh and eighth row below A in Table 4 tell us that w2 must be the actual world in (18a), and w1 in (18b). Hence, answer A implicates that the house in Shakespeare Avenue does not have a garden in (18a), and does have a garden in (18b). Probably, the most natural continuation would in both cases be a clarification request of the form: ‘And does it also have a garden? ’ The effect of the possibility of clarification requests has not been accounted for in our normal optimal answer model. Hence, we can detect at least two shortcoming of the models proposed in this paper: they cannot deal with the possible suspension of normality assumptions, and they cannot account for the effects of clarification requests. A first step towards a solution to the first problem was taken in (Benz, 2009). The effect of clarification requests was studied in (Benz, 2008). In both cases, a fully satisfying solution has to wait for future research.

5

Conclusion

In the introduction, we characterised our paper as a preparatory study for a theory of relevance implicatures in discourse. What it achieves is an isolation of the crucial parameters and a list of normality assumptions which are needed for the construction of normal optimal answer models. That this is far from a discourse interpretation theory for implicatures needs no further explanation. What our study tells us is which parameters we have to look for when interpreting text in order to obtain its implicatures. What is most clearly needed is a theory of the interaction of discourse structure, as defined by rhetorical relations, and conversational implicatures which are defined by background decision problems. We must leave this to future research. That multi-attribute utility theory should play a major role in the analysis of relevance implicatures is not too surprising in view of the theory’s importance to applied decision theory. There, the elicitation of the decision maker’s preferences over attributes is a central step in setting up the model of a decision problem. And in view of the importance of decision making in everyday life, it is also not too surprising that language users have developed the means for efficiently communicating these preferences. Attribute-value functions play two important roles in our model. First, as they represent the attributes to which the agents react. This means, they tell us what the relevant properties of the epistemically possible worlds are. This led us to identify possible worlds with the attribute-value functions which we find by

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the multi-attribute utility analysis of a given discourse. The second important application of attribute utility functions concerned the representations of desires. We assumed that a sentence like “John likes vanilla ice” is adding a new attribue and setting the value of John’s utility function to a default value. Here again, our paper is only clearing some ground where a comprehensive theory must be developed in the future. The last section showed how to derive systematic classifications of dialogue situations based on the normal optimal answer approach. We also showed how to calculate the associated lists of optimal answers. This section reveals some limitations of our approach. In particular, there is no account for the effect of clarification request, or for the possibility of suspending a normality assumption. A satisfying solution, again, has to wait for future research. Although there are clear limits to the model, the underlying normality assumptions, which are the main concern of this paper, are not affected by them. These assumptions must play an important role in any discourse interpretation theory which not only accounts for semantic and rhetoric information but also for Gricean implicatures.

References Benz, A.: Utility and Relevance of Answers. In: Benz, A., J¨ ager, G., van Rooij, R. (eds.) Game Theory and Pragmatics, Palgrave, Mcmillan, China (2006) Benz, A.: Implicatures of Irrelevant Answers and the Principle of Optimal Completion. In: Bosch, P., Gabelaia, D., Lang, J., et al. (eds.) TbiLLC 2007. LNCS, vol. 5422, pp. 95–109. Springer, Heidelberg (2009) Benz, A.: On the Suspension of Implicatures. In: Proceedings of the 10th Symposium on Logic and Language, LoLa10, Budapest, pp. 73–79 (2009) Benz, A., van Rooij, R.: Optimal assertions and what they implicate: a uniform game theoretic approach. Topoi - an International Review of Philosophy 27(1), 63–78 (2007) French, S.: Decision Theory — an introduction to the mathematics of rationality. Ellis Harwood, Chichester (1988) Grice, H.P.: Logic and Conversation. In: Grice, H.P. (ed.) Studies in the Way of Words, pp. 22–40. Harvard University Press, Harvard (1989) Groenendijk, J., Stockhof, M.: Dynamic predicate Logic. Linguistics & Philosophy 14, 39–100 (1991) Hanson, S.O.: The Structure of Values and Norms. Cambridge University Press, Cambridge (2001) Keeney, R.L., Raiffa, H.: Decisions with Multiple Objectives —Preferences and Value Tradeoffs. Cambridge University Press, Cambridge (1993) Lewis, D.: Convention. Blackwell Publishers Ltd., Oxford (2002)

Strategic Vagueness, and Appropriate Contexts Kris De Jaegher1 and Robert van Rooij2 1

2

1

Utrecht School of Economics, Universiteit Utrecht Institute for Logic, Langauge, and Computation, Universiteit van Amsterdam

Introduction

This paper brings together several approaches to vagueness, and ends by suggesting a new approach. The common thread in these approaches is the crucial role played by context. In Section 2, we treat game-theoretic rationales for vagueness, and for the related concepts of ambiguity and generality. Common about these rationales is that they are based on the assumption of a conflict of interest between speaker and listener. We review this literature using a single example. We argue that the most plausible application to vagueness in natural language of these models is one where the listener only imperfectly observes the context in which the speaker makes her utterances. Yet, it is clear that not all vagueness can be accounted for by conflicts of interest. This is why the rest of the paper looks at the case of common interest. Section 3 argues that being vague by saying that someone is bald makes sense in a context where precision is of less importance; in a context where precision is of more importance, one can then refer to someone as completely bald. This make sense because the longer and therefore more costly to utter expression ‘completely bald’ is then used less often. Vagueness is thus seen as an application of Horn’s pragmatic rule that (un)marked states get an (un)marked expression. Section 4 tackles the Sorites paradox, which apparently leads to the violation of standard axioms of rational behaviour, and shows that this paradox arises from the use of vague predicates in an inappropriate context. If, as suggested by the Sorites paradox, fine-grainedness is important, then a vague language should not be used. Once vague language is used in an appropriate context, standard axioms of rational behaviour are no longer violated. Section 5 finally takes a different approach from the previous sections, and following prospect theory assumes that context directly enter agents’ utility functions in the form of reference points, with respect to which agents think in gains and losses. The rationale for vagueness here is that vague predicates allow players to express their valuations, without necessarily uttering the context, so that the advantage of vague predicates is that they can be expressed across contexts.

2

Vagueness and Games of Conflict

Game theorists have tried to justify generality, vagueness and ambiguity in language as ways to solve conflicts of interest between a sender and a receiver. We A. Benz et al. (Eds.): Language, Games, and Evolution, LNAI 6207, pp. 40–59, 2010. c Springer-Verlag Berlin Heidelberg 2010 

Strategic Vagueness, and Appropriate Contexts

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Table 1. U (ti , aj ) a1 a2 a3 t1 3,3 1,0 0,2 t2 1,0 0,3 -1,2

start by treating such a sender-receiver game with a conflict of interest (a variant of a game treated by Farrell, 1993), and its equilibria in the absence of what is interpreted as generality, vagueness and ambiguity. Consider the signalling game with the payoff structure in Table 1. The sender (she) observes whether the state of nature is t1 or t2 . Each of these states occur with probability 12 . The receiver (he) can choose among action a1 , a2 or a3 . After having observed the state of nature, and prior to the receiver taking one of the actions, the sender can send a signal ‘s1 ’, a signal ‘s2 ’ to the receiver, or no signal at all. Sending a signal comes at no cost whatsoever to the sender. The meaning of the signals is completely conventional, but we focus on separating equilibria where each signal is sent more often in one particular state, thus justifying the labels of the signals. All aspects of the game are common knowledge among the sender and the receiver. Using a story adapted from Blume, Board and Kawamura (2007), we may consider the sender as Juliet, who may be of two types, namely one who loves Romeo (the receiver), and one who is merely fond of him as a friend. a1 means that Romeo acts as if Juliet is in love with him, a2 means that Romeo acts as if Juliet is a friend, and a3 means that Romeo acts as if Juliet could be either a friend or in love with him. Without any knowledge, Romeo prefers the latter action, but this is Juliet’s least preferred outcome. Both types of Juliets prefer a1 to a2 . We start by looking at the standard case where any randomisation of the sender in her strategies is unrelated to the randomisation of the receiver (excluding correlated equilibria — see below), and where a signal sent by the sender is always received and never misinterpreted by the receiver (excluding noisy equilibria — see below). It is easy to check that the signalling equilibria has two Nash equilibria. In a first equilibrium, the pooling equilibrium, Juliet does not send any signal, and Romeo does a3 . This Nash equilibrium is also a perfect Bayesian equilibrium (cf. Fudenberg & Tirole, 1991) because we can find beliefs for each player that underpin this equilibrium. Romeo believes that anything Juliet says does not contain any credible information. His beliefs are not disconfirmed as he never actually observes Juliet talking, and he might as well keep his beliefs. Juliet from her side correctly believes that anything that she might say will be met with action a3 . More correctly, there is in fact a range of pooling equilibria, where given that signals are costless to send, Juliet may talk in any uninformative way, for instance by always saying the same, or mixing between signals in a manner that has little correlation with her own state. The second Nash equilibrium is a separating equilibrium, and takes the form of a mixed equilibrium. Juliet always sends signal ‘t1 ’ (‘I love you’) in state t1 ; however, in state t2 , she sends ‘t2 ’ (‘I like you’) with probability 12 , and ‘t1 ’ (‘I love you’) with probability 12 . Put otherwise, Juliet is always honest when

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she is in love, but is half of the time honest and half of the time dishonest when she is merely fond of Romeo. Romeo always acts as if Juliet is a friend when she says that she likes him; however, when she says ‘I love you’, with probability 12 he acts as if she loves him, and with probability 12 he acts as if she could either be a friend or someone who loves him. To see that these are mutual best responses, note first that Juliet’s expected payoff of telling that she loves Romeo when she is merely fond of him is now 12 × 1 + 12 × (−1) = 0; 0 is also her expected payoff of sending signal ‘t2 ’. It follows that Juliet is indifferent between saying ‘I love you’ and ‘I like you’. Second, note that when Romeo sees Juliet telling that she loves him by Bayes’ rule will (somewhat prosaically) estimate the probability 1 that she indeed does to be 1 + 21 × 1 , and will estimate that in fact she is merely fond of him to be

1 1 2×2 1 1 1 + × 2 2 2

2

2

2

. It follows that Romeo’s expected utility of acting as if 1

1

×1

Juliet loves him when she tells him she does equals 1 + 12 × 1 × 3 + 1 +2 1 ×2 1 × 0 = 2. 2 2 2 2 2 2 This equals Romeo’s payoff 2 of acting as if Juliet might either love him or like him. Similarly, his expected payoff of acting as if Juliet is merely a friend equals 1 1 1 2 2×2 1 1 1 ×0+ 1 1 1 × 3 = 1. It follows that Romeo is indifferent between acting + × + × 2 2 2 2 2 2 as if Juliet loves him, and acting as if she might either love him or like him. In these circumstances, Romeo might as well take each action half of the time. As argued by Lipman (2006), one could see Juliet as being vague, as ‘I love you’ does not always mean then that Juliet really does love Romeo. But a particular feature of such a mixed equilibrium is that Juliet is completely indifferent about what to say when she is merely fond of Romeo, and Romeo is indifferent about what to do when Juliet tells him that she loves him. And yet, each player is assumed to mix between his or her strategies in a very particular way, in order to keep the other player indifferent. A justification given for the mixed equilibrium (Harsanyi, 1973) is that we can see this as a simple representation of a more complex model where in fact we have a population of Juliets, whose payoffs differ so that they can be interpreted as varying according to their degree of trustworthiness; and a population of Romeos, whose payoffs differ so that they can be interpreted as varying according to their degree of trustfulness. Each Juliet knows her own trustworthiness, but does not observe the trustfulness of the Romeo facing her; she merely knows how trustfulness is distributed among Romeos. Similarly, Romeo only knows how trustworthiness is distributed among Juliets. So, rather than mixing taking place, the most trustworthy Juliets in the population are honest, and the most trustful Romeos trust the Juliets. Yet, such a justification of the mixed equilibrium adds new aspects to the game. Sticking to the original game between the individual Romeo and Juliet in the game in Table 1, we now investigate several ways in which the players could still communicate, without the unsatisfactory aspects of a mixed equilibrium. 2.1

Strategic Generality

Let us say that a sentence is specific when it is true only in a few circumstances. A sentence is general when it is true in many circumstances. It is standard to

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43

assume in pragmatics that it is better to be more specific. However, in case of a conflict of interest, it can be beneficial to be general. To see why, take the example in Table 1, and introduce a third state t3 about which the interests of sender and the receiver fully coincide (namely, a4 is then the best action), as represented in Table 2. Concretely, in terms of the above story, in state t3 Juliet neither likes nor loves Romeo, and action a4 means that Romeo acts as if she neither likes him nor loves him. Then a separating equilibrium exists where in both state t1 and state t2 , Juliet tells Romeo ‘Either I love you or I like you’, upon which Romeo acts as if she might either love him or like him; in state t3 Juliet tells Romeo that she does not like him and she certainly does not love him. Table 2. U (ti , aj ) t1 t2 t3

a1 a 2 3,3 1,0 1,0 0,3 -2,-2 -2,-2

a3 0,2 -1,2 -2,-2

a4 -2,-2 -2,-2 1,1

A more sophisticated version of this argument is found in Crawford and Sobel (1982). In their model, the sender observes one state out of a continuous range of states of nature, and the receiver can pick an action from a continuous range of actions. The discrepancy between the sender’s optimal action and the receiver’s optimal action, where the sender always prefers a higher action, measures the degree of conflict between the two players. Crawford and Sobel (1982) show that communication can still take place between sender and receiver in spite of the conflict of interest between them if the sender uses a finite number of signals, where each signal is used for a range of the continuum of states of nature, so that signals partition the continuum of states. In the efficient separating equilibrium, this partition is less fine-grained the higher the degree of conflict between sender and receiver. Intuitively, as her signals become less and less finely tuned, eventually the sender prefers to tell the truth. 2.2

Strategic Vagueness and Strategic Ambiguity

Yet, it is sometimes possible to do better still than with strategic generality, and in the game of Table 2 let the sender and receiver still communicate about t1 and t2 as well. In particular, this is possible when the sender uses what could be described either as vague or as ambiguous sentences rather than general sentences. To show this, it suffices to show that players can still communicate in the game in Table 1 without playing a mixed equilibrium. Noisy signalling interpreted as vagueness/ambiguity. We first look at noisy signalling (Farrell, 1993; Myerson, 1991; De Jaegher, 2003a,b; Blume, Board and Kawamura, 2007; Blume and Board, 2009). Juliet sends noisy signals, which may simply remain unheard (errors of detection) or may be misinterpreted (errors of discrimination). Denote by μ(˜i|˜j) the probability that

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Romeo perceives signal ˜i when the sender sent signal ˜j = t˜1 , t˜2 , where ˜i = t˜1 , t˜2 , 0 (where 0 denotes not receiving any signal). Consider a noisy signalling system where μ(t˜1 |t˜2 ) = μ(0|t˜2 ) = 14 , μ(t˜2 |t˜2 ) = 12 , and μ(t˜1 |t˜1 ) = μ(0|t˜1 ) = 12 . It is easy to check now that a separating equilibrium exists where Juliet honestly send signal t˜1 (t˜2 ) in state t1 (t2 ). Romeo does a1 (a2 ) when perceiving signal t˜1 (t˜2 ), and does a3 when not receiving any signal. To see this, note that ‘t˜2 ’ is only received in state t2 , so that Romeo indeed does a2 . When perceiving signal ‘t˜1 ’, 1 1 ×1 ×1 Romeo’s expected payoff of doing a1 equals 1 × 21 + 21 × 1 × 3 + 1 × 21 + 41 × 1 × 0 = 2. 2 2 2 4 2 2 2 4 This equals his fixed payoff 2 of taking action a3 . Romeo’s payoff of taking 1 1 ×1 ×1 action a2 equals 1 × 21 + 21 × 1 × 0 + 1 × 21 + 41 × 1 × 3 = 1. It follows that Romeo 2 2 2 4 2 2 2 4 is just still willing to act as if Juliet loves him when receiving noisy signal ‘t˜1 ’. Juliet’s expected payoff when sending noisy signal ‘t˜2 ’ in state t2 equals 1 1 1 4 × 1 + 2 × 0 + 4 × (−1) = 0. Juliet’s expected payoff when sending noisy signal ‘t˜1 ’ in state t2 equals 12 ×1+ 12 ×(−1) = 0. It follows that, when she is merely fond of Romeo, Juliet is just still willing to send the noisy signal ‘t˜2 ’. Juliet’s expected payoff when sending noisy signal ‘t˜1 ’ in state t1 equals 12 × 3 + 12 × 0 = 1.5; her expected payoff of sending noisy signal ‘t˜2 ’ equals only 14 ×3+ 12 ×1+ 14 ×0 = 1.25. But note now that this noisy signalling equilibrium perfectly replicates the outcome of the mixed equilibrium (with noiseless signalling) described above, and suffers from similar drawbacks: both Romeo and Juliet are in fact indifferent about what to do. Yet, other levels of noise can be found such that each player strictly prefers to follow the separating equilibrium. For instance, take the case μ(t˜1 |t˜2 ) = 0.2, μ(0|t˜2 ) = 0.28, μ(t˜2 |t˜2 ) = 0.52, and μ(0|t˜1 ) = 0.55, μ(t˜1 |t˜1 ) = 0.45. Note that when perceiving signal ‘t˜1 ’, Romeo’s expected payoff of doing a1 equals 1 1 2 ×0.45 2 ×0.2 × 3 + 1 ×0.45+ × 0 = 2.08. His payoff when taking action 1 1 ×0.45+ 1 ×0.2 ×0.2 2

2

1

×0.45

2

2

1

×0.2

2 2 a2 equals 1 ×0.45+ × 0 + 1 ×0.45+ × 3 = 0.92, his payoff when tak1 1 2 2 ×0.2 2 2 ×0.2 ing action a3 is fixed at 2. It follows that Romeo strictly prefers to do action a1 . When not receiving any signal, Romeo’s expected payoff of doing a1 equals 1 1 2 ×0.55 2 ×0.28 × 3 + 1 ×0.55+ × 0 = 1.99. His expected payoff of doing a2 1 1 ×0.55+ 1 ×0.28 ×0.28 2

2

1

×0.55

2

2

1

×0.28

2 2 equals 1 ×0.55+ × 0 + 1 ×0.55+ × 3 = 1.01. This is both smaller than 1 1 2 2 ×0.28 2 2 ×0.28 the fixed payoff 2 of doing a3 . Juliet’s expected payoff when sending noisy signal ‘t˜2 ’ in state t2 equals 0.2 × 1 + 0.52 × 0 + 0.28 × (−1) = −0.08. Her expected payoff when sending noisy signal ‘t˜1 ’ in state t2 equals 0.45 × 1 + 0.55 × (−1) = −0.1. It follows that she strictly prefers to send noisy signal ‘t˜2 ’. Juliet’s expected payoff when sending noisy signal ‘t˜1 ’ in state t1 equals 0.45 × 3 + 0.55 × 0 = 1.35; her expected payoff from sending noisy signal ‘t˜2 ’ in this state equals only 0.2 × 3 + 0.52 × 1 + 0.28 × 0 = 1.12. In general, a range of noisy signalling systems exist that allow for pure-strategy, strict separating equilibria. Note that, as is the case in the treated examples, these noisy signalling systems may leave the players better off than with an equilibrium without noisy signalling as treated in Section 1. Noisy signalling systems are found by plugging general noise levels μ(˜i|˜j) into the linear

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constraints telling that Juliet should be honest, and that Romeo should follow the lead of the perceived signals (or do a3 when not receiving any signal). Given that μ(t˜1 |ti )) + μ(t˜2 |ti ) + μ(0|ti ) = 1, and given that all constraints are linear, the set of noisy signalling systems allowing for a separating equilibrium can be represented as a polyhedron in a four-dimensional space. A more sophisticated version of this argument is found in Blume, Board and Kawamura, (2007) who extend Crawford and Sobel’s (1982) model (see above) to the case of noise. The authors show that noisy communication allows for separating equilibria that are Pareto superior to those presenting strategic generality. In such a model with a continuous range of actions, appropriate noise leads the receiver to revise the action response to any given signal downwards, thus aligning the sender’s and the receiver’s interest. This argument, and the example above, merely show that if players happen to use one out of a particular set of noisy signalling systems, they can still effectively communicate. Yet, if multiple noisy signalling systems are available, communication can only be effective if Romeo observes that Juliet is using a particular noisy signalling system. Otherwise, Juliet could pretend to be using such a system, and still always send a clear ‘I love you’ signal. Put otherwise, the noisy signalling system must ostentatiously be such that the sender cannot control how the receiver will interpret her signals. Blume and Board (2009) assume that the receiver observes the degree of noisiness chosen by the receiver, and that the sender maximises her utility with respect to the level of noise. In terms of the example above, given that the space of noise levels allowing for communication takes the form of a polyhedron, and given the linear structure of Juliet’s expected payoffs, she will pick one of the corner points of the polyhedron. In particular, it can be shown that Juliet will pick a noisy signalling system as close as possible to the one replicating the mixed equilibrium. De Jaegher (2003a) argues that the level of noise in the noisy signalling system can itself be considered as a signal, in constituting a handicap signal. When Romeo receives a noisy ‘I love you’ signal, Romeo makes the reasoning that only a Juliet who really loves him would be willing to incur the cost of her message sometimes getting lost. For an example, take a case where there are only errors of detection, with μ(0|t˜2 ) = 13 , μ(t˜2 |t˜2 ) = 23 , and μ(0|t˜1 ) = 23 , μ(t˜1 |t˜1 ) = 13 . Obviously, Romeo will always take action a1 (a2 ) upon a t˜1 (t˜2 )) signal. When not receiving any sig1 1 ×2 ×1 nal, Romeo’s expected payoff of doing a1 equals 1 × 22 + 13 × 1 ×3+ 1 × 22 + 31 × 1 ×0 = 2. 1

2

×2

3

2

3

2 1

3

×1

2

3

His expected payoff of doing a2 equals 1 × 22 + 13 × 1 × 0 + 1 × 22 + 13 × 1 × 3 = 1. 2 3 2 3 2 3 2 3 The fixed payoff of doing a3 equals 2. It follows that Romeo is just still willing to do a3 . Juliet’s expected payoff when sending noisy signal ‘t˜2 ’ in state t2 equals 23 × 0 + 13 × (−1) = − 13 . Her expected payoff when sending noisy signal ‘t˜1 ’ in state t2 equals 13 × 1 + 23 × (−1) = − 13 . It follows that she is just still willing to send noisy signal ‘t˜2 ’. Juliet’s expected payoff when sending noisy signal ‘t˜1 ’ in state t1 equals 13 × 3 + 23 × 0 = 1; her expected payoff from sending noisy signal ‘t˜2 ’ in this state equals only 23 ×1+ 13 ×0 = 23 . Thus, not receiving a sufficient amount of

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information leads Romeo to taking a costly action. By making her signals more noisy the ‘higher’ is her type, Juliet makes it more likely that Romeo takes an action that is costly to her the higher is her type. This again aligns the players’ interests. How can such noisy talk be linked to natural language? In the words of Blume et al. (2007): “In a given context, the meaning, and hence the correct interpretation of a vague word may depend on the language habits of the utterer. Here, when Juliet says ‘I love you’, Romeo understands that she is probably in love with him, but also that these words are occasionally used to express mere fondness.” But this rationale very much resembles the one for a mixed equilibrium: Juliet sometimes says ‘I love you’ when she is merely fond of Romeo. Yet what we are looking for is an utterance ‘t˜1 ’ sent only in state t1 , but which Romeo sometimes mistakenly interprets as referring to state t2 . Perhaps ambiguity is a potential example. Suppose that Juliet when she loves Romeo ironically tells him ‘Dreadful guy!’. The idea is then that such irony is risky, and that outside of Juliet’s control it will sometimes be misunderstood. Unfortunately, this example could be referred to as one of strategic ambiguity rather than strategic vagueness. In general, it is very difficult to make any concrete link between noisy signalling and natural language. After all, in Farrell’s original example (1993), one way in which Juliet could still credibly communicate with Romeo is by sending an unreliable carrier pigeon which sometimes fails to arrive. In the next section, we argue that correlated equilibria give a more plausible account of vagueness. Correlated equilibria and vagueness. Instead of looking at a noisy signalling equilibrium, let us now explore the possibility of correlated equilibria. In a correlated equilibrium, each player observes random events in nature, and lets his or her strategies depend on these events. The random events observed by the players may be correlated, in turn making the players’ strategies correlated (Aumann, 1974). In the Romeo-Juliet game, consider the following example. Fully independently of whether state t1 or state t2 occurs, let Juliet observe with probability P (tI,j ) an event tI,j , and with probability P (tII,j ) an event tII,j , where P (tI,j ) + P (tII,j ) = 1. Before signalling takes place, Romeo and Juliet go on a date, and event tI,j means that Juliet had an excellent time, whereas in event tII,j Juliet merely had a nice time. Denote by tI,r the event that Romeo had an excellent time, and by tII,r the event that Romeo had a nice time. Denote by μ(i|j) the probability that Romeo perceives event i = tI,r , tII,r when Juliet has observed event j = tI,j , tII,j . Thus, in this model, it need not always be the case that Romeo had an excellent time when Juliet did. Consider now an equilibrium where Juliet says ‘I love you’ in events (t1 , tI,j ), (t1 , tII,j ), and (t2 , tI.j ), and says ‘I like you’ in event (t2 , tII,j ). Put otherwise, Juliet always says ‘I love you’ if she loves Romeo. However, she also says this if she merely likes Romeo but just had an excellent time. Romeo acts as if Juliet loves him (a1 ) when observing (t1 , tII,r ), and acts as if Juliet may either like him or love him (a3 ) when observing (t1 , tI,r ). Also, he acts as if Juliet merely considers him as a friend (a2 ) whether observing (t2 , tI.r ) or (t2 , tII,r ).

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Specifically, consider the case P (tI,j ) = 0.5, P (tII,j ) = 0.5, μ(tII,r |tI,j ) = 0.5, μ(tI,r |tII,j ) = 0.5. When receiving signal ‘I love you’ and observing tII,r , Romeo’s expected payoff of doing a1 equals 1 2 ×(0.5×0.5) 1 1 ×(0.5∗0.5+0.5×0.5)+ 2 2 ×(0.5×0.5) 1 2 ×(0.5×0.5+0.5×0.5) 1 1 2 2 ×(0.5×0.5+0.5×0.5)+ 2 ×(0.5×0.5)

1 2 ×(0.5×0.5+0.5×0.5) 1 1 2 ×(0.5×0.5+0.5×0.5)+ 2 ×(0.5×0.5)

×3+

×0 = 2. In the same case, his expected payoff of do1

×(0.5×0.5)

2 ing a equals ×0+ 1 ×(0.5×0.5+0.5×0.5)+ × 1 2 2 ×0.5×0.5 3 = 1. His expected payoff of doing a3 is fixed at 2. It follows that Romeo weakly prefers to do a1 . When receiving signal ‘I love you’ and observing tI,r , Romeo, similarly weakly prefers to do a3 . Juliet’s expected payoff when sending signal ‘t1 ’ in state t2 upon tII,j , equals 0.5 × 1 + 0.5 × (−1) = 0. Her payoff when sending signal ‘t1 ’ in state t2 upon tI,j , equals (0.5) × 1 + (0.5) × (−1) = 0. Juliet’s expected payoff when sending signal ‘t2 ’ in state t1 equals 0. It follows that Juliet weakly prefers to send signal ‘t1 ’ in state t2 upon tI,j , and signal ‘t2 ’ in state t2 upon tII,j . Her payoff of sending signal ‘t1 ’ in state t1 upon cue tI,j or cue tII,j equals 0.5 × 3 + 0.5 × 0. Her payoff of sending ‘t2 ’, independent of whether she observes tI,j or tII,j , is 1. By checking how often each action gets done in each state, it is easy to see that this correlated equilibrium perfectly replicates the noisy equilibrium with μ(t1 |t2 ) = 0.25, μ(0|t2 ) = 0.25, μ(t2 |t2 ) = 0.5, and μ(0|t1 ) = 0.5, μ(t1 |t1 ) = 0.5. At the same time, this correlated equilibrium perfectly replicates the mixed equilibrium. In fact, the correlated equilibrium treated here is a limit case where the random events observed by the players are independent. When Romeo has an excellent time, this in fact tells nothing about whether or not Juliet had an excellent time. It is tempting to infer that there must also be correlated equilibria where Romeo is more likely to have had an excellent time when Juliet had an excellent time. In such a candidate equilibrium, Juliet says that she loves Romeo in state t1 , but she also says this in state t2 when she had an excellent time (tI,j ). Because of this, when Romeo receives a signal ‘t1 ’ (‘I love you’) and when he had an excellent time (tI,r ), Romeo does a3 , as it is then quite possible that Juliet says ‘I love you’ because she had an excellent time, and not because she loves Romeo. Romeo would only do a1 when receiving a signal ‘t1 ’ (‘I love you’) if he had merely a nice time (tII,r ). As it is relatively likely that Juliet also merely had a nice time, the signal ‘t1 ’ is now convincing. Juliet must really love Romeo, as it seems she merely had a nice time, and still says ‘I love you’. Yet, it can be checked that such an equilibrium cannot exist. For Juliet to tell ‘I love you’ in state t2 when having had an excellent time (tI,j ), it must be that she expects it to be relatively likely that Romeo had a nice time, and will thus do a1 . But if the events observed by Romeo and Juliet are correlated, Romeo will on the contrary be more likely to have had an excellent time as well. For this reason, if each player can observe only two realizations of a random event, then the only correlated equilibrium is an equilibrium that replicates the mixed equilibrium, and is a limit case where the two players’ random events, and with them their strategies, are fully independent.

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At the same time, the analysis of noisy equilibria shows that additional correlated equilibria, where their strategies are truly correlated, must exist. By the so-called revelation principle (Myerson, 1991, Section 6.3), all correlated equilibria of sender-receiver games can be replicated by investigating equilibria where there is a mediator between sender and receiver who possibly garbles the sender’s messages before sending them on to the receiver.1 In the presence of such a mediator, an equilibrium is found if the sender honestly reports her state to the mediator, and if the receiver follows the advice of the mediator. The expected payoffs obtained by the players in such a mediated equilibrium can also be obtained in a correlated equilibrium where sender and receiver observe correlated events. The noisy signalling system described in Section 2.1 performs the same role of a mediator. The existence of a strict noisy equilibrium treated in Section 2.1 shows that a strict correlated equilibrium also exists. However, the argument above shows that such an equilibrium takes on quite a complex form, as a random event with two realizations for each player does not suffice. The interpretation of correlated equilibria in terms of vagueness is the following. Contrary to what is the case in noisy equilibria, the signals used themselves are not noisy. However, the sender uses the signals in a certain context (e.g. Juliet says “I love you” both if she loves Romeo, and if she does not love him but just had an excellent time), and the receiver only observes an imperfect cue of this context. But it is exactly the asymmetric information about the context in which a signal is used that allow the interests of sender and receiver to be still aligned. Put otherwise, the sender’s signals are still credible because it is vague in which context they are used. We end this section by noting that it is obvious that not all vagueness can be seen as an attempt to solve conflicts of interest. It can easily be seen that in a sender-receiver game of the type in Table I, but where the interests of sender and receiver fully coincide, the players could not possibly benefit from using a vague rather than a precise language (Lipman, 2006). For this reason, in the rest of this paper we focus on situations where the interests of the sender and the receiver fully coincide.

3

Absolute Adjectives and Adverbs

In this section, we assume that sender and receiver have common interests, but that due to computational limitations, the sender finds it advantageous to think and talk in coarse-grained categories. Thus, just as in Crawford and Sobel (1982), 1

In general, however, the set of correlated equilibria is larger than the set of noisy equilibria with one-sided communication. For instance, the electronic mail game (Rubinstein, 1989), treated to illustrate noisy signalling in De Jaegher (2003b) is a sender-receiver game where the sender not only sends a signal, but also takes an action. In this case, the set of correlated equilibria can only be approached by means of noisy signalling if the sender and the receiver engage in a conversation of noisy signals. Intuitively, the players can only correlate their actions if the receiver sends noisy confirmations of the sender’s messages. For when such mechanisms replicate the set of correlated equilibria, see Forges (1986).

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49

the sender partitions the continuum of states, but now due to computational limitations rather than to a conflict of interest. Consider adjectives like ‘full’, ‘flat’, or ‘straight’. Just like the meaning of ‘tall’, also the meaning of these adjectives is vague. These adjectives are also perfectly acceptable in comparatives: there is nothing wrong with saying that one surface is flatter than another, or that one bottle is fuller than another. In this respect they differ from adjectives like ‘pregnant’ and ‘even’, and are on a par with other gradable adjectives like ‘tall’. However, as observed by Unger (1975) and also discussed by Rothstein & Winter (2004) and Kennedy & McNally (2005), while with relative adjectives one can easily say something like ‘John is tall, but not the tallest/ but somebody else is taller’, this cannot be done (so naturally) with (maximal) absolute adjectives. What this contrast shows is that sentences with absolute adjectives generate entailments that sentences with relative adjectives lack: it is inconsistent to say that something is flatter than something that is flat. Thus, from ‘The desk is flatter than the pavement.’ we conclude that the pavement is not flat. It seems natural to propose that semantically speaking, an object is flat if and only if it is a maximal element with respect to the ‘flatter than’-relation: M x ∈ F latM iffdef ∀y : x ≥M F lat y, i.e., x ∈ max(>F ). But this straightforward analysis immediately gives rise to a problem: it falsely predicts that an absolute adjective like ‘flat’ can hardly ever be used. It gives rise to another problem as well: it cannot explain why absolute adjectives combine well with adverbs like ‘absolutely’, ‘completely’, and ‘hardly’. A proposal that is both compatible with the natural way to give meaning to absolute adjectives, but that can still account for both of these problems was suggested by Lewis (1979): what is a maximal element with respect to a comparative relation like ‘flatter than’ depends on the level of fine-grainedness. Of course, once we look at things from a courser grain, we loose some information. But this need not always be a bad thing, and can even be beneficial. This is, for instance, the case when we reduce some information (the noise) in an audio signal. Hobbs (1985) suggests that thinking and talking of objects in terms of coarser granularities can be advantageous given our computational limitations, i.e., given that we are only boundedly rational: Our ability to conceptualize the world at different granularities and to switch among these granularities is fundamental to our intelligence and flexibility. It enables us to map the complexities of the world around us into simple theories that are computationally tractable to reason in. Suppose we have two models, M and N , that each give rise to a comparative N ordering ‘flatter than’: ‘>M F ’ and >F ’. Assume that the domains of these models are the same, but that this ordering is more fine-grained in model M than in N model N , i.e. ∀x, y ∈ I : x >M F y → x >F y. In that case, the set of maximal elements of this ordering in M, i.e., the flat objects in M, is a subset of the set of maximal elements of this ordering in M, i.e., the flat objects in N . Let us N assume that in fact max(>M F ) ⊂ max(>F ). Let us now assume that although the denotation of ‘flat’ depends on the fine-grainedness of the model, it still has an independent ‘meaning’: a function from a level of fine-grainedness/model to the maximal elements of the ‘flatter

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than’-relation. If we would limit ourselves to the models M and N , the meaning N of ‘flat’ would be just {max(>M F ), max(>F )}. If ‘flat’ just denotes in each model the maximal elements of the ‘flatter than’relation, modification by means of an adverb like ‘absolutely’ or ‘completely’ does not seem to make much sense: ‘flat’ and ‘completely flat’ would have the same denotation in each model, and thus they would even have the same ‘meaning’. But why, then, would we ever use the adverb? The solution to this problem, we propose, also explains why bare adjectives like ‘flat’ have a ‘vague’ meaning. The explanation is really just the same as Krifka’s (2004) proposal for why round numbers are more vague than others. His explanation crucially makes use of the so-called Horn’s division of pragmatic labor: (un)marked forms get an (un)marked meaning. This principle has recently been given a game-theoretic explanation: if we assume that φc and φ have the same meaning, i.e. [[φc ]] = [[φ]] = {m, m }, but that (i) the marked form, φc , is slightly more costly than an unmarked form φ, and (ii) state m is more probable than m , it can be explained (by Pareto optimality, Parikh; evolution, van Rooij 2004 and De Jaegher 2008; forward induction, van Rooij, 2008)2 why the mapping that associates φ with m and φc with m is pragmatically the most natural equilibrium. Let us see how this explanation works for our case. First we say that ‘flat’, ‘f ’, and ‘completely flat’, ‘cf ’, have the same meaning: [[f ]] = [[cf ]] = {max(>M F ), max(>N F )}. It seems natural to assume that ‘completely flat’ is the marked form, because it is longer than ‘flat’.3 The difference in probability of the two elements N in [[f ]] naturally follows from the fact that max(>M F ) ⊂ max(>F ). But that is all we need: the marked expression ‘completely flat’ gets the more precise but also the more unlikely meaning max(>M F ), while the unmarked ‘flat’ is interpreted as max(>N F ). As an immediate consequence we also see that the bare adjective ‘flat’ is thus interpreted pragmatically in a rather vague, or coarse-grained, way.

4

Semi-orders and Bounded Rationality

In this section we will show that the reason why a predicate is vague is closely related with the reason why agents are only boundedly rational (in a certain way). We will see that a particular type of bounded rationality can be modeled in terms of so-called ‘semi-orders’ and that these orders are also appropriate to model vagueness. In this section we axiomitize such orderings and show how this is related with a pragmatic take on the Sorites paradox, related with the pragmatic approach to vagueness as explained in Section 4. In the economic theories of individual and collective choice, preference relations are crucial. In the theory of ‘revealed preference’ it is standard to derive a preference relation in terms of how a rational agent would choose among different sets of options. Let us define with Arrow (1959) a choice structure to be a 2 3

For a recent laboratorium experiment, see De Jaegher, Rosenkranz and Weitzel (2008). Of course, for this explanation to go through, we have to assume that ‘flat’ is not in competition with another costly expression with the same meaning like ‘flat, roughly speaking’.

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triple I, O, C, where I is a non-empty set of options, the set O consists of all finite subsets of I, and the choice function C assigns to each finite set of options o ∈ O a subset of o, C(o). Arrow (1959) stated the following principle of choice (C) and constraints (A1) and (A2) to assure that the choice function behaves in a ‘consistent’ or ‘rational’ way:4 (C) ∀o ∈ O : C(o) = ∅. (A1) If o ⊆ o , then o ∩ C(o ) ⊆ C(o). (A2) If o ⊆ o and o ∩ C(o ) = ∅, then C(o) ⊆ C(o ). If we say that x > y, iffdef x ∈ C({x, y}) and y ∈ C({x, y}), one can easily show that the ordering as defined above gives rise to a strict weak order. A structure I, R, with R a binary relation on I, is a strict weak order just in case R is irreflexive (IR), transitive, (TR), and negatively transitive (NTR). Definition 1. A strict weak order is a structure I, R, with R a binary relation on I that satisfies (IR), (TR), and (NTR): (IR) ∀x : ¬R(x, x). (TR) ∀x, y, z : (R(x, y) ∧ R(y, z)) → R(x, z). (NTR) ∀x, y, z : (¬R(x, y) ∧ ¬R(y, z)) → (¬R(x, z)). If we now define the indifference relation, ‘∼’, as follows: x ∼ y iffdef neither x > y nor y > x, it is clear that ‘∼’ is not only reflexive and symmetric, but also transitive and thus an equivalence relation. It is well-known that in case ‘>’ gives rise to a (strict) weak order, it can be represented numerically by a real valued utility function u such that for all x, y ∈ I: x > y iff u(x) > u(y), and x ∼ y iff u(x) = u(y). The transitivity of the ‘indifference’-relation defined in terms of the preference relation as induced above is sometimes problematic. A famous example to show this is due to Luce (1956): A person may be indifferent between 100 and 101 grains of sugar in his coffee, indifferent between 101 and 102, ..., and indifferent between 4999 and 5000. If indifference were transitive he would be indifferent between 100 and 5000 grains, and this is probably false. That the observed indifference relation is non-transitive might be interpreted in two different ways: (i) on a descriptive approach one can say that the choice behavior simply does not need to satisfy Arrow’s constraints, or (ii) alternatively, one takes the axiom to define rationality, and an agent who does not seem to obey them can then at best be only boundedly rational. Of course, the nontransitivity of the relation ‘∼’ extends beyond the case of preference: sounds a and b can be judged equally loud, just as sounds b and c, but it need not be the case that sounds a and c are judged equally loud. This non-transitivity is a problem, because it immediately leads to the ‘Sorites paradox’, the famous problem induced by vague expressions. 4

Arrow (1959) actually stated (C) and (A): If o ⊆ o and o ∩ C(o ) = ∅, then C(o) = o ∩ C(o ). But, obviously, (A) is the combination of (A1) and (A2).

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Consider a long series of cups of coffee ordered in terms of the grains of sugar in it. Of each of them you are asked whether the coffee is sweet or not. We assume that you are always indifferent between two subsequent cups. Now, if you decide that the coffee in the first cup presented to you — the one with 100 grains of sugar — does not taste sweet, i.e., taste bitter, it seems only reasonable to judge the second cup of coffee to be bitter as well, since you are indifferent between the two different cups. But, then, by the same token, the third cup should also be bitter, and so on indefinitely. In particular, this makes the last cup of coffee taste bitter, which is in contradiction with our intuition that the coffee in this last cup is sweet. This so-called Sorites reasoning is elementary, based only on our intuition that the coffee in the first cup is not sweet, the last one is, and the following inductive premise, which seems unobjectable: [P] If you call one cup of coffee bitter, and this cup is indistinguishable from another cup, you have to call the other cup bitter too. Thus, for any x, y ∈ I : (P (x) ∧ x ∼P y) → P (y). If we assume that it is possible that ∃x1 , . . . , xn : x1 ∼P x2 ∧ . . . ∧ xn−1 ∼P xn , but P (x1 ) and ¬P (xn ), the paradox will arise. Strict weak orders, however, do not allow for the possibility that ∃x1 , . . . , xn : x1 ∼P x2 ∧ . . . ∧ xn−1 ∼P xn , but P (x1 ) and ¬P (xn ). Fortunately, there is a well-known ordering with this property. According to this ordering the statement ‘x P y’ means that x is significantly or noticeably greater than y, and the relation ‘P ’ is irreflexive and transitive, but need not be almost connected. The indistinguishability relation ‘∼P ’ is reflexive and symmetric, but need not be transitive. Thus, ‘∼P ’ does not give rise to an equivalence relation. The structure that results is what Luce (1956) calls a semi-order. A structure I, R, with R a binary relation on I, is a semi-order just in case R is irreflexive (IR), satisfies the interval-order (IO) condition, and is semitransitive (STr). Definition 2. A semi-order is a structure I, R, with R a binary relation on I that satisfies the following conditions: (IR) ∀x : ¬R(x, x). (IO) ∀x, y, v, w : (R(x, y) ∧ R(v, w)) → (R(x, w) ∨ R(v, y)). (STr) ∀x, y, z, v : (R(x, y) ∧ R(y, z)) → (R(x, v) ∨ R(v, z)). As we noted above, semi-orders can be understood as models of bounded rationality or limited perceptive ability. Just as weak orders, also semi-orders can be given a measure theoretical interpretation. If P is the predicate ‘bitter’, one can think of R as the semi-order relation (significantly) ‘more bitter than’, and say that ‘x P y’ is true iff the bitterness of x is higher than the bitterness of y plus some fixed (small) real number . In the same way ‘x ∼P y’ is true if the difference in bitterness between x and y is less than . In case  = 0, the semi-order is a strict weak order. Semi-orders are appropriate to represent vagueness, but by themselves do not point to a solution to the Sorites paradox at all. The standard reaction to the

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Sorites paradox taken by proponents of fuzzy logic and/or supervaluation theory is to say that the argument is valid, but that the inductive premise [P] (or one of its instantiations) is false. But why, then, does it at least seem to us that the inductive premise is true? According to the standard accounts of vagueness making use of fuzzy logic and supervaluation theory, this is so because the inductive premise is almost true (in fuzzy logic), true in almost all complete valuations (in supervaluation theory), or that almost all its instantiations are true. Most proponents of the contextuallist solution follow Kamp (1981) in trying to preserve (most of ) [P] by giving up some standard logical assumptions, and by making use of a mechanism of context change. But we do not believe that context change is essential to save natural language from the Sorites paradox. Our preferred ‘solution’ is radically pragmatic in nature and completely in line with Wittgenstein’s Philosofische Untersuchungen.5 The first observation is that the meaning of vague adjectives like small and tall are crucially context dependent: for Jumbo to be a small elephant means that Jumbo is being small for an elephant, but that does not mean that Jumbo is small. For instance, Jumbo will be much bigger than any object that counts as a big mouse. One way to make this explicit is to assume with Klein (1980) that every adjective should be interpreted with respect to a comparison class, i.e. a set of individuals. The truth of a sentence like John is tall depends on the contextually given comparison class: it is true in context (or comparison class) o iff John is counted as tall in this class. The idea of the radically pragmatic solution to the Sorites paradox is that it only makes sense to use a predicate P in a context – i.e. with respect to a comparison class –, if it helps to clearly demarcate the set of individuals within that comparison class that have property P from those that do not. Following Gaifman (1997), we will implement this idea by assuming that any subset of I can only be an element of the set of pragmatically appropriate comparison classes O just in case the gap between the last individual(s) that have property P and the first that do(es) not must be between individuals x and y such that x is clearly P -er than y. This is not the case if the graph of the relation ‘∼P ’ is closed in o × o.6 Indeed, it is exactly in those cases that the Sorites paradox arises. How does such a proposal deal with the Sorites paradox? Well, it claims that in all contexts o in which P can be used appropriately, [P1 ] is true, where [P1 ] is ∀x, y, o : (P (x, o)∧x ∼P y) → P (y, o)). If we assume in addition that the first element x1 of a Sorites series is the absolute most P -individual, and the last element xn the absolute least P -individual, it also claims that in all contexts o in which it is appropriate to use predicate P in combination with x1 and xn , ‘P (x1 , o)’ is true and ‘P (xn , o)’ is false.7 Thus, in all appropriate contexts, the premises of the 5 6 7

See in particular sections 85-87. Notice that also in discrete cases the relation ‘∼P ’ can be closed in o × o. In just depends on how ‘∼P ’ is defined. But don’t we also feel that in case o is an inappropriate context, the first element should still be called a P -individual, and the last one a ¬P -individual? According to this account, this intuition is ‘accounted’ for by noting that this indeed comes out in all appropriate subsets of o.

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Sorites argument are considered to be true. Still, no contradiction can be derived, because using predicate P when explicitly confronted with a set of objects that form a Sorites series is inappropriate. Thus, in contrast to the original contextualist approaches of Kamp (1981), Pinkal (1984), and others, the Sorites paradox is not avoided by assuming that the meaning (or extension) of the predicate changes as the discourse proceeds. Rather, the Sorites paradox is avoided by claiming that the use of predicate P is inappropriate when confronted with a Sorites series of objects. Our pragmatic solution assumes that the appropriate use of a predicate P must clearly partition the relevant comparison class. Moreover, we want ‘P -er than’ to generate a semi-order rather than a strict weak order. In the following we will suggest how to generate semi-orders in terms of the way predicates partition appropriate comparison classes. Let us start just like Arrow (1959) with a choice structure: a triple I, O, P , where I is a non-empty set of options, the set O of comparison classes, and for each predicate a choice function P assigns to each o ∈ O a subset of o, P (o). We will adopt Arrow’s condition (C) and his constraints (A1) and (A2). If we say with Klein (1980) that that x >P y, iff there is an o ∈ O such that def x ∈ P (o) and y ∈ 0 − P (o), it is easy to see that the relation will be a strict weak order if O consists of all subsets of I (of cardinality 2 and 3). The idea how to generate a semi-order ‘P ’ is to assume that O does not consists of all these subsets. Instead, we just start with all subsets of I that consist of two elements that are not indifferent and close this set of subsets of O under the following closure conditions:  (P1) ∀o ∈ O : ∀x ∈ O : o ∪ {x} ∈ GAP → o ∪ {x} ∈ O. n with o ∈ GAP iffdef ∃n−1 o ⊂ o : card(o ) = n − 1 ∧ o ∈ O. (OR) ∀o ∈ O, {x, y} ∈ O : o ∪ {x} ∈ O or o ∪ {y} ∈ O. (P2) ∀o ∈ O, x ∈ I : o ∈ GAP2 → o ∪ {x} ∈ O. with on ∈ GAP2 iffdef ∃n o ⊂ o : card(o ) = n − 1 ∧ o ∈ O. Constraint (P1) says that to any element o of O one can add any element x ∈ I to it that is in an ordering relation with respect to at least one other element, on the condition that o∪{x} satisfies the GAP -condition. The intuition behind this condition is that only those subsets of I satisfy GAP if there is at least one gap in this subset w.r.t. the relevant property. It is easy to show that (P1) guarantees that the resulting ordering relation will satisfy transitivity and will thus be a strict partial order.8 If constraint (OR) is added, the resulting ordering relation also satisfies the interval ordering condition. Constraint (P2), finally, guarantees that the ordering relation also satisfies semi-transitivity, and thus is a semiorder. As far as we can see, the closure conditions stated above are such that they generate all and only all appropriate contexts, i.e., just those subsets of I for which there is such a sufficiently large gap such that also vague predicates can clearly partition the context without giving rise to the Sorites paradox. 8

The proof of this result, and the others mentioned below, are all closely related with similar proofs for slightly different choice structured discussed in Van Rooij (2009).

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In particular, O does not necessarily contain all subsets of O. This is essential, because otherwise the resulting ordering relation would also satisfy (NTR) which is equivalent to ∀x, y, z : x P y → (x P z ∨ z P y). If that were the case, the resulting ordering relation would not only be a semi-order, but also a weak order, which is what we do not want. It suffices to observe that no constraint formulated above forces us to assume that {x, y, z} ∈ O if x P y, which is all that we need.

5

A Behavioural Approach to Vagueness: Prospect Theory

In the previous sections, context was each time used to account for vagueness. Yet, context did not directly enter the agent’s valuations. We now treat a model where context as seen by the agent, or the agent’s reference point, directly enters his or her valuation function. Let x be the temperature of a room in degrees Celsius. An agent has a valuation function v(x), with v  (x) > 0. If v ≥ V , the agent considers the room as warm. If v < V , she considers it as cold. The problem with such a simple model is that it is highly implausible that there exists such a context-independent cut-off point. This would for instance mean that our agent considers any room of 18o as cold, whether she has just walked into this room from freezing temperatures outside, or from another room heated at 21o . Let us try to extend our model such that the meaning of ‘warm’ and ‘cold’ becomes context-dependent. Consider the set of all possible alternatives (= in alternative room temperatures) open to agent, denoted as I, where x denotes a typical alternative in I. Let an agent currently compare the alternatives in a set o, with o ⊆ I. Let |o| be the number  of elements in o. Define the average value 1 of the alternatives in set o as |o| x∈o v(x) . Define now as the agent’s psychic valuation the value: 1  u(xi , o) =def v(xi ) + α[v(xi ) − v(x)] |o| x∈o The coefficient α ≥ 0 measures the extent to which the agent is subject to the context in his valuation of xi . For α = 0, we have a standard utility function, leading to a context-independent cut-off point. For α approaching infininity, we have an agent who strictly thinks in gains and losses with  respect to a value 1 reference point, which in this case takes on the value |o| x∈o v(x). The latter is a simplified version of the psychic valuation described by Kahneman and Tversky (1979) in their prospect theory. Assume now that if u ≥ U , the room is considered as warm. If u < U , it is considered cold. Any α with 0 < α < 1 now suffices to account for a concept ‘warm’ that has no clear context independent boundaries. To see why, consider the example in Figure 1. The choice sets under consideration of the agent are first the subset o , and second the subset o of I. The numbers are real temperatures measured in Celsius. Let v(x) = x, thus v(18o ) = 18o . Consider u as how warm or cold the agent subjectively feels the

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real temperature to be. Let u ≥ 19o be warm, and let u < 19o be cold. Let α = 0.5. Consider first u(18o , o ). As the average real temperature in this set of alternatives is 18o , u(18o , o ) = 18, and the agent considers 18o to be cold. The agent may be seen as recently having been in rooms heated at 16o , 18o and 20o . A room heated at 18o is then felt to be a cold room. Next, consider u(180 , o ). Here, the average actual temperature in u is 16o . As this average temperature is rather low, u(18o , o ) = 19o . Thus, when the alternatives under consideration are the ones in o , 18o feels like 19o , and is considered as warm. The agent now may be seen as having recently walked between outside temperatures of 14o and 16o , and a room heated at 18o . The room heated at 18o now feels warm. These subjective feelings may also be seen as a form of satisficing, where satisficing behaviour depends on one’s reference level. When the choice is between 14o , 16o and 18o , 18o will do. When the choice is between 16o , 18o and 20o, 18o is considered as unsatisfactory. D 20 18 20

18 D

16

O 14

O

16 14

Fig. 1. Temperatures and reference points

Prospect theory can thus help us to account for the fuzzy boundaries between concepts such as warm and cold. The reason why these boundaries are fuzzy may be that the ad hoc boundary depends on the decision maker’s reference point. Consider then a sender and receiver with common interests, in that the receiver would genuinely like to know the extent to which a sender feels something to be warm. Then temperatures will not do, because depending on the sender’s reference point, he may feel 18o to be warm or cold. Fuzzy language instead is able to convey the extent to which the sender feels warm. There is a further aspect of vagueness reflected in the Sorites paradox, namely that small differences (e.g. in temperature) do not matter. This has been theoretically argued by Luce (1956), and has also been observed in psychological experiments. What could prospect theory say about small differences? It seems intuitive that the extent to which small differences are perceived again depends on the agents’ reference point. The difference between 18o and 19o may seem whimsical to an agent who has just experienced 0o . But this difference may not seem whimsical to an agent who has only experienced 18o , 19o and 20o. Unfortunately, the reference-dependent utility function above cannot account for such

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an effect. To see this, consider two alternatives xi and xj , where first these alternatives are considered within a set o , and next they are considered within a set o . It is clear that u(xi , o ) − u(xj , o ) = u(xi , o ) − u(xj , o ). Because of the linear structure of the reference-dependent utility function, the reference, measured by the average in the set under consideration, cancels out. Let us instead assume that the consumer does not directly value the difference between v and the reference point (the average of the v’s under consideration), but instead values a function f of this difference: u(xi , o)

=def

v(xi ) + αf [v(xi ) −

1  v(x)] |o| x∈o

and let u(xi ) ≈ u(xj ) if |u(xi ) − u(xj )| < . In line with prospect theory, where agents have smaller marginal valuations the further away from the reference point, f is assumed to have an s-shaped form, as represented in Figure 2. The origin in Figure 2 is determined by the reference point. Gains with respect to the reference point are measured above the horizontal axis, losses are measured below the horizontal axis. f



v(xj ) =

1 |O  |

O 

v(x)

Fig. 2. Prospect theory psychic valuation function

√ As an example, let α = 0.5, and let f (·) be equal to + if v(x) is larger than √ the reference, and equal to - if v(x) is lower. Assume again that v(x) = x. Let  = 1.2. Consider now first u(17o , o ) − u(16o , o ) when o = {0o , 16o, 17o }. Then u(17o , o ) − u(16, o ) = 1 + 12 × (60.5 − 50.5 ) = 1.11 < 1.2. When compared to 0o , the difference between 16o and 17o is considered whimsical by the agent. Next, consider u(17o, o ) − u(16, o ) when o = {16, 17, 18}. Then u(17o , o ) − u(16o , o ) = 1 + 12 × (10.5 ) = 1.5 > 1.2. When compared to 18o , the difference between 16o and 17o is considered as significant. The relevance to the Sorites paradox of the point that whether or not small differences matter depends on one’s reference point, is the following. When asked to consider a series of Sorites arguments, saying that 0.1o does not matter, we are contemplating the full range of cups of coffees, from very cold ones to very hot ones. Given this reference point, 0.1o of difference indeed does not matter.

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However, when we are making an actual decision on whether a cup of coffee is warm of cold, our reference point will not be the whole range of coffees, but will probably be a small range of coffee temperatures, such as they are produced by our coffee machine. Given such a reference point, small differences may still matter. Thus, the Sorites paradox may arise because the paradox induces us to take e.g. a clearly cold cup of coffee as a reference point, and then to think about the difference between hot and slightly less hot cups of coffee. However, when facing actual cups of coffee, our reference point will never differ that widely from our actual experience, so that small differences may still matter.

6

Conclusion

In Section 2, we have started this paper by reviewing game-theoretic rationales for vagueness (and the related concepts of ambiguity and generality), which see vagueness as solving conflicts of interest between speaker and listener. We conclude from this review that it is difficult to link these game-theoretic rationales for vagueness to concrete instances of natural language. Also, it does not seem plausible that all instances of vagueness would be linked to conflicts of interest. Given these conclusions, two areas for further research arise, and have been treated in this paper. In one approach (Sections 3 and 4) one continues to assume, just as in game theory, that agents are rational, even though they find it difficult to distinguish between different states of the nature. Pragmatically, in contexts where they find this most difficult, it makes most sense to make use of a vague language. Section 5 treats a behavioral approach where contexts, in the form of reference points, directly enter the agents’ preferences. Laboratory experiments have shown that agents form such reference points. Analyzing language use under such reference-dependence is an open field for future research.

References Arrow, K.: Rational choice functions and orderings. Economica 26, 121–127 (1959) Aumann, R.: Subjectivity and correlation in randomized strategies. Journal of Mathematical Economics 1, 67–96 (1974) van Benthem, J.: Later than late: on the logical origin of the temporal order. Pacific Philosophical Quarterly 63, 193–203 (1982) Blume, A., Board, O.J., Kawamura, K.: Noisy talk. Theoretical Economics 2, 395–440 (2007) Blume, A., Board, O.: Intentional vagueness working paper. University of Pittsburgh (2008) Crawford, V.P., Sobel, J.: Strategic information transmission. Econometrica 50, 1431– 1451 (1982) De Jaegher, K.: Error-proneness as a handicap signal. Journal of Theoretical Biology 224, 139–152 (2003a) De Jaegher, K.: A game-theoretic rationale for vagueness. Linguistics and Philosophy 26, 637–659 (2003b)

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De Jaegher, K.: The evolution of Horn’s rule. Journal of Economic Methodology 15, 275–284 (2008) De Jaegher, K., Rosenkranz, S., Weitzel, U.: Economic laboratory experiment on Horn’s rule. Utrecht University (2008) (working paper) Dummett, M.: Wang’s paradox. Synthese 30, 301–324 (1975) Farrell, J.: Meaning and credibility in cheap-talk games. Games and Economic Behavior 5, 514–531 (1993) Forges, F.M.: An approach to communication equilibria. Econometrica 54, 1375–1385 (1986) Fudenberg, D., Tirole, J.: Game Theory. MIT Press, Cambridge (1991) Gaifman, H.: Vagueness, tolerance and contextual logic. Columbia University (1997) (manuscript) Graff, D.: Shifting sands: An interest relative theory of vagueness. Philosophical Topics 28, 45–81 (2000) Harsanyi, J.C.: Games with randomly disturbed payoffs: a new rationale for mixedstrategy equilibrium points. International Journal of Game Theory 2, 1–23 (1973) Kahneman, D., Tversky, A.: Prospect theory: an analysis of decision under risk. Econometrica 47, 263–291 (1979) Kamp, H.: Two theories of adjectives. In: Keenan, E. (ed.) Formal Semantics of Natural Language, pp. 123–155. Cambridge University Press, Cambridge (1975) Kamp, H.: The paradox of the heap. In: M¨ onnich, U. (ed.) Aspects of Philosophical Logic, pp. 225–277. D. Reidel, Dordrecht (1981) Keefe, R.: Vagueness without context change. Mind 116, 462 (2007) Klein, E.: The semantics of positive and comparative adjectives. Linguistics and Philosophy 4, 1–45 (1980) Krantz, D., Duncan Luce, R., Suppes, P., Tversky, T.: Foundations of Measurement, vol. 1. Academic Press, New York (1971) Lipman, B.L.: Why is language vague? Boston University (2006) (working paper) Luce, R.D.: Semiorders and a theory of utility discrimination. Econometrica 24, 178– 191 (1956) Myerson, R.B.: Game Theory – Analysis of Conflict. Harvard University Press, Cambridge (1991) Pinkal, M.: Consistency and context change: The Sorites paradox. In: Landman, F., Veltman, F. (eds.) Varieties of Formal Semantics, Proceedings of the fourth Amsterdam Colloquium. Groningen-Amsterdam Studies in Semantics, Foris Publications, Dordrecht (1984) Raffman, D.: Vagueness without paradox. The Philosophical Review 103, 41–74 (1994) Raffman, D.: Vagueness and Context-Relativity. Philosophical Studies 81, 175–192 (1996) van Rooij, R.: Revealed preference and satisficing behavior. In: Bonano, G., et al. (eds.) Proceedings of LOFT 2008, Amsterdam University Press (2009)

Now That You Mention It Awareness Dynamics in Discourse and Decisions Michael Franke and Tikitu de Jager ILLC, Universiteit van Amsterdam

Abstract. We model unawareness of possibilities in decision making and pragmatic reasoning. A background model is filtered through a state of limited awareness to provide the epistemic state of an agent who is not attending to all possibilities. We extend the standard notion of awareness with assumptions (implicit beliefs about propositions the agent is unaware of) and define a dynamic update for ‘becoming aware.’ We give a propositional model and a decision-theoretic model, and show how pragmatic relevance reasoning can be described in the latter. An utterance can be relevant even if semantically uninformative, if it brings relevant alternatives to awareness. This gives an explanation for the use of possibility modals and questions as hedged suggestions, bringing possibilities to awareness but only implicating their degree of desirability or probability.

Our aim in this paper is to introduce the notion of unawareness into formal pragmatics and to show how this helps extend existing decision-theoretic approaches. It is a rather commonplace observation that when using language on an everyday basis, we do not attend to all conceivable possibilities all of the time. Quite sensibly, we do not devote attention and consideration to far-fetched and bizarre contingencies that are of little practical effect. Less sensibly, but inevitably, we also occasionally fail to recognise genuinely relevant possibilities: we forget, we overlook, we miss connections and fail to draw conclusions. Recent research in rational choice theory converged on the realization that the epistemic attitudes we have to such overlooked possibilities are not well represented by either belief (as standardly understood) or by uncertainty. To cope with unawareness, we therefore need richer representations of an agent’s cognitive state to encode not only the agent’s beliefs but also which possibilities she conceives of and which she is unaware of. These insights are relevant to formal semantics and pragmatics. Palpably, awareness of possibilities can change in the course of a conversation. More crucially even, if unawareness is frequent and object to change in conversation, then 

The authors are listed in alphabetical order. We would like to thank: Sven Lauer and two anonymous referees for their attention to the manuscript; Robert van Rooij and Remko Scha for discussion of the notions; Anton Benz for the invitation to present these ideas at the zas, Berlin; and audiences in Amsterdam and Berlin for the valuable feedback we have received.

A. Benz et al. (Eds.): Language, Games, and Evolution, LNAI 6207, pp. 60–91, 2010. c Springer-Verlag Berlin Heidelberg 2010 

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it is natural that conversationalists might deliberately bring about changes in each others’ states of awareness. Here the distinctly pragmatic dimension of awareness dynamics enters the picture. But raising awareness differs from more traditional ‘semantic information flow’: interlocutors can make one another aware of a new possibility without having to express any belief-like attitude regarding it, and thus without giving any direct information in the usual sense. Our suggestion is that this perspective gives us access to a wide range of conversational phenomena that cannot be analysed in terms of information-giving. We will try, as far as possible, to extend standard techniques from decision-theoretic pragmatics to accommodate unawareness and awareness dynamics. This will help to capture some of the relevant pragmatic reasoning surrounding unawareness in discourse. In the next two sections we will give first an informal then a formal introduction to unawareness. Section 3 works out an example of pragmatic reasoning based on awareness dynamics in detail, and Section 4 describes the dynamics of awareness updates and their interaction with factual information growth. We site our work in the intersection between linguistics and rational choice theory in Section 5, before Section 6 concludes.

1

Unawareness in Conversation

Example 1: Little Bo Peep. Little Bo Peep has lost her keys and doesn’t know where to find them. She’s checking her pockets, the nail behind the door, beside the telephone, and so on, but the keys are nowhere to be found. Frustrated and pouting, Bo slams onto the sofa. From his corner Little Jack Horner helps her out: jack: Did you leave them in the car when you came in drunk last night? Little Bo slaps her forehead (and his, impudent little scamp) and goes out to the car. Several days later Bo finds her keys in the sugar jar. Nobody should have trouble understanding at an intuitive level the kinds of changes Little Bo Peep’s epistemic state undergoes in this example. Bo didn’t think of the car as a possible place to look for the keys, but when Jack mentioned it to her, her oversight struck her as foolish and she promptly took the proper action. Should anybody be worried about a situation as commonplace as this? Peculiarities show once we take a closer look. Interestingly, Jack manages to change Bo’s epistemic state in quite a significant way, simply by asking her a question. Since under any standard semantic analysis questions are uninformative, there is some explaining to be done here. Before we get sidetracked by considering rhetorical questioning or possible pragmatic analyses, though, consider the following alternatives to Jack’s helpful observation: (1) Jack: Do you think it’s possible the keys are in the car? (2) Jack: The keys might be in the car. (3) Jack: [Not paying attention] Hey, this tv show is really funny, this guy is looking everywhere for his keys and they were in his car the whole time!

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(4) Advertiser on tv: Do you forget your keys in the car? You need the ExtendaWristLock KeyChain! (Patent pending.) Order now and pay just $19.99! (5) Passing motorist: Honk honk! Bo’s response to any of these might quite naturally be the same: she slaps her forehead at her foolishness and goes immediately to check the car. While the first two should be amenable to pragmatic explanation, this clearly won’t do for the others. Intuitively what’s going on here is that Bo is failing to consider a possibility, which when brought to her attention she realises should not be ruled out. We will say that Bo is unaware of the possibility that the keys might be in the car. A formal notion of unawareness has been explored very fruitfully first in theoretical computer science (see [FH88]) and has recently been applied variously in rational choice theory, i.e., game and decision theory (e.g., [Fei04, Fei05, HR06]; see also Section 5). 1.1

Three Slogans about Unawareness

So what is unawareness of a contingency? Here are three basic interrelated properties of the notion in question1 . Slogan 1: Unawareness is not uncertainty. Little Bo Peep’s behaviour (pouting, sitting on the couch) does not indicate uncertainty about whether the keys are in the car. At the point of the story where Bo gives up the search, it is implausible to assume that she puts any credence, however small, on the possibility that the keys are in the car; if she did, she would surely have gone and checked. Judging from her behavior, if anything, it seems as if Bo believes that the keys are not in the car. We will say that Bo has an implicit belief that the keys are not in the car and argue that, firstly, implicit beliefs are different from the more familiar explicit beliefs (see the next two slogans) and that, secondly, unawareness of a possibility typically goes together with an implicit belief we call an assumption. Implicit beliefs are typically assumptions of ‘normality’ (we discuss them in more detail, including the connection to closed-world reasoning, in Section 5.2). Slogan 2: Unawareness is not introspective. Although Bo’s behavior indicates an implicit belief, she does not explicitly (or consciously) believe that the keys are not in the car. In fact, she holds no explicit beliefs about the car, not even the tautological belief that the keys are either in the car or elsewhere. A selfreferential way to get to grips with this fundamental intuition is to say that she is 1

There are subtle but crucial differences, both conceptually as well as formally, between our approach here and other approaches in computer science or rational choice theory. These differences can best be appreciated after we have exposed our formalism, so a discussion and comparison is postponed to Section 5.

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unaware of her own implicit beliefs. This failure of (negative) introspection2 leads us to a definition of awareness in terms of the language Bo uses: if she were asked to describe all her beliefs about the keys she would not mention the car at all. (If prompted she might start enumerating tautologies: “The keys aren’t on the table, so I suppose logically speaking they’re either on the table or not on the table”; she would never, however, extend this sequence with “The keys are either in the car or not in the car.”) The formal model we give in Section 2 will distinguish syntactically between the agent language Bo would use spontaneously to describe her explicit beliefs and the language we as modellers use to describe her implicit beliefs. Slogan 3: Unawareness is easily overturned. Bo’s implicit belief is very fragile, in a way that her explicit beliefs are generally not: it does not take any particularly convincing argument to overturn it. Such awareness dynamics often have what we might call a ‘forehead-slap property’: in becoming aware of her implicitly held belief, Bo realises the mistake she made in overlooking a certain (intuitively: relevant) possibility. We will concern ourselves exclusively with unawareness of this kind in this paper: unawareness through inattentiveness or absent-mindedness. Indeed, as the alternative (5) in Example 1 shows, overturning this kind of unawareness need not even involve anything linguistic or intentional. But where language is concerned, we argue that the mere mentioning of some possibility that an agent is unaware of is sufficient for this unawareness to be overturned. This is why it is not possible to talk to Bo about her implicit beliefs: if Jack were to ask her the question in (1), we might imagine her answer to be something like: “Well, now that you ask me I do think it’s possible that the keys are in the car, but I wouldn’t have if you hadn’t asked me.” 1.2

Pragmatic Reasoning about Unawareness

We have seen how an agent can become aware of previously overlooked possibilities. Pragmatics enters the picture because one agent may deliberately make another aware of some contingency. Just as in cases of information-giving, pragmatic reasoning starts when the hearer asks “Why did the speaker choose to tell me that ?” In the case of an awareness update, then, the hearer may think “Why is he directing my attention to that particular possibility?” The fact that the speaker chooses her utterance under constraints of cooperative conversation licenses the same kinds of pragmatic inferences that we are used to calculating for more conventional utterances. The account of pragmatic inference we make use of in this paper is decisiontheoretic. Cooperative conversation takes place against the background of a commonly understood goal represented by the utility component of a decision problem, and expected utility measures progress towards that goal. Our aim in 2

She does not know, but does not know that she does not know. [MR94] and [DLR98] show that unawareness must be more than simply failure of negative introspection, if we are to capture the properties we want. Still, the notion is a good intuitive starting point.

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this paper is not to provide an extensive justification of such an account; rather we take it largely for granted as a starting point, and extend it with a representation of awareness. Our intention, then, is to keep the pragmatic component as standard as possible; we will argue that the representation of a decision problem needs to incorporate only slightly more structure to allow unawareness effects, but the rest of the account is entirely standard. In Bo’s case, of course, pragmatic reasoning is not strictly necessary. Jack’s question brings to her attention exactly the possibility she needs to be aware of, just as the non-communicative “Honk” in (5) of a passing car would have done.3 However conversational agents with some pragmatic sophistication can achieve more than this. Here is an example where the hearer gains information from pragmatic reasoning about an awareness update: Example 2: Bob the Baker. Bob (who is an expert baker) is visiting his friend Farmer Pickles (who isn’t). pickles: I was going to bake a cake but I haven’t got any eggs! bob: Did you think of making shortbread instead? pickles: I didn’t, in fact I didn’t even know that you don’t need eggs to make shortbread! Thanks, Bob! We can highlight the pragmatic component of this update by contrasting it with one in which Pickles, idly flicking through a recipe book, sees a page headed “Shortbread.” In both cases he becomes aware of the possibility of baking shortbread, but in the second case he would have to check the recipe to find out whether it needs eggs or not. The pragmatic reasoning involved here is at an intuitive level entirely standard: Bob’s utterance must be intended to improve Pickles’s situation (on the assumption that Bob is cooperative); the utterance draws attention to a possibility Pickles has overlooked; this can only be helpful if that possibility is a ‘useful’ one, i.e., if the recipe for shortbread does not require eggs; this, in turn, is a fact which Bob (an expert baker) can be expected to know. The schema here is standard: utterance ϕ is only (cooperatively) felicitous if the speaker believes that ψ, and this particular speaker can be expected to know whether ψ or not; therefore the hearer concludes from the utterance of ϕ that ψ holds. Only one link in the chain is unusual: that drawing attention to a possibility can be ‘useful’ in a decision-theoretic sense. To represent this, we need first a representation of the dynamics of awareness.

2

Formalising Unawareness

Let us start by presenting a simple propositional model of awareness and its dynamics, just enough to model Little Bo Peep’s predicament, in Section 2.1. We will then extend this basic propositional treatment to decisions under growing awareness in Section 2.3. In both sections, we will develop the basic formal 3

Pragmatic reasoning may still play a role, though, in her decision not to answer his question. She may decide that the question was intended to make her aware, rather than to elicit information that Jack himself is interested in.

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notions alongside a few intuitive examples; in the decision-theoretic case in particular the definitions will undergo some revision as we introduce complications. For the reader’s convenience, all final definitions are collected at the end of each section, in Definitions 1 and 2 in Section 2.2 (for the propositional case) and Definitions 3–7 in Section 2.3 (for decision problems). 2.1

The Propositional Case

We start with a set P of proposition letters, representing as usual statements about how the world might be. For Bo these express the location of the key: “(they’re in her) pocket”, “(they are on the) nail”, “(. . . ) phone”, “(. . . ) car”, and “(. . . ) sugar-jar” and so on. A possible world w is associated with a valuation function vw : P → {0, 1} as is usual. Since in fact (and according to Bo’s most fundamental beliefs) the keys can only be in one place at a time, we only need to look at a few of the combinatorially possible worlds, and we can give them the same names as the propositions themselves: “pocket” is the world where the proposition “pocket” is true and none of the others is. We’ll call this full set W : W = {pocket, nail, phone, car, sugar-jar}. In a standard model, then, Bo’s epistemic state would be the set of worlds she has not ruled out by observation or other trustworthy information sources. We call this her information set σ. We would standardly assume that Bo’s initial information set σ0 , before her search starts, is simply W , i.e., her information rules out none of the worlds in question (this may of course change as she learns more). However according to our observation of Bo’s behaviour, her epistemic state as our story opens seems instead to be {pocket, nail, phone} (these are the places that she goes on to check before giving up in frustration). We will capture this by filtering her information set σ through an awareness state α, which contains the proposition letters she is unaware of, and the assumptions she holds about their valuation. (Strictly speaking, this might be better named an “unawareness state”; we will use the terms interchangeably.) Formally, an awareness state α is a pair U, v where U ⊆ P is the set of unmentionables (proposition letters the agent is unaware of; we will mention them frequently but the agent herself cannot) and v : U → {0, 1} is a valuation function giving the assumptions the agent holds. In Bo’s case, we initially have α0 with U0 = {car, sugar-jar} and v0 = {car → 0, sugar-jar → 0}, i.e., she assumes (in typical ‘default’ fashion) that the keys are not in the car and not in the sugar jar (see also Section 5.2 for further justification of these assumptions). Taking a different perspective, an awareness state α specifies a set of worlds Wα = {w ∈ W ; v ⊆ vw }, those worlds in W which agree with the assumptions. This latter, equivalent view of awareness states facilitates the definition of filtering through awareness: we write σα for Bo’s information set filtered through her awareness, and define σα = σ ∩ Wα . Taken together, σ captures the complete factual information an agent like Bo has: σ would be her epistemic state if she was aware of all relevant contingencies; Wα is the set of worlds she entertains given her (possibly limited) awareness; and σα is the subset of

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these worlds that her information does not rule out, the ones which generate her beliefs. As our story opens, Bo has no factual information, but she is unaware of some propositions: σ0 α0 = Wα0 = {pocket, nail, phone}. But, if this is Bo’s epistemic state, she should believe that the keys are not in the car. That’s true from the modellers perspective (her implicit belief) but her explicit beliefs shouldn’t mention the car at all. As we argued for in connection with the slogan “Awareness is not introspective”, we rely on a syntactic notion to capture this: a belief formula ϕ can be explicit with respect to an epistemic state under unawareness σα only if ϕ does not use any proposition letters in U. These are the unmentionables according to Bo’s awareness, and her explicit beliefs must not mention them.4 Let us now try to model the different kinds of updates given in the story: factual and awareness. Given Bo’s initial information set σ0 = W and awareness Wα0 = σ0 α0 = {pocket, nail, phone}, she begins to systematically investigate the three places she is aware of as possible hide-outs for the keys, and eliminates them one by one. Now σ1 = {car, sugar-jar} but she has not gained awareness of anything new: α1 = α0 , so σ1 α1 = ∅! This explains Bo’s frustration: as far as she can see, she is in the inconsistent state. However inconsistency with unawareness is not as destructive as in the standard picture: it’s quite natural for Bo to realise that there is (or that there must be) some possibility she has missed. Her frustration arises because nothing in the situation gives her any guidance as to what this might be, so there is no reasonable action she can take to get out of the trap she’s entered.5 But, then comes Jack’s offhand question from his corner, and the scales fall from Bo’s eyes! That is, on hearing an expression mentioning the proposition letter “car”, Bo becomes aware of it: it disappears from her unawareness state. So α2 = {sugar-jar} , {sugar-jar → 0} and Wα2 = {pocket, nail, phone, car}; σ1 α2 = {car}, and it is easy to see why Bo immediately runs to check the car. 2.2

Summary of the Model

For convenience we collect here the formal features of this model. Definition 1: Propositional unawareness. Let P be a set of proposition letters and W a set of worlds, where each world w ∈ W is associated with a valuation function vw : P → {0, 1}. An epistemic state for an agent is a pair σ, α with σ an information set (a subset of W representing the worlds that her 4

5

This approach is explicitly syntactic: what matters is not the extension of the proposition letters, but whether they themselves (as syntactic elements) appear in U. This is what allows us to exclude a tautology such as “The keys are either in the car or not in the car” from Bo’s explicit beliefs. Another natural reaction would be to search again the places she has already looked. This shows another way that inconsistency might not be fatal: if some of the information leading to it turns out to be incorrect. However this perspective requires belief revision as opposed to update, and has little to do with awareness.

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information has not ruled out) and α is an awareness state. The awareness state α = U, v specifies her unmentionables U ⊆ P and assumptions v : U → {0, 1}, that is, the proposition letters she is unaware of and the truthvalues she unconsciously assumes they hold. The state α gives rise to a set of worlds Wα = {w ∈ W ; v ⊆ vw }, the worlds entertained by the agent. An information state under unawareness σα (to be read as “σ filtered through α”) is simply σ ∩ Wα . Let us also be explicit about the syntactic sublanguages we use, although these will feature only implicitly in the rest of the paper. Definition 2: Syntax and belief statements. The language we define contains two belief operators for each agent: Bi (for implicit belief) and Be (for explicit belief). An awareness state α = U, v defines an agent language Lα , the language inductively defined using only the mentionable proposition letters P \ U and the explicit belief operator Be . Implicit belief corresponds to belief in a standard model: Bi (ϕ) holds for an agent in epistemic state σα iff σα supports ϕ. However explicit belief has a stronger requirement: Be (ϕ) holds in σα iff Bi (ϕ) holds and ϕ ∈ Lα . (Under this definition all explicit beliefs are implicit; we will often use “implicit belief” loosely where “strictly implicit belief” would be more correct.) 2.3

Decision Problems and Awareness Dynamics

Strictly speaking, the propositional treatment of Bo’s growing awareness is a rather crude oversimplification: names such as “pocket” or “car” could at the same time represent states of the world (“(they’re in her) pocket”) or actions that Bo might wish to execute (“(search in her) pocket”). So, for example, when we concluded that in the epistemic state σ1 α2 = {car}, where Bo is aware of the car as the only open possibility, she would go check the car, we have silently succumbed to this equivocation between states and actions. But when the identification of propositions and actions is unwarranted, an extension of the analysis of awareness dynamics to decision problems is called for; not least because unawareness of propositions shows first and foremost in the agent’s behavior. The basic picture. A decision problem is usually conceived as a tuple S, A, P, U  where S is a set of relevantly distinct states of the world, A a set of possible actions, P a probability distribution over S, and U : S × A → is a utility function giving a numerical desirability for each action in each state. There is a sense in which this definition already implicitly includes unawareness, in its limited set S and, more palpably even, in the limited actions A under consideration: common sense dictates that when modelling a particular decision problem we do not include in S every potentially relevant distinction of the state of the world that may affect the outcome of the agent’s choice of action, but only certain distinctions that the agent can entertain herself (given her awareness-limited vocabulary); similarly, and more obviously even, we do not want to include in A all conceivable actions

Ê

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but only the ones that the agent is aware of as relevant to the task at hand. One of the main ideas we wish to stress in this paper is that a classical decision problem should be seen as an agent’s limited subjective conceptualization of a decisionmaking situation: Slogan 4: Decision problems represent subjective awareness. A decision problem, which by definition includes only a small set of states and possible actions and thus restricts attention to only a small facet of reality, represents the agent’s subjective assessment of the relevant factors of the situation, given her state of awareness. Here is a simple example for the kind of subjective unawareness represented in decision problems. At the beginning of her search, Bo is aware of the nail, the phone and her pocket as places where her keys might be. Her decision problem δ = S, A, P, U  which comprises her limited awareness at this point of the story contains exactly these states: S = {nail, phone, pocket} . We assume that the actual state is “sugar-jar” but this is a state that Bo is neither entertaining, nor considering possible at the outset. Instead, Bo considers all and only the states in S possible. This is represented in the decision problem δ by assuming that the probability distribution P , which captures Bo’s beliefs, has full support, i.e., assigns some non-zero probability to all states in S. (By definition it can assign no probability outside the states given by the decision problem.) The actions Bo can take in this key-search scenario correspond oneto-one with the possible states (“(the keys are in her) pocket” and “(search in her) pocket”) and so we can, for modelling purposes, use the same names for states and actions: A = S. (Whence the constant equivocation in the exposition of the propositional case.) And, of course, since we assume that Bo wants to find the keys, her utility function should be something like (formally speaking: some positive linear transformation of):  1 if s = a U (s, a) = 0 otherwise. Taken as a whole, then, the decision problem δ represents Bo’s own subjective assessment of the decision situation under her own limited awareness. It is obvious how this model of Bo’s epistemic state would treat factual information flow. If Bo learns (for instance, by checking) that the keys are not on the nail, she would revise her probabilistic beliefs (by a simple Bayesian update with the proposition “¬nail”). But what about extending Bo’s awareness? Suppose, whatever her probabilistic beliefs P might be, that she becomes aware of the car as a possible hide-out of the keys and of the corresponding action “car.” Most straight-forwardly, we would like to update Bo’s decision problem δ so as to include a state and action “car.” This much is easy. But what should Bo’s probabilistic beliefs be after she becomes aware of the new contingency? And what would her utilities be in the new updated decision problem?

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Clearly, we would not want to specify these features by hand with every update. We would prefer a model which fully determines the outcome of an awareness update. This is where the idea of filtering that we used in the propositional case applies: in order to model how an agent’s epistemic state changes under growing awareness we assume that there is a structure in the background, called a background model, which represents the agent’s epistemic state under full awareness; unawareness is then modelled by an awareness state as a restriction, or filter, on the background model; the outcome of the filtering process is (or gives rise to) a decision problem, which is interpreted as the agent’s assessment under limited awareness in line with the above slogan. Awareness updates are then fairly simple updates of the awareness state (basically: adding or removing elements from sets), which however may have rather far-reaching repercussions in the agent’s decision problem. Here is a first simplified attempt at implementing this architecture for Bo’s decision problem. We assume in the background another decision problem δ ∗ = S ∗ , A∗ , P ∗ , U ∗  which represents Bo’s decision problem under full awareness. According to our slogan this should also represent subjective awareness; indeed, it represents the features the modeller is aware of as possibly relevant. So, for the background model in Bo’s case we have chosen S ∗ = A∗ = {nail, phone, pocket, car, sugar-jar} (taking advantage again of the naming convention conflating states, propositions and actions) and appropriate beliefs P ∗ and utilities U ∗ . We should consider δ ∗ the equivalent of the information set σ in the propositional case: δ ∗ contains all the factual information that Bo would have under full awareness. This background structure δ ∗ is then filtered through an awareness state, as before. Propositional awareness states had no component to represent awareness of actions, while Bo’s restricted awareness is both a restriction on the set of states S and on the set of possible actions A. Consequently, we need to enrich the notion of an awareness state to include a component A, analagous to U, which represents the actions the agent is unaware of: our new awareness states will be triples U, v, A where A is a subset of A∗ giving the actions the agent does not consider. For example, Bo’s initial awareness state α0 , before she starts her search, has her aware of “nail”, “phone” and “pocket” as the possible states and possible actions, so that: U0 = {car, sugar-jar} v0 = {car → 0, sugar-jar → 0} A0 = {car, sugar-jar} . Just as in the propositional case we can define Sα as the set of states from S ∗ that are compatible with the assumptions of α. Filtering δ ∗ through this awareness state gives us the restricted decision problem δ that we started with. In general, filtering in this case comes down to this: if δ ∗ = S ∗ , A∗ , P ∗ , U ∗  is a decision problem and α = U, v, A is an awareness

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state then the filtered decision problem δα is the decision problem S, A, P, U  with S = Sα∗

A = A∗ \ A P = P ∗ (· | Sα∗ )

U = U ∗ (S × A). The set Sα of states being entertained drives the agent’s probabilistic beliefs under limited awareness by updating P ∗ , the agent’s beliefs under full awareness, with all implicit assumptions the agent is making due to her unawareness. This is exactly what the beliefs in P ∗ (· | Sα ) represent. Suppose that in δ ∗ the probabilities are 0.24 for each of “nail”, “phone”, “pocket” and “car”, and 0.04 for “sugar-jar.” In her initial state of unawareness Bo consequently holds states “nail”, “phone” and “pocket” possible, each with 1 probability 13 because P ∗ (nail | {nail, phone, pocket}) = 0.24 0.72 = 3 . If she becomes aware of new possibilities without eliminating the existing ones by searching (if Jack helps her out before her search begins, for instance) these probabilities decrease: to 14 if she becomes aware of the car, and to the limit value of 0.24 under full awareness. Awareness dynamics are now easy to define. If Bo becomes aware of the proposition and action “car” we simply remove this proposition from U, the corresponding assumption from v (thus enlarging the set of entertained states Sα ), and also the corresponding action from A. The background model makes sure that utilities and probabilities are defined in the updated decision problem which is retrieved from filtering through this updated awareness state. Effectively, this filtering process allows an easy implementation of deterministic awareness updates on decision problems: we, as modellers, specify the limit-stage of the agent’s growing awareness to the extent that it is important for modelling purposes. We also have a simple structure that captures which bits and pieces the agent is aware of. Simply removing the parameters that the agent becomes aware of from her unawareness state in the most straight-forward fashion produces, via the background decision problem, the new and updated decision problem with all (numerical) information specified correctly. Individuating states, unawareness without assumptions. Bo’s key-search example is fairly simple because propositions, states and actions correspond oneto-one and so the awareness update involved a nearly trivial extension of the idea of filtering from the propositional case to a richer decision-theoretic structure. Bob’s shortbread suggestion in Example 2, on the other hand, requires further scrutiny of the notion of a state and another slight revision to the notion of an awareness state. Here is why. Let’s first consider the background model for Farmer Pickles’ epistemic state in Example 2. What would his decision problem look like if he were aware of all contingencies that we as modellers are interested in? First of all, Pickles considered baking a cake a possible action and he is made aware of a further

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possible action, namely baking shortbread. We should maybe allow Pickles to abstain from all baking, but further actions clearly do not play a role, so that for the background decision model δ ∗ = S ∗ , A∗ , P ∗ , U ∗  we should assume that A∗ = {cake, shortbread, abstain} . But what should Pickles’ assessment of the relevant states be (from the modeller’s perspective)? Pickles knows that there are no eggs available, so this is not something that the model needs to distinguish. But there is a relevant piece of subjective uncertainty that we would like to model and that is whether the recipe for shortbread requires eggs or not. So, when fully aware, Pickles would make a distinction between two relevant states of affairs, one in which baking shortbread requires eggs and another one in which it does not: S ∗ = {eggs, no-eggs} . It is not significant at the moment whether Pickles has any beliefs as to which state is more likely, but we should assume that he does not rule out any state completely. So again we assume that P ∗ has full support on S ∗ . As for utilities, it is natural to assume that U ∗ is a function that orders state-action pairs as follows: ·, cake , eggs, shortbread ≺ ·, abstain ≺ no-eggs,shortbread In words: since there are no eggs, baking a cake is as bad as baking shortbread if this does indeed require eggs; it’s better not to bake anything; but most preferred, of course, is baking shortbread when it in fact does not require eggs. So, how do we represent Pickles’ epistemic state as a decision problem when he is unaware of shortbread as an option for baking? The obvious answer to simply leave out the action “shortbread” in his representation of the situation leaves us puzzling why, if Pickles is unaware of shortbread as an action alternative, his decision problem would nevertheless distinguish a state where the shortbread recipe specifies eggs from a state where it does not. Rather, an intuitively attractive representation of Pickles’ decision situation before becoming aware of shortbread should only have one state: there is indeed no decision-relevant epistemic uncertainty at all. But it is also not the case that we should simply leave out either one of the two states in S ∗ in the representation of Pickles’ initial state. For, unlike in Bo’s example, there is no indication that Pickles holds any assumption about whether shortbread requires eggs. This suggests two slight amendments. We should firstly alter the definition of an awareness state U, v, A to allow v to be a partial function from U to truth-values. This way we can represent which unmentionables an agent holds assumptions about, as well as what the assumptions are. Secondly, we should also define a reasonable grouping mechanism that specifies which states (or worlds) of the background model together form a state in the decision problem under an agent’s limited awareness. Let us implement these changes in the final model. The final model. Take, as before, a set P of proposition letters and a set W of possible worlds, and associate with each world in w ∈ W a valuation function

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vw : P → {0, 1}. Background models and filtered models will be defined in terms of these worlds. This will keep conceptually distinct worlds as the minimal modelling units in background and filtered model from states as they occur in proper decision problems, even when worlds and states are to be identified, e.g. under an empty or otherwise trivial awareness state. (This will become clear below.) Definition 3: Background Models. A background model is a quadruple: W, A, P, U  where – – – –

W is a set of worlds; A is a set of actions; P ∈ Δ(W) a probability distribution on current worlds; U : W × A → is a utility function.

Ê

A background model captures the agent’s epistemic state under full awareness, just as an information state did in the propositional case. Definition 4: Awareness States. An awareness state α is a triple U, v, A such that U ⊆ P is a set of unmentionables, v : U → {0, 1} is a (possibly partial ) valuation function on the set of unmentionables and A ⊆ A is a set of actions. The unmentionables U are propositions that the agent is unaware of; the assumptions v capture her implicit beliefs or assumptions (where an agent need not hold assumptions about all unmentionables); the actions A are likewise those she is unaware of. Based on an agent’s awareness state we can define the set of worlds that the agent entertains, i.e., the set of worlds not ruled out by her assumptions. Since the function v may be partial, the set of entertained worlds or outcomes is no longer necessarily the set of worlds the agent can distinguish given her awarenesslanguage. In particular, she may entertain possibilities, because she does not hold any assumption that would rule them out, but still not be able to distinguish these possibilities in her limited vocabulary. (Think of Pickles in Example 2 who could not distinguish a state where shortbread requires eggs from one where it does not, because he is unaware of this distinction, but nevertheless held no assumptions about the recipe for shortbread.) We therefore also define how an agent’s limited awareness aggregates worlds into states: here we should consider as a single state all those entertained worlds that agree on everything the agent can distinguish in her language. (The aggregation relation will define states in the agent’s decision problem, see below.) Definition 5: Entertaining and Aggregation. Let α = U, v, A be an awareness state. The set of worlds that an agent in α entertains, i.e. the set of worlds the agent does not rule out by an assumption, is the set Wα = {w ∈ W ; v ⊆ vw }. Furthermore, the agent considers equivalent by reason of unawareness two worlds w, w ∈ Wα , iff vw (p) = vw (p) for all p ∈ P \ U.

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Obviously this is an equivalence relation on Wα , for which we write ≡α . Two worlds are equivalent in this sense if the agent is not aware of anything that would distinguish them. We will define the states in a decision problem by aggregation using this relation: a state is simply an equivalence class under ≡α . A background structure and an awareness state together give us the agent’s subjective assessment of her situation. This includes limited awareness and possible implicit beliefs. We capture this in the notion of a filtered model. Definition 6: Filtered Models. Given a background model M = W, A, P, U and an awareness state α = U, v, A, the filtered model Mα is a tuple W  , A , P  , U   with: W  = Wα A = A \ A P  = P (· | Wα ) U  = U(W  × A ). A filtered model is the same kind of object as a background model; the only direct effect of filtering is to restrict attention to a sub-part of the background model. These models are, in a sense, decision problems that just contain more unnecessary state distinctions than the classical variety. We can obviously read off a decision problem in its classical guise from any such model, be that filtered or background. The only noteworthy elements in the following construction are the formation of states by aggregation, and the definition of the utilities, where we need to compute expected utilities to factor in the agent’s uncertainty. Definition 7: Decision Problems. Let M = W, A, P, U  be a background model and α = U, v, A an awareness state. Let Mα = W  , A , P  , U   be the filtered model as defined above. The agent’s decision problem δ(Mα),   ˆ defined on the filtered model, is of the classical form S, A, P , U where: S = W  /≡α A = A = A \ A  P  (w | W  ) Pˆ (s) = w∈s

U (s, a) =



P  (w | s) × U  (w, a).

w∈s

In words: S is the set of equivalence classes on W  given by the aggregation relation; A is simply the actions being entertained; Pˆ is the filtered probability distribution P  but interpreted on states (that is, on sets of worlds); and U gives the expected utility of a in s, under epistemic uncertainty about which world w from s obtains.

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Example: Bob and Pickles. Let’s see how background models, filtering and aggregation in decision problems work for the simple case in Example 2, where Pickles becomes aware of the possibility of baking shortbread. The first thing to do is to define the background model and a relevant set of proposition letters to capture Pickles’ possibly restricted awareness. We have already done most of this in Section 2.3. The background model, which captures Pickles’ epistemic state of full (relevant) awareness, is the tuple M = {W, A, P, U }. Here W = {w1 , w2 } contains two possible worlds, where the one awareness-relevant proposition “eggs” (that shortbread requires eggs) is false in w1 and true in w2 . There are three actions (“sb” of course standing for shortbread): A = {cake, sb, abstain} . Pickles’s probabilistic beliefs are given by a probability distribution P on W, which we leave parametrised for the time being:   w1 → p P = w2 → 1 − p and his utilities by a function that maps world-action pairs to real values, for instance as follows: sb cake abstain w1 1 w2 0

0 0

.5 .5

This background model is filtered through an unawareness state that captures Pickles’ neglect of action “sb” and his consequential inattentiveness to the question whether shortbread requires eggs. Since he also arguably does not hold an implicit belief about this latter question, his initial unawareness is represented in the state α0 in which U0 = {eggs} v0 = ∅ A0 = {sb} . Without assumptions, Pickles then entertains all worlds that are in the background model: Wα0 = W. But that does not mean that he can distinguish them all. According to the aggregation relation ≡α the two worlds in W are indistinguishable, because they differ only with respect to an unmentionable proposition. Consequently, filtering the background model M through α0 yields a filtered model Mα0 with the components: W  = Wα0 = {w1 , w2 } A = A \ A = {cake, abstain} P  = P (· | Wα0 ) = P U  = U(W  × A ).

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Given the aggregation relation ≡α0 , this filtered model gives rise to a decision problem in which the worlds w1 and w2 are aggregated into one state, and probabilities and utilities are amended accordingly. The decision problem δ(Mα0 ) we obtain has just one state, in particular it has the components: S = W  /≡α0 = {s1 } A = {cake, abstain} Pˆ (s1 ) = 1 U (s1 , cake) = 0 U (s1 , abstain) = .5 This simple example demonstrates how worlds, as the basic state distinctions in background and filtered models are aggregated into more coarse-grained state distinctions to obtain standard decision problems. The example also once more demonstrates how even our background models are, in essence, aggregates of those components we, as modellers, deem relevant for consideration. The explicit modelling of an agent’s limited awareness lets us model transitions in awareness states in a natural way: when Pickles becomes aware of shortbread as an action, and of the question whether shortbread requires eggs, his unawareness is overturned into a trivial awareness states α1 for which U1 = ∅ v1 = ∅ A1 = ∅. Filtering of the background model M through α1 just yields M again, as it should be. Notice also that the decision problem that M gives rise to under α1 is also exactly M itself: δ(Mα1 ) = M. To summarise briefly, we have characterised unawareness as a filter, in effect as restrictions of an agent’s subjectively accessible language to describe his decision situation. Awareness dynamics arise by alleviating these restrictions. A fixed background model assures that this process is furthermore well-defined and, in a sense, deterministic. This much assigns a role to awareness in specifying agents’ epistemic states and decision problems, but it certainly leaves open a number of questions about the role of awareness in conversation. Most importantly, we have to investigate the relationship between awareness dynamics and factual information growth. There are at least two important aspects to this, and we will treat each aspect separately in the following sections. For one, if one agent (Bob) makes another agent (Pickles) aware of a contingency even without directly conveying factual information, there is often room for a pragmatic inference that does result in factual information growth (that shortbread does not require eggs). We will deal with this issue in Section 3. But, for another, not all contributions in conversation strictly either only convey factual information or raise awareness. Some utterance may do both. Here the question arises how in general factual information flow and awareness dynamics interact. This is the topic of Section 4.

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Awareness Dynamics, Decisions, and Pragmatics

Unawareness from inattentiveness seems fairly natural and wide-spread. It is therefore not surprising to find that inattentiveness also figures in everyday conversation. In the above example, it is fairly natural to assume that Bob makes his friend Pickles deliberately aware of a contingency that (Bob believes) Pickles had overlooked. Bob’s question “Did you think of making shortbread instead?” does not convey any factual information, and arguably first and foremost serves to direct Pickles attention to the possibility of baking shortbread. Pickles awareness dynamics are then sufficiently captured by the above simple formalisation: when Bob mentions shortbread, we assume that Pickles becomes aware of the action “sb” and of the question whether shortbread requires eggs, to which he himself does not know the answer. The result of this awareness update is the trivial awareness state α1 (full awareness and no assumptions): after Bob’s remark about shortbread, Pickles represents his decision problem as Mα1 which is identical to M. The decision problem M has Pickles uncertain whether the recipe for shortbread contains eggs of not. This uncertainty, however, can be overturned by a pragmatic inference that crucially relies on the idea that Bob, an expert baker who knows whether shortbread requires eggs, is cooperative much in the sense of [Gri89] and —enter awareness— has ostensibly made Pickles aware of shortbread. Here is a semi-formal account of this pragmatic reasoning. Let’s first of all ask ourselves what Pickles would do in the initial decision problem when he was still unaware of shortbread. Since eggs are unavailable baking a cake would seem stupid. In formal terms it has an expected utility of zero in his decision problem δ0 = δ(Mα0 ), where expected utility of an action (in a classical decision problem δ) is defined as:  def EUδ (a) = P (s) × U (s, a) s∈S

In contrast, abstaining from all baking has expected utility .5, so that in δ0 this is clearly the preferred option. But now compare this with Pickles’ decision problem under full awareness, δ1 = δ(Mα1 ) = M. In δ1 the highest expected utilities of baking a cake and abstaining from baking remain zero and .5 respectively. But we also have a new competitor, namely baking shortbread, for which we calculate: EUδ1 (sb) = 1 × p + 0 × (1 − p) = p This means that under full awareness baking shortbread will be preferred to abstaining from baking (with expected utility .5) iff p > .5. In other words, barring pragmatic considerations Pickles will bake shortbread only if he thinks it is more likely that the recipe for shortbread does not require eggs.6 But even if his subjective probability favored the possibility that shortbread does require 6

Naturally the specific utilities chosen don’t matter for the general point that there is some threshold beyond which p is high enough to justify baking.

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eggs (p < .5) he could still revise these beliefs based on the following pragmatic reasoning: if Bob knows that Pickles faces the decision problem in question (including unawareness of shortbread) and if furthermore Bob is also helpful and cooperative, then his conversational move (deliberately bringing shortbread to awareness) can only be motivated if p > .5, for otherwise Bob’s effort would be futile, or only lead Pickles to choose an action that is even worse. If furthermore Bob is an expert baker who knows for sure whether shortbread requires eggs, Pickles is safe in concluding that it does. 3.1

Decision-Theoretic Relevance

This intuitive formulation suggests that considerations of relevance drive Pickles’ pragmatic inference. Pickles must be able to explain Bob’s question as relevant to the purposes of the conversation, otherwise he would have to conclude that Bob was not being cooperative. If those purposes are, roughly, getting something tasty baked, it’s hard to imagine how Bob could have relevantly intended his question as a literal request for information. However if it is given the natural interpretation of deliberately and ostensibly bringing a possibility to awareness, the prospect looks much better. If our promise of a formal solution is to be fulfilled, though, we need a formal notion of relevance that is appropriate for the decision-theoretic setting. As it happens one already exists for purely informational updates, which we can adapt with very minor changes to the current setting. The measure in question is called the value of sample information. We use a variant of that defined in [RS61]; it is used for pragmatic inference implicitly in [Par92] and explicitly in [Roo01], and has become a standard tool in decision-theoretic pragmatics. Let us first define the value of sample information for factual information uptake not involving expanding awareness. Towards this end, extend the definition of expected utility of an action in a decision problem to a set B ⊆ A of actions by taking the average: 1  def EUδ (B) = EUδ (a). |B| a∈B

We write BA(δ) for the set of actions with maximal expected utility in δ: BA(δ) = {a ∈ A ; ∀a ∈ A : EUδ (a ) ≤ EUδ (a)}. def

If δ is a decision problem, write δ[ϕ] for δ updated with factual information ϕ. So, δ and δ[ϕ] are the same, except that where δ has P (·), δ[ϕ] has P (· | [[ϕ]]). The value of sample information ϕ in the original decision problem δ, written VSIδ (ϕ), is given by def

VSIδ (ϕ) = EUδ[ϕ] (BA(δ[ϕ])) − EUδ[ϕ] (BA(δ)). Formally, we compare the expected utilities (given the information ϕ) of two sets of actions: those the agent considers best before she learns ϕ, and those she prefers after she learns ϕ.

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We will motivate this measure presently, but let us first extend the formal notion to cover also cases of changing awareness. The definition carries over almost as it stands. If M, α is a decision-theoretic epistemic state with possible unawareness, then we define the expected utility EUM,α (a) simply by taking EUδ (a) where δ = δ(Mα) is the decision problem that Mα gives rise to according to Definition 7; the definition of “BA(·)” is extended in the same way. If we now interpret the update (M, α)[ϕ] as the obvious ‘awareness update’ of removing relevant items from α,7 we can write EUM,α[ϕ] (BA(M, α)) for the consequences (judged in terms of awareness of ϕ) of the actions the agent considered best before ϕ was brought to her attention. This then gives us a new notion, call it “Value of Epistemic Change” which deals with both factual information and awareness changes: def

VECM,α (ϕ) = EUM,α[ϕ] (BA(M, α [ϕ])) − EUM,α[ϕ] (BA(M, α)). Obviously, VEC conservatively extends VSI: if ϕ conveys only factual information and leaves awareness unchanged, VEC comes down to VSI as defined above. 3.2

In Defense of VEC

The intuition behind this decision-theoretic notion of relevance is the following. The relevance of factual information in a decision problem can be measured as the change in the agent’s expected utility with the information, compared to without it. In particular, we compare the expected utility as it presents itself under some new information ϕ of those actions that appeared optimal before ϕ was obtained. If some previously optimal action fails to be optimal in the light of ϕ, then this means that the agent has learned something relevant, because she may have made a mistake, as she may have chosen an action that appears suboptimal under ϕ. Conversely, information that does not cause the agent to change her mind about her best action is irrelevant.8 In line with this intuition, one convenient feature of VEC is that information never has negative value and that information has strictly positive value only if it reveals some apparently optimal action to not in fact be so; that is, if it removes something from the set of best actions. (To see this, consider the alternatives: (i) the set of best actions stays the same, in which case its value doesn’t change, or (ii) something is added to the set, in which case the new action must have the same expected value as the previous elements —or they would be removed— so the average does not change.) A caveat. It is easy to confuse the non-negativity of the VEC measure with the intuitive idea that information can be ‘hurtful’ or unwelcome. A putative 7 8

The update could also both alleviate unawareness and convey factual information in the way defined in Section 4. This can easily be rejected as overly simplistic, since information making an agent more certain of a choice she has made can intuitively be highly relevant. The value of sample information does not incorporate the higher-order notions of uncertainty that doing this intuition justice would require, however we feel that this omission is harmless for the paradigm cases of unawareness that we treat.

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counterexample concerns a man who maintains both a wife an a mistress, without either knowing of the other. If both his wife and his mistress happen to be in the same caf´e, then learning that his wife is there (true but incomplete information which leads him to enter the caf´e), is ‘hurtful’ in the sense that his action without the information —not entering the caf´e— gave him a better utility (in the actual world) than his action with it. To argue that the VEC of such hurtful partial information ought to be negative, is to fail to take seriously enough the notion of expected utility that VEC makes use of. If going to the caf´e is our agent’s best action on learning only that his wife is there, then the probability that his mistress is there as well must be very low (assuming, as seems natural, a very large negative utility for meeting them both there). The expected utility of his action (given the information) is positive, even though the actual utility (given which world is actual) turns out negative. Reasoning similarly to this ‘counterexample’, when lottery ticket number 62,354,201 wins a cool million we might say “I should have bought ticket 62,352,201.” While this is true enough in hindsight (and, equally with the benefit of hindsight, our attempted bigamist ‘should not’ have gone to the caf´e) it has nothing to do with the decision-theoretic reasoning under uncertainty that leads to the original decision. Finally, there is of course always the possibility for a malicious speaker to manipulate the hearer: by giving less information than she knows (as in the case of the bigamist), or by raising awareness of possibilities that she knows to be ruled out (think of a murderer diverting attention during the investigation: “It might have been the butler”). However such behaviour falls far outside the rubrick of (neo-)Gricean pragmatics, since it violates the fundamental assumption of cooperativity. Therefore, the value VEC seems a good formal measure of the practical relevance of factual information and alleviated unawareness. 3.3

Reasoning about Relevance of Awareness

Armed with an explicit definition of relevance we can at last formalise the pragmatic reasoning we attribute to Pickles when he concludes that the recipe for shortbread should not require eggs. The formalisation rests on the assumption that although Pickles does not know which of the two possible worlds is actual (w1 in which shortbread does not require eggs, or w2 in which it does) he knows that Bob the Baker does know which world obtains. Pickles can then reason as follows: “Before Bob made me aware of shortbread, I would have abstained from baking, because this had highest expected utility. But then, Bob’s utterance, which does arguably not convey any direct information, changed my epistemic state from (M, α0 ) to (M, α1 ). The value of this epistemic change will only be positive, if in the decision problem after the update, which is M itself, the action “abstain” no longer is an action that maximizes expected utility. The only action that could possibly replace “abstain” as best action in M is baking shortbread. But baking shortbread will only be the preferred action if p > .5.” So, if Pickles accepts Bob’s authority on the question whether shortbread requires eggs, then based on a presumption of relevance of Bob’s

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awareness raising, Pickles can conclude that p > .5, or even that p = 1 if Pickles trusts that Bob knows about these matters with certainty. This reasoning sketch is largely simplified and presupposes, among other things, that Pickles believes that his epistemic states, before and after Bob’s utterance, are common ground between the two. But these simplifying assumptions do not invalidate the important part of this reasoning, namely that reasoning about the relevance of an induced pure awareness update can convey factual information.

4

Information Dynamics under Awareness

So far we have discussed the effects of factual information growth and of changing awareness, and even the pragmatic reasoning surrounding these dynamic changes as such, but we have not yet addressed the exact relation between awareness updates and the uptake of factual information, if this happens at the same time, say, when an utterance both raises awareness and explicitly conveys information. This is what we will do presently. Again we proceed in stages from simple to complex, starting with the propositional case. 4.1

The Propositional Case

There are two fundamental ideas to the treatment of information dynamics under awareness. Firstly, we have already argued in the introduction, in particular in the slogan “Awareness is easily overturned”, that unawareness from inattentiveness is lifted spontaneously whenever agents process linguistic information that contains mention of an unaware contingency. That is why we will assume that an agent who processes an utterance of some natural language sentence ϕ, be that for information uptake or anything else, will involuntarily become aware of all linguistic elements (proposition letters and actions) used in ϕ (or rather: a formal representation thereof in propositional logic) even before she can engage in any further processing. A second key feature of information dynamics under unawareness is that information uptake can only take place, so to speak, within the window of awareness: more concretely, if, for the propositional case, an agent in the epistemic state σ, α is aware of all proposition letters in ϕ, an informative update with the information in ϕ will be an update on the filtered state σα only (that is, only worlds being entertained are eliminated, not worlds from the background information set that are excluded by assumptions). This is fairly natural once appreciated: an agent who learns factual information can process this information only in the light of her (possibly limited) awareness. (Things become more complicated when the awareness state itself changes, a complication taken up in Section 4.3.) These considerations lead to the following treatment of information updates for the propositional case. We will write σ, α [ϕ] for updating an epistemic state σ, α with a propositional formula ϕ. This update can be considered a

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sequential update first of the awareness state, for which we will write α[ϕ], and subsequently an update of σ with ϕ under the agent’s updated awareness α[ϕ]. If ϕ is a propositional formula, write P(ϕ) for the proposition letters occurring in ϕ and [[ϕ]] for the set of worlds where ϕ is true. Then we define propositional update with awareness as follows: Definition 8: Epistemic update with (propositional) awareness. Let σ0 , α0  be an epistemic state, where σ0 ⊆ W is an information set and α0 is an awareness state. Then def σ0 , α0  [ϕ] = σ1 , α1  where α1 = α0 [ϕ] is given by def

U, v [ϕ] = U \ P(ϕ), v(U \ P(ϕ)) , and σ1 is given by

σ0 \ ((σ0 α1 ) ∩ [[¬ϕ]]).

For emphasis: updating σ0 to σ1 uses the new awareness state α1 , rather than the old one; first we make all proposition letters in ϕ mentionable and then we eliminate all entertainable worlds that are incompatible with ϕ. 4.2

Updates for Decision Problems

The main features of information dynamics under awareness carry over from the basic propositional case to the richer decision-theoretic models fairly straightforwardly. An epistemic state is now the pair M, α where M is a background model and α is an awareness state. Updating an epistemic state with a formula ϕ proceeds analogously to the propositional case by first making the agent aware of all linguistic elements featured in ϕ, where this might now include actions as well, and subsequently updating the background model through ‘the awareness window’ of the filtered model M(α[ϕ]) with the information [[ϕ]]. This boils down to eliminating from the background model all worlds and outcomes where ϕ is not true that are visible in the awareness window after the agent became aware of all contingencies mentioned in ϕ. Let A(ϕ) be all the actions mentioned in ϕ and define: Definition 9: Epistemic update with (decision-theoretic) awareness. Let M0 , α0  be the epistemic state of some agent, where α0 = U, v, A. Then def

M0 , α0  [ϕ] = M1 , α1  where α1 = α0 [ϕ] is given by def

U, v, A [ϕ] = U \ P(ϕ), v(U \ P(ϕ)), A \ A(ϕ) , and M1 is derived from M0 = W, A, P, U  (indices omitted for readability) as follows: W1 = W \ (Wα1 ∩ [[¬ϕ]]) A1 = A P1 = P (· | W1 ) U1 = U(W1 × A1 )

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For clarity: the only non-trivial updates of the background model are the elimination of worlds and outcomes, which is but exactly the same procedure as in the propositional case. The restrictions to probabilities and utilities are simply required to keep the structure well-defined.9 4.3

Old Information in the Light of New Awareness

The perhaps most fundamental idea behind out treatment of updates with factual information by agents with limited awareness is that factual information can only be evaluated (at the time it is observed) within the ‘window of awareness’ of the agent. But that may mean that assumptions can block the elimination of worlds which the agent might or might not want to rule out later as well when the implicit belief is given up by growing awareness. Here is a simple example to illustrate the sequential interaction of awareness and information updates. Suppose for simplicity that P = {p, q} and that our agent is unaware of p, assuming it to be true, and aware of q but uncertain about it; this gives us four possible worlds W = {pq, pq, pq, pq} (identifying them sloppily with their valuations). Then σ0 α0 = {pq, pq}; if the agent now learns that q is true, she will erase the world pq and her information set will become (according to the definitions we’ve given) σ1 = {pq, pq, qp}. Now this means that within her awareness window she has come to believe q, because σ1 α0 = {pq}; this is an explicit belief by our definition, but, surprisingly, one that is not necessarily stable under awareness updates, because when the agent becomes aware of her implicit assumption about p, a mere awareness update that removes p from the set of unmentionables brings with it the world pq which has not been ruled out by the previous information update. So, taken together, when an agent processes factual information her implicit beliefs might in fact block correct information uptake. In order to rule out worlds that have not been ruled out by an informative update, because these worlds were hidden behind an implicit belief, the agent has to, in our system, reprocess or reconsider the previous factual information in the light of her extended awareness. The reader’s response at this point may be: “But then you have defined information updates in the wrong way.” Indeed, it is tempting to give up the idea that information is processed only in the light of awareness and instead assume that information percolates, perhaps secretly, all the way up to the background model. This would save us quite some trouble, not only in the definition of information uptake, but also in dispensing with the “reprocessing” of factual information. However there is an important distinction between observing that q holds and merely hearing reported that q holds, and one that turns on unawareness. If 9

A different route could also be taken: instead of removing worlds from the information set entirely, simply adjusting their degree of credence, assigned by P , to zero. Which is appropriate depends on whether you think possibilities ruled out by information are still entertained or not, which might even vary depending on the application under consideration.

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our agent assumes p holds, she does not think to check whether a report of q is conditional on this assumption or not. The speaker, in turn, might hold the same assumption and might herself not be willing to commit to the truth of q if she is made aware of p. The point is clearest in the case of lawlike conditionals (see also Section 5.2). If I hear “If she has an essay to write she will study late in the library” and I am assuming the library is open, it is simply unclear whether the speaker makes the same assumption or is trying to tell me something stronger (that the student is so fanatical she will find a way to sneak in anyway, for instance). Were this not a case of unawareness I could always ask the speaker for clarification, but the distinction hinges on possibilities I am not yet entertaining; it is only in retrospect, when they have been brought to my attention, that I realise the potential ambiguity of the speaker’s intent. This complicates the picture of epistemic update in conversation considerably. Rather than simply carrying around an epistemic state, agents must carry at least a rough memory of the updates that brought them to that state, in order to be able to reinterrogate that memory in the light of new possibilities. Of course this is a more realistic picture of real conversation, but it is a significantly less tractable one. However it raises one very interesting possibility: that a speaker might come to repudiate a statement she has previously accepted, or even made herself, without having in the strict sense learned anything new in the interim. This is what suggests awareness dynamics a prima facie plausible explanation of the context- and sometimes apparently speaker-sensitive meaning of especially conditionals and modals, a possible future application to which we will come back below.

5

Related Work and Some Reflection

The purpose of this paper is explicitly a dual one: to propose a model of a sort of unawareness that has not yet been closely investigated in the rational choice literature, and also to make linguists aware of unawareness, so to speak. This section compares our contribution with other relevant models of unawareness, elaborates briefly on some of the major differences and gestures tentatively towards further applications of unawareness models in linguistics. 5.1

Formal Awareness Models

The classical reference for the notion of unawareness is [FH88], whose original motivation was developing inference systems that did not suffer from the problem of logical omniscience (that an agent knows all logical consequences of the facts that she knows). Fagin and Halpern point out that there are several distinct reasons to want to do away with logical omniscience such as ‘strict’ unawareness of possibilities, computational limitations and resource bounds, lack of knowledge of inference rules, or issues of attention and focus. Different modelling choices result from different conceptualizations of unawareness which in turn depend on the intended application of the unawareness model.

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This is then also the primary difference between the models presented here and the majority of unawareness models that have sprouted recently in rational choice theory.10 Apart from a general interest in modelling reasoning about this notion (see for instance [DLR98, Hal01, HMS06, HMS08a, MR94, MR99]) and in including unawareness into game theoretic solution concepts (see for instance [Fei04, Fei05, HR06, HMS07, Ozb07]), most applications have focused on reanalyzing in the light of possible unawareness certain fairly strong game-theoretic predictions about rational behavior: [Fei04], for instance, shows how the possibility of unawareness helps establish cooperation as a rational solution in the prisoners dilemma; [HMS06] shows how possible unawareness has otherwise rational agents wholeheartedly engage in speculative trade despite the well-known class of “No-Trade Theorems” (for example [MS82]). The source of unawareness that we have been concerned with in this paper is inattentiveness. This is because we believe that it is this kind of unawareness that plays a key role in certain aspects of conversation. The crucial feature of unawareness from inattentiveness is the ease with which it is overturned. To appreciate the difference between unawareness from inattentiveness and that resulting from, for instance, a lack of conceptual grasp, suppose a teenager is presenting a poorlyreasoned argument in favour of unprotected sex, and you mention the possibility of aids; the instant awareness update along the lines we have described is easy to imagine. Treating unawareness from lack of conceptual grasp is like imagining the same conversation as if it it were held in the ’70s, when the disease was unidentified and the acronym not yet invented: the new possibility being brought to painful awareness was not forgotten but simply had not yet been imagined. It should be clear that where linguistic generalizations about extending awareness through dialogue are concerned it is the former, not the latter type of awareness dynamics that we should focus on. This is then the main difference in conceptual interpretation of unawareness between our models and the collection of models entertained in economic theory. The notional difference further cashes out in two major differences in the modelling. The first difference between our linguistically-inspired models and the ones studied for economic applications is that the latter do not consider and spell out assumptions. Recall that in introducing the notion of assumptions as implicit beliefs, we referred to the intuition that in the initial example unaware Bo Peep behaves as if she believes the keys are not in the car. Interestingly, it seems to us that the motivation for explicit modelling of assumptions of agents is not exclusively linguistic. For instance, when Heifetz et al. seek to explain how unawareness overturns the “No-Trade Theorem” [HMS07], they also need to assume (implicitly) a particular “as-if” behavior of unaware sellers and buyers, namely behavior as if certain favorable or unfavorable contingencies were believed to be true or false. We suggest that the notion of an assumption might be an interesting enrichment of existing unawareness models and briefly enlarge on the idea below in Section 5.2. 10

The online unawareness bibliography maintained by Burkhard Schipper (http://www.econ.ucdavis.edu/faculty/schipper/unaw.htm) is a good starting point for readers interested in exploring the rational choice literature further.

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The second major difference stems from our goal to apply an unawareness model to cooperative conversation. We are interested in describing systematically the effects of awareness updates on decision problems, which requires specifying numerical probabilities and utilities for the newly-introduced possibilities. The main idea to achieve this end is filtering through an awareness state.11 The problem of changing awareness has also been addressed, typically in gametheoretic settings where it is natural to assume that observing a player make a move that you were unaware of overturns your unawareness. [HMS08b] gives a game-theoretic model and a variant of rationalizability for games with possibly unaware players, and [Fei04, Fei05, HR06] have taken similar equilibrium-based approaches. However, the emphasis in these efforts is on non-cooperative game theory, whose solution concepts do not, strictly speaking, supply vanilla awareness updates irrespective of rationality considerations. The demands of linguistic pragmatics, based as it is on a fundamentally cooperative notion of interaction, are quite different: we would like to pin down pure awareness dynamics first and show how pragmatic reasoning can take place on top of it. Seen in this light, the model of [Ozb07] deserves special mention. Ozbay gives in a non-cooperative setting a signalling game model with an equilibrium refinement somewhat similar to our notion of relevance as VEC. In the model an aware sender can make an unaware receiver aware of certain contingencies by her choice of signal, but the beliefs the receiver adopts when becoming aware are not determined, but subject to strategic considerations. Ozbay offers a refined equilibrium notion according to which the receiver should adopt beliefs under extended awareness that prompt him to choose a different action from the one that he would have chosen under his initial unawareness. While this kind of constraint on belief formation seems to be what pragmatic reasoning based on a notion of relevance as VEC provides, it is unclear whether this should apply in all cases of (possibly) conflicting interests. It should, to our mind, apply for the cooperative case, and we have spelled out this kind of reasoning based on the example of Bob and Pickles. In short, although our work is based on standard models of rational choice theory, our notion of unawareness is not quite the same. The linguistic application, and in particular the structural requirements imposed by decision-problem representations, have led us to develope a significantly different model based on similar, but subtly different, intuitions. 5.2

Assumptions and Associations

Given that the present approach differs from the bulk of unawareness models in that it features assumptions and is directed towards awareness dynamics in cooperative conversation, let us briefly reflect on both aspects, asking firstly where assumptions come from and how these are to be motivated. Clearly, not all assumptions are equally sensible. A forgiving reader might not complain that 11

We introduced this approach in [FJ07], in a preliminary and in many ways unsatisfactory model which nonetheless contains the seeds of the present account.

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we haven’t sufficiently motivated Bo’s implicit belief that the keys are not in her car, but should certainly object if we had her unconsciously assuming that they were in fact hiding in the sugar-jar. We have appealed to intuitions of normality, without really making precise what we mean by this. Clearly ‘normality’ is sensitive to the details of the decision-making context; it is probably normal to assume the library is open when checking the remaining to-do list for an almostfinished essay, and equally reasonable to assume it is closed when plotting to break in at midnight for some clandestine reading. The library example is not chosen at random: law-like conditionals such as “If she has an essay to write she studies late in the library” were used in a now classic experiment in psychology of reasoning, the ‘suppression task’ [Byr89], which shares many characteristics with the notions of awareness. The basic observation is that subjects asked to accept the truth of the conditional (as a premise in a logical argument) seem to implicitly hedge it with a normality assumption: “If she has an essay to write (and nothing unexpected happens) she studies late in the library.” Stenning and van Lambalgen give an explanation of the data in terms of closed-world reasoning, where they represent the implicit hedge as a ‘dummy’ proposition which is assumed false if there is no evidence that it is true [SL08]. While the details do not concern us here (the parallel with unawareness is incomplete, although provocative), the closed-world reasoning is a perfect fit for our notion of assumptions. In other words, if our examples are to be intuitively satisfactory, assumptions should have a closed-world flavour: unusual events do not occur and the status quo is maintained, unless explicit reason is given to believe otherwise. This formulation in turn suggests a loose probabilistic constraint on our assumptions due to unawareness. That is, it should generally be the case that the probability mass hidden by a particular assumption (an ‘unusual event’) is relatively small compared to the probability mass on the worlds being entertained (including, although not limited to, the ‘status quo’). In other words, while becoming aware may qualitatively overturn an assumption, it should generally replace certainty that p only with uncertainty, not with near-certainty that ¬p. We do not believe that this is a ‘hard’ semantic (or even pragmatic) constraint on acceptable states of awareness. However if we recall that our notion of unawareness is linked to absent-mindedness and cognitive limitations of attentiveness it seems that we should expect our cognitive apparatus (superbly evolved, as it seems to be, for problem-solving) to be reasonably good at prioritising attention, keeping focussed on the most probable and most utility-relevant contingencies and letting only the marginal ones slip beneath the surface. Taking this cognitive perspective also solves a conceptual problem with awareness dynamics that we have been carefully ignoring so far. Consider again the case of Bob and Pickles. If Bob tells Pickles he could bake shortbread (making Pickles aware of a possible action), nothing in the formal setup we’ve given so far explains how Pickles gets to entertain the state distinction whether shortbread requires eggs. Still, intuitively he should: when becoming aware of the action

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“baking shortbread” he should also become aware of certain natural outcomes of that action, together with natural preconditions leading to these outcomes.12 Although clearest in this case, the problem is not confined to actions and outcomes. The reality is that some possibilities are cognitively closely associated, so that becoming aware of one may bring on awareness of the other. However, very little formal or precise can be said in the present framework about this process of association in its full complexity. Hearing a possibility mentioned at least brings the possibility itself to awareness and mentioning a possible action certainly calls to mind certain stereotypical outcomes and their preconditions. But beyond this we cannot say much more. That is why in this paper we’ve been careful not to make associations do any explanatory work. However, for the Pickles example we must at least rely on the association between the action “bake shortbread” and shortbread-related propositions such as “shortbread requires eggs.” As in the case of assumptions, we may gesture at the adaptive nature of our cognitive capabilities in support of the idea that the right associations will spring to mind when they are needed. Even more so, in conversation often enough interlocutors share associations and can relatively reliably predict what springs to mind in listeners when hearing certain expressions in a given context. Apart from the formal definition of bringing to awareness propositions that are explicitly mentioned, however, the details of this association process must remain somewhat vague. For linguistic applications we will have to have common sense supply the ‘right’ associations that our formal models then readily implement. 5.3

Unawareness in Linguistics

In the linguistic literature awareness-related notions and intuitions are often appealed to but have never been treated as a distinct phenomenon amenable to a unified formal treatment. For instance, in his seminal paper [Lew79], David Lewis gave a unifying account of a wide range of accommodation effects in terms of an evolving “conversational score.” Awareness effects as we have described them make a somewhat uncomfortable fit with this picture, since unawareness updates (if we are correct) proceed not by accommodation but by something akin to inherent salience or attention-focusing effects. However one class of observations given by Lewis fits the awareness story very comfortably: his Example 6, on relative modality. Lewis is concerned here with modals such as “can” and “must”, and their apparent restriction, in normal usage, to a subset of all ‘metaphysical’ possibilities. There is a large literature on this subject, of course, but certain features recur again and again: a restricted set of possibilities that are ‘in play’ at any given 12

The formal distinction between actions and propositions is of course a theoretical fiction which a shift to a first-order model (with possibilities of defining unawareness of terms such as “shortbread”, whether occurring in descriptions of actions or states of affairs) could alleviate. A first-order unawareness model has recently come on the market [BC07] however it’s not yet clear how to combine this approach with implicit beliefs based on (possibly false) assumptions.

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moment, against which modal statements should be evaluated, and the possibility to add hitherto unconsidered possibilities into this set as a conversation progresses. The similarity to the unawareness picture is clear, so we should say something instead about the differences. It might be thought that our ‘worlds being entertained’ correspond directly to a Stalnakerian context set [Sta78]: the possibilities not ruled out by presuppositions in force. However there is a crucial difference between our assumptions and the presuppositions that this parallel would conflate them with: an assumption is typically something an agent would repudiate if she were made aware of it. This is implicit in our slogan “Unawareness is easily overturned”: it is only when overturning unawareness also overturns an implicit belief (that is, when an assumption is given up as unfounded) that the epistemic update is as it were visible to the observer, since an awareness update that simply ratifies an implicit belief does not result in a change of behaviour. Nevertheless the notions of assumption and presupposition are closely linked, and the exact relation between them remains a problem for further study. It seems, for instance, that assumptions can sometimes be ‘converted into’ presuppositions. Suppose you make a na¨ıve statement due to unawareness of some contingency p. I am aware of p, and see that you seem to have neglected it, but even so I agree with your statement (suppose for example that I explicitly assign very low probability to p). If I choose not to object, it seems that all assertions in our further conversation are contingent on p being false, but in two quite different ways: we might say that you are assuming, while I am presupposing. Whether this is in fact the right distinction is unclear (the possibility of uncertainty about the awareness basis from which a speaker makes assertions complicates matters), but certainly the issue deserves further investigation. Another difference to the standard approach is the inadvertency of an awareness update: the agent who undergoes such an update cannot choose rather to remain unaware, and no pragmatic reasoning can undo the immediate effects it produces. An interesting topic for further research is the interplay between such automatic updates and the explicit negotiation about which possibilities are ‘on the table’ displayed in sentences like “Assume that doesn’t happen — what then?” Awareness dynamics may thus also be crucial for accounts of the intricate acceptability patterns of (sequences of) conditionals [F01, Gil07, Wil08, Mos07], and also for accounts of discourse effects and acceptability of modal statements. As for the latter, for instance, Eric Swanson’s treatment of the language of subjective uncertainty [Swa06b], elaborated in [Swa06a], in terms of so-called “coarse credal spaces” is very closely analogous to the aggregated states and outcomes in our decision-theoretic formulation. Concluding here on a rather speculative note, it seems that unawareness could help shed some new light on several phenomena treated classically at the semanticspragmatics interface. The perspective from unawareness raises the hope that minimal addition of an agent’s unawareness conserves otherwise classical and simple formalisms. Much of this, however, has to be left for future consideration.

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Conclusion

Let us finally take stock, and repeat the central ideas of this paper. Initially, we have used three slogans to give intuitions about unawareness: 1. Unawareness is not uncertainty (it cannot be represented formally by uncertainty; it typically takes the form of implicit beliefs). 2. Unawareness is not introspective (it must be represented intensionally; the modeller’s language is not the agent’s language). 3. Unawareness is easily overturned (it stems from absent-mindedness or inattentiveness; mere mention of possibilities, whatever the linguistic setting, suffices). In particular the third slogan shows how our notion differs from the version common in the rational choice literature; as far as we can see, this characteristic is key for a linguistic application of the idea. We’ve modelled unawareness in terms of filtering a background model through a set of unmentionables (which define a limited agent language) and assumptions, and distinguished between implicit and explicit beliefs. These are key concepts we would like to see generally adopted, whatever the specific implementation. We also suggested that decision problems be considered a subjective representation of the relevant features of the situation, and that models of unawareness be used whenever that subjective notion of relevance may undergo revision over time. Background models and filtering produce numerically precise and potentially quite complex revisions of decision problems by way of simple updates to awareness states. Finally, we have argued that awareness dynamics are a natural feature in conversation. We have offered a perspicuous example of pragmatic reasoning centered on a conversational move intended first and foremost to bring a possibility to awareness. Beyond that, we have gestured, still rather tentatively, at the further relevance that a notion of unawareness may have for formal semantics and pragmatics.

References [BC07]

Board, O., Chung, K.-S.: Object-Based Unawareness. In: Bonanno, G., van der Hoek, W., Woolridge, M. (eds.) Proceedings of the Seventh Conference on Logic and the Foundations of Game and Decision Theory (2007) [Byr89] Byrne, R.M.J.: Suppressing Valid Inferences With Conditionals. Cognition 31(1), 61–83 (1989) [DLR98] Dekel, E., Lipman, B.L., Rustichini, A.: Standard State-Space Models Preclude Unawareness. Econometrica 66(1), 159–173 (1998) [Fei04] Feinberg, Y.: Subjective Reasoning — Games with Unawareness. Research Paper No. 1875. Stanford University (November 2004), https://gsbapps.stanford.edu/researchpapers/library/RP1875.pdf

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Feinberg, Y.: Games with Incomplete Awareness. Stanford University (May 2005), http://www.stanford.edu/%7Eyossi/Files/Games%20Incomplete%20DP.pdf [FH88] Fagin, R., Halpern, J.Y.: Belief, Awareness and Limited Reasoning. Artificial Intelligence 34, 39–76 (1988) [F01] von Fintel, K.: Counterfactuals in a Dynamic Context. In: Kenstowicz, M. (ed.) Ken Hale: A Life in Language, pp. 123–152. MIT Press, Cambridge (2001) [FJ07] Franke, M., de Jager, T.: The relevance of awareness. In: Aloni, M., Dekker, P., Roelofsen, F. (eds.) Proceedings of the Sixteenth Amsterdam Colloquium, pp. 91–96 (2007) [Gil07] Gillies, A.S.: Counterfactual Scorekeeping. Linguistics and Philosophy 30, 329–360 (2007) [Gri89] Grice, P.H.: Studies in the Ways of Words. Harvard University Press, Cambridge (1989) [Hal01] Halpern, J.Y.: Alternative Semantics for Unawareness. Games and Economic Behavior 37, 321–339 (2001) [HMS06] Heifetz, A., Meier, M., Schipper, B.C.: Interactive Unawareness. Journal of Economic Theory 130, 78–94 (2006) [HMS07] Heifetz, A., Meier, M., Schipper, B.C.: Unawareness, Beliefs and Games. Tech. rep. 6. Bonn Econ Discussion Papers (2007) [HMS08a] Heifetz, A., Meier, M., Schipper, B.C.: A canonical model for interactive unawareness. Games and Economic Behavior 62, 304–324 (2008) [HMS08b] Heifetz, A., Meier, M., Schipper, B.C.: Dynamic Unawareness and Rationalizable Behavior (April 2008) (unpublished manuscript) [HR06] Halpern, J.Y., Rˆego, L.C.: Extensive Games with Possibly Unaware Players. In: Proceedings of the Fifth International Joint Conference on Autonomous Agents and Multiagent Systems, pp. 744–751 (2006); Full paper unpublished; a preliminary version appeared, http://www.cs.cornell.edu/home/halpern/papers/aamas06.pdf [Lew79] Lewis, D.: Scorekeeping in a language game. Journal of Philosophical Logic 8, 339–359 (1979) [Mos07] Moss, S.: On the Pragmatics of Counterfactuals. MIT, Cambridge (2007) (unpublished manuscript) [MR94] Modica, S., Rustichini, A.: Awareness and Partitional Information Structures. Theory and Decision 37, 107–124 (1994) [MR99] Modica, S., Rustichini, A.: Unawareness and Partitional Information Structures. Games and Economic Behavior 27, 265–298 (1999) [MS82] Milgrom, P., Stokey, N.: Information, trade and common knowledge. Journal of Economic Theory 26, 17–27 (1982) [Ozb07] Ozbay, E.Y.: Unawareness and strategic announcements in games with uncertainty. In: Samet, D. (ed.) Proceedings of TARK XI, pp. 231–238 (2007) [Par92] Parikh, P.: A Game-Theoretic Account of Implicature. In: Moses, Y. (ed.) Proceedings of TARK IV, pp. 85–94 (1992) [Roo01] van Rooy, R.: Revelance of Communicative Acts. In: van Benthem, J. (ed.) Proceedings of the 8th Conference on Theoretical Aspects of Rationality and Knowledge (TARK 2001), pp. 83–96 (2001) [RS61] Raiffa, H., Schlaifer, R.: Applied Statistical Decision Theory. MIT Press, Cambridge (1961)

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Stenning, K., van Lambalgen, M.: Human reasoning and cognitive science. MIT Press, Cambridge (2008) [Sta78] Stalnaker, R.: Assertion. In: Cole, P. (ed.) Syntax and Semantics, vol. 9, pp. 315–332. Academic Press, New York (1978) [Swa06a] Swanson, E.: Interactions with Context. PhD thesis. Massachusetts Institute of Technology (2006) [Swa06b] Swanson, E.: Something ‘Might’ Might Mean. University of Michigan (2006) (unpublished manuscript) [Wil08] Robert, J., Williams, G.: Conversation and Conditionals. In: Philosophical Studies, vol. 138(2), pp. 211–223 (March 2008)

The Role of Speaker Beliefs in Determining Accent Placement James German1 , Eyal Sagi2 , Stefan Kaufmann2,3 , and Brady Clark2 1

1

Laboratoire Parole et Langage 2 Northwestern University 3 University of G¨ ottingen

Introduction

In English and other languages, the distribution of nuclear pitch accents1 within a sentence usually reflects how the meaningful parts of the sentence relate to the context. Generally speaking, the nuclear pitch accent can only occur felicitously on focused parts of the sentence, corresponding to information that is not contextually retrievable or given.2 In most contemporary theories, focus is formally represented by an abstract syntactic feature ‘F ’. Those parts of the sentence that are given tend to resist F -marking and thus nuclear accentuation.3 In short, there is a more or less tight coupling between (i) the contextual information status of parts of the sentence; (ii) the focus structure of the sentence (represented by the distribution of syntactic F -marking); and (iii) the actual accent placement in the phonological form. 1.1

Focus Projection and the Grammar

A commonly encountered view holds that the relation between information status, F-marking, and accent is governed by rules that may be complicated, but are nonetheless unequivocal, even deterministic, so that relative to a particular context exactly one placement of the accent is felicitous. One can easily adduce data for which this is indeed the case, such as the question-answer pairs in (1).4 In each of B’s responses, the part which corresponds to the ‘wh’-phrase of the question is the focus of the answer and the only natural location of the nuclear 1 2

3 4

This is typically defined as the last pitch accent in an intermediate phrase (Pierrehumbert, 1980; Beckman and Pierrehumbert, 1986). We follow Schwarzschild’s suggestion to abandon the use of new as a collective technical term for the complement of given material, since doing so would intimate a homogeneity that is not there. There are cases in which other constraints overrule this tendency and force prosodic pominence on given material (see Schwarzschild, 1999, for examples and discussion). As usual in the literature, the location of the nuclear pitch accent is typographically indicated by capitals. This convention glosses over certain details of the intonation contour and is therefore not always appropriate. It is sufficient for our purposes, however.

A. Benz et al. (Eds.): Language, Games, and Evolution, LNAI 6207, pp. 92–116, 2010. c Springer-Verlag Berlin Heidelberg 2010 

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pitch accent. Consequently, the same syntactic string has to be pronounced differently in response to different questions: While each of the answers in (1-a-c) is felicitous in the context of its question, it cannot be felicitously replaced with either of the others. (1)

a. b. c.

A: Who did John praise? B: John praised MARY. A: Who praised Mary? B: JOHN praised Mary. A: What did John do to Mary? B: John PRAISED Mary.

In general, though, the correspondence between accent placement and the questions an utterance can felicitously answer is not so tight. Selkirk (1996) noted that (2), in which the accent is located on the word ‘bats’, can answer any of the questions in (2-a-e). (2)

Mary bought a book about BATS. a. What did Mary buy a book about? b. What kind of book did Mary buy? c. What did Mary buy? d. What did Mary do? e. What’s been happening?

Depending on which question the sentence is used to answer, different constituents of it are in focus (those which correspond to the ‘wh’-phrases of the respective questions), and while all of these focused constituents contain the accented word ‘bats’, they also contain additional, unaccented material in all cases except (2-a). Selkirk and others have argued that the relationship between accent and focus is mediated through an abstract syntactic feature F . This feature must originate from an accented word, but may percolate to other constituents, subject to certain syntactic rules of focus projection (Chomsky, 1972). In Selkirk’s system, F -marking may spread (i) from an internal argument to its head, and (ii) from a head to the constituent it projects. Thus in (3) the word ‘bats’ must be accented if it is to be F -marked at all, since for syntactic reasons the F -marking could not project to ‘bats’ from any other location in the sentence. However, the F -marking can project from ‘bats’ to each of the constituents listed as F -marked in (3-b-e). The corresponding questions from (2) are given on the right. (3)

a. b. c. d. e.

Mary bought a book about [BATS]F . Mary bought a book [about BATS]F . Mary bought [a book about BATS]F . Mary [bought a book about BATS]F . [Mary bought a book about BATS]F .

(2-a) (2-b) (2-c) (2-d) (2-e)

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In most accounts of focus projection, the grammatical rules fully determine where the accent must fall in order to realize a given focus structure.5 Once the latter is fixed, there is no room for variation, let alone speaker choice. 1.2

The Role of Speaker Choice

Two steps lead from context to accent: Contextual information status maps to focus structure, which in turn guides and constrains the placement of nuclear pitch accents. Both of these steps are frequently treated as though they were governed by deterministic grammatical principles. This seems to be an oversimplification at both levels, however. First, regarding the relationship between information status and focus structure, some authors explicitly assume that focus is part of speakers’ communicative intentions, thus representing an active choice. For Roberts (1996), for example, the focus of a sentence indicates which question or issue the speaker takes to be the one currently under discussion; in this sense, focus is a means of keeping interlocutors’ common ground and communicative goals in alignment. For Schwarzschild (1999), focus structure is determined by a particular type of anaphoric relationship between parts of an utterance and the contents of the discourse context, which he calls Givenness. Here, speakers have some freedom in the choice of anaphoric relationships. Second, under certain conditions, syntactic rules constraining accent placement (such as those presented in Selkirk, 1996) are violated. German et al. (2006) showed that speakers tend to avoid placing nuclear accents on prepositions, even in contexts in which those prepositions are new and the only alternative is to place an accent on given material. Thus when uttering (4-B) in the context of (4-A), speakers prefer to place the nuclear accent on the direct object ‘game’ (5-a) rather than the preposition ‘in’ (5-b), even though the preposition is the only new information in the clause. (4)

A. I noticed that Liz and Sally really like to play their game. B. Unfortunately, Paul wrecked the tent that they play their game in.

Interestingly, though, the avoidance of accented prepositions was only partial in their study. While the overall preference was for patterns like (5-a), speakers also produced the pattern in (5-b), which accords with Selkirk’s focus projection rules. The design of the experiment ensured that this variability could not be attributed to speaker or experimental error. (5)

5

a. b.

. . . that they play their GAME in. . . . tent that they play their game IN.

Kadmon (2001) and Winkler (1997) contain recent major overviews of the literature on focus projection.

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German et al. account for this finding by introducing an OT-style markedness constraint into the grammar which militates against forms with nuclear accents on prepositions. To explain the variability in outcomes, they follow Anttila (1997) and Boersma and Hayes (2001) in proposing that this markedness constraint interacts probabilistically with other constraints governing the distribution of focus in the general case. This treatment may provide as good an account of the variation as one can expect from the constraints considered by German et al. (2006), but there are reasons to doubt that it actually explains what is going on. The observation is that speakers can use a form (the one with the accent on the direct object) with a focus structure with which it is not conventionally associated. Modifying the grammatical principles to accommodate this fact would seem to imply that the form in question can in some sense mean the same as the one that is conventionally associated with the focus structure in question. But intuitively, the deviating form is “pressed into service,” so to speak, despite the fact that it does not mean the same. In this paper, we propose instead to treat focus projection rules such as those presented in Selkirk (1996) as just one factor among several that influence accent placement, and ask whether and under what circumstances it may be safe or even advantageous for speakers to violate those rules. Specifically, we conjecture that the observations of German et al. are not due to random variation after all, but rather to factors which were not represented in their model and, consequently, not controlled for in their experiment.6 In a nutshell, our proposal is that speakers’ beliefs about hearers’ expectations play a role in determining when to use certain accent patterns. On the one hand, speakers’ tendency to avoid accenting prepositions is due to a cost associated with the effort involved in using such a form. On the other hand, in certain contexts the hearer can “guess” the information structure of a sentence independently of the accent pattern it carries. In cases like (4), the speaker’s choice comes down to a tradeoff: If there is a substantial risk that the hearer would choose the wrong interpretation without the information carried by the accent pattern, then the speaker will pay the extra cost and accent the preposition. If the risk of miscommunication is low, however, the speaker will tend to avoid accenting the preposition. We formalize this tradeoff in a signaling game and explore the predictions resulting in terms of either Nash equilibrium strategies or Pareto-Nash dominant strategies (Parikh, 2001, 2010). Importantly, which strategy dominates is predicted to depend on the prior probabilities of the various information-structural interpretations under consideration. In contrast to earlier treatments of information structure and accent placement, our model does not deal with the structure 6

There are precedents for the view that pragmatic factors may override the rules of grammar, for instance in binding theory. Thus Chomsky (1981) argues that “... these contexts [e.g. contexts that license Principle C violations] do not constitute counterevidence to principle (C); rather they indicate that principle (C) may be overridden by some condition on discourse, not a very startling fact.”

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of the grammar directly, but with the extent to which speakers and hearers are bound by the grammar in negotiating their respective communicative goals and preferences. Our application differs from previous game-theoretic treatments (such as those in Parikh, 2010; Benz et al., 2006) in that it is not merely concerned with pragmatic enrichment or strengthening, but with a case in which winning strategies may step outside the form-meaning mappings licensed by the grammar. In Section 2, we discuss in some more detail the main assumptions and intuitions underlying our proposal. Section 3 presents the formal version of our model as well as its application to the key problem that this paper addresses. Section 4 discusses the main results and implications, and section 5 concludes with some brief remarks about how the approach might be broadened and carried forward.

2

Contextual Factors in Accent Placement

What are the factors that may be driving speakers’ choices in cases like (4-B)? Central to our proposal is the assumption that extant theories of focus and accent placement are essentially right (e.g., that the accent “belongs” on the preposition in (4-B)), and that the role of information structure in determining accent placement is part of the knowledge speakers and hearers bring to their interactions. In terms of its communicative function, the placement of accents is significant for a variety of reasons. For instance, it has been argued that by placing prosodic prominence on those parts of an utterance which introduce new information, speakers draw hearers’ attention to those parts and facilitate their understanding (Schmitz, 2005). Aside from this facilitating role, accent placement is also an aide in synchronizing speakers’ and hearers’ respective beliefs about the common ground and the goals of the ongoing interaction. Seen this way, accent placement is a grounding device (Clark and Wilkes-Gibbs, 1986; Clark, 1996; see Thompson, 2009 for an overview and references). Under what circumstances, then, would we expect the speaker to use a form that deviates from the grammatically “correct” one for the focus structure she has in mind? One such circumstance would be if that intended focus structure is highly expected independently of specific linguistic cues in the utterance itself – if, for example, the context already makes the intended focus structure highly salient or likely. In general, the more of a need the speaker feels to provide additional cues for her intended focus assignment, the stronger the incentive to use the form that the grammar licenses. Conversely, if the context provides strong cues for the intended focus assignment, accent placement loses its significance as a grounding device. Our model predicts that it is precisely in those situations that other factors – such as prosodic preferences – may outweigh focus structure in determining speakers’ choices. On this view, it is not surprising that participants in German et al.’s study produced (5-a), rather than (5-b), in the context

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of (4-A): Speakers feel free to use such “mismatched” forms whenever they can do so without risk of miscommunication. Thus, one factor that we take to play an important role in determining speakers’ choices is the hearer’s uncertainty about the intended focus structure. In the formal model, this parameter is encoded as the hearer’s subjective probability distribution over the various focus structures the speaker may have in mind. Following standard practice in the theory of signalling games, we assume that this probability distribution is common knowledge between both interlocutors.7 For the above example, this predicts that the speaker’s license to produce the “mismatched” contour in (4-B), with the accent placed on the direct object, in a situation in which she takes the direct object to be given, increases with the degree to which she believes its givenness to be already expected by the hearer. A case like (4-B), where the expression ‘their game’ directly repeats an expression from the previous utterance, leaves little uncertainty about the speaker’s intended interpretation. Thus (4) is a very clear example of the situation we are describing: No miscommunication is likely to result from accenting the direct object. This reasoning leads to predictions about the conditions under which speakers are allowed to produce accent patterns that do not match the intended focus structure. It does not yet explain, however, when and why speakers will actually do so. The fact that an incongruent accent placement is unlikely to result in miscommunication is not in itself a reason for preferring it over its congruent alternatives. To address this question, we assume that in addition to the hearer’s beliefs about the speaker’s intended focus structure, three further factors play a role, all related to the effort involved in production and interpretation. In the model, they are represented as costs incurred by the interlocutors. The first factor represents the speaker’s effort in production. Here we follow German et al. as well as much of the literature on functional theories of grammar in invoking preferences against the production of particular forms (Croft, 1990; Eckman et al., 1986; Haiman, 1985, among others). Specifically, in our example, we assume that the congruent form with accentuation on the preposition ‘in’ is dispreferred due to a general tendency against nuclear accentuation on certain function words (Ladd, 1980; Selkirk, 1995; German et al., 2006). Similarly to the optimality-theoretic approach, we assume that this tendency is always operative – 7

This may seem to be an oversimplification, since the speaker does not really have access to the hearer’s actual beliefs and may be mistaken about them. But recall that our goal is to model the factors which motivate the speaker’s choice of a form, at the time she makes her utterance. Although the speaker takes the listener’s perspective into consideration in making her decision, her choice can only be informed by what she takes to be the hearer’s beliefs, not by the hearer’s actual beliefs. Therefore, although one could devise a more complicated model which allows for the possibility that the speaker is wrong about the hearer’s actual probability distribution over various focus structures, this extension would not contribute substantially to the analysis we are concerned with.

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thus the cost is always incurred by speakers who produce the accent on the preposition – but may be outweighed by other forces. The remaining two factors arise from a mismatch between the grammatically determined focus-to-accent mapping and the actual choices made in production and interpretation. On the one hand, the speaker incurs a cost whenever she chooses an accent pattern that is not the one specified by the grammar for the focus structure she has in mind. Essentially, the speaker prefers, all else being equal, to adhere to the grammar, and will not deviate spuriously. Similarly, the hearer incurs a cost whenever he chooses, as an interpretation, a focus structure that is not grammatically consistent with the accent pattern that the speaker has produced. Like the speaker, then, the hearer prefers not to deviate from the grammar, but may do so when other considerations apply. We might view this extra cost, particularly as it concerns the hearer, as a processing cost incurred by extra inferences required to decide whether such a mismatch should be permitted in the given situation. A different, though perhaps not unrelated role of this cost would be in perpetuating the grammatical system throughout the population as well as diachronically. Without some formal reflex of such “grammaticality-bias” in the model, there may be no particular advantage to any one pairing between forms and meanings. Such a framework would furnish an account that we consider bizarre, namely that the grammatical role played by nuclear accent is itself subject to variation: Depending on the contextually given probabilities, accent sometimes marks non-givenness (as is standardly the case) and sometimes givenness. Intuitively, one of these uses is the norm from which the other deviates. The cost we stipulate is intended to account for this intuition. Overall, interlocutors share the goal of successful communication, and both will assign a positive value to combinations of actions that lead to the hearer’s choosing the information structure that the speaker had in mind. Speakers and hearers also prefer choices which adhere to the grammatically specified correspondence between accent patterns and information structure, and costs are incurred whenever this correspondence is contravened. Finally, speakers may incur an additional cost for using particular forms. For our examples, it is sufficient to assume that, all else being equal, utterances that include nuclear accents on prepositions incur a greater cost than ones with nuclear accents on full nouns. Apart from the costs and benefits, our model assumes that interlocutors share certain beliefs regarding which focus structure a speaker is likely to convey for a given sentence in a given context. Formally, this is a function from (explicit) contexts to functions from sentences to probability distributions over possible focus structures. In the cases we discuss, the context and textual content of the sentence are known. Thus, for practical purposes, this feature can be reduced to a single, mutually accessible probability distribution over those focus assignments that are sensible given the syntax and lexical content of the sentence being uttered. All of this fundamentally assumes that the grammar provides a fixed mapping between information structure and accent placement. Yet we have not yet

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specified which version of such a grammar we are assuming. In fact, for the purposes of our analysis, a few minimal assumptions suffice. We take the core aspects of Schwarzschild (1999) as the foundation of our simplified theory. Specifically, we assume that (i) all non-F -marked nodes in the syntax are interpreted as given, and (ii) each given node introduces a presupposition that there is an antecedent in the context with which it co-refers.8 Finally, we assume that nuclear accentuation introduces F -marking, which relieves the constituents in question of the givenness presupposition.9 In the German et al. study, the vast majority of productions broke down in two basic categories: one in which the only nuclear accent in the embedded clause falls on the direct object, as illustrated in (5-a), and one in which it falls on the stranded preposition, as in (5-b). (5)

a. b.

. . . that they play their GAME in. . . . that they play their game IN.

There were several prosodic variations in the material preceding the embedded clause (the most common being whether the head of the relative clause received an accent), but these differences did not substantially affect the predictions that Selkirk (1996) and Schwarzschild (1999) make for the embedded clause itself. Specifically, these theories predict that a pattern like (5-a) can realize a number of focus assignments, including the ones in (6): In (6-a) the direct object ‘their game’ is treated as the only F -marked element, whereas in (6-b) the F -marking projects to the entire verb phrase. (6)

. . . they play their GAME in a. . . . they play [their game]F in b. . . . they [[[play]F [their game]F ]F [in]F ]F

While (5-a) is thus consistent with a number of focus structures, all of them have in common that the direct object ‘their game’ is treated as F -marked. Since that is the feature we are most interested in, we ignore the differences and collapse all of these cases into one. In contrast, for a pattern like (5-b) the theories predict that the preposition ‘in’ must be interpreted as the only F -marked constituent in its clause; in particular, the pattern is predicted to be inconsistent with F -marking on the direct object. Thus we draw the relevant distinction in terms of F -marking on the direct object vs. the preposition and adopt the notation in (7). It should be reiterated, however, that this is merely a shorthand notation and that the alignment between accent placement and F -marking is more complicated. 8

9

Schwarzschild in fact allows for a more inclusive notion of inferability, formally modeled in terms of entailment under existential closure. In the minimal contexts we consider, this relation does not add anaphoric possibilities beyond those available by coreference. This does not mean that the element must not have a contextually salient antecedent, but merely that it carries no presupposition to that effect.

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(7)

a. b.

. . . they play [their game]F in . . . they play their game [in]F

(5-a) (5-b)

With these preliminaries in place, we turn to the specification of our formal game-theoretic model.

3

Formal Model

For our purposes, a language consists of two non-empty sets F (of forms) and M (of meanings). Since we are interested in a language which comes with a conventional interpretation, the formal model should also include a mapping of some kind between F and M , such as a relation R in F × M , which constrains the interpretations available for each form. But since our main point is that speakers can and do “step outside” the bounds imposed by the conventional interpretation, the conventional interpretation should be capable of interacting with and being overruled by other forces. To this end, we assume that the interpretation is given as part of the payoff structure – specifically, in terms of costs of production and interpretation. Our example involves the speaker’s choice between the two forms in (5-a) and (5-b). Here we label them fNP and fnp , indicating the placement of the nuclear pitch accent on the noun phrase or on the preposition, respectively. The two meanings we are concerned with are mn and mg , corresponding to the focus-structural status of the noun phrase (see (7-a) and (7-b) above). Thus under the familiar grammatical constraints on accent placement, the pairings mn , fNP  and mg , fnp  are congruent, whereas mn , fnp  and mg , fNP  are “mismatches.” In our game-theoretical model, a speaker strategy is a function σ mapping meanings to forms, and a hearer strategy is a function τ mapping forms to meanings. On each occasion of use, the speaker and the hearer choose strategy profile. Definition 1 (Strategies and strategy profiles). Let a language L = F, M  be given. The set of speaker strategies for L, Ss , is the set of functions σ : M → F . The set of hearer strategies for L, Sh , is the set of functions τ : F → M . The set of strategy profiles is S = Ss × Sh . The speaker utters the form which her strategy assigns to the meaning she wants to convey, and the hearer uses his strategy to map the form he receives to a meaning. We assume that the form the speaker utters and the one the hearer perceives are identical, thus there is no noise. Since σ and τ are functions, once they are fixed, the outcome of the exchange is determined by the speaker’s intended meaning.  R Definition 2 (Costs and benefits). Let L = F, M . A function Cps : F → assigns to each form in F a cost incurred by the speaker for uttering it. Two

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functions Cms , Cmh : (F × M ) → R assign to each form-meaning pair f, m a cost incurred by the speaker and the hearer, respectively, for producing and interpreting f as conveying m. The benefit of successful communication is given by a function B : (M × M ) → R, such that for each m, m ∈ M, B(m, m ) > 0 if m = m , 0 otherwise. In our example, the costs are represented by variables as follows. The production s for placing the accent on the preposition, costs for the forms in question are Cnp s and CNP for placing it on the noun phrase. These production costs arise due to prosodic constraints governing the respective forms and are independent of the information structure. Furthermore, C✗s and C✗h represent the speaker’s and hearer’s respective costs of producing and processing a “mismatched” interpretation of a nuclear pitch accent (i.e., mapping the accented constituent to given information). In contrast, C✓s and C✓h are the respective costs of producing and processing the “canonical” pairings which map the accented constituent to new s s information. Based on the above discussion, we take it that generally Cnp > CNP , s s h h C✗ > C✓ , and C✗ > C✓ . As we will see, the relative magnitude between these pairs of costs is more important than their absolute values. Successful communication is rewarded by a benefit which we stipulate is positive if the meaning the hearer extracts is the same as the one the speaker intended to convey (thus communication is successful), and zero otherwise. For all possible outcomes, the benefit is the same for both interlocutors. Thus the game is one of coordination. This choice rules out many real-life situations, such as ones in which the speaker has an interest in misleading the hearer. The predictions of the model would change considerably in such cases, but we exclude them here because such situations lie beyond the purview of this paper. For each linguistic encounter, the benefits and costs associated with the chosen strategy profile jointly determine its utility: Definition 3 (Utility). Given Cps , Cms , Cmh and B, a utility function U : M × F × M → R for L is defined as follows: For all m, m ∈ M and f ∈ F , U (m, f, m ) = B(m, m ) − Cps (f ) − Cms (f, m) − Cmh (f, m ) Now, neither of the interlocutors knows the other’s choice of strategy, and the hearer only has probabilistic information about the speaker’s intended meaning. Therefore the outcome is not predictable with certainty. However, since the hearer’s beliefs about the speaker’s intentions are common knowledge, both participants are able to calculate the expected utility of each strategy pair σ, τ – the weighted sum of the utilities for each of the meanings the speaker may intend, where the weights are the hearer’s subjective probabilities of those meanings. Definition 4 (Expected utility). Let L = F, M  be a language, U a utility function for L and P : M → [0, 1] a probability distribution over the meanings in L such that for each m ∈ M , P (m) is the hearer’s prior probability that the

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speaker intends to convey m. The expected utility for L given U and P is a function EU : S → R defined as follows, for all σ, τ ∈ S:  P (m) × U (m, σ(m), τ(σ(m))) EU (σ, τ) = m∈M

With the definitions so far, we have secured all the ingredients for a game in the formal sense. Definition 5 (Game). Given a language L, a utility function U as defined above, and a probability distribution P over meanings in L, a (two-player) game for L is a triple  = {s, h}, S, EU , where s, h are speaker and hearer, respectively; S is the set of strategy profiles for L; and EU is the expected utility function for L given U and P . The most fundamental and commonly used notion in making this prediction is that of a Nash Equilibrium. In a game of coordination like ours, there is always at least one Nash Equilibrium; in general, there may be more than one. Definition 6 (Nash Equilibria). The set of Nash equilibria in a game  is the set N E() = {σ, τ|∀σ  [EU (σ  , τ) ≤ EU (σ, τ)] ∧∀τ  [EU (σ, τ  ) ≤ EU (σ, τ)]} The Nash equilibrium has been employed in linguistic analyses by Lewis (1969), Dekker and van Rooij (2000), and others. However, it has some limitations which have prompted some authors to look for refinements and alternatives. Parikh’s (2001, Section 4.4; 2010, Section 3.3.5) proposal is to filter the Nash equilibria in a given game by the criterion of Pareto dominance in order to eliminate “local minima” and retain only those that are closer to our intuitive notion of “best choice.” Overall, the question of appropriate solution concepts for various kinds of games is still open (cf. van Rooij, 2004:506; Parikh, 2006). Here we adopt Parikh’s strategy of Pareto-dominance as the criterion for the normative model. More specifically, we adopt the notion of weak Pareto-dominance, which ensures that if there is at least one Nash Equilibrium in the game, then there is a (not necessarily unique) Pareto-dominant one. Therefore, since our games are guaranteed to have Nash Equilibria, at least one of them has to be Paretodominant. Definition 7 (Pareto-Nash Equilibria). The set of Pareto-Nash Equilibria in a game  is the set P N E() = {σ, τ|∀σ , τ  [EU (σ, τ) ≤ EU (σ , τ  ) → EU (σ, τ) = EU (σ , τ  )]} With these formal notions in place, let us now examine our example more closely. Recall that the set of meanings is {mn , mg } (where the noun phrase is new and

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Table 1. Speaker and hearer strategies with costs incurred by each move Speaker strategies costs   s mn →fNP CNP C✓s σ1 : s mg →fnp Cnp C✓s 

mn →fNP σ2 : mg →fNP  σ3 :  σ4 :

mn →fnp mg →fNP mn →fnp mg →fnp

  

Hearer strategies costs   fNP →mn C✓h τ1 : fnp →mg C✓h 

s CNP C✓s s CNP C✗s

fNP →mg τ2 : fnp →mg

s Cnp C✗s s CNP C✗s

τ3 :

s Cnp C✗s s Cnp C✓s

τ4 :

 

fNP →mg fnp →mn fNP →mn fnp →mn

  

C✗h C✓h C✗h C✗h C✓h C✗h

given, respectively) and the set of forms is {fNP , fnp } (where the accent is placed on the noun phrase or on the preposition). Table 1 lists all possible speaker and hearer strategies together with the associated costs for speaker and hearer for each possible move.

4

Results

In this section we present an analysis of the model in terms of the dominance relationships between the expected utilities of strategy sets under various conditions.10 We have limited our detailed analysis to just the first two rows and columns – that is, to the strategy profiles involving σ1 , σ2 , τ1 and τ2 . There are several reasons for this. First of all, we feel that the relationships between these strategy sets most clearly illustrate the intuitions behind the phenomenon we are modeling. σ1 , τ1 , for example, represents the “canonical” situation in which the speaker and hearer fully observe the rules of the grammar, thereby maximizing the benefit from successful communication and minimizing the costs from grammatical mismatches, while σ2 , τ2  represents what is in many ways the most interesting deviation from this pattern: The speaker avoids the extra 10

The relationships between the various strategy sets take the form of conditions on dominance based on the variables that the model includes. In certain cases, these conditions are mathematically non-trivial, and may seem somewhat abstract in comparison to the concrete communicative processes that we are trying to model. It should be noted that we do not mean to imply that the variables in our model, to the extent that they have a psychological reality, take on a precise numerical value that one could measure with any precision. Nevertheless, the mathematical inequalities do serve to elucidate certain broad tendencies that are likely to hold if the factors we consider have any psychological reality at all, and these are discussed where appropriate.

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cost associated with accenting the preposition even when the NP is given, and the hearer interprets all forms as having a given NP. Strategy sets involving σ3 , σ4 , τ3 and τ4 deviate in other, sometimes interesting ways. It should be noted, in fact, that under certain conditions, the set of Nash equilibria and even Pareto-dominant strategies is not limited to the first four strategy sets. In the discussion that follows our analysis (Section 4.2), we mention such cases and discuss their implications for our model. For reasons of space, however, we leave these out of the analysis itself, and we leave it to the reader to carry out the associated mathematical proofs. 4.1

Costs and Cost Differentials

First of all, in comparing strategy profiles we can dispense with using costs s s directly (e.g., Cnp and CNP ) and operate with just the differences between them. The resulting rankings are the same because for any given strategy profile, the expected utility is just the weighted sum of the utility terms, where for each s s pair of related costs (e.g., Cnp and CNP ), if one is incurred with probability x, then the other is incurred with probability (1 − x). Thus for instance, the total form cost incurred for any strategy set is described by the term in (8-a), which is equivalent to (8-b). (8)

a. b.

s s M × Cnp + (1 − M ) × CNP s s s CNP + M (Cnp − CNP )

s Since the lower cost term (here, CNP ) is constant across strategy sets, it can be ignored, for it will always be subtracted out of any comparison or inequality between two strategy sets. In other words, pairs of terms like (8-a) will henceforth be replaced by terms like (9), where Dps is the difference between the two costs.

(9)

M × Dps

With this in mind, we can write the payoff matrix as in Table 2. (For readability, the matrix is spread over two rows.) Table 2. Payoff matrix τ1 Pg Dps

τ2

σ1 σ2 σ3 σ4

B− Pn B − Pg Dms −Pn Dps − Dms Pg B − Dps − Pn Dms τ3

Pg B − Pg Dps − Pn Dmh Pg B − Pg Dms − Dmh Pg B − Pn Dps − Dms − Pg Dmh Pg B − Dps − Pn Dms τ4

σ1 σ2 σ3 σ4

−Pg Dps − Dmh Pg B − Pg Dms − Dmh B − Pn Dps − Dms − Dmh Pn B − Dps − Pn Dms − Dmh

Pn B − Pg Dps − Pg Dmh Pn B − Pg Dms Pn B − Pn Dps − Dms − Pn Dmh Pn B − Dps − Pn Dms − Dmh

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Fact 1. σ1 , τ1  dominates σ1 , τ2  whenever Pg < 1.11 In descriptive terms, this just means that whenever the speaker is using a strategy that is sensitive to her intended meaning and accords with the rules of focus projection, then it is always better if the hearer uses a strategy that is sensitive to form and also accords with those rules. Fact 2. σ1 , τ1  dominates σ2 , τ1  whenever Pg > 0 and Dps < B + Dms . Since we are assuming that Dms is positive, under the further assumption that Dps < B this condition will always be met. In fact, there is good reason to assume that Dps < B as a general fact. If the cost of accenting a preposition were greater than the benefit that could be gained from successful communication, then it would always be better to remain silent than to produce such a form. This is not what is observed, however. Speakers in the German et al. study, especially, did accent prepositions, so the cost of doing so cannot be higher than the maximum benefit that can be attained in this way. Intuitively, this result suggests that as long as that the hearer is using a strategy that is sensitive to form and conforms to the rules of focus projection, it is better for the speaker to mark her intentions in a way that also conforms to the grammar. Avoiding the cost of accenting the preposition will never sufficiently offset the risk of unsuccessful communication in such a case. h Fact 3. σ2 , τ2  dominates σ2 , τ1  whenever Pg > (Dm /2B) + 0.5.

As long as Dmh < B, the right side of the inequality ranges between 0.5 and 1.12 This means that Pg must be at least as large as 0.5 in order for the condition to hold. In addition, for a fixed B, the minimum condition on Pg increases linearly as a function of Dmh . 11

Proof. Notice first that (i) and (iv) are equivalent: (ii) is obtained by substitution from the payoff matrix, the rest follows by simple algebra (recall that B − Pg B = Pn B). EU (σ1 , τ1 ) > EU (σ1 , τ2 ) B−

> Pg B −

Pn B +

Pn Dmh

>0

(iii)

Dmh )

>0

(iv)

Pn (B +

12

Pg Dps

(i)

Pg Dps

−

Pn Dmh

(ii)

Clearly (iv) is true if and only if both factors on the left-hand side are true; i.e., iff Pn > 0 (equivalently, Pg < 1) and B + Dmh > 0. The latter is assumed.   We omit the proofs of subsequent results; they are obtained in a similar fashion. See the discussion above regarding Dps and B in the preceding paragraphs. A similar argument applies here. If Dmh were greater than B, then hearers would never deviate from the grammar of focus projection by interpreting an accented NP as given. They would adhere rigidly to the grammar even at a very high risk of miscommunication, in spite of any and all contextual evidence in favor of an ungrammatical interpretation. While the production data does not corroborate this assumption in the same way as for Dps , on an intuitive level, this is precisely what we are assuming licenses a speaker to contravene the rules of focus projection in the German et al. data.

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Since the speaker always accents the NP in σ2 , a hearer using τ2 will always incur a mismatch cost, regardless of the speaker’s actual intention. This result then suggests that the probability of the NP being given (Pg ) has to be high enough so that the increased chance of successful communication is sufficient to offset the hearer’s cost of always deviating from the grammar. If that probability is too low, or Dmh is too high, then it would be better for the hearer to interpret accented NPs as new, and accept miscommunication in all cases where the speaker intends the NP to be given. Fact 4. σ2 , τ2  dominates σ1 , τ2  whenever Dps > Dms + Dmh and Pg > 0. If the hearer is using an insensitive interpretive strategy, then it is preferable that the speaker use a uniform marking strategy whenever the cost of accenting the preposition is higher than either the speaker or hearer mismatch costs. This relationship is less intuitive than the others, since it raises the question of why the speaker would bother to mark the focus structure with accent placement if the hearer is not attending to form. However, it makes more sense from the standpoint of cases where the NP is given. In those cases, using σ2 always avoids the cost of accenting the preposition. However, since σ2 and τ2 each incurs a mismatch cost when the NP is given, the cost avoided (Dps ) has to offset the costs incurred (Dms + Dmh ). Fact 5. σ1 , τ1  is a Nash equilibrium. This follows straightforwardly from Facts 1 and 2 and the definition of a Nash equilibrium. When Pg = 0 or Pg = 1, then this will be weakly true, since EU (σ1 , τ1 ) will be equal to EU (σ2 , τ1 ) and EU (σ1 , τ2 ) respectively in those cases. Fact 6. For Pg > 0, σ2 , τ2  is a Nash equilibrium whenever Pg > (Dmh /2B)+0.5 and Dps > Dms + Dmh . This is just the conjunction of the conditions in Results 3 and 4. Note that when this stronger condition is met, then σ1 , τ1  and σ2 , τ2  are both Nash equilibria. When it is not met, then σ1 , τ1  is the only Nash equilibrium, since, except when Pg = 1 or Pg = 0, σ2 , τ2 ’s competitors, σ1 , τ2  and σ2 , τ1 , are always dominated by σ1 , τ1 . Fact 7. σ2 , τ2  strictly dominates σ1 , τ1  whenever Pg > (B + Dmh )/(B + Dps − Dms ). Notice that since Pg ≤ 1, this can hold only if Dmh ≤ Dps − Dms . This is equivalent to Dps ≥ Dms + Dmh , which we know from Result 4 is a prerequisite for σ2 , τ2 ’s being equilibrium strategy in the first place. In addition, since we are assuming that B > Dmh and B > Dps , this condition implies that Pg must be at least greater than 0.5. Beyond that, the condition on Pg varies with (i) the difference between Dmh and the term (Dps − Dms ), and (ii) the magnitude of these two terms as a proportion of B. When Dmh and (Dps − Dms ) are relatively small as a proportion of B, then their difference will have little effect on the minimum condition for Pg ,

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and that condition will be close to 1.0. By contrast, when those terms are large as a proportion of B, then their difference will have a large effect on the condition. When the difference is very small, then Pg must be close to 1.0, but as the difference increases, σ2 , τ2  may dominate σ1 , τ1  at smaller values of Pg . This relationship is intuitively plausible, first of all, from the standpoint of the relative size of the factors as a proportion of the benefit for successful communication. When the difference between the cost of accenting a preposition (Dps ) and the speaker mismatch cost (Dms ) is very small, the switch from a strategy that avoids accenting prepositions (σ2 ) to one that avoids mismatch costs in exactly the same cases (σ1 ) is virtually an even trade, and there is little motivation to do so except when the probability is very high that the NP is given. This relationship also makes sense from the standpoint of speaker costs versus hearer costs. If the differential just described (i.e., Dps − Dms ) is not much bigger than the cost a hearer incurs for a mismatch, then there is little motivation to use a strategy that incurs such a cost, except when it is very likely that the NP is given. By contrast, when the cost of accenting a preposition is very high as a proportion of B, the cost of both speaker and hearer mismatches are very low as a proportion of B, then it is desirable to use a pooling strategy that avoids production costs whenever it is even moderately more likely that the NP is given than not. 4.2

Discussion

Besides the equilibria described in Results 5, 6 and 7, there are several other interesting cases that lie outside of the four strategy profiles discussed above. To begin with, σ2 , τ3  is equivalent to σ2 , τ2 , and therefore dominates its row in exactly the same set of cases as σ2 , τ2 . This is intuitively clear when one considers that σ2 only generates fNP , which τ2 and τ3 both treat in the same way. The conditions under which σ2 , τ3  dominates in its column, however, are rather specific and unintuitive, and it is not of much use, therefore, to discuss the conditions under which σ2 , τ3  forms a Nash equilibrium. To the extent that it does, however, it cannot possibly be Pareto-dominant in any cases that σ2 , τ2  cannot also be. Moreover, when σ2 , τ2  and σ2 , τ3  are both (weakly) Pareto-dominant, it is not possible to distinguish between them behaviorally. In fact, one might speculate that speakers and hearers do not care which hearer strategy is being employed in such cases, since half of the strategy cannot possibly be relevant for the outcome. It is also noteworthy that σ2 , τ4  is not only equivalent to σ2 , τ1 , but also dominates its column under relatively weak sets of conditions. When the conditions described in Result 3 are not met, such that σ2 , τ1  dominates σ2 , τ2  (and σ2 , τ3 ), σ2 , τ4  may be a weak Nash equilibrium. Note, however, that according to Result 1, σ1 , τ2  is always dominated by σ1 , τ1  and can never itself be a Nash equilibrium. This implies even when σ2 , τ4  is a weak Nash equilibrium, it will be Pareto-dominated by σ1 , τ1 . Finally, σ3 , τ3  represents a surprisingly strong strategy set in our model under a range of conditions. It is a general result (whose proof we leave to the reader) that it dominates its own row whenever 0 < Pg < 1. It also

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dominates σ1 , τ3 , and σ4 , τ3  whenever B > Dms , which we are assuming anyway. Finally, it dominates σ2 , τ3 , and therefore represents a Nash equilibrium, whenever B < Dms + Dps . Moreover, since σ2 , τ3  is equivalent to σ2 , τ2 , σ3 , τ3  also Pareto-dominates σ2 , τ2  in those cases. Note however, that EU (σ3 , τ3 ) > EU (σ1 , τ1 ) whenever Pg > (Dms + Dmh )/2Dps + 0.5. In other words, like σ2 , τ2 , Pg must be somewhat higher than 0.5 before σ3 , τ3  even competes with σ1 , τ1  for Pareto-dominance. This result suggests that using a strategy set like σ2 , τ2 , that is insensitive to both intentions and forms, may not be the only rational alternative for avoiding costly forms. Under certain conditions, it may be better to use a strategy set that actually reverses the mapping associated with the rules of focus projection. This implies that both speaker and hearer mismatch costs are incurred for every possible outcome. This extra cost is offset, however, by the fact that communication is always successful. Furthermore, although σ3 , τ3  does not completely avoid extra form-based costs the way that σ2 , τ2  does, those costs play less of a role when the NP is very likely to be given, since the more costly form is the less likely to be used in such a case. It is not possible to know from the German et al. results whether speakers who accented given NPs were using σ2 , τ2 /σ2 , τ3  or σ3 , τ3 . As already mentioned, σ2 , τ2  and σ2 , τ3  are indistinguishable from a behavioral standpoint, so there is no data that could ever distinguish between them. σ3 , τ3 , on the other hand, predicts a distinct set of behavioral outcomes, which could, in principle, distinguish it from σ2 , τ2  and σ2 , τ3 . We leave this to future research. In many ways, however, there is something very counterintuitive about σ3 , τ3 , even if our model suggests that is sometimes the most rational outcome. Does it make sense that speakers and hearers would, or ever do, temporarily negotiate a set of strategies that literally flips the grammar on its head? Temporarily resorting to a pooling strategy, on the other hand, is easier to imagine, and more closely resembles various other human behaviors (such as laziness) that have a stronger presence in popular discourse and folk psychology. Perhaps it is not unreasonable, then, to suppose that there are other biases involved that our model does not represent. In other words, while speakers and hearers may temporarily accept a slight deviation from the rules of the grammar when there are obviously costly forms to be avoided with relatively minimal risk of miscommunication, they may assign a disproportionately high cost to strategy sets that deviate too far from those rules.

5

Conclusions and Future Work

We conclude this paper with a brief discussion of areas in which our model goes beyond existing theories of both accent placement and game-theoretic pragmatics, some other phenomena where a similar approach would seem promising to us, and a suggestion of ways to test the predictions of the model.

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109

Novel Aspects of the Model

Accent placement. As explained above, our model concerns the rationalistic factors that influence a speaker’s choice of accent placement. It does not seek to address the rules of the grammar that relate accent placement to information structure. However, some aspects of our proposal indirectly concern the architecture of that grammar. First, while we agree with the suggestion of German et al. (2006) and others that certain forms may be dispreferred despite being wellformed, we argue that such dispreferences should not be encoded in the grammar, but should be explained in terms of factors that lie outside of the grammar. Second, we argue that a mutually accessible probability distribution over possible speaker intentions (i.e., focus assignments) plays a key role in interlocutors’ selection and interpretation of accent placement. Much previous work has neglected to reconcile the explicit assumption that speakers freely choose an intended focus structure with the implicit assumption that intuitions about felicity are a reflection of grammatical constraints. In our model, we straightforwardly adopt the first assumption and propose that intuitions about felicity may be explained by the fact that interpretation is guided by a mix of forces, including mutual beliefs and expectations. This offers a way to reconcile two widespread but seemingly contradictory assumptions in the theory of focus and accent placement. Game-theoretic pragmatics. Game theory has been particularly useful for modeling the ways in which interlocutors enrich the conventional meaning of forms. In Parikh’s (2001) analysis of scalar implicatures, for example, strategic inferences make it possible for an utterance of ‘Some of the boys went to the party’ to convey the truth-conditionally stronger meaning of the sentence ‘Some of the boys went to the party, and not all of the boys went to the party’. Notice that the latter meaning entails the former. In fact, it is typical of game-theoretic analyses that the meanings at issue are monotonically related in some way.13 By comparison, our analysis does not assume any particular relationship between the grammatically determined interpretation of an accent pattern and the interpretation that results from strategic inference. Our model does not require, in other words, that the pattern of F -marking in (10-a) and those in (10-b) and (10-c)14 be related in any particular way. (10)

13

14

a. b. c.

. . . they play their game [IN]F . . . they play [their GAME]F in . . . they [[[play]F [their GAME]F ]F [in]F ]F

Consider Parikh’s (2001) analysis of relevance implicatures, for example, in which a sentence like ‘It’s 4pm’ is enriched with the meaning of ‘Let’s go for the talk’. Even though the latter is not “semantically related” to the former in the same way that ‘Some of the boys went to the party’ is related to ‘Not all of the boys went to the party’ (Parikh, 2001, p. 93), the inferred content is monotonically added (via logical conjunction) to the conventional meaning of the utterance. Recall that, because of focus projection, accentuation on the direct object is grammatically consistent with multiple patterns of F -marking.

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Instead, the alternatives to the grammatically licensed interpretation arise merely because they correspond to different ways of assigning F -marking to the underlying syntactic representation. Certain alternatives then emerge as more relevant to the game structure because the context renders them more probable than others. 5.2

Extending the Analysis

There are additional cases in which weak prosodic restrictions may be interacting with discourse-related constraints. The ones we address in this section differ from our own in a number of ways, yet we feel that there is an underlying similarity among them that warrants a common treatment. Tone Compression in German. Languages vary in the way they treat complex intonation contours applied to monosyllables (Ladd, 1996, 132-4). In English, rise-fall-rise contours may be associated with a single syllable, as in (11): (11)

Sue?! L+H* L-H%

English works differently in this respect from German. Examples (12) and (13) illustrate the high-fall-rise intonation contour that marks questions in German (Ladd, 1996, 133). (12)

¨ Ist das Ihre TUTE? H* L-H% ‘Is this your BAG?’

(13) #Ist das Ihr GELD? H* L-H% ‘Is that your MONEY?’ In (12), the three tones associated with this contour are realized over two syllables,15 such that no syllable carries more than two tones. In (13), by comparison, all three tones are compressed onto a single syllable. According to Ladd (1996), (13) has a “phonetically degraded” quality, even in contexts in which it is expected to be pragmatically appropriate (e.g., someone has left some money on the table). Ladd suggests that in such cases, a speaker is likely to substitute an alternative form, such as (14), in which the three tones are realized over at least two syllables. (14)

Ist das IHR Geld? H* L-H% ‘Is that YOUR money?’

Crucially, however, (13) and (14) do not seem to carry the same meaning. In terms of our earlier framework, (14) marks the expression ‘Geld’ as given, and is 15

Note that ‘T¨ ute’ is pronounced  .

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predicted to be most appropriate when money has been explicitly mentioned in the discourse, while (13) is more appropriate otherwise. In short, the form-based preference for (14) over (13) appears to outweigh the speaker’s desire to mark the status of ‘Geld’ in the contextually most appropriate way. This suggests an interesting twist on our analysis of stranded prepositions. In that analysis, we claimed that certain focus structures may be conveyed in spite of being inconsistent with what is required by focus projection rules. By comparison, it does not seem likely that (14) can be used to convey the meaning of (13). On the contrary, (14) intuitively seems to require that the listener accommodate the fact that ‘Geld’ is given. This suggests that the relevant tradeoff is not between form-based costs and mismatch costs as in our earlier example, but between form-based costs and the utility that the speaker associates with each of the possible meanings. If the speaker assigns roughly equal utility to each way of assigning a status to ‘Geld’, for example, with perhaps a slight preference for treating ‘Geld’ as non-given as in (13), then there is an increased potential for factors other than context to influence the speaker’s choice. In this particular case, the preference for avoiding tone compression is sufficient to favor the pattern in (14). If, however, the speaker were to associate a much higher utility with treating ‘Geld’ as non-given as compared with treating it as given, then the speaker will prefer (13), and any preference between the two forms is unlikely to affect his or her decision.16 Since we are assuming for the present case that the form-meaning association given by the grammar is fixed, there are effectively only two strategy sets to consider, one in which the speaker uses (13) to communicate (successfully) that ‘Geld’ is non-given, and one in which the speaker uses (14) to communicate that ‘Geld’ is given. In that sense, the problem may in fact reduce to a simple decision problem for the speaker between which of the two strategy sets has the higher expected utility. Nevertheless, problems like this have important aspects in common with our above analysis and suggest a certain generality to the approach we are taking. Prosodic Promotion in English. Pitch accent assignment is often discussed in terms of its tendency to encode information structure. Across speakers, utterances, and even phrases within utterances, however, there is substantial variability in the density of accent assignment. Pierrehumbert (1994) proposes to account for this variability through a phonological process called prosodic promotion. Normally, pitch accents that do occur, tend to occur on prominent sylla16

It is interesting to note that the preferred form in this case involves a nuclear accent on ‘Ihr’ ‘your’, which is a possessive pronoun and therefore a function word. To the extent that German et al.’s findings for prepositions generalize to other function word categories, this would be somewhat unexpected. In the end, however, the preference for (14) is observed independently on the basis of phonological well-formedness (however impressionistic), so this does not pose a problem for our approach. It does suggest, however, that in this particular context, the phonological preference against tone compression outweighs any similar preference against accenting possessive pronouns.

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bles. Prosodic promotion, however, allows syllables that would normally be too weak to carry a pitch accent to be “promoted” to a higher level of prominence so that they may carry one. This process, she argues, is not only sensitive to “discourse factors,” but it is “generally available to strengthen prosodically weak elements if the speaker for any reason wishes to accent them.” Notice, however, that prosodic promotion may have consequences for the prosodic marking of information structure. In general, the distribution of accents in an utterance encodes information about the focus assignment because accents are (i) required to occur within a focus and (ii) excluded or at least minimized elsewhere (Schwarzschild, 1999; Truckenbrodt, 1995; Williams, 1997; Sauerland, 2005; B¨ uring, 2008). In the most canonical examples, the focused constituent includes a nuclear accent close to its right edge, while the post-focal material is free of pitch accents or deaccented up to the end of the utterance. The result is a kind of discontinuity in the pitch accent distribution that allows the hearer to identify the rightward extent of the focus. When accents are freely assigned as a result of prosodic promotion, however, then the discontinuity may disappear, thereby obscuring the location of the right edge of the focus. Applied to our earlier example, the location of the last accent would normally distinguish the focus assignment in (10-a) from those in (10-b) and (10-c). (10)

a. b. c.

. . . they play their game [IN]F . . . they play [their GAME]F in . . . they [[[play]F [their GAME]F ]F [in]F ]F

If prosodic promotion applies at the level of the syllable, however, then all words, including ‘game’ and ‘in’, will receive an accent as in (15), and all information about the focus assignment will be lost. In short, prosodic promotion may be detrimental to communication. (15)

THEY PLAY THEIR GAME IN

Consider also that accents are themselves associated with cost and effort, as is assumed, for example, by the wide range of proposals that link accentuation with markedness (Schwarzschild, 1999; Beaver, 2004; Clark and Parikh, 2007; Hirschberg and Ward, 1991). In addition, it is widely held that pitch accents represent a type of morpheme (Liberman and Sag, 1974; Liberman, 1975; Gussenhoven, 1983; Ladd, 1980; Pierrehumbert and Hirschberg, 1990; Pierrehumbert, 1994). On standard assumptions of both neo-Gricean and gametheoretic analysis, then, an increase in the number of accents would result directly in an increase in morpho-syntactic complexity, which in turn is associated with an increase in production effort and processing costs. If prosodic promotion results in utterances that are not only more costly to produce, but potentially carry less information about the speaker’s intention, then what factors would lead a speaker to use it? One factor may be the need to overcome situational impediments to the speech signal, as in cases of ambient noise or unreliable channels. In such a case, a speaker applies prosodic promotion as a way to increase not only the overall amplitude of the utterance, but the

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acoustic distinctness of individual syllables and phonemes. This tends to insure that the lexical content of the speaker’s utterance is recovered, even if other aspects of its meaning are forfeited. In terms of the model we have outlined, this suggests that in cases where prosodic promotion would obscure the relationship between accent placement and focus assignment, the speaker must decide whether the risk of miscommunicating the lexical content outweighs both (i) the risk of miscommunicating focus, and (ii) the extra cost incurred by any additional pitch accents. Generally, this will be a function of the probability of successful communication of the lexical content given the degree of the impediment, as well as the prior probability that is assigned to the speaker’s intended focus assignment given the context. When the probability of unsuccessful communication of the lexical content is high, and the context includes strong cues or the intended focus structure, then the speaker is likely to accept the added costs and risks associated with prosodic promotion. By contrast, when that probability is low, and the context includes only weak cues to focus structure, then the speaker will prefer to minimize accentuation so that focus structure is maximally encoded by the accent distribution. 5.3

Testing the Model

Our model predicts that accent placement should be sensitive to how likely the various parts of an utterance are to be treated by the speaker as given. Throughout the literature on information structure and reference resolution, the tendency for an expression to be interpreted as given or new is linked to various psycho-attentional properties of its potential antecedents. Often, these are collapsed into a single notion of either salience or accessibility, though specific proposals suggest a relevant role for the recency of the antecedent (Arnold, 1998), the morphosyntactic status of the antecedent (Prince, 1981; Gundel et al., 1993; Grosz et al., 1995), or the embedding relationship between adjacent discourse segments (Grosz and Sidner, 1986; Nakatani, 1997). In the context of our model, such factors are predicted to play an important role in the way that probability distributions are assigned to focus structures in the shared beliefs of the speaker and hearer. Moreover, to the extent that they can be manipulated in a concrete way, it should in principle be possible to test the predictions of the model under a varied set of conditions. If, for example, expressions with more recent antecedents are treated as given in those focus structures that have the highest prior probability, then our model predicts that accentuation of the preposition in examples like (4) will actually be less likely when the antecedent is very recent. Notice that this presupposes that the speaker’s intention is fixed across all contexts. For the German et al. study, it was assumed that the speaker’s intended focus assignment was known precisely because the antecedent utterance was very recent. In effect, the subjects were playing the role of both speaker and hearer simultaneously. When the same type of reading task is used for a study which varies the recency of the antecedent, however, then the interpretation that the speaker intends to represent may actually covary with the factor being manipulated. In

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other words, it is no longer reasonable to assume that the speaker’s intended focus assignment is fixed across contexts in such a case. Thus, a study like the one we describe will ultimately require a more sophisticated approach, specifically, one that controls both (a) the focus assignment that the speaker hopes to convey, and (b) the focus assignment that the speaker believes to have the highest prior probability in the mind of the hearer.

Acknowledgments We are grateful to the editors for their support and extreme patience. Parts of this work were presented at Games and Decisions in Pragmatics II in Berlin, Germany, 2006. We thank the audience there for helpful comments. SK gratefully acknowledges the generous support of the American Council of Learned Societies and the Lichtenberg-Kolleg at the University of G¨ ottingen, Germany.

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German, J.S.: Prosodic Strategies for Negotiating Reference in Discourse. PhD thesis, Northwestern University (2008) Grice, H.P.: Studies in the Way of Words. Harvard University Press, Cambridge (1989) Grosz, B., Joshi, A., Weinstein, S.: Centering: A framework for modeling the local coherence of discourse. Computational Linguistics 21, 203–226 (1995) Grosz, B., Sidner, C.: Attention, intentions, and the structure of discourse. Computational Linguistics 12, 175–204 (1986) Gundel, J., Hedberg, N., Zacharsky, R.: Cognitive status and the form of referring expressions in discourse. Language 69, 274–307 (1993) Gussenhoven, C.: Focus, mode and nucleus. Journal of Linguistics 19, 377–417 (1983) Haiman, J.: Natural Syntax, Iconicity, and Erosion. Cambridge University Press, Cambridge (1985) Hirschberg, J., Ward, G.: Accent and bound anaphora. Cognitive Linguistics 2, 101–121 (1991) Kadmon, N.: Formal Pragmatics: Semantics, Pragmatics, Presupposition and Focus. Blackwell Publishers, Oxford (2001) Ladd, D.R.: The Structure of Intonational Meaning. Indiana University Press, Bloomington (1980) Ladd, D.R.: Intonational Phonology. Cambridge University Press, Cambridge (1996) Lewis, D.: Convention. Harvard University Press, Cambridge (1969) Liberman, M.: The Intonation System of English. PhD thesis, MIT. Reprinted by Garland Press (1975) Liberman, M., Sag, I.: Prosodic form and discourse function. In: Papers from the 10th Regional Meeting of the Chicago Linguistic Society, pp. 416–426 (1974) Nakatani, C.: Discourse structural constraints on accent in narrative. In: van Santen, J., Sproat, R., Olive, J., Hirschberg, J. (eds.) Progress in Speech Synthesis, pp. 139–156. Springer, New York (1997) Parikh, P.: The Use of Language. CSLI Publications, Stanford (2001) Parikh, P.: Pragmatics and games of partial information. In: Benz, A., et al. (eds.), pp. 101–122 (2006) Parikh, P.: Language and Equilibrium. The MIT Press, Cambridge (2010) Pierrehumbert, J.: The Phonology and Phonetics of English Intonation. PhD thesis, MIT. Distributed, Indiana University Linguistics Club (1980) Pierrehumbert, J.: Alignment and prosodic heads. In: Proceedings of the Eastern States Conferences on Formal Linguistics, vol. 10, pp. 268–286 (1994) Pierrehumbert, J., Hirschberg, J.: The meaning of intonation and the interpretation of discourse. In: Cohen, P., Morgan, J., Pollack, M. (eds.) Intentions in Communication, pp. 271–311. The MIT Press, Cambridge (1990) Prince, E.: Toward a taxonomy of given-new information. In: Cole, P. (ed.) Radical pragmatics, pp. 223–256. Academic Press, New York (1981) Roberts, C.: Information structure in discourse: Towards an integrated formal theory of pragmatics. In: OSU Working Papers in Linguistics, vol. 49 (1996) (papers in Semantics) van Rooij, R.: Signalling games select Horn strategies. Linguistics and Philosophy 27, 493–527 (2004) Sauerland, U.: Don’t interpret focus! Why a presuppositional account of focus fails and how a presuppositional account of givenness works. In: Maier, E., Bary, C., Huitink, J. (eds.) Proceedings of Sinn und Bedeutung, vol. 9, pp. 370–384 (2005) Schmitz, H.-C.: Optimale Akzentuierung und Active Interpretation. PhD thesis, University of Bonn (2005)

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Evolutionarily Stable Communication and Pragmatics Thomas C. Scott-Phillips Language Evolution and Computation Research Unit School of Psychology, Philosophy and Language Sciences University of Edinburgh

In the past 20 or so years there has been much research interest in the evolution of cooperation in humans (Axelrod, 1995; Boyd & Richerson, 1992; Fehr & Fischbacher, 2003; Milinski et al., 2002; West et al., 2006). The foundational problem addressed by this work is how cooperation can remain evolutionarily stable when individuals have incentives to freeride; that is, to take but not contribute from the public good (Hardin, 1968). There is an analogous problem associated with the evolution of communication: how can signalling remain evolutionarily stable when individuals have incentives to be dishonest? This gametheoretic question is the defining problem of animal signalling theory (Maynard Smith & Harper, 2003; Searcy & Nowicki, 2007). The main goals of this chapter are to explore the various possible solutions to this problem and to ask which most likely applies to human communication. In addition to this it will also, using insights from pragmatics, provide some insight as to the nature of the problem and hence clarify some of the relevant issues. It is somewhat remarkable that the question of the evolutionary stability of human communication has historically received little interest relative to the attention given to the evolution of cooperation and the burgeoning literature on the evolution of language. In the last 15–20 years both have expanded dramatically. Language evolution in particular has grown from a niche interest into a well-recognised academic discipline in its own right, with regular conferences, an ever-increasing number of papers on the topic (Google Scholar returns1 13,800 hits for the search language evolution in 1990, increasing almost monotonically to 54,400 in 2005), and special issues of relevant journals (e.g. Lingua, volume 117(3), 2007; Interaction Studies, volume 9(1), 2008). It would be reasonable to assume that solutions to the problem of evolutionarily stable communication in humans would be a central explanandum for such a discipline, but that is not the case: very few papers have made this question a central focus (exceptions include Knight, 1998; Lachmann et al., 2001; Scott-Phillips, 2008; Szmad & Szathmry, 2006). There has thus been only limited progress beyond speculative discussion, and the contrast with developments in the evolution of cooperation is striking. This chapter begins with the observation that although there are important equivalences between the problems of cooperation and communication, it can be misleading to think about the latter exclusively in terms of the former, as that masks the fact that there are in fact two problems associated with the 1

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evolutionary stability of human communication, rather than the one in cooperation. These problems are: (i) how can we know that a signal means what we take it to mean?; and (ii) how can we have trust in that meaning? This division is made all the more clear once we recognise that it maps directly onto a distinction that is central to pragmatics; that between communicative and informative intent. Indeed, the relationship between these two interdependent aspects of communication shines valuable light on the matters at hand. Thus while this chapter in general seeks to explore how evolutionary considerations can inform pragmatic concerns, there are also valuable lessons that pass in the opposite direction. How can these two problems be solved? The possible answers to this question are critically evaluated and classified. Accordingly, we see that one answer in particular fits with our instinctive ideas about how social contracts work: unreliable and dishonest communication is deterred because a reputation for such behaviour is socially maladaptive. Despite its intuitiveness, this idea has not been empirically tested, and as such represents a potentially fruitful topic for future research.

1

The Problems of Evolutionarily Stable Communication

The problem (note the singular — the inconsistency with the title of this section will become clear shortly) may be simply stated: if the signaller can gain more from an unreliable or dishonest signal than a reliable or honest one then we should expect just such signals to evolve. If the receiver’s payoff to responding to such a signal is negative, as seems reasonable, then we should expect the receiver to evolve not to attend to the signal. Now that the receiver is not attending to the signal there is no possible benefit to the signaller, and so they will evolve not to produce the signal at all (if nothing else, there are likely to be metabolic and opportunity costs associated with signal production (Maynard Smith & Harper, 1995), and hence a pressure not to incur such costs if there is no consequent payoff). The system has now collapsed in much the same way as it does in Aesop’s fable of the Boy Who Cried Wolf, in which the shepherds learnt not to attend to the boy’s calls, since they were so frequently dishonest (Maynard Smith, 1982). That makes the problem sound like a conceptually straight-forward one, and in many ways it is. However for human communication the matter is more complex, since there are two (analogous) problems rather than one. At one level communication is an inherently cooperative act: there must be some agreement about what signals refer to what phenomena in the world — there must be a shared agreement on the mappings between ‘meaning’ and form. (I put ‘meaning’ in scare quotes only because it is not clear what it might mean for an animal to have meanings in any recognisable sense of the term, a point that is expanded on below.) At another level that signal must be something that the audience can place their trust in, so that they are not misinformed in any way. And then, of the course, at a third level the goals to which communication is applied may

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be more or less cooperative: two individuals with a shared goal will use communication for cooperative ends, but two individuals with mutually incompatible goals will use it antagonistically. Importantly, however, for it to be even used antagonistically it must already be cooperative in the first two senses. An explanatory analogy between the first and third levels is with a game of tennis (or indeed any competitive sport). To even be able to play tennis with each other we must both recognise the rules of the game and play within them; refusal to do so means that we cannot even play a meaningful game at all. In the context of communication we can call this communicative cooperation: interlocutors must agree upon the meaning of a signal. However, once we have agreed to play by the rules of tennis we will, if we are intent on winning the game, play as uncooperatively as possible, pushing the ball to the corners of the opponent’s court and generally trying to force errors in their play. This is material cooperation (or rather: non-cooperation), and within communication it is entirely optional. This distinction between the first and third levels has been previously outlined (Hurford, 2007). My suggestion is that we also recognise another type of cooperation involved in communication, nestled between these two: the honest use of signals. Interestingly there is, for the pragmatician, an obvious term for this type of cooperation: informative cooperation. The reason it is obvious is that it recognises the distinction, central to pragmatics, between an individual’s informative intent and their communicative intent. To outline: the former refers to the speaker’s intention to inform the listener of something, and the latter to the speaker’s intention that the listener recognise that they have an informative intention (Grice, 1975; Sperber & Wilson, 1995). Pragmatics thus recognises that when a speaker produces an utterance they do not just intend that the listener understand whatever it is they are talking about, but also that they intend that the listener understand that the utterance is an act of communication designed to achieve an informative intention. We can thus distinguish between the communicative layer, which is about the fact that there is a coherent communicative act, and which requires a reliable mapping between meaning and form; and the informative layer, which is about the fact that the content of the utterance is a reliable guide to the world, and which requires honesty on the part of the speaker. Correspondingly, we have two types of cooperation necessary for communication: communicative cooperation and informative cooperation. We also have a third, entirely optional type: material cooperation. For example, when I lie to my colleague I am reliable but dishonest; but when she argues with me and in doing so prevents me from doing my work she may be both reliable and honest but is materially uncooperative. As necessary conditions, the first two layers demand evolutionary explanation. Indeed, in many respects the evolutionary stability of cooperative enterprises is the defining problem of social evolutionary theory (Axelrod & Hamilton, 1981; Frank, 1998; Maynard Smith, 1982; West et al., 2007). There are two problems to be addressed, then: one regarding how signaller and receiver can agree upon a shared ‘meaning’ for a given signal (communicative cooperation); and another about whether the signaller uses that meaning in an honest way (informative cooperation). To distinguish between the two

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problems, and to disambiguate between two terms that have previously been used synonymously, I suggest that the former problem be termed the problem of signal reliability, and the latter the problem of signal honesty. The difference is depicted in figure 1. These two problems are formally equivalent; that is, they have the same logical structure. As a result the possible solutions are identical too. Of course, this does not mean that the two problems need actually have the same solution — it is perfectly possible that the problem of reliability will be solved differently to the problem of honesty in any particular case. Before we ask about the possible solutions to these problems, I want to comment briefly on why these distinctions have not previously been recognised by animal signalling theorists. One key difference between humans and other animals is that humans exercise what has been termed epistemic vigilance (Sperber & Wilson, 2008): once we comprehend utterances we can evaluate whether or not we consider them true. This distinction between comprehension and acceptance does not, in general, seem to apply to other animals; once informed, they act (but see below). Importantly, the distinction maps directly onto the previously identified distinction between communicative and informative cooperation. Communicative cooperation is a matter of whether or not signals are reliable (that is, whether individuals share the same signal-form mappings), and once it is achieved then signals become comprehensible. Similarly, informative cooperation is a matter of whether or not signals are honest, and once that is achieved then receivers can accept them as true, and are thus worth attending to. When receivers do not, or rather cannot, exercise epistemic vigilance then these two problems collapse into one. There are, of course, some occasions in which non-humans do seem to exercise epistemic vigilance to at least some degree (Cheney & Seyfarth, 1990). In such cases, we have two problems to solve rather than the usual one studied by animal signaling theorists. What term should we use to refer to the situation when honesty and reliability collapse into a single problem? Both reliability and honesty seem to depend upon a coherent notion of the meaning of signals: reliability is a problem about a disjoint between the meaning-form mappings held by different individuals, while the notion of honesty seems to presuppose that a signal has a propositional meaning whose truth-value can be assessed. However it is at best unclear whether it is coherent to talk about animal signals having meanings in the same way that human utterances do. Despite this, I will not suggest an alternative term, for at two reasons. First, a suitable alternative is not forthcoming; and second, the two terms are in such widespread use in the animal signalling literature, with little if any apparent confusion, that redefinition seems both unwise and unlikely to succeed. On the contrary, the use of anthropomorphic gloss is a common strategy in behavioural ecology and social evolution (Grafen, 1999). Of the two, honesty seems the more preferable, if only because it seems to be the more common. This is perhaps because it is the more theoretically interesting: it is hard to see what payoffs could be attained through unreliable communication (if one cannot be understood then why should one signal at all?), but the potential payoffs to dishonesty are clear.

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Fig. 1. The twin problems of (a) reliability; and (b) honesty. In both cases the girl has said “banana” having thought of an apple, and this fails to correspond to the boys mapping of the sound (which is as per the convention in English). However the reasons for this failure are different in each case. In (a) the girl has a different (in fact, the precise opposite) mapping from sounds to meaning than the boy, and this makes her unreliable. In (b) she has the same mappings as the boy but has chosen to communicate a different meaning than the one she has thought of, and this makes her dishonest.

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corresponding evolutionary gloss problem Do interlocutors have the same meaning- reliability form mappings as each other? Does the signal carry information that is honesty worth the receivers attention? Is communication being used to achieve munone tually beneficial goals?

This section has introduced and discussed a number of other terms, and so it seems appropriate to summarise them and their relationships to each other. Table 1 does this. The nature of the problems of evolutionarily stable communication should now be clear, and we can thus ask about possible solutions.

2

Solutions to the Problems of Evolutionarily Stable Communication

We turn now to possible solutions. Inclusive fitness theory (Grafen, 2006; Hamilton, 1964), or kin selection (Maynard Smith, 1964), is the most significant contribution to evolutionary theory since Darwin. Haldane’s quip that he wouldn’t give up his life for one of his brothers, but that he would for both of them, or eight of his cousins, nicely captures the basic idea: that since many of my genes are shared with my relatives, it is in my own genetic interests to help them. This insight is captured by Hamilton’s simple rule, that altruistic behaviours will favoured if the cost incurred by the actor is outweighed by the benefit to the recipient times the degree of relatedness between the two individuals: br > c. If this inequality is satisfied then dishonest or unreliable behaviour should be unexpected. Accordingly, there are many instances of kin-selected communication in nature, most obviously among eusocial insects. A related point is that spatial organisation is important: the individuals with which an organism communicates are not chosen at random, but instead tend to be those that are nearby. The degree to which populations disperse themselves is measured in terms of viscosity, a notion that is closely tied to that of kin selection: if viscosity is high, meaning that there is limited dispersal, then over time individuals in the same area tend to be related to one another, and can hence ensure stable communication due to Hamilton’s rule. The most famous explanation of how non-kin can maintain stable communication is the handicap principle. Although introduced to animal signalling theory in the 1970s (Zahavi, 1975), and almost simultaneously to economics (Spence, 1973), the idea goes back much further, at least to 19th-century sociological discussions of the conspicuous consumption of the leisure class (Veblen, 1899). Indeed, such expenditure serves as a nice illustration of the basic idea: the purchase of expensive, conspicuous objects (Ferraris, Tiffany jewellery, etc.) advertises to onlookers that the purchaser can afford to make such purchases, and

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therefore must be well-off; the cost of the objects is a handicap that only the most affluent can afford. As a further example, while writing this article I came across the following passage in a newspaper article about stock market trading in the City of London: “Certain clients even expected such behaviour [drinking and drug taking, often to excess, during working hours] from their brokers, viewing their antics as proof that they were so good at their job, they were given free rein to behave as they pleased” (“Tricks of the traders”, 2008). Similarly, large tails make peacocks less dexterous and slower than they would otherwise be. Only the highest quality peacocks can afford such a handicap, and hence the peacock tail acts as a reliable indicator of quality. Consequently the peacock tail has become the exemplar par excellence of the handicap principle. The idea was originally met within evolutionary biology with some skepticism, with a number of models and arguments produced that purported to show that it was unlikely to work (e.g. Maynard Smith, 1976), but that changed once a formal proof of its stability was published (Grafen, 1990). There was no such similar skepticism within economics; on the contrary, its proponent was awarded the Nobel prize in part for his articulation of the idea. Although initially paradoxical, once grasped the logic of the handicap principle is often recognised as an ingenious solution to the problem of evolutionary stability. Perhaps for this reason, it has sometimes been assumed (e.g. Knight, 1998) that it is the only process by which we might stabilise communication, and hence that if communication is to remain stable then signals must incur costs over-and-above those necessary to actually produce the signal in the first place. This distinction between the costs that are necessary to produce the signal and any additional costs that are paid as a handicap is captured by the terms efficacy costs and strategic costs respectively (Maynard Smith & Harper, 1995). The handicap principle is in essence a statement that communication can be stabilised by the payment of strategic costs. However, although they are sufficient, it is not the case that strategic costs are necessary for stability. On the contrary, such a claim is false both theoretically and empirically: several alternative processes have been identified by animal signalling theorists (Maynard Smith & Harper, 2003), and there are many instances of signalling in nature in which no strategic costs are paid: the status badges of male passerines (Rohwer, 1975; Whitfield, 1987); the display of an already fertilised ovipositor by female fruit flies (Maynard Smith, 1956); the hunting calls of killer whales (Guinet, 1992); and, of course, human language are just a few of the many examples (for more, and more details on the listed examples, see Maynard Smith & Harper, 2003). One further point should be emphasised, as it will be of critical importance later: it is the signal itself that must incur the strategic costs. The handicap principle is unstable if the costs are transferred onto some other associated behaviour. That is, there must be a causal relationship between signal form and the cost incurred. For example, there is no strategic cost associated with the size of a male passerine’s badge of status (Rohwer, 1975; Whitfield, 1987), the size and colouration of which correlates with the bird’s resource holding potential (a composite measure of all factors that influence fighting ability (Parker, 1974)).

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However low-status birds that have large badges will incur the costs of getting into fights they cannot win (Rohwer & Rohwer, 1978). To call such a scenario a handicap seems to render the notion of a handicap far too general, a point that will be expanded upon below, where the formal difference between this scenario and the peacock’s tail is made clear. Despite this, the notion of a handicap has been used in this more general sense. For example, the two previous suggestions about how the handicap principle might be relevant to the evolution of language, discussed in the next section, are precisely examples of where the costs are not associated with the signal itself but instead with its social consequences. What are the alternatives to the handicap principle? Indices are causal associations between signal meaning and signal form. This link precludes even the possibility of unreliability or dishonesty. An example is the roar of red deer, where formant dispersion is reliably (negatively) correlated with the deer’s size (Reby & McComb, 2003) as an inevitable consequence of the acoustics of the deer’s vocal apparatus. The deer’s larynx descends upon vocalisation, and the comparative evidence suggests that this is the result of a selection pressure to exaggerate one’s size (Fitch & Reby, 2001). However that process seems to have gone as far as it can without compromising other aspects of the deer’s anatomy (ibid.). As a result it is actually impossible for the deer’s roar not to carry reliable information about its size and hence its social dominance; deer can lower their larynx no further, and hence the formant dispersion of their vocalisations is unfakeable. Other examples of indices include male jumping spiders, who expose the ventral surface of their abdomen as an indicator of their current condition (Taylor et al., 2000) and snapping shrimps, who advertise their claws to each other as a way to avoid physical conflict (Versluis et al., 2000) (again, for more examples and more details on these examples, see Maynard Smith & Harper, 2003). To state the idea of an index in formal game-theoretic terms, signals can be free of strategic costs and evolutionarily stable so long as the efficacy cost of the signal is a function of the trait in question (e.g. a function of size, in the red deer example) (Hurd, 1995; Lachmann et al., 2001; Szmad & Szathmry, 2006). There is some potential for confusion here. Indices and handicaps are supposed to be mutually exclusive, yet on the one hand handicaps are stable only if the strategic costs associated with the handicap are tied to signal form, but on the other hand indices are defined by a causal relationship between signal meaning and signal form. So we have a chain of associations from strategic costs to meaning (handicaps), and from meaning to form (indices). Strategic costs are thus associated with form, and the difference between a handicap and an index becomes unclear. To retain the distinction we must be more precise in our terminology: a handicap is indexical of signal cost, while what we would normally call an index is indexical of signal meaning. Moving on, are there any non-indexical solutions to the problem of evolutionarily stable communication? That is, what explanations are available when a signal is free of strategic costs and is not indexical of meaning? Several possibilities have been identified, but there is an open question about how best to categorise them. One classification (Maynard Smith & Harper, 2003) suggests a three-way

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division between coordination games, repeated interactions, and punishment. Coordination games are those in which in which some common interest overrides any conflicting motivations the participants might have (Silk et al., 2000). The classic example is the ‘War of the Sexes’: the husband wants to go to the pub for the evening and the wife wants to go to the theatre, but they share an overriding common interest that whatever they do they want to do it together. A realworld example is courtship in fruit flies, who mate only once in their lifetime. If a male attempts to court a female after this mating she will display her ovipositor to him and thus advertise that his efforts are futile. He then ceases courtship immediately (Maynard Smith, 1956). In this way the female’s signal saves both of them wasting time. Formally such games can be settled only if there an asymmetry in the relationship such that one player or the other backs down, and if this asymmetry is known to both players (Maynard Smith, 1982). In a repeated interaction the longer-term payoffs of honesty may outweigh the short-term payoff of dishonesty (Silk et al., 2000), and hence the problem should not arise. Repeated interactions are more likely in viscous populations, a point highlighted in the literature on cooperation but not much considered with respect to communication (but see Grim et al., 2006; Skyrms, 1996). Indeed, repeated interactions are a candidate explanation for both communication and cooperation (it is, after all, the basic logic behind reciprocity, in which individuals trade what would otherwise be altruistic and hence evolutionarily unstable behaviours (Trivers, 1971)). Finally there is punishment, in which one individual actively punishes another for unreliable/dishonest signalling (Clutton-Brock & Parker, 1995). This will of course act as an incentive against such behaviour, but this only really moves the problem on to a different locus, since we must now ask why punishing behaviour will evolve if it is itself costly. Indeed, it seems to be a prime candidate to fall foul of the tragedy of the commons, since all individuals get an equal share of the payoff (stable communication) but can let others pay the costs of punishing that are necessary to obtain it. On the surface this three-way distinction seems a reasonable one, but in fact it is not at all clear how we should distinguish between the three. If, for example, an individual abuses a system of mutually-beneficial repeated interactions and their partner then refuses to re-enter into the relationship, should we classify that refusal as the collapse of an arrangement of repeated interactions or as punishment? At a functional level the two behaviours are identical: they are an adaptive reaction to a partner that has abused a previously established relationship. Any criteria we use to distinguish between them must therefore be to do with mechanisms rather than functions. Such distinctions are desirable, but are not suggested here, since it would require a full review of possible mechanisms, a project that is outside the purview of the present article. At the same time, we should recognise when behaviours are functionally equivalent. For that reason I suggest a highorder classification of deterrents. In general, deterrents refer to the situation where reliable/honest communication is cost-free, but where dishonesty is costly. It can be shown that not only is such an arrangement stable, but that where it occurs costly signals will be selected against (Gintis et al., 2001; Lachmann et al., 2001).

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That is, with deterrents the costs are paid by those who deviate from the ESS. This differs from the handicap principle in the following important sense: there, costs are paid as part of the ESS. This is the reason why the handicap principle should not be extended to scenarios in which the costs are paid socially rather than in production; the two are formally quite different. In one, handicaps, costs are incurred by honest signallers; while in the other, deterrents, costs are incurred by dishonest signallers. This is a fundamental difference that is not captured by present terminology. In fact, it has become standard to use the term handicap to refer to both scenarios. The suggestion here is that they be distinguished. We thus have a three way classification of the basic functional outcomes by which communication between non-relatives may be stabilised (Scott-Phillips, 2008). Each of these, and particularly deterrents, could be subdivided using mechanistic criteria, but that matter is not discussed here: – indices, in which signal form is tied to signal meaning; – handicaps, in which signal cost is tied to signal form, and hence acts as a guarantee that is incurred by reliable/honest signallers; – deterrents, in which costs are incurred by signallers who deviate from reliability/honesty. These possibilities are mutually exclusive and are provisionally taken to be exhaustive — additional suggestions are not forthcoming. The present trichotomy covers scenarios in which unreliability/dishonesty is either precluded (indices), expensive (handicaps) or unwise (deterrents). The possibility of further alternatives is not discounted, but it is hard to see what form they might take. We now turn to the question of which one most likely applies to human communication, and natural language in particular.

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Three of the solutions discussed above can be discounted an explanation of why linguistic communication is evolutionarily stable. First, kin selection has been proposed as a partial explanation of the stability of human communication (Fitch, 2004), but there is, of course, an obvious flaw — that we freely speak to non-relatives. This is the reason why kin selection can only be a partial explanation of honesty and reliability; the suggestion is only that contemporary language evolved “primarily in a context of kin selection” (Fitch, 2004, p.275). Exactly what is entailed by this needs to be more fully developed before the idea can be properly evaluated. Second, linguistic form is famously unrelated to meaning (Saussure, 1959): dog, chien and Hund all refer to the same set of canine animals, despite no similarities in form. (Onomatopoeia is a rare exception.) Third, handicaps may be excluded because, as discussed above, the notion of a handicap should be restricted to those scenarios in which there are strategic costs associated with signal production, yet human utterances do not seem to carry such costs. Despite this “crippling problem” (Miller, 2000, p.348), researchers have still looked to the handicap principle as an explanation of stability in language.

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On example is the suggestion that the sort of ritualised performance witnessed in many societies acts as a costly signal of one’s commitment to the group, and hence performers are trusted as in-group members (Knight, 1998; Power, 1998). However there is nothing in this model to stop an individual paying the costs to enter the in-group and then once accepted behaving dishonestly or even unreliably. This is because the costs of the performance are not causally tied to the individual’s subsequent utterances. A second example is the hypothesis that politeness phenomena act as a handicap (van Rooij, 2003), in that they reduce the speaker’s social standing relative to the listener, place them in the listener’s debt or otherwise incur socially relevant costs. For example, the utterance “I don’t suppose there’d be any possibility of you. . . ” can be read as an announcement that the speaker is prepared to incur some social cost in order to ensure that the desire which follows the ellipsis is satisfied. Let us accept, for the sake of argument, that this argument is correct. This does not make politeness a handicap, since the costs incurred are not paid as part of the signal. If politeness does place us in social debt then this would be an example of a self-imposed deterrent rather than a handicap: it imposes a social obligation on us to return the favour in some way, and we do not renege on this because the threat of social exclusion deters us from doing so. The difference between deterrents and handicaps is implicitly acknowledged by this paper, which discusses how the costs can be due to either by the signal (handicaps) or the receiver(s) (deterrents). However the term handicap is then used to refer to both scenarios — and as such offers a good example of how such usage has become standard. By deduction, then, we are drawn towards deterrents as a solution to the two evolutionary problems of reliability and honesty. There is an intuitiveness to both ideas: unreliable communication, for example if one says “dog” to refer to feline pets, is deterred because it means that one will not be understood, and hence cannot achieve one’s communicative goals; and dishonest communication will result in a loss of trust and the consequent social costs. In fact, deterrents are what we should logically expect to find in humans. In general, when indices are not available, and when the expected gains from dishonesty or unreliability outweigh the costs, then costly signals must be employed to ensure stability (Lachmann et al., 2001). Deterrents will be used only if verification is both possible and cheap — which is precisely what epistemic vigilance gives us. A similar finding is that signalling can be stable if unreliability is costly (Gintis et al., 2001), and it should also be noted that deterrents allow signals to take an arbitrary form (Lachmann et al., 2001). The fact that utterances are cheap yet arbitrary is too often taken to be paradoxical: “resistance to deception has always selected against conventional [arbitrary –TSP] signals — with the one puzzling exception of humans” (Knight, 1998, p.72, italics added). This is, as the examples discussed above show, simply not true. Instead, once we remove the requirement that costs be causally associated with signal form, as we do if we place the onus of payment on the dishonest individual, then the signal is free to take whatever form the signaller wishes. This paves the way for an explosion of symbol use.

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What keeps humans from dishonesty and unreliability? There is an obvious candidate: reputation. For this to work it must be possible for individuals to modify their future behaviour in the light of other’s behaviour. This is a task that the human memory performs with ease, often subconsciously (see Pentland, 2008) but we should nevertheless recognise it as a crucial prerequisite. Emotions like anger ensure that we do not repeatedly trust those that have cheated us (Ekman, 1992; Tooby & Cosmides, 1990), and the empirical literature contains many illustrations of our sensitivity to untrustworthy behaviour. For example, we are more likely to recall the identities of cheaters than cooperators (Chiappe et al., 2004; Mealey et al., 1996; Oda, 1997). We are well attuned to the detection of unfakeable physical cues of dishonest behaviour, for example a lack of eye contact and a large number of unfilled pauses in speech (Anolli & Ciceri, 1997; Scherer et al., 1985), and these appear to be cross-cultural (Bond Jr. et al., 1990). In fact such cues may even be seen not only when we are deceptive but also in our everyday appearance: when presented with a number of faces and asked to recall them later, experimental participants are more likely to recall the identities of individuals who later defected in a game of prisoner’s dilemma, even when they do not have access to this information (Yamagishi et al., 2003). We are also very sensitive to our own reputational status within the social group in general, and are keen to maintain our standing: cooperation can be maintained in various economic games once reputational effects are added, but not otherwise (Milinski et al., 2002; Piazza & Bering, 2008; Wedekind & Milinski, 2000). This is true even if we experience only subtle cues of a potential loss of reputation, such as stylised eyespots on a computer (Haley & Fessler, 2005). Such effects have also been found in more ecologically-valid conditions: an honesty box for tea, coffee and milk in a University common room received greater contributions when the small picture above it was a pair of human eyes rather than a flower (Bateson et al., 2007). This attentiveness to one’s own reputation and to cues that it may be affected by current behaviour should not be a surprise, since a loss of reputation will mean exclusion from the local group, a heavy penalty for a social species like ourselves. Indeed, the emerging consensus from the burgeoning literature on the evolution of cooperation is that reputational effects are crucial to stability (Fehr, 2004); without such effects scenarios like the tragedy of the commons are far more likely to arise (Milinski et al., 2002). A similar story seems to hold in primate societies (Gouzoules & Gouzoules, 2002). Note also that this effect is likely to snowball once language in some form or another is off the ground, since individuals then become able to exchange information about the honesty and reliability of others (Enquist & Leimar, 1993). This may explain why so much of our conversational time is dedicated to gossip (Dunbar, 1997). An important implication of the hypothesis that unreliability and dishonesty are deterred by the threat of poor reputations is that the second-order problem of how deterrents are implemented does not arise. Nobody is asked to bear the brunt of the costs of punishing others, because social exclusion is not itself costly to enforce. On the contrary, it is the most adaptive response to individuals with a reputation for unreliability or dishonesty.

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Concluding Remarks

In this chapter I have sought to review the various ways in which communication can be evolutionarily stable, and ask which most likely applies to linguistic communication. Language is a more complex case than most if not all animal signals, since it sets two problems rather than one. The first is reliability: we must agree upon the meaning of signal. The second is honesty: why should signallers be honest if dishonesty pays? These two terms are often used synonymously, but the case of language makes it clear that they are separate problems. They correspond to two different layers of communication, analogous to the well-recognised distinction within pragmatics between communicative and informative intent. A third layer, to do with whether or not communication is used to achieve mutually beneficial goals, is also identified. This material cooperation is, of course, not necessary for stable communication: we can antagonise and argue with our interlocutors, but still maintain stability. One way in which the evolutionary problems of communicative and informative cooperation can be solved is for there to be a causal relationship between meaning and form. This ensures that the signal cannot be faked, and is termed an index. Alternatively, the signals may be costly, and if there is a causal relationship between that cost and the signal’s meaning then we have handicaps. Finally, there may be some costs associated with dishonesty or unreliability that outweigh the potential benefits. These costs act as deterrents. Note that these deterrents are often a consequence of the environmental make-up rather than pro-active punishment, since such enforcement would only replace the first-order problems of dishonesty and reliability with an analogous second-order problem, under the reasonable assumption that this enforcement is itself costly. In humans, deterrents seem the most likely solution to both problems — we are deterred from unreliable and dishonest communication because that would give us a bad reputation, with obvious evolutionary consequences. There is scope for empirical investigation of this proposal. One way in which this could be done would be to use the economic games that have been profitably used to study the effects of reputation in the evolution of cooperation in humans (e.g. Axelrod, 1995; Milinski et al., 2002), but with the independent variable as honesty in a communication game rather than cooperation in a prisoner’s dilemma or some other cooperative game. Investigation of whether and how humans might differ from other primates in this regard would also be useful. One matter that has not been discussed is the informational value of utterances. Conversation is sometimes thought of as an exchange of information, which is kept stable through reciprocity (e.g. Ulbaek, 1998). This would imply that we keep track of who we have given information to, punish those who do not provide information in return, and compete to listen to others. These predictions do not seem to be correct; on the contrary, we compete to speak rather than to listen (Dessalles, 1998, 2007). In general, speaking appears to be a selfish rather than an altruistic act (Scott-Phillips, 2007). One reason for this is to gain a better reputation: the scientist who presents good work at a conference will go up in his colleagues’ esteem, for example. As such, then, this story also involves

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reputation, in this case the attainment of good reputation. This is the other side of the bad reputation that will follow if we speak unreliably or dishonestly. It hardly bears stating that the honesty and reliability of utterances are central to pragmatics. Without reliability communication cannot take place at all, and honesty is so crucial that Grice saw fit to make it one of his four maxims. He also desired a naturalistic basis for his ideas: “I would like to think of the standard type of conversational practice not merely as something that all or most do in fact follow but as something that it is reasonable for us to follow, that we should not abandon” (1975, p.48, italics in original). This chapter has sought to explore how animal signalling theory can be applied to language so as to provide an important part of that foundation: evolutionary stability.

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Simulating Grice Emergent Pragmatics in Spatialized Game Theory Patrick Grim Group for Logic & Formal Semantics Department of Philosophy, Stony Brook University

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Introduction

How do conventions of communication emerge? How do sounds or gestures take on a semantic meaning, and how do pragmatic conventions emerge regarding the passing of adequate, reliable, and relevant information? My colleagues and I have attempted in earlier work to extend spatialized game theory to questions of semantics. Agent-based simulations indicate that simple signaling systems emerge fairly naturally on the basis of individual information maximization in environments of wandering food sources and predators. Simple signaling emerges by means of any of various forms of updating on the behavior of immediate neighbors: imitation, localized genetic algorithms, and partial training in neural nets. Here the goal is to apply similar techniques to questions of pragmatics. The motivating idea is the same: the idea that important aspects of pragmatics, like important aspects of semantics, may fall out as a natural results of information maximization in informational networks. The attempt below is to simulate fundamental elements of the Gricean picture: in particular, to show within networks of very simple agents the emergence of behavior in accord with the Gricean maxims. What these simulations suggest is that important features of pragmatics, like important aspects of semantics, don’t have to be added in a theory of informational networks. They come for free.1 Sections 2 and 3 outline some of the background of the current work: emergence of cooperation in spatialized game theory and the emergence of a simple semantics among agents in a simulated environment of wandering food sources and predators. Section 4 applies the techniques developed there to the case of pragmatics. The focus of the current work is the emergence of the Gricean maxims themselves. Communicative exploitation of those maxims in conversational Oimplicature and inference is a further matter, and the simulations used here do not generally take us that far. In the case of scalars, however, Section 5 shows that coordinated behavior in the general region of Gricean implicature does appear even for agents and environments as simple as those modeled here. Color illustrations and animations for this chapter can be found at http://www.pgrim.org/pragmatics 1

My research collaborators in this work include Peter Ludlow, Dustin Locke, and Aaron Bramson at the University of Michigan and Brian Anderson at Stony Brook.

A. Benz et al. (Eds.): Language, Games, and Evolution, LNAI 6207, pp. 134–159, 2010. c Springer-Verlag Berlin Heidelberg 2010 

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Cooperation in Spatialized Game Theory

In classical game theory, we deal with the interactions of small numbers of idealized agents. In evolutionary game theory we are dealing with whole populations, applying principles of replicator dynamics pioneered by R. A. Fisher in theoretical biology. The result is a model beautiful for its simplicity and mathematical ease. In both theoretical biology and game theory, however, replicator dynamics brings with it what might be termed global assumptions: assumptions of randomly mixed interactions within a large population, and assumptions of randomly distributed reproduction. It is spatialized game theory that is the necessary background for the results below. The focus of spatialized game theory is the dynamics of local action within a network structure rather than global interaction across a population. Here agents do not interact randomly with all other agents in the population. They interact (or interact with increased probabilities) with specific agents— their neighbors in the network structure. Local rather than global interaction of this type is addressed in theoretical biology under the term ‘viscosity’. Figure 1 offers an example of spatialized game theory using the simple reactive strategies in the iterated Prisoner’s Dilemma: the eight strategies that turn on merely an initial move (cooperate or defect) and what the other player did on the previous round. The eight Boolean combinations of play for initial move, reaction to cooperation, and reaction to defection include All-Defect, AllCooperate, Cooperate-then-All-Defect, Tit for Tat, and Suspicious Tit for Tat (like Tit for Tat except starting with an initial defection). In this illustration a randomization of strategies is embedded in the spatialized lattice familiar from cellular automata. All play is local—individual cells play only with immediate neighbors touching them on the sides or the diagonal. After a series of 200 games, players look to see if any neighbor has achieved a higher score. If so, they update to the strategy of their most successful neighbor. The agents in such a model are of course as egoistic as any in classical game theory. The general question of interest is how conditional cooperation, in the form of a dominance by Tit-for-Tat, might develop. Updating by imitation in this kind of network turns out to be a powerful mechanism for the emergence of cooperation. Figure 1 shows the evolution of a randomized array, with early success by All Defect (animation at http://www.pgrim.org/pragmatics). Because of local interaction and local imitation, however—because of ‘viscosity’—communities of mutually reinforcing Tit for Tat then expand, eventually occupying the entire array (Grim 1995, 1996; Grim, Mar & St. Denis 1998).2 It is worthy of emphasis that spatialization is crucial to these results. If one of the mechanisms at issue is made global rather than local, the effect is broken: 2

This first example employs synchronous updating. Each of our agents simultaneously calculates whether any neighbor has done better, if so updating to the strategy of the most successful neighbor. Results are much the same, however, if we randomly update only a small proportion of cells at each turn. See Grim, Wardach, and Beltrani 2006.

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Fig. 1. Conquest by TFT in a spatialized environment. Typical evolution of a randomized array of the 8 reactive Prisoner’s Dilemma strategies, where cells copy the strategy of their most Successful neighbor. TFT in black, All-D in white. Generations 1, 5, 10 and 15 shown. A full animation of the evolution of the array can be seen at http://www.pgrim.org/pragmatics

cooperation no longer proves dominant. Suppose, for example, that each agent updates its strategy not by imitating the strategy of its most successful neighbor but by imitating the strategy of the most successful cell in the entire array. With such an alteration, competitive play remains local but updating becomes global. In that case Tit for Tat is driven to extinction and we have a clear conquest by All-Defect (Figure 2).3 3

Although spatialized lattices of this form are my main tool throughout, it is clear that a study of the dynamics of interaction across the entire range of possible network structures is needed. With regard to cooperation in particular, it turns out that the results outlined here generalize nicely across network structures. See Grim 2009a, 2009b.

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Fig. 2. Conquest by All-Defect with a global assumption. Typical evolution of a randomized array of the 8 reactive Prisoner’s Dilemma strategies, using global replacement: in each generation 5% of the array is replaced with the strategy of the most successful cell in the array.

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Studies in Communication

It is clear that game-theoretic cooperation can emerge in distributed networks. Can patterns of simple signaling emerge in a similar way? Can a dynamics this simple produce simple forms of communication? Though the modeling techniques are very similar, the games at issue in this exploration go well beyond the simple Prisoner’s Dilemma. The basic model again uses an array of individuals in a two-dimensional lattice.4 In order to keep things simple, individuals in this model don’t move: they are embedded in the array something like coral in a reef. What does move are food sources and predators, each of which travels in a random walk, cell by 4

Here again the ultimate goal would be an understanding of communication across various networks.

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cell, across the array. It should be emphasized that the food sources are food sources—individuals feed when a food source lands on them, but the food sources themselves are not consumed and do not disappear. Food sources are like a school of fish, perhaps, continuing their random walk and offering nourishment for the next individual down the line. In much the same way, predators in the model are never sated: they continue to pose a danger in their random walk. Each of the individuals embedded in the array has a small behavioral repertoire. It can open its mouth, hide, stay in neutral, make a sound 1 (heard by itself and immediate neighbors), or make a sound 2. Each individual also has a limited range of perception. It knows when it is fed—when its mouth is open and a food source lands on it—and when it is hurt—when it is not hiding and a predator lands on it. Each agent can hear and distinguish sounds made by itself or immediate neighbors; it knows when someone just made sound 1, for example, or someone just made sound 2. The behavior of each individual is dictated by a simple strategy code. An agent’s code may dictate that it opens its mouth only when it hears sound 1, for example, or hides only when it hears sound 2. An agent’s code might also dictate that it never opens its mouth. When an individual opens its mouth and food is on it, it gains a point for ‘feeding’. When a predator hits an individual that isn’t hiding, that individual is ‘hurt’ and loses a point. But opening one’s mouth, hiding, and making a sound all exact an energy cost. There are no costless actions in this model, and there are no costless signals. This model differs from others in the literature in a number of ways. One difference is a thorough spatialization; in this model, all action is local action. Some of Cecilia and Paolo Di Chio’s work puts a similar emphasis on spatialization (Di Chio & Di Chio 2007). A second difference is the fact that the emergence of signaling is ecologically situated. The question is how arbitrary sounds take on a meaning, but the context is an environment with some built-in significance of its own: the importance for individual agents of capturing food and avoiding predators. Many of the models in the literature employ something like a Lockean theory of meaning, in which meanings correspond to something in individual heads. In contemporary guise, those Lockean meanings often take the form of representation matrices, with convergence on the same matrix in different individuals taken as the measure of ‘successful communication.’ The model offered here is much closer to a theory of meaning as use. Here there is no matching of matrices in different heads; the measure of communication is simply and solely successful behavioral coordination across a community. A final distinctive feature of the model is its emphasis on individual gains, tied directly to individual success in capturing food and avoiding predators. Unlike many models—unlike even Lewis’s signaling games—there is no ‘mutual reward’ for successful communication or mental matching, for example. Prashant Parikh’s models share this emphasis on individual gains (Parikh 2001, 2006, Parikh & Clark 2007).

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What my collaborators and I were interested in was the possibility that patterns of simple signaling might emerge as a result of a situated economics of information: that communication might emerge in response to environmental pressures on the basis of individual gains. We investigated three forms of strategy updating in three stages of the work. In initial studies, strategy change was by simple imitation, just as in the cooperation studies above (Grim, Kokalis, Tafti & Kilb 2000). In a second series of investigations, we cross-bred our strategies using genetic algorithms, but restricted our genetic algorithms—like everything else— to purely local instantiation. Genetic algorithms are generally applied globally; individuals ranking highest on some fitness function are selectively cross-bred and their offspring are re-introduced randomly into the population as a whole. We replaced that global algorithm with a local one in which strategies are ‘bred’ as hybrids between the strategy of a cell and that of its most successful neighbor; local success results in local hybrid reproduction (Grim, Kokalis, Tafti & Kilb, 2001). In more recent models, our individuals instantiate simple neural nets, doing a partial training on the behavior of successful neighbors (Grim, St. Denis & Kokalis 2003; Grim, Kokalis, Alai- Tafti, Kilb & St. Denis 2004). On the basis of lessons learned regarding the pitfalls of perfect worlds (Grim, Kokalis, AlaiTafti, & Kilb 2000) and following hints in the work of Martin Nowak (Nowak & Sigmund 1990, 1992), we worked throughout with a stochastically imperfect world. In 5% of cases individuals open their mouths and in 5% of cases they hide regardless of strategy or input. Results using all three forms of strategy updating showed an emergence of simple signaling in a spatialized environment of wandering food sources and predators. The following sections summarize the main points using results from the more recent work with neural nets. 3.1

The Dynamics of Feeding, Predation, and Simple Signaling

For even creatures as simple as those outlined, there is a behavioral strategy that would seem to qualify as an elementary form of signaling. Consider a spatialized community of individuals which share the following behavioral strategy: – They They – They They

make sound 1 when they successfully feed open their mouths when they hear a neighbor make sound 1 make sound 2 when they are hurt hide when they hear a neighbor make sound 2

We have termed these ‘perfect communicators.’ Our food sources migrate in a random walk from cell to cell, never being entirely consumed. So suppose a community of ‘perfect communicators’, and suppose that one individual successfully feeds. Because it’s a ‘perfect communicator’, it makes sound 1. All its neighbors are perfect communicators as well; when they hear sound 1, they open their mouths. The food source will migrate, and one of those neighbors will successfully feed. That cell will then make sound 1, alerting its neighbors. The result in a community of ‘perfect communicators’ is a chain reaction in which the food source is successfully exploited on each round.

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This advantage, of course, demands that we have a full community of ‘perfect communicators’. For a single ‘perfect communicator’ there is no advantage to such a strategy at all: given the energy cost for sounding and opening its mouth, in fact, there is a significant disadvantage. The dynamics are slightly different for predators. If an individual is hit with a predator and gives an alarm call, his neighbors hide and therefore are not hit by the wandering predator. Those neighbors therefore do not pass on the alarm call. Because there is no alarm call, someone does get hit by the predator on the next round. Even ‘perfect communication’ therefore affords protection against predators only on alternate rounds. Because of that difference in dynamics, and in order to avoid a bias toward one environmental pressure rather than another, we worked throughout with twice as many predators as food sources across the array. Since our individuals have two sounds at their disposal, of course, there are two versions of ‘perfect communicators’. One uses sound 1 for food and sound 2 for predators; the other uses sound 2 for food and sound 1 for predators. 3.2

Emergence of Communication in Spatialized Arrays of Neural Nets

In our simplest work with neural nets, our individuals are instantiated as perceptrons: two-layer feed-forward nets of the form shown in Figure 3.5 This structure has two distinct lobes. The left takes sound as input and outputs whether an individual opens its mouth, hides, does neither or both. The right lobe takes as input whether it has been fed or hurt on a particular round, outputting any sound the agent makes.6 We use a bipolar coding for inputs, so that ‘hear sound 1’, for example, takes a value of +1 if the individual hears sound 1 from an immediate neighbor on the previous round, and takes a value of −1 if it does not. Each input is multiplied by the weight shown on arrows from it, and the weighted inputs are then summed at the output node. To that is added the value (positive or negative) of the bias, which might alternatively be thought of as a third weight with a constant input of 1. If the total at the output node is greater than 0, we take our output to be +1, and the individual opens its mouth, for example; if the weighted total is less than 0, we take our output to be −1, and the individual keeps its mouth closed. Here as before an element of noise is built in: in a random 5% of cases each individual will open its mouth regardless of weights and inputs. On the other side of the lobe, individuals also hide in a random 5% of cases. 5

6

Accessible but technically complete introductions to neural nets are unfortunately difficult to find. Perhaps the best is Fausett, 1994. We are grateful to Laurene Fausett for personal correspondence regarding construction of the model. A ‘two-lobe’ structure for signaling, it turns out, has been invented or reinvented numerous times in the literature. See de Saussure 1916; MacLennan 1991; Oliphant & Batali 1997; Cangelosi & Parisi 1998; Nowak, Plotkin & Krakauer, 1999; Nowak, Plotkin, and Jansen 2000.

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Fig. 3. Perceptron structure of each agent embedded in the spatialized environment of wandering food sources and predators with partial training on the behavior of successful neighbors

We code our behavioral strategies in terms of the outputs they give for possible pairs of inputs. Figure 4 shows the possible inputs at ‘hear sound 1’ and ‘hear sound 2’ for the left lobe of the net, with samples of output pairs for ‘open mouth’ and ‘hide’ for these pairs of inputs. The left-lobe behavior of a given strategy can thus be coded as a series of 8 binary digits. With a similar pattern of behavioral coding for the right lobe, we can encode the entire behavior of a net with 16 binary digits. The sample space consists of 38,416 strategies in this sample space, but there are only 2 that qualify as ‘perfect communicators’.

Fig. 4. Possible pairs of inputs at ‘hear sound 1’ and ‘hear sound 2’ for the left lobe of each agent, with samples of output pairs for ‘open mouth’ and ‘hide.’ This strategy opens its mouth whenever it hears sound 2, and never hides. With a similar coding for the right lobe, the behavior of each net can be coded as 16 binary digits.

We populate our array with neural nets carrying twelve random weights. After each 100 rounds, our individuals look around to see if there are more successful neighbors. If so, they do a partial training on the behavior of their most successful neighbor. We use the standard delta rule as the learning algorithm for our perceptrons. For a set of four random inputs, the cell compares its outputs with those of its target neighbor. At any point at which the behaviour of the

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Fig. 5. Emergence of two dialects of perfect communicators, shown in solid black and white, in a randomized array of perceptrons with partial training on successful neighbors. All other behavioral strategies are coded using shades of gray for backgrounds and central dots. Centuries 1, 10, 50, 100, 200 and 300 shown. A full animation of the development can be seen at http://www.pgrim.org/pragmatics

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training cell differs from its target, we nudge each of the responsible weights and biases one unit positively or negatively. Within the limits of our value scale, use of bipolar values for target and input allow us to calculate this simply as wnew = wold + (target × input) and bias new = bias old + target. Our training run consists of only four random sets of inputs, with no provision against duplication. Training will thus clearly be partial: only four sets of inputs are sampled, rather than the full 16 possible, and indeed the same set may be sampled repeatedly. The learning algorithm is applied using each set of inputs only once, moreover, leaving no guarantee that each weight will be shifted enough to make the behavioural difference that would be observable in a complete training. Partial training is quite deliberately built into the model in order to allow numerical combinations and behavioural strategies to emerge from training which might not previously have existed in either teacher or learner, thereby allowing a wider exploration of the sample space of possible strategies (for details see Grim, St. Denis, Kokalis, 2002 and Grim, Kokalis, Alai-Tafti, Kilb, & St. Denis, 2004). We start, then, with an array of perceptrons with randomized weights and biases. Figure 5 shows the evolution of such a randomized array over the course of 300 generations (animation at http://www.pgrim.org/pragmatics). Here

Fig. 6. Emergence of two dialects of perfect communicators in an array of perceptrons through partial training on successful neighbors in a spatialized environment of wandering food sources and predators

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‘perfect communicators’ are coded in pure black and white, though it should be noted that they appear here entirely by partial training. The initial display contains no ‘perfect communicators’ at all: they emerge by the mechanism of partial training on successful neighbors, spreading by the same means. Figure 6 graphs the result.7 The primary lesson of the work in communication is that simple forms of signaling can emerge on the basis of entirely individual gains, geared solely to environmental success, in distributed networks of local interaction. If these simple forms of communication count as ‘semantics’, what this shows is that quite minimal conditions are sufficient for the emergence of simple semantics.

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The Emergence of Pragmatics

Can the reach of this kind of modeling be extended beyond semantics? In what follows the same tools are applied to aspects of pragmatics in an attempt to understand the emergence of the Gricean maxims. Here as in the semantic case we wanted the information our agents deal with to be real information, rather than simply some arbitrary fitness function. In line with the studies in semantics, we wanted the pragmatics of communication to be contextually situated—realistically tied to individual success in the environment. Here we return to a two-dimensional lattice, but with an array of agents who have already fixed on a signaling system—an array of agents for whom a system of communication is already in place. The environment for these studies is not one of wandering food sources and predators, but it does have an essentially spatialized character. This environment, like that of the semantic studies, is an environment of random events of local significance. Here events come in three colors, playing across the array— these are rains of manna, perhaps, or threatening storms, or clouds of locusts. The important thing for each individual is that it act appropriately with regard to the environmental events around it. If an agent finds itself in a magenta cloud (portrayed as a shade of gray in Figure 7, but visible in color at the url for this chapter), it will gain points by acting in an appropriately magenta way. It will lose points if it does not. If an agent finds itself in a bright red cloud, it is red behavior that is advantageous and failing to act in an appropriately red way that will lose points. Color acting is by no means free, however—one cannot hedge one’s bets by always acting magenta, red, and yellow simultaneously. If an agent acts red when there is no red cloud it loses points as well. If all agents could detect their own environmental contingencies at all times, the model would be much less interesting—everyone would do well. In our model, however, agents cannot always tell what is happening; only a random 10% of the agents on each round are able to detect their current environment. The others, for that round, are blind. 7

In a second series of studies we used more complex neural nets, capable of handling the full range of Booleans, with partial training by backpropagation. The results are just as strong. See Grim, St. Denis, Kokalis, 2002 and Grim, Kokalis, Alai-Tafti, Kilb & St. Denis, 2004.

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Fig. 7. The environment of random events of local significance used in the pragmatics studies

Our assumption, however, is that a system of communication is already in place. Our agents already speak ‘red’, ‘yellow’, and ‘magenta’, in the following sense: They can give ‘red’, ‘yellow’, and ‘magenta’ signals heard by their immediate neighbors. If an agent receives a ‘red’ signal from a neighbor, he acts red; if he receives a ‘magenta’ signal, he acts magenta, and so forth. If you as an agent happen to be in the 10% that can see your immediate environment on a given round, then, you have all the information you need to do well. If you have a communicating neighbor who is in that 10%, you derive the same benefit. It is here that the Gricean maxims come in. Suppose that you as an agent have an immediate neighbor who is among the 10% who can see what’s happening. We’ll call him ‘Seymour’. Because a system of communication is already in play between you and Seymour, and because Seymour can see what’s happening, you are going to do fine. You are going to do fine, that is, as long as he’s following the appropriate pragmatic maxims: You will have the information you need on the condition that Seymour observes something like a maxim of quantity: as long as he gives you enough information. Agents start out with a set probability of transferring important information—a probability, if they see ‘red’, of signaling red to their neighbors. We labeled our probability intervals 1 to 10, corresponding to 0.1 intervals between 0 and 1.

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You will have the information you need on the condition that Seymour observes a maxim of quality: as long as he is telling you the truth. Agents start out with a probability interval between 1 and 10 of sending a correct message if they send a message at all: a probability of sending ‘red’ in a red environment rather than ‘magenta’ or ‘yellow’, for example. You will have the information you need on the condition that Seymour observes a maxim of relation: on the condition that he is giving you relevant information. I will detail the modeling of relevance in a moment. The general question at issue is whether pragmatic behavior in accord with Gricean maxims will emerge in a communicative network of this kind. Here as in the semantic case, it should be emphasized, the model works entirely in terms of (1) individual gains and losses— gains from acting appropriately in one’s environment and avoiding inappropriate action, and (2) local action in the distributed network itself—the local action of sending and receiving signals with immediate neighbors and of updating on the behavior of successful neighbors. In the pragmatic studies that follow, our behavioral updating is simply by imitation. One of the lessons from the semantic studies seems to be that it is patterns of mutual advantage in an environment that are of importance, regardless of how agents learn from their neighbors. We expect much the same to hold in the case of pragmatics, but more complicated mechanisms of localized genetic algorithm and neural net training are left to further work. In the following sections communicative behavior in accord with each of the maxims is developed successively; we then show that these behaviors can coevolve as well. 4.1

Quantity

We begin with the pragmatics of quantity. What our model gives us is an environment of changing but spatialized events. On each round a random 10% of agents can detect those events. All individuals in the array are primed with appropriate signals—‘red’ for a red environment, ‘yellow’ for a yellow, and the like—but our agents start with different probabilities of signaling at all. Some transfer information whenever they have it. Some signal with only 60% probability, or 10%, or never signal at all. On each round, all agents gain and lose points on an individual basis. After 100 rounds, they look to see if any neighbor has gained more points— individual points gained on the basis of information received, of course. If so, they imitate the quantity probability in sending of their most successful neighbor—the probability of that neighbor signaling to others. What happens if we start with an array of agents with randomized ‘quantities’ ? In Figure 8 we use the darkest shades to code those with the least probability of signaling, using the brightest shades to code those with the highest probability (in the url, coding is done with darker and lighter shades of blue). Over the course of a few hundred generations the array comes to be increasingly dominated by those following a Gricean maxim of quantity: by those with the highest probability of signaling. Here signaling does not have to be free: the results hold even with a cost for signaling. Figure 9 shows a graph for a typical run.

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Fig. 8. The Emergence of Quantity. Agents with lower probabilities of signaling are coded in darker shades. Evolution to pragmatic maxim of quantity by imitation of successful neighbors over 100 generations. Initial randomization and generations 5, 10, and 100 shown.

4.2

Quality

In order to run the model for quality we assume that quantity is already in place: all those who detect a particular environmental condition signal it to their neighbors. They may or may not signal accurately, however. Our agents come with a probability between 1 and 10 of accurate signaling, otherwise sending a decidedly incorrect signal—‘red’ instead of ‘yellow’, for example, or ‘yellow’ instead of ‘magenta’. We start with a randomization in terms of quality. Those with the lowest quality are coded with the darkest color; those with the highest are coded with the brightest (Figure 10; in the url, coding is done with darker and lighter shades of green). Over the course of 100 generations the array comes to be dominated by those with following a Gricean maxim of quality: the population converges on truth-telling (Figure 11).

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Fig. 9. The Emergence of Quantity. Percentages of population shown over 200 generations.

Fig. 10. The Emergence of Quality. Agents with lower probabilities of accuracy in signaling are coded in darker shades. Evolution to pragmatic maxim of quantity by imitation of successful neighbors over 100 generations. Generations 1, 5, 10 and 90 shown.

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Fig. 11. The Emergence of Quality. Percentages of population shown over 100 generations.

4.3

Relevance

Information regarding environmental conditions is relevant, on this model, to those who are actually in an event environment. Environments have fringes, however, and agents come with a ‘degree of relevance’ in their reports (Figure 12). Some agents signal precisely when an event is happening in their immediate neighborhood. Others report when something is happening in a slightly larger area, and which therefore has a lower probability of actually occurring where they are. Others report on a still wider area, offering information that is still less relevant.

Fig. 12. Relevance neighborhoods

Here we assume both quantity and quality to be in place: reports are forthcoming from all agents, truthful as to the character of the event reported, but of varying relevance. We start with an array randomized as to relevance, coloring those with least relevance in darker shades, those with most relevance in brighter shades (in the url coding is done in shades of red). The result, as shown

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Fig. 13. The Emergence of Relevance. Agents with lower relevance in signaling are coded in darker shades. Evolution to pragmatic maxim of relevance by imitation of successful neighbors over 100 generations. Generations 1, 5, 10 and 100 shown.

Fig. 14. The Emergence of Relevance. Percentages of population shown over 100 generations.

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in Figures 13 and 14, is convergence to the highest degree of relevance across the array. The population fixates on maximized relevance. 4.4

Co-emergence of the Maxims

What if we combine all three of these pragmatic parameters? Here we start with an array randomized for all pragmatic virtues: randomized for quality, quantity, and relevance. Shadings are combined: black or near-black indicates a combination of poor pragmatic values, a light grey indicates an individual virtuous in terms of all three pragmatic maxims. In the animation accessible at http://www.pgrim.org/pragmatics, we have simply combined the blue, green, and red codings of the pragmatic virtues above. On each turn, each agent gains and loses points by acting on the information he receives. But of course the quality of that may information may vary—he may not receive important information from a neighbor, he may receive incorrect information, or he may receive irrelevant information. At the end of 100 rounds, if a neighbor has garnered more points from acting on received information in the environment, each agent updates on just one aspect of that neighbor’s pragmatic character in sending information: He randomly adopts the neighbor’s quality rating, quantity rating, or relevance rating. That means, of course, that he may update on the wrong thing—the neighbor’s success may be due to something other than the trait he imitates. The fact that only one character will change each time, and that they are all interacting, makes the result significantly slower in this simulation. Nonetheless, over 600 generations or so, convergence is to the highest value in all pragmatic parameters (Figures 15 and 16). In the models for pragmatics as in those for signaling and cooperation, it should be emphasized, it is local spatialized action that is the key. In correlate work in evolutionary game theory, using assumptions of global interaction, Nicholas Asher was able to produce pragmatic conventions of sincerity only with very strong assumptions regarding the immediate advantage to a speaker of telling truth (Asher, Sher & Williams 2002; private correspondence). In the dynamics of localized action and imitation, on the other hand, a full range of pragmatic conventions emerge in terms of environmentally significant events, assuming no immediate advantage to the speaker, and despite signaling costs. Although simulations should never be treated as conclusive, they can be wonderfully suggestive. What these results suggest is that fundamental phenomena of pragmatics emerge spontaneously in communicative networks. Those fundamental phenomena don’t need to be added, they don’t need to be cultivated, and they don’t need to be assumed. In any informational network of a certain sort—one in which individuals are striving merely for individual informational advantage—behavior in accord with Gricean maxims of quality, quantity, and relation will come for free.

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Fig. 15. Co-Emergence of the Maxims together over 600 generations. Quality, quantity and relevance are coded from dark to light in three different shades. Initial randomization and generations 10, 100, and 600 shown. A better color version can be seen at http://www.pgrim.org/pragmatics

Fig. 16. Co-Emergence Quantity, Quality, and Relation over 600 generations

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These results seem to go some way in fulfilling what Grice himself wanted: “. . . I would like to be able to show that observance of the Cooperative Principle and maxims is reasonable (rational) along the following lines: that anyone who cares about the goals that are central to conversation/communication (such as giving and receiving information, influencing and being influenced by others) must be expected to have an interest, given suitable circumstances, in participation in talk exchanges that will be profitable only on the assumption that they are conducted in general accordance with the Cooperative Principle and the maxims. Whether any such conclusion can be reached, I am uncertain. . . ” (Grice 1989, 28-29) On the other hand, there are aspects of the simulational work that run directly counter to Grice. Grice’s approach to meaning is heavily intentional, and his work in pragmatics follows suit. Following Grice’s lead, a great deal of good and solid work in pragmatics assumes highly reflective and highly cognitive agents— agents capable of identifying the limits of literal meaning, capable of logical deduction, and capable of building cognitive models of their interlocutors. What these simulations suggest is that something much simpler and much less cognitive may be going on far beneath that level. The agents in these simulations are too simple to form or recognize intentions. They have no explicit capacities for logical inference, and are far too simple to build cognitive models of their interlocutors. Despite that, something develops in networks of those simple agents that looks a lot like signaling. Behaviors develop in networks of those simple agents that also seem to be in accord with the Gricean maxims. With more complicated agents, of course, these same dynamics may play out in more complicated forms involving intentions, logical inference, and cognitive modeling. But what the simulations suggest is that cognitive complexity of that sort isn’t strictly necessary: information maximization in communication networks produces fundamental phenomena of pragmatics even with far simpler cognitive agents. These results also offer some suggestions regarding the conceptual categories in play. Grice followed Kant in speaking of ‘maxims’. But the term ‘maxims’ itself may suggest too high a cognitive level. Grice also followed Kant in dividing his maxims into quantity, quality, relation and manner, but even Grice himself seemed to recognize the artificiality of these categories. There are undoubtedly important distinctions to be drawn between forms of linguistic convention, but the lines should perhaps not be drawn where Grice drew them. In terms of dynamics within information systems, even the distinction between semantics and pragmatics may not be as hard and fast as it has sometimes been assumed to be. The background work summarized in sections 1 and 2 concerns the emergence of signaling systems, read as semantic. The newer work outlined above assumes a semantic system in place, and reads further results as an emergence of pragmatics. What if we combined the simulations; what if we did the two together? We

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haven’t yet done that work, but my prediction is that the results will be much the same. My prediction is that information maximization will emerge both in terms of coordinated use of signals and in pragmatics of appropriate, relevant, and accurate signaling. Would that be the emergence of two things simultaneously—semantics and pragmatics? One could look at it that way. But I also think that one could take the result as an indication that there is a single core phenomenon at issue, of which semantics and pragmatics are somewhat artificially separated ‘aspects’. The core phenomenon, and the engine that drives both, is environmental information maximization in communication networks.

5

Pragmatic Implicature and Inference

What the simulational results offer so far is the emergence of something like Gricean pragmatic conventions. Assuming a semantics of basic signals, the dynamics of local interaction, self-advantage, and imitation of successful neighbors is sufficient to produce conventions of quality, quantity, and relevance. But there is something important that this does not yet give us. It does not yet give us Gricean inference on the part of the hearer—reliance on conventions in play in order to draw particular inferences beyond what is said. It also does not yet give us Gricean implicature on the part of the speaker—the use of conventions in play in order to encourage particular inferences beyond what is said. Can the modeling techniques outlined here take us that next step? The answer appears to be ‘Yes and No,’ but an interesting ‘Yes and No’. The reason for the ‘No’ answer is that most Gricean inferences and implicatures are characterized in highly cognitive terms. Some of Grice’s examples seem to depend crucially on cognitive modeling of the other agent—an aspect rightly emphasized by Asher and his collaborators (Asher, Sher, & Williams 2002; Asher & Lascarides 2003). Some of the examples depend on an agent’s explicit recognition of the limits of literal meaning, or of conventions as conventions, or explicit recognition of patterns of logical entailment. Can the simulation techniques above be used to model those? No, because we are working with agents at far too low a cognitive level. In order to extend network techniques to most cases of Gricean inference and implicature we would need networks of more complex agents. Gricean inference and implicatures are a mixed bag, however, and there are some phenomena in at least the general area that do show up even in simulations of agents this simple. One of those phenomena is a close relative of scalar inference and implicature—the inference and implicature from ‘some students came to the party’ to ‘not all came’, for example, or ‘there are five apples in the basket’ to ‘there aren’t seven.’ What is at issue in such cases is a particular kind of coordination between speaker and hearer. The speaker’s ‘five applies’ implies ‘not more.’ He doesn’t say ‘there are five apples in the basket’ when there are ten, even though it’s true that there are five apples in the basket when there are ten. The hearer, on the

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other end of the coordination, hears ‘five apples’ and infers ‘and not more’. In simplest terms, the speaker doesn’t use a scalar lower than the highest justified. He doesn’t say ‘three apples’ when there are five, for example. The hearer, on his side, doesn’t think ‘he said five apples, so there might be ten;’ he doesn’t act on a scalar higher than that conveyed. Whether or not it fully qualifies as implicature, this kind of coordination between speaker and hearer can emerge even with agents as simple as those used here. That is the ‘Yes’ of the ‘yes and no’. This aspect of scalar implicature seems to be simpler than many of the things in the Gricean mixed bag; interestingly, it was also scalar implicature that Gerhard J¨ ager explored using the very different (and non-spatialized) ‘best response’ dynamic (J¨ ager 2007). A further simulation shows the co-emergence of coordinated speaker and hearer behavior along the lines of scalar implicature. The environment is again one of randomized spatialized events, but the events differ in severity: there are storms of grades 1, 2, 3, and 4, for example. If you as an agent are prepared for a storm of grade 2, you are prepared for a storm of either grade 1 or 2, but you are still open to damage if hit with a 3 or 4. If you are prepared for a storm grade 3 and you only get hit with a 1, on the other hand, you have wasted precious resources—the effort expended in needless preparation for levels 2 and 3. The ideal in such an environment is to act in a way precisely appropriate to the storm: to act 1 and 2 in a 2 environment; 1, 2 and 3 in a 3 environment. In this simulation you as an agent get a positive point for every degree you are correct about—a positive point for both 1 and 2 if you act ‘2’ in a two environment, for example. You get a negative point for every degree you are wrong about—a negative point for acting 3 if the storm is only a 2, for example. If the storm is a 4 and you only act 1, you get a positive point for your ‘1’ but negative points for the 2, 3, and 4 that you missed. Here as before, only 10% of the population on any round can directly observe the relevant environment—hence the need for communication. In modeling the emergence of pragmatic conventions, we started with an established semantics for red, magenta, and the like. Individuals who heard ‘red’ automatically acted red. Here we start with an established scalar semantics: If you hear ‘2’ you act both 2 and 1. If you hear 3 you will act 3, 2, and 1. We also start with established pragmatic conventions of the forms evolved above. Speakers who observe a storm always send a signal, they send a signal regarding the immediate environment, and—most important for our purposes—they never send a signal that isn’t true. Those specifications still leave something important open—something about scalars. Our agents are both speakers and hearers. As speakers they never speak anything but the truth, but we start with an initial population in which speakers may say something that is true but does not convey the full scalar height. Storms of degree 4 are also storms of degree 3, 2, and 1. As speakers, therefore, our agents start with a probability interval between 1 and 10 of giving a signal less than the full observed scalar. As hearers, our agents always act 1 and 2 on hearing the signal ‘2’. But they may also act on a scalar higher than that heard: they

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Fig. 17. Emergence of scalar coordination between speakers and hearers over 200 generations. Speaker’s tendency to understate scalars is in darker green, hearer’s tendency to act on higher scalar than announced is in darker blue, combined here as darker gray. Generations 2, 11, 50 and 200 shown.

Fig. 18. Emergence of scalar coordination between speakers and hearers over 350 generations

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may act not only 1 and 2 but 3, for example. As hearers, our agents start with a probability interval between 1 and 10 of acting not only on what they hear but on a scalar higher than that they hear. In our initial array of randomized probabilities, then, a scalar signal does not necessarily indicate that the truth isn’t higher. A heard scalar doesn’t entail that you shouldn’t act higher. There is as yet no scalar coordination between speakers and hearers. Indeed even within each agent there need be no coordination between speaking behavior and hearing behavior. We start with a situation in which the coordination characteristic of scalar implicature and inference doesn’t yet exist. Our agents code speaker probability in one color scale (blue, in the color version at the url for this chapter), with lighter shades indicating a lower probability of signaling below the observed scalar. Agents’ hearer probabilities are coded in another color scale (green), with lighter shades indicating a lower probability of acting above the scalar message received. As before, all gains are purely individual. After 100 rounds of gains and losses, individuals see if any neighbor has done better in terms of over-all score. If so, they imitate just one of that agent’s features, at random: that agent’s probability of giving a scalar less than the observed reality, or that agent’s probability of acting on a scalar higher than that heard. Figures 17 and 18 show the evolution of the array. In such an environment, speakers could routinely understate the severity of observed storms. Hearers could routinely act above a signal that was likely to be low, following a general strategy of over-reacting to assumed under- statement. But that is not the direction in which this kind of array evolves. It evolves systematically to a coordination between speakers and hearers in which speakers have the lowest probability of sending a signal less than the observed scalar height, and in which hearers have the lowest probability of acting as if what they heard might be less than the full scalar height. As indicated above, it is unclear whether this qualifies as Gricean implicature in the full sense. It does show, however, that extremely simple speakers and hearers in information networks can develop precisely the communicative coordination that is characteristic of scalar implicature and inference. It is interesting to note, by the way, that speakers and hearers do not converge to this coordination at the same rate. It is the hearers who converge first, learning not to over-react. In large part because of the dynamics of payoffs, speakers are significantly slower in learning not to under-state.

6

Conclusion

What simulations suggest is that maximization in spatialized information networks, even networks of very simple agents, is sufficient to produce aspects of semantics, of pragmatics, and even coordinated communication behaviors suggestive of scalar implicature and inference. Some of the core phenomena at issue in semantics, pragmatics, and implicature appear in the fundamental dynamics

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of information maximization, even for agents far below the cognitive level of those that appear in Grice. Even if this captures some of the core phenomena at issue in pragmatics, it cannot be said to capture it all. There are things in the mixed bag of Gricean pragmatics that seem to demand a higher cognitive level—the level of agents that can explicitly recognize logical inferences, that can explicitly reflect on conventions as conventions, or that can cognitively model other agents. Those things will not appear in networks of agents this simple. Those other aspects might appear, however, with a similar dynamics, in spatialized networks of more complex agents. All of the phenomena illustrated here operate in terms of individual information maximization across spatialized arrays of communicative agents. It is to be expected, I think, that information maximization of that kind will use all the tools available to it. Information maximization in arrays of simple agents exploits the full cognitive abilities of those agents, even where—as here—those cognitive abilities are severely limited. Information maximization in arrays of more complex agents will have a wider range of cognitive abilities to exploit, and it is predictable that it will do so. The next step along these lines would therefore be an exploration of the dynamics of maximization in information networks involving more complex agents. That could tell us whether a similar dynamics in the context of more complex agents might be sufficient for further aspects of pragmatics. If particular aspects of language use will appear in networks of agents only where those agents have particular cognitive abilities, such a research trajectory would also give us a new typology and a new understanding of different communicative phenomena: an understanding in terms of the level of networked cognitive abilities that different communicative phenomena require.

References Asher, N., Sher, I., Williams, M.: Game Theoretic Foundations for Gricean Constraints. In: van Rooij, R., Stokhof, M. (eds.) Proceedings of the 13th Amsterdam Colloquium, Institute for Logic, Language and Computation, Universiteit van Amsterdam (2002) Asher, N., Lascarides, A.: Logics of Conversation. Cambridge University Press, Cambridge (2003) Cangelosi, A., Parisi, D.: The emergence of a ‘language’ in an evolving population of neural networks. Connection Science 10, 83–97 (1998) de Saussure, F.: Cours de Linguistique Generale (1916); trans. Harris, R.: 1983 as Course in General Linguistics. Duckworth, Lubbock (1983) Di Chio, C., Di Chio, P.: Evolutionary Models of Language. In: Pietarinen, A.-V. (ed.) Game Theory and Lingusitic Meaning, Elsevier, Amsterdam (2007) Fausett, L.: Fundamentals of Neural Networks. Prentice Hall, Upper Saddle River (1994) Grice, P.: Studies in the Way of Words. Harvard University Press, Cambridge (1989) Grim, P.: The Greater Generosity of the Spatialized Prisoner’s Dilemma. Journal of Theoretical Biology 173, 353–359 (1995)

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Grim, P.: Spatialization and Greater Generosity in the Stochastic Prisoner’s Dilemma. BioSystems 37, 3–17 (1996) Grim, P.: Philosophical Implications of Interaction and Information Networks. In: Evolution, Game Theory & the Social Contract Conference, Beckman Center for the National Academies, University of California, Irvine (March 2009) (2009a) Grim, P.: Network Simulations and Their Philosophical Implications: Models for Semantics, Pragmatics, and Epistemology. In: Models and Simulations, Charlottesville Virginia, vol. 3 (March 2009) (2009b) Grim, P., Mar, G., St. Denis, P.: The Philosophical Computer: Exploratory Essays in Philosophical Computer Modeling. MIT Press, Cambridge (1998) Grim, P., Kokalis, T., Alai-Tafti, A., Kilb, N.: Evolution of Communication in Perfect and Imperfect Worlds. World Futures: The Journal of General Evolution 56, 179–197 (2000) Grim, P., Kokalis, T., Alai-Tafti, A., Kilb, N.: Evolution of Communication with a Spatialized Genetic Algorithm. Evolution of Communication 3, 105–134 (2001) Grim, P., Denis St., P., Kokalis, T.: Learning to Communicate: The Emergence of Signaling in Spatialized Arrays of Neural Nets. Adaptive Behavior 10, 45–70 (2002) Grim, P., Kokalis, T., Alai-Tafti, A., Kilb, N., Denis St., P.: Making Meaning Happen. Journal for Experiemntal and Theoretical Artificial Intelligence 16, 209–244 (2004) Grim, P., Wardach, S., Beltrani, V.: Location, Location, Location: The Importance of Spatialization in Modeling Cooperation and Communication. Interaction Studies: Social Behavior and Communication in Biological and Artificial Systems 7, 43–78 (2006) J¨ ager, G.: Game Dynamics Connects Semantics and Pragmatics. In: Pietarinen, A.-V. (ed.) Game Theory and Lingusitic Meaning. Elsevier, Amsterdam (2007) MacLennan, B.: Synthetic Ethology: an approach to the study of communication. In: Langton, C.G., Taylor, C., Farmer, J.D., Rasmussen, S. (eds.) Artificial Life II, SFI Studies in the Sciences of Complexity, vol. X, pp. 631–655. Addison Wesley, Redwood City (1991) Nowak, M., Plotkin, J., Krakauer, D.: The evolutionary language game. Journal of Theoretical Biology 200, 147–162 (1999) Nowak, M., Plotkin, J., Jansen, V.: The evolution of syntactic communication. Nature 404, 495–498 (2000) Nowak, M., Sigmund, K.: The evolution of stochastic strategies in the prisoner’s dilemma. Acta Applicandae Mathematicae 20, 247–265 (1990) Nowak, M., Sigmund, K.: Tit for tat in heterogeneous populations. Nature 355, 250–252 (1992) Oliphant, M., Batali, J.: Learning and the Emergence of Coordinated Communication. Center for Research on Language Newsletter 11(1) (1997) Parikh, P.: The Use of Language. CSLI Publications, Stanford (2001) Parikh, P.: Pragmatics and Games of Partial Information. In: Benz, A., J¨ ager, G., van Rooij, R. (eds.) Game Theory and Pragmatics. Palgrave Macmillan, New York (2006) Parikh, P., Clark, R.: An Introduction to Equilibrium Semantics for Natural Language. In: Pietarinen, A.-V. (ed.) Game Theory and Lingusitic Meaning, Elsevier, Amsterdam (2007)

Signaling Games Dynamics of Evolution and Learning Simon M. Huttegger1 and Kevin J.S. Zollman2 1

1

Konrad Lorenz Institute for Evolution and Cognition Research 2 Department of Philosophy, Carnegie Mellon University

Introduction

“Let us go down, and there confound their language, that they may not understand one another’s speech” (Genesis 11:1). The state of language confusion described in this passage may be understood as a state of maximal heterogeneity: every possible language is present in a population. It may also be viewed as a state of homogeneity, however; presumably, each possible language is spoken by a very small number of persons, inducing a uniform distribution over the set of languages. Should we expect individuals to stay at such a symmetric state? Or will they rather agree on one language, thereby breaking the symmetry of initial confusion (Skyrms, 1996)? These questions are basic for the origin of language. When individuals cannot communicate to a sufficiently high degree, how can they decide on signaling conventions? In the philosophical literature, such problems were formulated by Quine (1936), although they have already been considered before Quine. Similar questions have also sparkled interest in linguistics (Steels, 2001; J¨ ager and van Rooij, 2007) and in biology (e.g. communication at the microbiological level or animal signals). One would be interested to know if coherent signaling evolves under simplified conditions. Perhaps the most simple model one can think about was introduced by David Lewis (see Lewis (1969)). By using some concepts from game theory, Lewis introduced signaling games as a simplified setting to study the emergence of language conventions. On a larger scale, it should be emphasized that the evolution of language is an extremely complex issue where many more factors are involved than are captured in a signaling game. We think that studying very simplified models is nonetheless useful. Both experimental and theoretical approaches are confronted with the complexity of the problem of language evolution (Sz´ am´ado and Szathm´ ary, 2006). This makes results from simple mathematical models particularly important. Such simple models sharpen our intuitions as to what might be important features to look for in more complex models. Models like that of signaling games provide a general framework for studying the emergence of communication; making signaling games more complex is a result of giving them more structure. Properties of signaling games will thus reappear at a more structured level. Moreover, simple mathematical models of signaling provide insights into A. Benz et al. (Eds.): Language, Games, and Evolution, LNAI 6207, pp. 160–176, 2010. c Springer-Verlag Berlin Heidelberg 2010 

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specific processes that play an important role in language evolution. And, lastly, simple and tractable models allow us to identify key components of particular processes. In this paper we report several results on the dynamics of Lewis signaling games. A dynamical view of signaling games is indispensable since we are interested in the process of the emergence of communication. We spend a considerable part of this paper on the evolutionary dynamics of signaling games as given by the replicator equations and a perturbation thereof. These two models should be viewed as a baseline case for comparison with other studies. Accordingly, we shall be especially interested in finding differences between these two baseline cases and between them and more sophisticated dynamical models. These include structurally stable games, finite population models, and a number of models of learning in signaling games. We shall argue that the differences between all these models are such that the baseline models do not capture all possible dynamical behaviors. On the other hand, features like persistent and non-decreasing stochastic perturbations of evolutionary or learning dynamics appear to have qualitatively similar effects in a wide range of models. The interplay of various evolutionary and learning models that we describe in this paper may prove useful in studying more complex models of language evolution or other evolutionary problems.

2

Lewis Signaling Games

In his book Convention, David Lewis describes a situation for the emergence of conventional signaling. One individual, the sender, has some private information about the world and also has at her disposal a set of signals. Another individual, the receiver, observes the signal, but not the state, and takes some action. Each state has an appropriate action, and both parties are interested in the receiver taking the appropriate action given the state. Because of common interest, both parties are interested in coordinating on a convention to associate each state with a signal and each signal with the appropriate act. While there are many ways to specify this game, we will consider the easy circumstance where there are n states, n acts, and n signals. We may thus represent sender strategies and receiver strategies by n × n matrices M having exactly one 1 in each row, the other entries being 0. If M is a sender matrix, then mij = 1 means that the sender chooses signal j after having observed state i; if M is a receiver matrix, it means that the receiver chooses action j in response to signal i. Players may also choose what to do in response to the state or signal at random. We can expand the set of admissible matrices to those where for each entry 0 ≤ mij ≤ 1, and each row sums to 1.1 If a sender and receiver are matched to play the game they both gain a payoff of 1 if the receiver chooses the right action given the state; they gain 0 otherwise. 1

Representing randomization (or mixed strategies) in this way can either correspond to a single player intentionally randomizing, or alternatively to a population of players that don’t randomize but whose proportions are represented by the probabilities in the matrix.

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If each state is equally likely the sender (with strategy P ) and receiver (with strategy Q) will expect to get the following payoff: 1 π(P, Q) = pij qji . (1) n i,j The payoff function given above can of course be modified. The states need not be weighed equally or the interests of the players may not coincide completely. Such modifications lead to interesting games, and we will discuss the first one briefly below. It is easy to show that one-to-one strategies are of particular importance. A sender strategy P is one-to-one if no two states are mapped to the same signal, i.e. if the matrix P is a permutation matrix. Similarly, a receiver strategy is one-to-one if Q is a permutation matrix. A simple computation shows that if P is a permutation matrix and if Q is the transpose of P (Q = P T or qij = pji ), then π(P, Q) = 1, which is the maximal payoff. Such strategy pairs (P, Q) were, for obvious reasons, termed signaling systems by Lewis. Signaling systems can be viewed as simple languages. They are characterized by the property of yielding a maximum payoff to the players; i.e. no other strategy combination earns a payoff of 1. They are also the only strict Nash equilibria of signaling games. There is, however, a number of non-strict Nash equilibria which are part of Nash equilibrium components. If n = 3, one such Nash equilibrium component is given by ⎛ ⎞ ⎛ ⎞ 10 0 μ1−μ0 P = ⎝1 0 0 ⎠ , Q = ⎝ 0 0 1⎠ (2) 0 λ 1−λ 0 0 1 where 0 ≤ λ, μ ≤ 1 (Trapa and Nowak, 2000). At (P, Q), the players are always able to coordinate state 3 and act 3, but if state 1 or 2 occur they do not always achieve coordination. Here state 1 and 2 are “pooled” onto signal 1 and state 3 is communicated using two different signals (signals 2 and 3). As a result these equilibria are called partial pooling equilibria. There are also total pooling Nash equilibria. In these equilibria the sender sends the same signal regardless of state and the receiver takes the same action regardless of signal. Partial pooling equilibria like (P, Q) turn out to be particularly important for the emergence of communication in signaling games. Since signaling games have an uncountable number of Nash equilibria, the equilibrium selection problem becomes particularly pressing. Equilibrium refinement concepts like evolutionarily stable strategies and neutrally stable strategies exclude Nash equilibria which are not stable from an evolutionary perspective (Maynard Smith, 1982). In signaling games, signaling systems are the only evolutionarily stable strategies (W¨arneryd, 1991). But Nash equilibria such as (P, Q) are neutrally stable. This means that natural selection will not move a population away from a signaling system. (P, Q) is also stable relative to natural selection, but drift may cause a population to move away from a neutrally stable state. Analysis of signaling games in terms of other equilibrium concepts can also be given (Blume, 1994), but we think that an analysis from an evolutionary

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perspective is more revealing as to the problem of the emergence of communication. In this case, pinning down the evolutionarily and neutrally stable states does not get us very far. We are still confronted with a large number of possible evolutionary outcomes, and we do not know whether evolution leads to a state of communication. Moreover, concepts like that of an evolutionarily stable strategy appear to have no straightforward connection to models of learning in games. For these reasons, we think without a detailed analysis of various dynamic models our understanding of these games is incomplete at best and misleading at worst.

3

Evolutionary Dynamics of Signaling Games

The basic model of evolutionary game theory is given by the replicator dynamics (Taylor and Jonker, 1978; Schuster and Sigmund, 1983; Hofbauer and Sigmund, 1998). We imagine a population of individuals partitioned into several types. Each type corresponds to a strategy of the underlying game. For signaling games, a type may be characterized by a sender part P and a receiver part Q if we would like to study the evolution of communication within one population. Another possibility consists in analyzing a two-population model, with one sender population and one receiver population. A type in the sender population will, in this case, correspond to a sender strategy P , and a type in the receiver population to a receiver strategy Q.2 The replicator dynamics relates the growth rate of each type of individual to its expected payoff with respect to the average payoff of the population: types with above-average performance increase in relative frequency, while types with below-average performance decrease. In a biological context, payoffs can be interpreted as fitnesses. Thus, we sometimes speak of fitness instead of payoff, or average fitness instead of average payoff. If xi is the frequency of type i, x = (x1 , . . . , xn ) is the state of the population (being a probability vector) and u(xi , x) and u(x, x) are the payoffs to type i and the average payoff in the population at state x, respectively, then the replicator dynamics is given by x˙ i = xi (u(xi , x) − u(x, x)). (3) x˙ i denotes the time-derivative of xi . Notice that equation (3) is one possibility to formalize the dependency of a type’s growth rate to its performance relative to the population average; there are also others collectively known as payoff-monotonic dynamics. A system similar to (3) can be formulated for a two-population model (see Hofbauer and Sigmund (1998), Sections 10 and 11; in the context of signaling games, see Huttegger (2007b)). If a population’s initial condition is given by x, then (3) defines a unique orbit or solution curve φ(t) for t ∈ R with φ(0) = x. φ describes the evolution of the population in the state space of relative frequencies. If x˙ i = 0 for all i, then x is called a rest point of (3). This means that whenever x is the initial condition of a population, it will stay at x for all future times. A 2

Here will restrict ourselves to pure strategies – those where every entry is either 0 or 1.

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rest point x is called Liapunov stable if all nearby solutions stay close to x for all future times. A rest point x is called unstable if it is not stable. A rest point x is asymptotically stable if it is Liapunov stable and if there exists a neighborhood U of x such that φ(t) converges to x as t → ∞ whenever φ(0) ∈ U . The same notions can be defined for a set of points S instead of a rest point x as well. Moreover, we will say that almost all points converge to some set of points S under (3) if the set of points that do not converge to S has Lebesgue measure zero in the state space of relative frequencies. 3.1

Replicator Dynamics

Skyrms (1996) simulated the replicator dynamics of a binary Lewis signaling game, and Skyrms (2000) provides a mathematical analysis of a simplified binary Lewis signaling game, which does not include all 16 types (note that this is already a quite formidable number for a mathematical treatment of the dynamics). In simulations, population frequencies always converged to one of the two signaling systems. The same result was shown to hold analytically in the Lewis mini-game. These results suggested the optimistic conjecture that in every Lewis signaling game almost all initial population states will converge to one of the signaling games under the dynamics (3). Huttegger (2007a) and Pawlowitsch (2008) have shown independently that this is in general not the case, Pawlowitsch by utilizing connections between neutral stability and the replicator equations, and Huttegger by using techniques from center-manifold theory (Carr, 1981). Let us take a closer look at the dynamical properties of signaling games. Lewis signaling games have interior Nash equilibria. These equilibria represent states where all possible strategies are present. Huttegger (2007a) proves that these states are not stable for any signaling game. Indeed, interior equilibria are linearly unstable for the replicator dynamics (3). This implies that the set of points converging to an interior equilibrium has measure zero. Thus, for almost all initial populations, symmetry gets broken in the minimal sense that not all signaling strategies will survive under the evolutionary dynamics. Signaling systems are strict Nash equilibria of Lewis signaling games; hence they are asymptotically stable states for the replicator dynamics (both for the two-population replicator dynamics and the one-population replicator dynamics of the symmetrized signaling game). Asymptotic stability is a local concept; it does not give us global information about the properties of the dynamical system. In particular, asymptotic stability does not imply global convergence to one of the signaling systems (global in the almost-all-sense). Indeed, for n ≥ 3 it turns out that some of the continua of Nash equilibria that were described in Section 2 form sets that attract nearby points. Consider the connected set of Nash equilibria N given by (2). If we look at the dynamics (3) close to N we see that population frequencies sufficiently close to N converge to some point in N . When we look at the boundary of the set N , however, some of the Nash equilibria become dynamically unstable; i.e. there

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exist population frequencies arbitrarily close to such a Nash equilibrium that tend away from it. This implies that the set N is not asymptotically stable. We cannot find a neighborhood U of N such that any point in U converges to N . But each point x in the interior of N is Liapunov stable. Moreover—and this is the elephant in the kitchen—the interior of N attracts an open set of initial conditions. That is, the set of population frequencies converging to N has non-zero measure. Components of Nash equilibria such as N exist for all signaling games with n ≥ 3. This was shown by Huttegger (2007a) and by Pawlowitsch (2008). Pawlowitsch moreover links the existence of components like N to the concept of neutrally stable strategies, which was introduced by Maynard Smith (1982) as a generalization of evolutionarily stable strategies. Suppose a whole population adopts a certain strategy s of some game. Then s is neutrally stable if s is a Nash equilibrium and if there exists no strategy s that yields a higher payoff when played against itself than s yields when played against s . Thus, neutral stability implies that a strategy is robust against invasion by selection (but it is not robust against drift). Pawlowitsch (2008) finds an elegant characterization of neutrally stable strategies in Lewis signaling games: if P is a sender matrix and Q is a receiver matrix, then (P, Q) is neutrally stable if and only if (i) P or Q has no zero-column and (ii) neither P nor Q has a column with multiple maximal elements λ such that 0 < λ < 1. Thus, a signal can represent more than one event, but then these events cannot be represented by any other signal. Similarly, an event can be linked to more than one signal; in this case, however, the signals cannot be linked to any other event. In terms of the replicator dynamics (3), a neutrally stable strategy is a point in a component of strategies such as N ; i.e., if (P, Q) is neutrally stable and is contained in a component of other neutrally stable strategies, then this component attracts an open set of population frequencies. Whether the reverse statement is also true is an open problem. Signaling games with n = 2 are a special case. In such binary signaling games the existence of a component N that attracts an open set of population frequencies depends on the weights attached to the two events. If both weights are 12 , then no such component exists: almost all solution curves converge to one of the signaling systems. Once the weights are asymmetric, however, there exists a component N . Thus we may conclude that for the replicator dynamics (3) signaling systems do not evolve generically. Numerical simulations show that the size of the basins of attraction of signaling systems is decreasing in n; moreover, it is already nonnegligible for n = 3 (Huttegger et al., 2010). To understand the evolutionary dynamics of signaling games, a complete analysis of the replicator equations (3) is only a first step. The model of evolution as given by (3) can be extended and modified in various directions. Such explorations seem all the more necessary since the situation of having components of Nash equilibria is quite peculiar, as we shall explain now.

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3.2

Selection-Mutation Dynamics

From the point of view of dynamical systems, the continua of rest points corresponding to these Nash equilibrium components are not structurally stable (see Guckenheimer and Holmes 1983 or Kuznetsov 2004).3 Structural stability refers to small perturbations of a system of differential equations like (3) (small relative to the functions constituting the differential equations and their partial derivatives). The system is structurally stable if such small perturbations do not change the qualitative properties of the solution trajectories. The solution trajectories of the original and the perturbed system are topologically equivalent. A system that is not structurally stable is called degenerate. Systems with continua of rest points are always degenerate. This follows from the fact that continua of rest points are associated with zero-eigenvalues of the corresponding Jacobian matrix (the sign of the eigenvalues determines the qualitative nature of the solution trajectories near rest points). Perturbing the system will push the real part of zero-eigenvalues into the positive or the negative reals. This implies that the qualitative nature of the flow will change close to continua of rest points. Depending on the perturbation, the dynamics might change in many different ways. Thus, it is essential to choose a plausible perturbation of the dynamical system. Hofbauer and Huttegger (2007, 2008) argue that the selection-mutation dynamics provides a plausible and (to some extent) tractable perturbation of the replicator equations (3) (for more information on this dynamics see B¨ urger 2000; Hofbauer 1985; Hofbauer and Sigmund 1998; see also Huttegger et al. 2010). One version of the selection-mutation dynamics is given by x˙ i = xi (u(xi , x) − u(x, x)) + ε(1 − mxi ),

(4)

where ε > 0 is a uniform mutation rate and m = n2n is the number of strategies for a signaling game with n signals. The first term on the right-hand side of (4) is the selection term, while the second term describes uniform mutation. The mutation term expresses the fact that a type might change into another type at each point in time, at a rate given by ε. If ε = 0, the selection-mutation dynamics coincides with the replicator dynamics. Hofbauer and Huttegger (2007, 2008) do not study the selection-mutation dynamics (4) directly. They instead focus on the two-population selection-mutation dynamics with a sender population and a receiver population. This enhances the tractability of the model and may be justified by assuming that the roles of sender and receiver are independent. Our remarks below refer to the twopopulation selection-mutation dynamics. There are two general results concerning the selection-mutation dynamics of signaling games. Both are statements about the location of rest points of the selection-mutation dynamics in comparison to the location of rest points 3

Notice that continua of Nash equilibria are generic; i.e., if we perturb payoffs in a way that respects the extensive form of the game, Nash equilibrium components persist (cf. Cressman 2003 and J¨ ager 2008).

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of the replicator dynamics. First, all rest points of the selection-mutation dynamics are close to Nash equilibria of the signaling game. This rules out rest points that are close to rest points of the replicator dynamics which are not Nash equilibria (Hofbauer and Huttegger, 2008). Second, perturbed signaling systems exist, are unique and asymptotically stable. By a perturbed signaling system we mean a rest point of the selection-mutation dynamics close to a signaling system. Note that the proof of its uniqueness is necessary to define a perturbed signaling system properly. For details of the proof and additional remarks concerning rest points of the selection-mutation dynamics in general consult Hofbauer and Huttegger (2008). Unfortunately, no general results are available for the existence and stability properties of possible rest points close to the attracting components of Nash equilibria that we described in the previous sections. Indeed, if N is such a component, then there are no general mathematical statements that would allow us to derive conclusions about the behavior of selection-mutation dynamics close to N . Hofbauer and Huttegger (2007, 2008) analyze the behavior of the selectionmutation dynamics close to N with the help of Taylor expansions in terms of the mutation rates, as well as index or degree theory (Hofbauer and Sigmund, 1998, Section 13.2) and Morse theory (Milnor, 1963). Their results do not give a clear-cut answer to the problem of the evolution of signaling systems. Whether perturbed signaling systems emerge depends the parameters involved, notably the ratio of the mutation rate of the sender population to the mutation rate of the receiver population and the probability distribution over the events. If all events are equiprobable (the distribution has maximum entropy), then communication is most important (Nowak et al., 2002, Box 2). As the entropy (the evenness) of the probability distribution decreases, communication becomes less important; always guessing the most probable event and ignoring signals is more attractive in this than in the equiprobable case. Hence, as the distribution becomes less even, the possibility of ending up in a state with suboptimal communication increases. If the receiver population’s mutation rate is sufficiently lower than the sender population’s mutation rate, then it also becomes more likely to end up in a suboptimal state under the selection-mutation dynamics. This can heuristically be explained by the receivers not being responsive enough to the experiments of the senders. For a precise mathematical formalization of these argument see Hofbauer and Huttegger (2008). It is important to notice that these results are specific for the perturbation (4), which is linear. Alternative perturbations could also include non-linear terms, which might create any finite number of perturbed rest points with all kinds of stability properties. Such alternative perturbations might, however, not have an equally clear empirical interpretation like the one given in (4). 3.3

Structurally Stable Signaling Games

J¨ ager (2008) studies games which he calls structurally stable. Structural stability in J¨ ager’s sense does not refer to perturbations of the dynamics, as in the previous

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subsection, but to perturbations in the payoffs of the players. In particular, he allows for the possibility of an uneven probability distribution for the set of events (like Nowak et al. 2002 and Hofbauer and Huttegger 2007) and requires that different signals incur differential costs. These features lead to a perturbation of the players’ payoffs. This does not destroy the existence of neutrally stable components, however. J¨ager (2008) shows that the replicator dynamics still converges to neutrally stable components of Nash equilibria from an open set of initial conditions. Given this result, it seems necessary to approach the problem of degeneracy in signaling games (i.e. the existence of components of Nash equilibria) from dynamical systems theory, as we outlined in the previous paragraph. 3.4

Finite Population Models

An alternative way to deal with degeneracy in signaling games which we described above is to consider finite population models. Pawlowitsch (2007) studies signaling games under the frequency-dependent Moran process (cf. Nowak et al. 2004). Her results show that selection never favors a strategy replacing a signaling system, whereas it favors some strategy to replace any strategy other than a signaling system (including neutrally stable strategies). It is important to notice that the model of Pawlowitsch also employs a kind of perturbation (given by weak selection). As is argued in Huttegger et al. (2010), a Moran-process without any kind of perturbation does yield qualitatively the same results as the replicator dynamics. Some models of finite populations also involve more population structure than is used in either the replicator dynamics or the Moran process. So called cellular automata models use grid structures where individuals are constrained to interact only with their neighbors. Zollman (2005) considers the 2-state/2signal/2-act signaling game with equiprobable states. He finds that although every individual adopts a signaling system strategy, both types of signaling system strategies persist. On the grid regions form, where individuals are perfectly communicating with those in their region, but are failing with those outside. Without mutation these states are stable, and with mutation they only undergo small persistent changes in the location of the borders. Wagner (2009) extends this model to include 3-state/3-signal/3-act signaling games and also to population structures other than the grid. He finds that population structure of various sorts significantly reduces the probability that either partial or total pooling equilibria will evolve.

4

Learning Models

Unlike population models that usually consider a large population of players playing a game against one another, models of individual learning usually consider two players playing against one another repeatedly. Each player chooses a play for each round by following a rule which uses the past plays and payoffs of the game. These models attempt to capture the process by which individuals come to settle on particular behaviors with one another.

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The literature is replete with different models of individual learning. In analyzing a wide variety of different learning rules scholars are usually interested in one of three questions. First, how little cognitive ability is needed to learn a signaling system? In the replicator dynamic model we found that at least some of the time a biological process, like natural selection, can result in the emergence of language. Can other simple dynamic systems which are implemented at the individual level result in the same outcome? Second, is the replicator dynamics an appropriate approximation for models of individual learning? If individual learning results in similar outcomes, we have some reason to suppose the replicator dynamics offers a good approximation.4 Finally, scholars are interested in determining the relationships between features of the learning rules and their ultimate outcomes. Do all models that have limited memory converge to signaling systems? What about all those that remember the entire history? With respect to the first question, it appears that very little cognitive ability is needed to result in signaling systems. In fact some very simple learning rules perform better than other more complex counterparts. This later fact also shows that no particular mathematical model (like the replicator dynamics) is likely to capture the range of possibilities presented in individual learning. This suggests that the study of learning in games represents an important avenue of research for those interested in the emergence of behavior in games. The last question, regarding the relationship between features of the learning rule and results, is complicated. We will postpone a detailed discussion until the end of this section. In the replicator dynamic models of signaling it is usually supposed that each individual is endowed with a contingency plan over all states or signals. In the one population model every individual had both receiver and sender contingency plans, while in the two population model individuals had only the relevant contingency plan (sender or receiver depending on their population). This model fits well with biological evolution, where individuals’ responses are determined by a heritable biological mechanism. A similar model is less plausible in the case of learning. Suppose that state a occurs and a player sends signal x to a counterpart receiver who acts correctly – both receive a reward. It would be unrealistic to suppose that the reward received would influence the sender’s propensity to send signal y in state b even though it did not occur. But this would often be the case if we modeled individuals as learning on entire strategies (full contingency plans for each state or signal). Instead, much of the learning literature restricts the learning to particular states or signals and models rewards as effecting only the behavior of the individual with regard to that state or signal.5 4

5

Since the replicator dynamics offers a sometimes mathematically simpler model than other learning rules, having it represent an adequate approximation can reduce the amount of analysis substantially. It should not be presumed that a strategy learning model is totally implausible, however. For instance, if I am able to observe many plays of a the game before adopting a new strategy, I might be able to observe contingency plans. Similarly, if I recognize the situation as strategic, I may attempt to formulate reasonable contingency plans and adopt them.

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Minimal Memory

We will begin our investigation by turning to the simplest learning rules, those that remember only the most recent round of play. One such learning rule, Win-stay/Lose-switch, was initially considered in a different context by Robbins (1952),6 and then later applied to in game theoretic situations by Nowak and Sigmund (1993). As its name suggests, players will remain with their most recent strategy when they “win” and switch to another strategy when they “lose.” For general game theoretic situations, much turns on what is classified as a win or loss, but since signaling games feature only two payoffs this need not concern us here. Barrett and Zollman (2009) consider Win-stay/Lose-switch and similar Winstay/Lose-randomize learning rules. They find that Win-stay/Lose-randomize will converge in the limit to perfect signaling both when learning is done on contingency plans and also when learning is done in individual states and signals. Interestingly such a result is not guaranteed for Win-stay/Lose-switch since the forced switch can make players miscoordinate forever. These learning rules require only limited knowledge of the situation and require no sophisticated reasoning. We might imagine a slightly more cognitively complex learning rule where individuals are capable of counterfactual reasoning, but still only consider the previous round. One such learning rule has an individual take the best response to the play of the opponent on the previous round. This requires more knowledge on the part of the player, since she must be capable of calculating what would have happened if she had acted differently.7 So-called “myopic best response” or “Cournot adjustment dynamics” has been considered extensively in the economics literature (cf. Fudenberg and Levine, 1998). In the case of 2-state/2-signal/2-act signaling games this learning rule has the same problem faced by Win-stay/Lose-switch: it can cycle forever. Beyond this fact, little is known about this learning rule and how it compares to the other short-memory learning rules. It is not always appropriate to assume that individuals have only a one period memory. We will now turn to a learning rule which is at the other extreme – it remembers the entire history of play. 4.2

Indefinite Memory

We will again return to considering learning rules which only consider their own actions and payoffs without engaging in counterfactual reasoning. So called Herrnstein reinforcement learning is one such learning rule. It was first introduced in the game theoretic literature by Roth and Erev (1995) and Erev and Roth (1998), but the underlying motivation traces to Herrnstein’s (1970) matching law – that the probability of an individual taking an action will be proportional 6 7

Robbins was considering a class of problems known now as bandit problems (cf. Berry and Fristedt, 1985). In signaling games, this learning rule would also require that the receiver by informed of the state after failure, so that she might calculate the best response.

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to the sum of the rewards accrued from taking that action. Herrnstein’s matching law is instantiated by defining the probability of an action a using the following formula: w  a (5) x wx wa is the total rewards from taking action a and the sum in the denominator is the total rewards for taking all actions over past plays. This function for taking past successes and translating them into current propensities for choosing an action is known as the “linear response rule.” As was done with the replicator dynamics, we will first consider the simplest case, two states, signals, and acts, with equiprobable states. In this case, it has been proven that a separate sender and receiver both employing reinforcement learning on individual actions will converge (almost surely) to signaling systems (Argiento et al., 2007). Unfortunately, the proofs for this case are difficult and generalizations have not been forthcoming. Almost all that is known about other cases is the result of simulation studies. Barrett (2006) found that for signaling games with more signals, states and acts reinforcement learning will often converge to the partial pooling equilibria described above. As the number of states, signals, and acts grow, the proportion that converge to one form of partial pooling or another grow as well, reaching almost 60% for eight state, signal, act games. Barrett did find, however, that those systems always achieved some success at information transfer. He observed no simulation that succeed less than half of the time, and a vast majority achieved relatively high success.8 Skyrms (2010) reports that failures similar to the replicator dynamics are observed when states are not equiprobable. In a two state, signal, act game with unequal state distributions total pooling equilibria are sometimes observed. Similar results are reported by Barrett (2006) regarding unequal state distributions for games with more signals, states, and acts. The story here is interesting. In the replicator dynamics it appears that the introduction of random shocks is sufficient to avoid the pitfalls of partial and total pooling equilibria (at least in some cases). Herrnstein reinforcement learning has persistent randomness, but the magnitude of that randomness decreases over time. Simulation results suggest that this randomness is insufficient to mimic the randomness obtained by the selection-mutation dynamics and thus insufficient to avoid partial pooling equilibria. Akin to Win-stay/Lose-switch, Herrnstein reinforcement does not use information about one’s opponent’s actions or about one’s alternative responses to those actions. One might modify Herrnstein reinforcement learning to consider such a case, where an individual attempts to “learn” the, possibly mixed, strategy of one’s opponent by observing past play.9 One assumes that the proportion of past plays represents an opponent’s strategy and then takes the best response 8 9

For instance in a four state, signal, act game he found that, of those that failed, all approached a success rate of 3/4. The term “learn” may be a bit of a misnomer since, if one is playing against a opponent who is also using this learning rule, there is no stable strategy to learn.

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to that strategy. So called “fictitious play” has been applied in many settings in game theory (cf. Fudenberg and Levine, 1998), but it has not been studied extensively in signaling games. There are also other, more complex learning rules which have likewise not been applied to this game (cf. Young, 2004). There have, however, been several other modifications to Herrnstein reinforcement that have been considered. They all retain the central idea that one’s play is determined only by the rewards one has received in the past and not by strategic considerations like those used in myopic best reply or fictitious play. 4.3

Similar Reinforcement Models

There are many different ways to modify Herrnstein reinforcement in order to introduce larger persistent randomness. Only a few have actually been studied and there has not been anything close to an exhaustive search of the possibilities. One might begin by modifying the way by which propensities are updated. It is usually assumed that the game being studied does not have negative payoffs so that propensities cannot become negative (and thus result in incoherent probabilities). Alternatively, one might allow for negative payoffs but truncate the propensities to remain above zero. Barrett (2006) investigates a collection of models where failure receives a payoff of less than zero and thus results in a “punishment” which decreases the probability of taking that action (rather than keeping it the same). Results of simulations involving different amounts of punishment suggests that this substantially decreases the basins of attraction of partial pooling equilibria and results in more efficient languages, although this depends on the magnitude of the different rewards and punishments. Games with unequal state distributions have not been studied with this model. In addition, Barrett (2006) considers a model where the propensities are subject to random shocks. Shocks are modeled as a number α which is drawn from some distribution with expectation of 1. On every round the propensities are multiplied by α resulting in random perturbations. Barrett finds that these shocks are sufficient to eliminate partial pooling equilibria in signaling games with more than two states, signals, and acts. Again, however, unequal state distributions have not been studied. Rather than modifying the updating rules, one can also modify the response rule. Skyrms (2010) considers a model where the probabilities are determined by a logistic (or exponential) response rule: eλwa  λw x xe

(6)

This exponential response rule alters the way that propensities are translated into probabilities over actions. The structure of this rule allows for small differences in propensities to have very little influence while larger differences have more significant influence. Skyrms (2010) finds that for reasonably small values of λ learners almost always learn to signal both for unequal state distributions and larger number of states, signals and acts. This occurs largely because, when

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λ is small, initial play is more random and later play is more deterministic than Herrnstein reinforcement learning, resulting in more early exploration. 4.4

More Radical Departures

The modifications considered so far preserved the underlying idea that weights are updated by addition (and potentially perturbed). Barrett and Zollman (2009) consider a model where the weights are updated by a weighted average instead of addition and propensities are calculated according to the exponential response in (6). They find that for particular parameter values individuals learn to optimally signal in games with three states, signals, and acts. This occurs largely because this learning rule approximates Win-stay/Lose-randomize by continually exploring until it succeeds and then locks into the strategy that produces that success. Barrett and Zollman (2009) also consider a yet more radical departure from Herrnstein reinforcement learning, the Adjustable Reference Point (ARP) learning model. ARP was first developed to explain human behavior in games by Bereby-Meyer and Erev (1998). We will avoid specifying the model here, but it is a reinforcement like model meant to capture four features absent in Herrnstein reinforcement: (1) what counts as success and failure can evolve based on past experience, (2) how one responds to “successes” and “failures” can differ, (3) more distant rewards and punishments have less effect than more recent ones, and (4) rewards in one domain can have effects on other domains as well. Barrett and Zollman find that the ARP model significantly outperforms Herrnstein reinforcement in converging to near-optimal signaling systems.10 They attribute this success to the persistent randomness introduced by feature (3) – its ability to forget the past. Their conclusion is largely based on the apparent success of other learning rules discussed above which also discard past experience.

5

Conclusions

Overall it does appear that some successful communication can emerge out of initial confusion. Both models of evolution and of individual learning often result in the emergence of somewhat successful communication. Such success is not always guaranteed, however. In signaling games with more than two states, signals, and acts, perfect communication is not guaranteed to emerge. Similarly the emergence of perfect signaling is not certain in games where the states are not equiprobable. These conclusions hold both for evolution and learning models. However, we did find that signaling can emerge with very little cognitive sophistication. Communication can emerge from natural selection alone, or from some very simple learning rules like Win-stay/Lose-randomize. Several similarities between the models of learning and evolution are apparent. The results for the replicator dynamics coincided with the results for Herrnstein 10

Because there is persistent randomness in ARP learning it will not ever converge to any pure strategy.

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reinforcement learning. The relationship between these two models is more significant than the similarities mentioned here, and so this result is not entirely surprising (cf. Beggs, 2005; Hopkins and Posch, 2005). The selection-mutation dynamics (for appropriate parameter values) converges to perturbed signaling systems. This coincides with the results obtained for the ARP learning model. However, many of the other learning rules always converge to a (non-perturbed) signaling system – we have no version of the replicator dynamics which models this result. Many of the learning rules which converged to signaling systems had an interesting feature: they began by exploring the space of possibilities, but then later began playing successful strategies with high probability. This feature is found in Win-stay/Lose-randomize and both reinforcement models with exponential response. Similarly, those that forget the past appeared to perform better than counterparts that did not, as was the case with ARP learning, Herrnstein reinforcement learning with random shocks, smoothed reinforcement learning, and Win-stay/Lose-randomize. These learning rules have large persistent randomness (at least early in the process). This feature is partially shared by the selection-mutation dynamics, which has persistent randomness throughout the process of evolution. The results from the extant literature on the evolution of communication suggests that this randomness is required in order for populations or individuals to converge on optimal signaling.

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Pragmatics, Logic and Information Processing Brian Skyrms University of California, Irvine

1

Logic

David Lewis wrote Convention in order to use game theory to answer Williard van Orman Quine’s skeptical doubts about the conventionality of meaning. Quine’s skepticism was directed at the logical positivists’ conventional theory of meaning in the service of a conventional theory of logic. According to the logical positivists, logical truths were true and logical arguments valid by virtue of the meanings of the terms involved. In a famous essay, “Truth by Convention,” Quine argued that Positivist accounts of convention (by explicit or implicit definition) required the pre-existence of logic. David Lewis replied that the existence of a convention can be thought of in a different way, as a strong kind of equilibrium or near-equilibrium in a signaling game played by a population. Lewis did not himself supply an account of how a population might get to signaling-system equilibrium — he did not think he had to in order to answer Quine — but the dynamics of evolution or of learning may supply such an account. Consider the simplest sender-receiver signaling game, of the kind introduced by Lewis in his book Convention. One player, the sender, observes which of two possible states of the world is actual. (Nature chooses the state with the flip of a fair coin.) The sender chooses one of two signals to transmit to the other player, the receiver. The receiver observes the signal, and chooses one of two possible acts. The sender and receiver have common interest in information transmission. Each gets a payoff of one if the right act is chosen for the state, and a payoff of zero otherwise. The signals have no prearranged meanings Any meaning that they acquire is a function of the information transmitted in an equilibrium of the game. An equilibrium in which there is perfect information transmission, where the players always get it right is a signaling system. There are two possible signaling systems in this game. In one signaling system the signals have just the opposite meaning that they do in the other, so we have a model in which meaning is perfectly conventional. (There are other equilibria where no information is transmitted.) A positive answer to the dynamical question raised by Lewis model of convention would show how the dynamics could, at least sometimes, spontaneously generate meaning. For this simplest example, there are strong affirmative answers. Evolution, as embodied by the replicator dynamics, always generates a signaling system.1 Naive reinforcement learning, in the form of Herrnstein’s matching 1

Proved in Hofbauer and Huttegger (2008).

A. Benz et al. (Eds.): Language, Games, and Evolution, LNAI 6207, pp. 177–187, 2010. c Springer-Verlag Berlin Heidelberg 2010 

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law, always generates a signaling system.2 In more complicated sender-receiver games, perfect signaling can still be generated spontaneously although it is no longer guaranteed to happen with probability one. This leaves open the question of whether these accounts in any way support the doctrine of the conventionality of logic, whose refutation was Quine’s ultimate goal. In so far as one’s view of logic is expansive — some positivists viewed all of mathematics as logic — the project may seem Quixotic. But the more modest goal of seeing whether we can get some logic out of information transfer in sender-receiver games, and if so how much, is something worth pursuing. In previous work,3 I advanced some tentative suggestions that involve modifications to the basic Lewis signaling setup. First, the sender may not observe the state of the world exactly. Rather her observation may rule out some possibilities while leaving a class of others viable. For example, a Vervet monkey may detect the presence of a ground predator — leopard or snake — without being able to see which it is. If this happens often enough and if, as is quite possible, the receiver’s optimal action given this information is different from both the optimal action for a leopard or for a snake, it is plausible that a special signal could evolve for this sender’s information state in exactly the same way as in the original signaling games. I call such a signal proto truth-functional because one way of giving its meaning is by the truth function “leopard or snake” — even though the signal itself is just a one-word sentence. Let us postulate a rich signaling environment in which lots of proto-truth functional signals have evolved. The second modification is to consider multiple senders, each in possession of a different piece of relevant information. For example, suppose one sender on the ground — seeing a movement of grass — sends the proto-truth function “leopard or snake,” and another sender, from the vantage point of a tree, sends the proto-truth function “no leopard.” Selection favors the receiver who takes the evasive action appropriate for a snake. Such a receiver has performed — or acts as if she has performed — logical inference. This story was put forward in a tentative and preliminary spirit and it leaves questions hanging. The proto truth-functions were assumed to have already evolved. Could they co-evolve with logical inference, or are they required to exist already? Where are the precise models? Where is their analysis in terms of evolutionary or learning dynamics? We would like to make some progress towards answering these questions, but first we need to show that considerations other than just logic are on the table.

2

Conversation

In 1967–68, H. Paul Grice delivered seven William James Lectures at Harvard University on Logic and Conversation. They were not published in their entirety until after his death, but the lectures themselves had a large impact. The text of the lectures circulated widely as thick typescripts. These lectures played a large 2 3

Proved in Argiento, Pemantle, Skyrms, and Volkov (2009). Skyrms (2000) (2004).

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part in reawakening linguists to pragmatics, which had been previously been eclipsed by issues of syntax. Grice was interested in how information over and above conventional meaning was transferred in conversation. Grice’s unifying idea was that conversation is fundamentally a cooperative enterprise, and that the presumption of cooperative intent can be used to extract information. If you tell me you have run out of gas and I say that there is a gas station around the corner, you can presume that the gas station is open or at least that I do not know it to be closed. That is so even though my statement would be literally true in either case. You assume that I am trying to cooperate, and that requires truth but also more than mere truth. If I ask you where Peter is, and you answer that he is either in Mexico or Zimbabwe, I can presume that you are not saying this simply because you know that he is in Mexico, although that would make it true. If you are trying to cooperate and know that Peter is in Mexico, you will say so. Grice and followers in this tradition derived various norms of conversation from the presumption of cooperation, which itself was ultimately elaborated in terms of common knowledge of cooperative intentions. He called these norms “maxims”; we have seen the most important ones — the maxims of quality and quantity — at work in the previous examples. Quality: 1. Do not say what you believe to be false. 2. Do not say that for which you lack adequate evidence. Quantity: 1. Make your contribution as informative as required. 2. Do not make your contribution more informative than is required. In adversarial proceedings, where cooperation cannot be presumed, the oath taken by witnesses is essentially to follow these maxims. In the basic Lewis signaling games, the foundation of cooperation is made explicit. It is the strong common interest assumed in the specification of the payoffs. The sender is reinforced if and only if the receiver is. This assumption specifies an environment most favorable to evolution of signaling systems. This is not to say that some information transmission may not be possible in the absence of complete common interest — a point to which we will return later. The presumption of common knowledge that we find in Gricean theory is, for the first time, made explicit in David Lewis’ account of conventions in signaling games. Although Lewis’ theory has affinities with Grice, his focus was quite different. Lewis was giving an account of conventional meaning, while Grice was interested in analyzing information transmission outside the bounds of conventional meaning. Since Lewis is talking about conventions rather than implicatures, the common knowledge is about social patterns of behavior and about the expectations of members of the relevant social group rather than about another participant in a conversation. Conventions, however, can be thought of as having

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crystallized out of the interplay of the pragmatics of individual conversations. Lewis fits together with Grice. In the approach pursued here, intentions, expectations, common knowledge, and rational choice are not required. They may very well be part of the story when they are present, but there is still a behavioral story when they are not. Payoff maximization is achieved — if it is achieved — by a trial and error process. Adaptive dynamics takes center stage. Strongly stable equilibria take the place of common knowledge. This setting is — I believe — a congenial setting for the radical pragmatic theory initiated by Grice and Lewis. It is, indeed, more radical than either. Lewis dispensed with Grice’s intentions and insisted that he didn’t need them. I also eliminate the common knowledge that Lewis relies upon to maintain equilibrium. It might be objected that I am implicitly assuming that the players know that know the structure of the signaling game model,4 but this is really not the case. Reinforcement learning, for instance, is applied to the signal sent in a given state, and to the act taken on receipt of a given signal—not to strategies for the whole game. Individuals or organisms involved need not, explicitly or implicitly, recognize any game structure or even that they are communicating.5

3

Logic and Conversation

When I was first telling my story about evolution of logic, Anil Gupta took me aside after a talk and said: “It’s not just logic that’s evolving, its logic together with conversational implicature and related pragmatic considerations. They only get separated later on in the evolution of language.” Barry Lam said the same thing quite independently and at about the same time in a paper written for me when he was an undergraduate. Surely they are right. My senders input all that they know into the “premises” that they are respectively sending. It is just that they don’t know everything. Their knowledge is limited by their imperfect observation of the state of the world. And my receiver is not performing — or acting as if he were performing — an arbitrary valid inference from the premises. Rather he is drawing the most specific valid inference from the premises, which is what he needs to take the optimal action. If this isn’t all exactly Grice, it is consonant with the spirit of Grice. Different maxims resonate with different parts of the story. On the other hand, it is immaterial to the account that the senders communicate everything that they observe. Even if each sender knew that the predator was a snake, and each, for whatever reason, chose to withhold some of the information, the account would work in the same way. It is important here that the senders “tell the truth” — that a sender does not send a “snake or leopard” signal while watching an eagle descending from the sky. Senders here need to obey the maxims of quality but not of quantity.6 4 5 6

Thanks to a referee for raising this point. See Skyrms (2008) (2009) and Argiento, Pemante, Skyrms and Vokov (2009). A sender who obeyed that maxim of quantity in this situation would send the specific information which would identify the predator.

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And what is important for the receiver is that he extracts the information from the signals that is relevant to his choice of action.

4

Information Processing

It is best to think of our two-sender, one-receiver model as an especially simple case of a problem of information processing. Multiple senders send signals that convey different pieces of information and the receiver can benefit from integrating this information. Let us consider some simple examples. 4.1

Inventing the Code

Suppose that there are four equiprobable states of nature, and that two individuals are situated to make incomplete observations of the state. The first sees whether it is in {S1, S2} or in {S3, S4} and the second sees whether it is in {S1, S3} or in {S2, S4}. Together they have enough information to pin down the state of nature, but separately they do not. Each sends one of two signals to a receiver who must choose one of four acts. The payoffs favor cooperation. Exactly one act is “right” for each of the states in that each of the individuals is reinforced just in case the “right” act for the state is chosen. I will not assume here, as I did in the story at the beginning of this chapter, that a convention has already been established for the signals used by the senders. We will make things a little harder and require that the content of the signals evolve together with the inference. You could think of sender one as waving either a red or a green flag and sender two as waving either a yellow or a blue one.7 A signaling system in this extended Lewis signaling game is a combination of strategies of the three players, two senders and one receiver, such that the receiver always does the right thing for the state. If we run simulations of reinforcement learning, starting with everyone out acting at random, the three individuals typically fall rapidly into one of the possible signaling systems.8 Consider the flow of information in these signaling-systems. Each sender’s signal conveys perfect information about her observation — about the partition of states of the world that she can see. The combination of signals has perfect information about the states of the world. Exactly one state corresponds to each combination of signals. And the receiver puts the signals together. The receiver’s acts contain perfect information about the state of the world. 4.2

Inventing the Categories and the Code

In the foregoing example, we postulated the categories that the senders can observe and thus those that could be embodied in their signals. For example, sender one can at best convey the information that the world is in one of the 7 8

The receiver will have to be able differentiate information from the two senders, since they are placed to make different observations. See Skyrms (2008) (2009).

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first two states or not. That is all that she can see. In a remarkable analysis, Jeffrey Barrett considers a model9 where the two senders and one receiver need to interact to spontaneously invent both the categories and the code in order to achieve a signaling system. In Barrett’s game there are four states and four acts, just as before, but each sender can observe exactly the true state of the world. Although each sender now has perfect information, each only has two signals available. There are two information bottlenecks. To achieve a signaling system our three individuals face a daunting task. Senders need to attach their signals to categories in such a way that these categories complement each other and jointly determine the state of the world. The receiver needs to extract the information from these signals. Receivers need to learn while senders are learning how to categorize and senders need to learn their complementary categorizations while receivers are learning to extract information from the combination of signals received. In a signaling system, sender 1 might send her first signal in states 1 and 2 and her second signal otherwise, and sender 2 might send her first signal in states 1 and 3 and her second otherwise. (These are just the categories imposed by observational restrictions in example 1.) But alternatively sender 1 might lump states 1 and 4 together for one signal and states 2 and 3 for another which, together with the same receiver’s strategy, would let the combination of signals peg the state of the world. To my considerable surprise, Barrett found that reinforcement learners reliably learned to optimally categorize and signal. The categories formed depended on the vicissitudes of chance — sometimes one set, sometimes another — but they always complemented one another in a way that allowed the receiver to do the right thing. Consider the flow of information in the signaling-system equilibria in Barrett’s game. Sender’s signals do not convey perfect precise information about their observations, but only partial information. Nevertheless, the combination of signals has perfect information about the states of the world. Exactly one state corresponds to each combination of signals. And the receiver puts the signals together. The receiver’s acts contain perfect information about the state of the world. Senders and receivers have, in a way, learned to obey the maxims of quality and quantity while learning to communicate. The receiver has also, in a way, learned to perform a most basic logical inference: from premises p, q to infer the conjunction p & q. This is all behavioral, so if you insist on framing these matters in terms of the mental life of agents, you will only say that they behave as if they are communicating and performing inferences. But I would say that they are communicating and performing inferences. 4.3

Extracting Relevant Information

Appropriate information processing depends on the character of the payoffs. Let us revisit example 1. The two senders again have their categories fixed by 9

Barrett (2006) (2007a,b).

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observation. Sender 1 can see whether the world is in one of the first two states or not; sender 2 can see whether the state is odd numbered or even numbered. We modify the example so that there are only two relevant acts with the following payoffs: Act 1 Act 2 State 1 0 1 State 2 1 0 State 3 1 0 State 4 0 1 Optimal signaling requires the receiver to do act 1 in states 2 and 3 and act 2 otherwise. Although there are only two acts now, the receiver cannot rely on only one sender, since neither has the sufficient information. The senders have information about their own categories — their own partitions of the states of the world — but the receiver needs information about a different partition. Reinforcement learners, starting with random exploration, learn optimal signaling here just as well and just as quickly as in the previous examples. Given optimal signaling, where players are always reinforced, each sender’s signal here carries perfect information about her observation and the combination of signals singles out the state of the world. But the receiver’s act only contains partial information about the state. It is “only as informative as is required” by the pragmatic considerations embodied in the reinforcement structure. The receiver has learned to extract the information that is relevant and to ignore that which is irrelevant. From the viewpoint of truth-functional logic, the receiver has had to learn how to compute the truth-value of the exclusive disjunction, “xor ”, from the truth values of its constituents. Sender 1 observes whether p is true; sender 2 observes whether q is true. The act that pays off is act 1 if p xor q, act 2 if not. If we look at this in terms of logical inference we can say that the receiver has — in a way — learned to infer p xor q from the premises p, not − q, but its denial from the premises p, q, and so forth. The inferences are not just valid inferences, but also the relevant valid inferences for the task at hand. Receivers can learn to compute other truth functions and to perform other inferences in just the same way. 4.4

Taking a Vote

So far, our senders have been infallible observers of the state of the world. They may not have seen everything, but what they think they have seen they have indeed seen. Senders’ strategies so far have been based on the truth, of not always the whole truth. In the real world there is observational error.10 If there is imperfect observation, it may make sense to ask for a second or third opinion. Consider the most basic Lewis signaling game, with two equiprobable 10

Nowak and Krakauer (1999) consider a different kind of error, receiver’s error in perceiving the signal. They suggest that minimizing this kind of error played an important role in the evolution of syntax.

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states, two signals and two acts, but with three senders. Each sender observes the state, but with some error — errors independent — and sends a signal to the receiver. Then the receiver chooses an act. It is not possible for signals to carry perfect information about the state. Error is endemic to the model. It is not possible for a signaling system to assure that the receiver always gets it right. But it is possible for an equilibrium to minimize the effects of error. The senders can convey perfect information about their fallible observations, and the receiver can pool this information to make the best choice. The optimal receiver’s strategy is then to take a vote. If the majority of senders “say” it is state one, then the receiver should do act one; if a majority of senders “say” it is state 2 then the receiver should do act 2. We could call this sort of equilibrium a “Condorcet signaling system.” Taking a vote allows a significant improvement over the payoffs attainable with only one sender. For example, with an error rate for observations of 10%, our receiver will have an error rate of less than 3%. With a few more senders the error the error rate can be driven very low, as the Marquis de Condorcet pointed out.11

5

Beyond Common Interest

The strong common interest in Lewis signaling games is by design. They were conceived as situations conducive to the existence of effective and stable norms of communication. Often signals are exchanged in completely cooperative situations, but not always, and the study of signaling should not be restricted to situations of common interest. Consider a one-sender, two-state, three-act signaling game with conflicting interests (sender’s payoffs first, receiver’s second): Act 1 Act 2 Act 3 State 1 3,10 0,0 11, 8 State 2 0,0 1,10 10, 8 If the receiver knows the state of nature, he will be best off taking act 1 (in state 1) or act 2 (in state 2), for a payoff of 10. If he is reasonably uncertain, he is better off taking act 3, which has a good payoff of 8, irrespective of the state. But the sender wants the receiver to take act 3, which gives her the best payoff in each state. Accordingly, it is in the sender’s interest to refrain from communicating information about the state. No signaling system is an equilibrium. But there are pooling equilibria where the sender always sends the same signal and the receiver always does act 3. Let us add another state where there is common interest. In the following example it is very important for both sender and receiver that act 2 be done in state 3. 11

For players learning the Condorcet equilibrium, and evolution leading to it, in our simplest example, see Skyrms (2009) and Rory Smead’s simulation results in the supporting matter.

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Act 1 Act 2 Act 3 State 1 3,10 0,0 11,8 State 2 0,0 1,10 10,8 State 3 0,0 10,10 0,0 Now we have a partial pooling equilibrium, where the sender sends one signal in states 1 and 2, and another signal in state 3 and where the receiver does act 3 on getting the first signal and act 2 on getting the second. In this situation there is a mixture of common and opposed interests. There is an equilibrium in which some information is transmitted from sender to receiver, but not all. We have a new example where one of the signals pools state 1 and state 2. Here this is not because the sender is unable to discern the state, but rather because she is unwilling to transmit specific information. We still have some information transmission even though the cooperative principle and the maxim of quantity are violated. A receiver who gets this signal, which is sent in both state 1 and state 2 but not otherwise, might be able to combine this information with that in a signal from a different sender with different payoffs, as in our original example of logical inference.

6

Deception

Fireflies use their light for sexual signaling. In the western hemisphere, males fly over meadows flashing a signal. If a female on the ground gives the proper sort of answering flashes, the male descends and they mate. The flashing “code” is species-specific. Females and males in general only use and respond to the pattern of flashes of their own species. There is, however, an exception. A female firefly of the genus Photuris, when she observe a male of the genus Photinus, may mimic the female signals of the male’s species, lure him in, and eat him. She gets not only a nice meal, but also some useful protective chemicals that she cannot get in any other way. One species, Photuris versicolor, is a remarkably accomplished mimic — capable of sending the appropriate flash patterns of eleven Photinus species. I would say that this qualifies as deception, wouldn’t you? It is not something we understand by thinking about mental states. It is not intellectually profitable to imagine oneself inside the mind of a firefly. Deception is a matter of information. The Photinus species have their signaling systems in place and encounters with Photuris are not sufficiently frequent to destroy it. More precisely, the signals carry information — not perfect information, but information nonetheless — to the effect that state of the world is “receptive female here.” This is all a question of frequencies, not intentions. Without the signal it is very improbable that there is a receptive female; with the signal it is probable overall that there is one. Photuris species systematically send signals that carry misleading information about the state of the world. As a consequence the receiving males are led to actions that benefit the senders, but lead to their own demise.

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How is deception possible? It is possible because signals need not be perfectly reliable in order to carry information — and because some level of deception is compatible with the existence of informative signaling,12 just as some level of error is compatible with informative signaling.13

7

Information Processing and the Cooperative Principle

The simplest Lewis signaling games specify an optimal environment for the evolution of information transfer by signals with conventional meaning. The cooperative principle is built into the model with the assumption of strong common interest — senders and receivers both paid off if and only if the receiver does the appropriate act for the state. A signaling system is the realization of optimal cooperation. The sender always knows the state of the world, and has no reason to withhold information or to engage in deception. In a signaling system the receiver need not perform any inference, but only has to pay attention to the state of the world indicated in the signal and do the act most beneficial to her. The optimal environment for signaling is not the only environment in which signaling can spontaneously arise. It is perhaps the first case to investigate, but it is not really the most interesting case. We have looked at a few simple generalizations that relax the idealizations of the optimal environment. Information transfer can occur when there are partially conflicting interests, where there are errors, and where there is deception. It can be arise where the receiver has to combine the partial information in distinct signals from different sources. This might involve making a logical inference or taking a vote. Information processing may indeed be a way of making up for failures of the cooperative principle, of observation, and in general of the idealized optimal model. What is left of the cooperative principle and of Grice’s maxims? They have a role to play in certain situations, but none of them are required for information transmission and processing. What is essential is contained in the pragmatics of sender-receiver interaction. This can studied from the standpoint of adaptive dynamics of evolution and of learning.

References Argiento, R., Pemantle, R., Skyrms, B., Volkov, S.: Learning to Signal: Analysis of a Micro-Level Reinforcement Model. Stochastic Processes and their Applications 119, 373–390 (2009) Barrett, J.A.: Numerical Simulations of the Lewis Signaling Game: Learning Strategies, Pooling Equilibria, and the Evolution of Grammar. Working Paper MBS06-09, University of California, Irvine (2006) Barrett, J.A.: The Evolution of Coding in Signaling Games. Theory and Decision (2007a), doi:10.1007/s11238-007-9064-0 12 13

For other examples where deception is compatible with informative signaling, see Bergstrom and Lachmann(1998); Searcy and Nowicki (2005). I will treat deception in more detail in a book—Signals: Evolution, Learning and Information—that is in preparation.

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Barrett, J.A.: Dynamic Partitioning and the Conventionality of Kinds. Philosophy of Science 74, 527–546 (2007b) Bergstrom, C.T., Lachmann, M.: Signaling Among Relatives III. Talk is Cheap. Proceeding of the National Academy of Sciences USA 95, 5105–5200 (1998) Grice, H.P.: Meaning. Philosophical Review 66, 377–388 (1957) Grice, H.P.: Logic and Conversation. In: Cole, P., Morgan, J.L. (eds.) Syntax and Semantics, vol. 3, pp. 41–58. Academic Press, ew York (1975) Grice, H.P.: Studies in the Way of Words. Harvard University Press, Cambridge (1989) Hofbauer, J., Huttegger, S.: Feasibility of Communication in Binary Signaling Games. Journal of Theoretical Biology 254, 843–849 (2008) Lewis, D.: Convention Cambridge. Harvard University Press, Mass (1969) Quine, W.V.O.: Truth by Convention. In: Lee, O.H. (ed.) Philosophical Essays for A. N. Whitehead, pp. 90–124 (1936) Nowak, M.A., Krakauer, D.: The Evolution of Language. Proceedings of the National Academy of Sciences of the USA 96, 8028–8033 (1999) Searcy, W.A., Nowicki, S.: The Evolution of Animal Communication: Reliability and Deception in Signaling Systems. Princeton University Press, Princeton (2005) Skyrms, B.: Evolution of Inference. In: Kohler, T., Gumerman, G. (eds.) Dynamics of Human and Primate Societies, pp. 77–88. Oxford University Press, New York (2000) Skyrms, B.: The Stag Hunt and the Evolution of Social Structure. Cambridge University Press, Cambridge (2004) Skyrms, B.: Presidential Address: Signals. Philosophy of Science 75, 489–500 (2008) Skyrms, B.: Evolution of Signaling Systems with Multiple Senders and Receivers. Philosophical Transactions of the Royal Society B. 364, 771–779 (2009)

Author Index

Benz, Anton

1, 14

Huttegger, Simon M.

Clark, Brady

92

J¨ ager, Gerhard

de Jaegher, Kris de Jager, Tikitu

40 60

Ebert, Christian

1

Franke, Michael

60

1

Kaufmann, Stefan

92

Sagi, Eyal 92 Scott-Phillips, Thomas C. Skyrms, Brian 177 van Rooij, Robert

German, James 92 Grim, Patrick 134

160

Zollman, Kevin J.S.

1, 40 160

117