Arising out of the authorâ€™s lifetime fascination with the links

Arising out of the author’s lifetime fascination with the linksbetween the formal language of mathematical models andnatural language, this short book comprises five essays investi-gating both the economics of language and the language of eco-nomics. Ariel Rubinstein touches on the structure imposed onbinary relations in daily language, the evolutionary develop-ment of the meaning of words, game-theoretical considerationsof pragmatics, the language of economic agents, and the rheto-ric of game theory. These short essays are full of challengingideas for social scientists that should help to encourage a funda-mental rethinking of many of the underlying assumptions ineconomic theory and game theory. A postscript contains com-ments by a logician, Johan van Benthem (University ofAmsterdam, Institute for Logic, Language and Computationand Stanford University, Center for the Study of Language andInformation) and two economists, Tilman Börgers (UniversityCollege, London) and Barton Lipman (University of Wisconsin,Madison).

A R I E L R U B I N S T E I N is Professor of Economics at Tel AvivUniversity and Princeton University. His recent publicationsinclude Modeling Bounded Rationality (1998), A Course inGame Theory (with M. Osborne, 1994) and Bargaining andMarkets (with M. Osborne, 1990).

Economics and Language

Five Essays

THE CHURCHILL LECTURES IN ECONOMIC THEORY

Economics andLanguage

Five Essays

ARIEL RUBINSTEINTel Aviv University

Princeton University

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE

The Pitt Building, Trumpington Street, Cambridge, United Kingdom

CAMBRIDGE UNIVERSITY PRESS

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10 Stamford Road, Oakleigh, Melbourne 3166, AustraliaRuiz de Alarcón 13, 28014 Madrid, Spain

© Ariel Rubinstein 2000

This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements,

no reproduction of any part may take place without the written permission of Cambridge University Press.

First published 2000

Printed in the United Kingdom at the University Press, Cambridge

Typeface Trump Mediaeval 9.75/12 pt System QuarkXPress® [SE]

A catalogue record for this book is available from the British Library

Library of Congress Cataloguing in Publication data

Rubinstein, Ariel.Economics and language / Ariel Rubinstein.

p. cm.Includes bibliographical references.

ISBN 0-521-59306-9 (hardbound)1. Economics–Language. 2. Game theory. I. Title.

HB62.R83 20003309.0194–dc21

ISBN 0 521 593069 hardbackISBN 0 521 789907 paperback

CONTENTS

0 Acknowledgments page viii

PART 1 ECONOMICS OF LANGUAGE

0 Economics and language 3

1 Choosing the semantic properties of language 9

2 Evolution gives meaning to language 25

3 Strategic considerations in pragmatics 37

PART 2 LANGUAGE OF ECONOMICS

4 Decision making and language 55

5 On the rhetoric of game theory 71

Concluding remarks 89

PART 3 COMMENTS

0 Johan van Benthem 93

0 Tilman Börgers 108

0 Barton Lipman 114

0 Index 125

vii

ACKNOWLEDGMENTS

This book emerged from the kind invitation of the fellowsof Churchill College to deliver the Churchill Lectures of1996. It is a pleasure for me to use this opportunity tothank Frank Hahn, for his encouragement during myentire career.

The book is an extended version of my ChurchillLectures delivered in Cambridge, UK in May 1996. Part ofthe material overlaps with the Schwartz Lecture deliveredat Northwestern University in May 1998.

Three people agreed on very short notice to provide theirwritten comments and to publish them in the book: Johanvan Benthem, Tilman Börgers and Bart Lipman. I thankthem wholeheartedly for their willingness to do so. Theidea to add comments to the book should be credited toChris Harrison, the Cambridge University Press editor ofthis book.

Several people have commented on parts of the text inits various stages. In particular, I would like to thank BartLipman, Martin Osborne and Rani Spiegler, who werealways ready to help, and Mark Voorneveld, who providedcomments on the final draft of the manuscript. My specialthanks to my partner in life, Yael. As so much else, the ideaof the cover is hers.

My thanks to Nina Reshef and Sharon Simmer, whohelped me with language editing; the remaining errors aremy own.

Finally, I would like to acknowledge the financial supportgiven me by the Israeli Science Foundation (grant no.06-1011-0673). I am also grateful to The Russell SageFoundation and New York University, which hosted meduring the period when much of the writing was completed.

viii

P A R T 1

ECONOMICS OF LANGUAGE

C H A P T E R 0

ECONOMICS AND LANGUAGE

The psychologist Joel Davitz once wrote: “I suspect thatmost research in the social sciences has roots somewherein the personal life of the researcher, though these roots arerarely reported in published papers” (Davitz, 1976). Thefirst part of this statement definitely applies to this book.Though I am involved in several fields of economics andgame theory, all my academic research has been motivatedby my childhood desire to understand the way that peopleargue. In high school, I wanted to study logic, which Ithought would be useful in political debates or in legalbattles against evil once I fulfilled my dream of becoming asolicitor. Unfortunately, I became neither a lawyer nor apolitician, and I have since come to understand that logicis not a very useful tool in these areas in any case.Nonetheless, I continued to explore formal models of gametheory and economic theory, though not in the hope of pre-dicting human behavior, not in anticipation of predictingthe stock market prices, and without any illusion aboutthe ability of capturing all of reality in one simple model. Iam simply interested in the reasoning behind decisionmaking and in the arguments people bring in debates. I amstill puzzled, and even fascinated, by the magic of the linksbetween the formal language of mathematical models andnatural language. This brings me to the subject of thislecture – “Economics and Language.”

0.1 Economics and language

The title of these lectures may be misleading. Althoughthe caption “Economics and Language” is a catchy title, itis too vague. It encompasses numerous subjects, most of

3

which will not be touched on here. This series of lectureswill briefly address five issues which fall under this generalheading. The issues can be presented in the form of fivequestions:

• Why do we tend to arrange things on a line and not in acircle?

• How is it that the utterance “be careful” is understoodby the listener as a warning and not as an invitation to adance?

• How is it that the statement “it is not raining very hard”is understood to mean “it is raining but not very hard”?

• Does the textbook utility function log (x111)x2 makesense?

• Is the use of the word “strategy” in game theory rhetori-cal?

All the issues discussed in these lectures lie somewherebetween economic theory and the study of language. Twoquestions spring to mind:

• Why would economic theory be relevant to linguisticissues? Economic theory is an attempt to explain regu-larities in human interaction and the most fundamentalnonphysical regularity in human interaction is naturallanguage. Economic theory carefully analyzes the designof social systems; language is, in part, a mechanism ofcommunication. Economics attempts to explain socialinstitutions as regularities deriving from the optimiza-tion of certain functions; this may be applicable to lan-guage as well. In these lectures I will try to demonstratethe relevance of economic thought to the study of lan-guage by presenting several “economic-like” analyses toaddress linguistic issues.

• Why would economic theory be a relevant subject ofresearch from the point of view of language? Becauseeconomic agents are human beings for whom language isa central tool in the process of making decisions andforming judgments. And because the other important“players” in Economic Theory – namely ourselves, theeconomic theorists – use formal models but these are

Economics of language

4

not simply mathematical models; their significancederives from their interpretation, which is expressedusing daily language.

0.2 Outline of the lectures

The book deals with five independent issues organizedinto two groups:

Part 1 is entitled “Economics of Language” and com-prises the core of this book. In Part 1, methods taken fromeconomic theory are used to address questions regardingnatural language. The basic approach is that languageserves certain functions from which the properties of lan-guage are derived.

In chapter 1, I assume that language is the product of a“fictitious optimizer” who operates behind a “veil of ignor-ance.” The substantive issue studied in this chapter is thestructure imposed on binary relations in daily language.The designer chooses properties of binary relations thatwill serve the users of the language. The three parts of thechapter discuss three distinct targets of binary relations:

(1) To enable the user of the relation to point out namelesselements.

(2) To improve the accuracy with which the vocabularyspanned by the relation approximates the actual termsto which the user of the language is referring.

(3) To facilitate the description of the relation by means ofexamples.

It will be shown that optimization with respect to thesethree targets explains the popularity of linear orderings innatural language.

In chapter 2, we discuss the evolutionary developmentof the meaning of words. The analytical tool used is avariant of the game-theoretic notion of evolutionary stablestrategy. Complexity considerations are added to the stan-dard notion of evolutionary stable equilibrium as an addi-tional evolutionary factor.

In chapter 3, I touch on pragmatics, the topic furthestfrom the traditional economic issues that are discussed in

Economics and language

5

these essays. Pragmatics searches for rules that explain thedifference in meaning between a statement made in a con-versation and the same statement when it is stated in iso-lation. Grice examined such rules in the framework of aconversation in which the participants are assumed to becooperative. Here, game-theoretical analysis will be usedto explain a certain phenomenon found in debates.

Part 2 is entitled Language of Economics and includestwo essays.

Chapter 4 deals with the Language of Economic Agents.The starting point of the discussion is that decisionmakers, when making deliberate choices, often verbalizetheir deliberations. This assumption is especially fittingwhen the “decision maker” is a collective but also hasappeal when the decision maker is an individual. Tools ofmathematical logic are used to formalize the assumption.The objective is to analyze the constraints on the set ofpreferences which arise from natural restrictions on thelanguage used by the decision maker to verbalize his pref-erences. I demonstrate in two different contexts that thedefinability constraint severely restricts the set of admis-sible preferences.

Chapter 5 focuses on the rhetoric of game theory. Muchhas been written on the rhetoric of economics in general;little, however, has been written on the rhetoric of gametheory. The starting point of the discussion is that an eco-nomic model is a combination of a formal model and itsinterpretation. Using the Nash bargaining solution as anillustration, I first make the obvious claim that differencesin models which seem equivalent result in significant dif-ferences in the interpretation of their results. The mainargument of the chapter is more controversial. I argue thatthe rhetoric of game theory is misleading in that it createsthe impression that game theory is more “useful” than itactually is, and that a better interpretation would makegame theory much less relevant than is usually claimed inthe applied game theory literature.

Though the book covers several distinct issues under theheading of “economics and language,” it by no meanscovers all the issues that might be subsumed under this


6

rubric. For example, I do not discuss the (largely ignored)literature labeled the “economics of language” which wassurveyed in a special issue of the International Journal ofthe Sociology of Language (see Grin, 1996). Grin (1996)defines the “economics of language” as “a paradigm oftheoretical economics and uses the concepts and tools ofeconomics in the study of relationships featuring linguis-tic variables; it focuses principally, but not exclusively, onthose relationships in which economic variables play apart.” This body of research does indeed revolve aroundtraditional “economic variables” and related issues suchas “the economic costs and benefits of multi-languagesociety,” “language-based inequality,” and “language andnationalism.” However, despite the similar headings,those issues are very far from my interests as expressed inthis book.

0.3 One more personal comment

While browsing through the literature in preparation forthese lectures, I came across a short article writtenby Jacob Marschack entitled the “The Economics ofLanguage” (Marschak, 1965). The article begins with adiscussion between engineers and psychologists regardingthe design of the communications system of a smallfighter plane. Following the discussion Marschak states:“The present writer … apologizes to those of his felloweconomists who might prefer to define their field morenarrowly, and who would object to … identification ofeconomics with the search of optimality in fields extend-ing beyond, though including, the production and distri-bution of marketable goods.” He then continues: “Beingignorant of linguistics, he apologizes even more humblyto those linguists who would scorn the designation of asimple dial-and-buttons systems a language.” I don’t feelthat any apology is due to economists … but I do feel asincere apology is owed to linguists and philosophers oflanguage. Although I am quite ignorant in those areas, Ihope that these essays present some interesting ideas forthe study of language.

Economics and language

7

R E F E R E N C E S

Davitz, J. (1976) The Communication of Emotional Meaning,New York: Greenwood Publishing Group

Grin, F. (1996) “Economic Approaches to Language andLanguage Planning: An Introduction,” InternationalJournal of the Sociology of Language, 121, 1–16

Marschak, J. (1965) “The Economics of Language,” BehavioralScience, 10, 135–40


8

C H A P T E R 1

CHOOSING THE SEMANTICPROPERTIES OF LANGUAGE

1.1 Introduction

This chapter will present a research agenda whose primeobjective is to explain how features of natural language areconsistent with the optimization of certain “reasonable”target functions. Rather than discuss the research agendain abstract, I will begin with the specific argument andreturn to the general discussion at the end of the chapter.

This chapter discusses binary relations. A binary rela-tion on a set V specifies a connection between elementswithin the set. Such binary relations are common innatural language. For example, “person x knows person y,”“tree x is to the right of tree y,” “picture x is similar topicture y,” “chair x and chair y are the same color,” and soon. I will avoid binary relations such as “Professor x worksfor university y” or “the Social Security number of x is y,”which specify “relationships” between elements whichnaturally belong to two distinct sets. I will further restrictthe term “binary relation” to be irreflexive: No elementrelates to itself. The reason for this is that the term“x relates to y” when x5y is fundamentally differentfrom “x relates to y” when xÞy. For example, the state-ment “a loves b” is different from the statement “a loveshimself.”

Certain binary relations, by their nature, must satisfycertain properties. For example, the relation “x is a neigh-bor of y” must, in any acceptable use of this relation,satisfy the symmetry property (if x is a neighbor of y, theny is a neighbor of x). The relation “x is to the right of y”

9

This chapter is based on Rubinstein (1996).

must be a linear ordering, thus satisfying the properties ofcompleteness (for every xÞy, either x relates to y or y to x),asymmetry (for every x and y, if x relates to y, y does notrelate to x), and transitivity (for every x, y, and z, if x relatesto y and y to z, then x relates to z). In contrast, the nature ofmany other binary relations, such as the relation “x lovesy,” does not imply any specific properties that the relationmust satisfy a priori. It may be true that among a particulargroup of people, “x loves y” implies “y loves x.” However,there is nothing in our understanding of the relation “xloves y” which necessitates this symmetry.

The subject of this chapter is in fact the properties ofthose binary relations which appear in natural language.Formally, a property of the relation R is defined to be a sen-tence in the language of the calculus of predicates whichuses a name for the binary relation R, variable names, con-nectives, and qualifiers, but does not include any individ-ual names from the set of objects V. I will refer to thecombination of properties of a term as its structure.

I am curious as to the structures of binary relations innatural language. I search for explanations as to why, out ofan infinite number of potential properties, we find thatonly a few are common in natural languages. For example,it is difficult to find natural properties of binary relationssuch as the following:

A1: If xRy and xRz (yÞz), and both yRa and zRa, then alsoxRa.

A2: For every x there are three elements y for which xRy.(In contrast, the relation “x is the child of y” on the setof human beings does satisfy the property that forevery x there are two elements y which x relates to.)

Alternatively, it is difficult to find examples of naturalstructures of binary relations which are required to betournaments (satisfying completeness and asymmetry)but which are not required to satisfy transitivity. Oneexception is the structure of the relation “x is locatedclockwise from y (on the shortest arc connecting x and y).”Is it simply a coincidence that only a few structures existin natural language?


10

The starting point for the following discussion is thatbinary relations fulfill certain functions in everyday life.There are many possible criteria for examining the func-tionality of binary relations. In this discussion, I examineonly three. I will argue that certain properties, shared bylinear orderings, perform better according to each of thesecriteria. Of course, other criteria are also likely to providealternative explanations for the frequent use of variouscommon structures such as equivalence and similarityrelations.

1.2 Indication-friendliness

Consider the case in which two parties observe a group oftrees and the speaker wishes to refer to a certain tree. Ifthe tree is the only olive tree in the grove, the speakershould simply use the term “the olive tree.” If there is nomutually recognized name for the tree and the two partieshave a certain binary relation defined on the set of trees intheir mutual vocabulary, the user can use this relation todefine the element. For example, the phrase “the thirdtree on the right” indicates one tree out of many by usingthe linear ordering “x stands to the left of y” when thegroup of trees is well defined and the relation “being tothe left of” is a linear ordering. Similarly, the phrase “theseventh floor” indicates a location in a building given thelinear ordering “floor x is above floor y.” There would beno need to use the phrase if it was known to be “the presi-dential floor.” On the other hand, the relation “line x onthe clock is clockwise to line y (with the smallest anglepossible)” does not enable the user to indicate a certainline on a number-less clock; any formula which is satis-fied by three o’clock is satisfied by four o’clock as well. Infact, the existence of even one designated line such as“twelve o’clock”, would enable the use of the relation tospecify all lines on the clock. The effect of using such adesignated element is equivalent to transforming thecircle into a line.

Thus, binary relations are viewed here as tools for indi-cating elements in a set whose objects do not have names.

Choosing the semantic properties of language

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We look for structures that enable the user to unambigu-ously single out any element out of any subset of V. We areled to the following definition:

Definition: A binary relation R on a set V is indication-friendly if for every A#V, and every element a[A, thereis a formula fa,A(x) (in the language of the calculus of predi-cates with one binary relation and without individual con-stants) such that a is the only element in A satisfying theformula (when substituting a in place of the free variablex).

All linear orderings are indication-friendly. If R is alinear ordering, the formula P1(x)5;y(xÞy→xRy) definesthe “maximal” element in the set A for A#V. The formulaP2(x)5;y(xÞy`2P1(y)→xRy) defines the “second-to-the-maximal” element, and so on. Note that in natural lan-guage there are “short cuts” for describing the variouselements. For example, the “short cuts” for P1(x) and P2(x)are “the first” and “the second.”

In contrast, consider the set V5{a,b,c,d} and the non-linear binary relation R, called “beat,” depicted in the fol-lowing diagram (aRb, aRc, dRa, bRd, bRc and cRd):

Referring to the grand set V, the element a is defined by“it beats two elements, one of which also beats two ele-ments.” The element b is defined by “it beats two ele-ments, which each beat one element.” And so on.However, whereas the relation R allows the user to defineany element in the set V, the relation is not effective in


12

Figure 1.1

defining elements in the subset {a,b,d}, in which case theinduced relation is cyclical.

We will now demonstrate that if V is a finite set and R isa binary relation, then R is indication-friendly if and only ifR is a linear ordering. We have already noted that if R is alinear ordering on V, then for every A#V and every a[A,there is a formula which indicates a. Assume that a binaryrelation R is indication-friendly. For any two elementsa,b[V, in order to indicate any of the elements in thetwo-member set A5{a,b}, it must be that either aRb orbRa but not both; thus, R must be complete and asymmet-ric. R must also be transitive since for every three ele-ments a,b,c[V, the relation must not be cyclical in orderto indicate each of the elements in the set A5{a,b,c}.

Conclusion 1: A binary relation enables the user to indi-cate any element in any subset of the grand set if and onlyif it is a linear ordering. Linear orderings are the most effi-cient binary relations for indicating every element in everysubset.

1.3 A detour: splitting a set

We will now make a brief detour from the world of binaryrelations in order to discuss unary relations. Assume thatthe user of the language can refer to a set of objects X (suchas “the set of flowers”). From time to time, the speakerwill wish to refer to subsets of X (either for the purpose ofconversing with another person or for storing informationin his own mind). However, he will be able to refer only toterms that appear in his language. Initially, the speaker canrefer to the set of all Xs (“pick all flowers”) or to the nullset (“don’t pick any flowers”). In order to extend his vocab-ulary the designer of the language is permitted to invent(given “hardware” constraints) one additional term for onesubset of X. The objective of the designer is to introduceone new term so that the speaker can refer to a new set inas accurate a manner as possible (on average).

Let us be more precise: Restrict X to be a finite set. If theterm “the set S” is well defined, then the user will have


13

four expressions available for referring to subsets of X: “allXs,” “the Ss,” “the not Ss,” and “nothing.” We will call thecollection of sets V(S)5{X,S,2S,À} the vocabulary spannedby S.

The basic idea is that language should be flexible enoughto function under unforeseen circumstances. The speakercan use the terms in V(S) but will eventually need to referto sets not necessarily contained in V(S). The term S willbe evaluated by the vocabulary’s ease of use with “leastloss.” It is assumed that when the speaker wishes to referto a set Z#X, he will use an element in V(S) which is“closest” to the set Z. In order to formally state this idea,we need to define the distance between two sets. Let thedistance between A and B, d(A,B) be the cardinality of theasymmetric difference between A and B (the set of all ele-ments which are in A and not in B or in B and not in A) –i.e. d(A,B)5|(A2B)<(B2A)|. One interpretation of thisdistance function fits the case in which the user whowishes to refer to the set B and employs a term A[V(S)lists the elements in B which are excluded from A orappended to A and bears a “cost” measured by the numberof elements which have to be excluded or appended (forexample, the sentence “You may eat only bread or any fruitwith the exception of apples and bananas” utilizes theterm fruit and three individual names: bread, apples, andbananas).

For a set B and a vocabulary V, define d(B,V)5minA[Vd(A,B), the distance of the set B from the closest setin the vocabulary V. By assigning equal “probabilities” apriori to all possible sets that the user might wish to referto, the problem for the designer becomes minS^z#Xd(Z,V(S))/2|X|. Essentially, the problem boils down to thechoice of the number of elements in the optimal set S.

To further clarify the nature of the problem, we willcarry out a detailed calculation for a four-element set X.Consider the case in which S contains two elements. Theuser can refer to any of the sets X, À, S or 2S, withoutincurring any loss. He can approximate any one- orthree-element set by using the set f or the set X, respec-tively, while bearing a “cost” of 1. If he wishes to refer to


14

one of the four two-element sets which are not S or 2S, heincurs a loss of 2. Thus, the average loss is [4(0)18(1)14(2)]/1651. A similar calculation leads to the conclusionthat a choice of S as À or X leads to an expected loss of 5/4(no loss for À and X, a loss of 1 for the eight sets of size 1 and3, and a loss of 2 for the six sets of size 2 – i.e. [2(0)18(1)16(2)]/1655/4). If S is taken to be a one-element set (or athree-element set), the average loss is only [4(0)16(1)16(1)]/1653/4. Thus, splitting X into two unequal subsetsof sizes 1 and 3 minimizes imprecision.

Despite the above results, our intuition is correct in thatthe optimal size of S should be half that of X. When the setX is “large,” the loss associated with choosing an S con-taining half of X’s members is “close” to being minimal.The following proposition is an exact statement of thisresult (its proof, omitted here, includes a combinatorialcalculation):

Claim: Let X be an n-element set. The difference betweenthe solution of the problem minS^z#Xd(Z,V(S))/2|X| and theexpected loss from the optimal use of the vocabularyspanned by an (n/2)-member subset of X, is in the magni-tude of 1/n1/2.

1.4 Informativeness

We now return to the world of binary relations on some setV. An additional function of binary relations on a set V isto transfer or store information concerning a specific rela-tionship existing between the elements of V. Consider thecase in which the grand set includes all authors of articlesin some field of research and the speaker is interested indescribing the relation “x quotes y in his article.” Thespeaker may describe the relation by listing the pairs ofauthors who satisfy the relation. Alternatively, he may usethose binary relations which are available in his vocabu-lary to describe the “x quotes y” relation. If he finds hisvocabulary insufficient to describe the relation, he will usea binary relation which is the best approximation. Forexample, if the relation “x is older than y” is well defined,


15

the speaker can use the sentence: “An author quotesanother if he is older than himself.” As this may not beentirely correct, he can add a qualifying statement such as“the exceptions are a who did not quote b (though b isolder), and c who did quote d (though d is not older).” Suchqualifying statements are the “loss” incurred from the useof an imprecise relation in order to approximate the “whoquotes whom” relation.

Our discussion envisages an imaginary “planner” who isable to design only one binary relation during the “initialstage of the world.” Of course, real-life language includesnumerous relations and the effectiveness of each dependson the entire fabric of the language. The assumption thatthe designer is planning only one binary relation is madesolely for analytical convenience.

The design of one binary relation allows the speaker toselect one of four binary relations. For instance, he canstate: “Every author is quoted by all others younger thanhim,” or “Every author is quoted by all others not youngerthan him,” and of course he can also state “Everyone quoteseveryone” and “No one quotes anyone,” which do notrequire familiarity with any binary relations. Given a rela-tion R, we will refer to these four relations as the vocabularyspanned by R and denote it by V(R). (Note that in definingthe vocabulary spanned by R, we ignore other possibilitiesfor defining a binary relation using R, such as statements ofthe type “xSy if there is a z such that xRz and zRy.”)

We assume that the speaker who wishes to refer toa binary relation S will use a relation in V(R) which isthe “closest” approximation. The loss is measured by thenumber of differences between the relation which thespeaker wishes to describe and the one he finds available inhis vocabulary. The distance between any two binary rela-tions R9 and R0 is taken to be the number of pairs (a,b) forwhich it is not true that aR’b if and only if aR0b. Note thataccording to this measure, any pair for which R9 and R0 dis-agree receives the same weight. Regarding the initial state,it seems proper to put equal weights on all possible“imprecisions.” The designer’s problem is to minimize theexpected loss resulting from the optimal use of his vocabu-


16

lary. It is assumed that from his point of view, all possiblebinary relations are equally likely to be required by thespeaker. Thus, the designer’s problem is minR^Sd(S,V(R)),where d(S,V(R))5minT[V(R)d(S,T) and d(S,T) is the distancebetween the relations S and T.

Stated in this way, the problem becomes a special case ofthe problem discussed in the previous section. The set X, inthe terminology employed in that section, is the set V3V25(v,v)|v[V6. A binary relation R is identified with the graphof R – i.e. the subset of X of all (a,b) for which aRb. Recallthat we concluded earlier that splitting the set X into twoequal sets is nearly optimal for the designer if he wishes toreduce the expected number of “imprecisions.”

We have one more step to go to reach the goal of thissection. In planning a binary relation on V, the designermay also consider the possibility that the relation willeventually be used in reference to a subset of V. This isanalogous to a binary relation being indication-friendly ifit allows the indication of any object out of any subsetof objects (see section 1.2). Hence, R has to be “optimal”for potential use in indicating a subset of V93V925(v,v)|v[V96 for every subset V9#V. If R is complete andasymmetric, for every subset V9#V, the induced relationR|V9 (defined by aR|V9b iff a,b[V9 and aRb) includes exactlyhalf the pairs in V93V92 5(v,v)|v[V96.

In summary, our fictitious planner wishes to design abinary relation which spans a vocabulary with the goal ofminimizing the expected inaccuracy of the term the userwill actually use. Viewing a relation as a set of pairs of ele-ments, the problem was linked to the optimization problemdiscussed in the previous section. There we concluded thatsplitting a set into two subsets allows “close to optimal” useof the induced vocabulary. Requiring the relation to be com-plete and asymmetric guarantees that for any subset of thegrand set, the restricted relation will be “close to optimal”as an aid to the user in specifying a relation on the subset.

Conclusion 2: In order to express binary relations asaccurately as possible on any subset of a set V using avocabulary spanned by a single binary relation on the set


17

V, a binary relation on V which is complete and asym-metric is close to optimal.

1.5 Ease of describability

In this section we discuss the third and last criterion bywhich binary relations are assessed in this chapter.Imagine that a hunter wishes to instruct his son on how tobehave in the forest. When he observes two potentialanimals a and b, should he pursue a or b? The instructionshave to be applicable to any pair of animals and must beclear. Thus, the set of instructions can be represented by acomplete and asymmetric binary relation R where aRbmeans that when the son simultaneously observes a and bhe should pursue a.

The son is aware of the sets of animals that are edibleand the structure of the relation R (i.e., the list of proper-ties satisfied). The son is acquainted with the structure ofthe relation either because it is instinctual or because hisfather has informed him of these properties. Therefore, allthat is left for the father to do, when transferring thecontent of R to his son, is to provide him with a list of“examples” – i.e., statements of the type “animal a shouldbe pursued when you see it together with animal b.” Theexamples should be rich enough to allow the son to inferthe entire relation from the structure and examples.

To illustrate, consider the case in which the relation R isa linear ordering and the number of elements in V is n. Theminimal number of examples that the father must providein this case is n21 (a1Ra2, a2Ra3 …, an21Ran).

This brings us to the main topic of our discussion. Weassume that providing examples is costly. (The complex-ities of the structure and process of making the inferencesare ignored here.) The following problem comes to mind:What are the structures of the complete and asymetricbinary relations (tournaments) for which the number ofexamples required for their description is minimal?

Definitions: We say that (f,{aiRbi}i[I) defines the binaryrelation R* on V when


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• f is a sentence in the language of the calculus ofpredicates with one binary relation named ai, biR and forall i, ai,bi[V

• R* is the unique binary relation on V satisfying the sen-tence f and for all i, it is true that aiR*bi.

The complexity of R*, denoted ø(R*), is the minimumnumber of examples that needs to be appended to R* inorder to enable a definition of R* – i.e., the minimum sizeof the set I on all possible definitions of R*.

As previously stated, our attention is limited to binaryrelations which are tournaments. The mathematicalproblem we wish to solve is:

minR*is a tournamentø(R*).

Note that for a given (finite) set of objects V5{a1,...,an}, the“structure” of any binary relation R* can be expressedby the sentence wR*(x1, …, xn)5'x1, …, xn(`{xiRxj|aiR*aj}).Thus, the optimization problem proposed above is equiva-lent to the following “puzzle-type” problem: Start with thegraph of a tournament in which the names of the verticeshave been erased. What is the minimum number of exam-ples required to recover the names of the vertices (up toisomorphism)?

Example 1: Consider the tournament R* on the setV5{a,b,c} where aR*b, bR*c and cR*a. The sentencewhich states that R is complete, asymmetric, and anti-transitive (;x,y,z(xRy and yRz→2xRz)) is consistent withtwo relations on V; hence, a single observation, aRb, isneeded to complete the definition of R*. Thus, ø(R*)51.

Example 2: Let R* be a linear relation. The relation isdefined by a sentence f expressing completeness, asymme-try and transitivity, and by n21 examples {aiR*ai11}i51, …, n21,where a1R*a2R*a3, …, an21R*an. Obviously, there is no defi-nition of a linear relation with less than n21 observations.Thus, ø(R*)5n21.

Example 3: Let V5{a,b,c,d} (n54) and let R* be the rela-tion satisfying aRx for all x and bRcRdRb. R* is defined by


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'wxyz[wRx`wRy`wRz`xRy`yRz`zRx] and the threeobservations aRb, aRc and bRc.

Example 4: Consider the relation R* on the set V5{a,b,c,d}described in figure 1.2. The structure of the relation R can beformulated by a sentence of the type 'v1v2v3v4w(v1,v2,v3,v4).Twenty-four different binary relations on V have this struc-ture. It is simple to verify that ø(R*)54.

Example 5 (Fishburn, Kim and Tetali, 1994):

This relation, R*, defined on the 5-element set V5{a,b,c,d,e}satisfies ø(R*)53. It is nicely defined by a sentence express-ing the property that for every x there are precisely two ele-ments “beaten” by x. The three observations aRb, aRc, andeRb, define the relation through the chain of conclusions{dRa, eRa}, {cRe, dRe}, {bRd, cRd} and, finally, bRc.

Examples 2 and 5 illustrate that for n53 and n55, a linearordering is not the most “economical” structure. Are thereany other binary relations with n.7 which are defined byless than n21 observations? I am not aware of a completeanswer to this question. However, we do know the follow-ing (this proposition was suggested to me by Noga Alon):

Proposition 1.3: For any e there exists n(e) such that forany n.n(e) and for any complete and asymmetric relationR on a set of n elements, ø(R).(12e)n.


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Figure 1.2

Thus, at least for large sets, linear orderings are “almost”optimal with respect to the criterion of minimizing thenumber of observations required for their definition.

Comment: Notice that in the above discussion, weallowed the relation to be defined by a formula whichdepends on the number of elements in the set V. In con-trast, the properties of linear orderings are expressed by aformula which does not depend on the number of elementsin the set V. This leads to the following conjecture:

Conjecture: Let w be a sentence in the language of the cal-culus of predicates which includes a single name of abinary relation R. There exists n* such that if |V|$n*;then, for any tournament R* which is defined on the set Vby the sentence w, ø(R*)$|V|21.

In other words, although one can define a relation for“small” sets with less than |V|21 examples, it is conjec-tured that a sentence must be accompanied by at least|V|21 elements in order to define a tournament when thesize of the set is “large enough.”

Comment: We conclude this section with an explanationof why the term “describability” – and not “learnability”– is used in this section. In our scenario the father choosesthe examples he presents to his son. The number ø(R*) isthe minimum number of examples the father has topresent in order to convey enough information for the sonto deduce the content of R*. When choosing the examples,the father knows the relation which he would like his sonto learn. On the other hand, had the son wished to acquirethe content of the relation R* by asking a list of “ques-tions” of the type “what should be chased, a or b?,” hewould not necessarily have asked first for the ø(R*)“examples” which could convey the relation R*. Forexample, he would need 2.5 inquiries “on average” inorder to infer a linear ordering defined on a three-elementset.


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1.6 Discussion

This chapter investigated the observation that in a naturallanguage, certain structures of binary relations appearmuch more frequently than others; in particular, we dis-cussed properties that are satisfied by linear orderings. Itwas argued that certain functions of binary relations innatural languages are better served by relations satisfyingthese properties. I believe that this is no more than aninteresting fact. A stronger interpretation of the resultsrequires establishing a connection between the optimalityconsiderations presented above and the realization of theoptimal solutions in the real world. Such a connectionwould require at least three premises:

(1) In order to function, natural languages include only asmall number of structured binary relations.

(2) Binary relations fulfill several functions in natural lan-guages.

(3) There are forces (evolution or a planner) which make itmore likely that structures which are “optimal” withregard to the functions of binary relations will beobserved in natural languages.

The first premise states that language inherently exhibitsfew of the properties of binary relations. Only if thenumber of possible structures is small can a user of the lan-guage deduce a relation’s structure from a small number ofinstances in which the relation is used. (It is amazing howfew observations of the type “a is better than b” are suffi-cient to teach a child that this relation is transitive.)

The second premise is the central one in this discussion.There are numerous potential criteria by which to measurethe functionality of binary relations. Three such criteriawere examined above. It was argued that certain properties(all shared by linear orderings) perform better according toeach of these criteria.

The third premise, which links the first two, states thateither there is a linguistic “engineer” who chooses theproperties of binary relations so that they function effec-tively or that evolutionary forces select structures which


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are optimal or nearly so with respect to the functions theyfulfill. This idea, which is popular in economics, has alsobeen noted by philosophers. For example, Quine states: “Ifpeople’s innate spacing of qualities is a gene-linked trait,then the spacing that has made for the most successfulinductions will have tended to predominate throughnatural selection” (Quine, 1969, p.126).

The approach adopted in this chapter is related to thefunctionality of the language approach discussed in lin-guistics (see, for example, Piatelli-Palmarini, 1970) and toattempts to explain the classification system in naturallanguage (see Rosch and Lloyd, 1978).

The discussion in this chapter is also related to classicalphilosophical discussions on “natural kinds.” The notionof a “natural kind” emerges from the philosophical inquiryinto the factors which confirm an inductive argument(see, for example, Goodman, 1972, Quine, 1969, Watanabe,1969. A key puzzle in this literature is the so called “riddleof induction”: Let us say that up to this moment, allobserved emeralds were green. Why does this observationimply that all emeralds are green rather than all emeraldsare “grue,” an alternative category which includes allobjects which were green up to this moment and blue fromnow on?

One possible answer to this question is that the induc-tive process relies on notions of similarity. Inductive argu-ments are made only with regard to categories of similarobjects. The category green contains similar elements;grue does not. A green element yesterday and a greenelement tomorrow are similar; in contrast, a grue elementyesterday is not similar to a grue element tomorrow.However, this only begs the question since one is left withthe problem of determining the natural similarity rela-tions. Here we run into similar difficulties. One possiblesolution is to argue that two objects are similar if “most”unary predicates coincide in satisfying the two objects.The difficulty with this criterion is raised by “The UglyDuckling Theorem” (see Watanabe, 1969, Section 7.6). Ifthe set of predicates is closed under Boolean operations,then the number of predicates which satisfies any possible


23

object is constant; thus, the existence of elementary unarypredicates cannot be the basis for explaining the existenceof specific similarity relations. This led Watanabe to con-clude that “if we acknowledge the empirical existence ofclasses of similar objects, it means that we are attachingnonuniform importance to various predicates, and thatthis weighting has an extra-logical origin” (Watanabe,1969, p. 376). Thus, there is no escape from assuming thata certain kind of predicate (like “green” and not like“grue”) has a preferred status called “natural kind” (seeQuine, 1969).

Within the class of properties of binary relations, linearorderings, more than other structures, appear to be of a“natural kind.” This chapter has attempted to providesome rationale as to why this is so.

R E F E R E N C E S

Fishburn, P., J.H. Kim and P. Tetali (1994) “TournamentCertificates,” AT&T Bell Labs, mimeo

Goodman, N. (1972) Problems and Projects, Indianapolis andNew York: Bobbs-Merrill

Piattelli-Palmarini, M. (ed.) (1970) Language and Learning: TheDebate between Jean Piaget and Noam Chomsky,Cambridge, MA: Harvard University Press

Quine, W.V. (1969), Ontological Relativity and Other Essays,New York: Columbia University Press

Rosch, E. and B. Lloyd (eds.) (1978) Cognition andCategorization, Hillsdale: Lawrence Erlbaum Associates

Rubinstein, A. (1996) “Why are Certain Properties of BinaryRelations Relatively More Common in NaturalLanguage?,” Econometrica, 64, 343-56

Watanabe, S. (1969), Knowing and Guessing, New York: JohnWiley


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C H A P T E R 2

EVOLUTION GIVES MEANING TOLANGUAGE

2.1 The lack of meaning of words in games

Let us imagine a community of fishermen who makedaily fishing trips onto the lake in small two-man boats.Occasionally, one fisherman notices a rock in the waterwhich requires his partner to take evasive action. In such asituation, the observer needs to warn his partner. It wouldappear that the observer has only to shout “Be careful” inorder to transmit the information about the obstacle. Butwhy is “Be careful” indeed interpreted by his partner in theway we understand the phrase rather than as “Here is afish” or as “What a beautiful day it is”? This is the ques-tion addressed in this chapter.

The situation is an interactive one in which the meaningof a statement depends on what the speaker thinks that thelistener understands, which in turn depends, in a circularway, on what the listener thinks that the speaker has inmind. Thus, the situation can be analyzed in a game-theo-retic framework. The basic situation will be viewed as asymmetric game. A player can either be an observer or alistener. A strategy is a plan of action for reacting to an

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The chapter is based on variations of Maynard Smith’s concept of an ESS(see Maynard Smith, 1982). The application of these ideas to the evolutionof communication protocols was explored by Blume, Kim and Sobel (1993)and Warneryd (1993), among others. The ideas in this chapter are closelyconnected to the work of Banerjee and Weibull (1993) and Warneryd(1998), who recognized the power of using “complexity considerations” inthe evolutionary analysis of “language formation.” For a formal introduc-tion of complexity considerations into game situations, see Rubinstein(1986). For studies of complexity considerations within an evolutionaryframework, see Binmore and Samuelson (1992) and Binmore, Piccione andSamuelson (1998).

obstacle in front of the boat and responding to the partner’scry while rowing the boat. Note (at this stage informally)that the situation has one “equilibrium” in which the cryis made in the case of an obstacle being spotted and isinterpreted correctly. However, the situation has another“pooling” equilibrium in which an observer ignores therock (remains silent) and is not “programmed” to respondto his partner’s shout if there is one.

The multiplicity of equilibria in this situation is one ofthe most “frustrating” challenges encountered by gametheoreticians. In fact, it is difficult to describe communica-tion using game-theoretic models owing to a fundamentalproperty of any game-theory model: The familiar solutionconcepts are “invariant” to the names of the actions.Consider a game and rename the actions: Whatever game-theoretic solution concept was used for the first game istransformed into an equilibrium of the game with theactions renamed!

One exception is Farrell’s solution concepts, which arerelated to “cheap talk” (see, for example, Farrell, 1993).Farrell assumes that words have a fixed external meaning.A listener operates under the assumption that unless aspeaker has a “good reason” to mislead him, he does not doso. Farrell argues for the instability of the pooling equilib-rium in which an observer remains silent and a listenerignores the cry “Be careful.” This equilibrium will be“destroyed” by a fisherman who nevertheless shouts “Becareful” when observing a rock, forcing the listener todecide whether to believe the observer or not. He reasonsas follows: “My partner has no reason to persuade me thatthere is an obstacle in my way unless indeed there is anobstacle because:

(1) If I believe him, the best course for me is to prevent acrash, an outcome that is beneficial for him as well.

(2) He gains no benefit from persuading me that there is arisk when there is none.”

Farrell’s argument is based on the underlying assumptionthat members of the community understand that the call“Be careful” is meant to indicate the existence of risk. The


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goal of this research agenda is to explain the existence of aterm with this meaning.

2.2 The basic evolutionary argument

The explanation discussed here is provided by evolution-ary game theory. The core idea rests on Maynard Smith’s(1982) notion of an Evolutionary Stable Strategy (ESS),according to which evolutionary forces select the moresuccessful modes of behavior. Language is taken to be abehavioral phenomenon and if it does not serve the needsof the population, evolutionary forces will act to improveits functioning.

If there were no term in the language which allowed onemember of the community to alert another, a mutationwould occur whereby a small group of mutants would use acertain costless “signal” called “Be careful” to indicatehazard and to correctly respond to such a signal. The muta-tion can be thought of as originating in nature or a collectivedecision by a small group of community members. Then, ifa mutant is matched with another in a boat, the informationregarding the hazard is transferred and utilized. Whether amutant does better than a nonmutant when he is matchedwith a nonmutant depends on the way that the cry “Becareful” is interpreted by a nonmutant. If, for example, non-mutants respond to this unfamiliar cry in panic, mutantswill, on average, do worse than nonmutated members of thepopulation. In that case, the success of a mutant paired withanother mutant would be outweighed by the significant lossincurred when he interacts with a nonmutant (this is themore common case when the mutated population is small).Thus, the explanation of the emergence of a proper meaningfor the cry “Be careful” depends on providing a persuasiveexplanation as to why unfamiliar “words” do not cause“nonmutants” to respond harmfully.

Evolutionary game theory has attempted to resolve thisissue. A simple and, in my opinion, very reasonable line ofreasoning is presented here from Warneryd (1993) andBanerjee and Weibull (1993). These authors assume thatthe respective evolutionary forces not only depend on the

Evolution gives meaning to language

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standard payoffs but also on “strategy complexity costs.”If, in equilibrium, no one shouts the phrase “Be careful,”then no one would waste his “mental resources” on beingprepared to respond to such a cry. It is assumed that evolu-tionary forces act to minimize the waste of mentalresources and eliminate the panic response to the message.Consequently, the term “Be careful” will remain availablefor mutants to use without suffering a loss from nonmu-tants’ harmful responses.

2.3 The basic situation

Let us now formalize the evolutionary argument outlinedin the previous section. We assume a “large” homogeneouspopulation whose members are occasionally paired andexamine a situation in which one of them is selected ran-domly to perceive information about a risk which “must”be transmitted to his partner.

Let W be a nonempty set. We will refer to W as the set ofwords, although nothing in the model requires the ele-ments of W to be verbal statements. The interactionbetween the members of the population is modeled as asymmetric game. A strategy in this game has the followingformat:

(1) When I perceive risk I will use the word w*[W orremain silent (“shhh,” where “shhh” is not a word in W).

(2) When I hear another person saying w[W, I will takeone of three possible actions:• “I will take action to avoid the hazard” (A), the

correct behavior in the case of a hazard.• “I will panic” (P), the worst possible action under any

circumstances.• I will take the default course of action, D, the best

course of action in the case of no hazard and thesecond best in the case of a hazard.

Thus, a “strategy” for a player in this game has two compo-nents: a choice of either “shhh” or w*[W and a functionr:W→{D,A,P}. Note that no decision is made when there isno hazard.


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It is assumed that there is no conflict of interest betweenany two fishermen so that we are dealing with a coordina-tion game. Given the existence of a hazard, the payoffsdepend only on whether or not the listener takes an actionto avoid the hazard (payoff 1), remains with the defaultaction (payoff 0), or panics (payoff 2M, where M is a“large” positive number).

Denote by u(s,s9) the expected payoff of a player whouses the strategy s while his partner uses the strategy s9,assuming that they have equal chances of being theobserver. This definition is extended to u(s, p) where p is aprobability measure on the set of strategies. A Nash equi-librium in this situation is a strategy s* satisfying that forno s, u(s,s*).u(s*,s*). There are two types of equilibriahere: In a “separating” equilibrium, a player as an observerof a hazard sends a message w* and on hearing w* hisresponse is A. In the pooling equilibria, no message is sentand the listener “plans” to respond to a message by ignor-ing it or panicking. The pooling equilibria are of course“bad” since valuable information is not being utilized.

This brief presentation raises the question whether thereare any evolutionary forces which can bring about theemergence of a “good” equilibrium in which informationis transmitted and used.

2.4 Evolutionary forces

The idea that evolutionary forces can explain the emer-gence of “words” in the natural language appeared in theeconomic literature as early as Marschak (1965). However,Marschak suggested only a direction for research. It is theliterature initiated by Maynard Smith and Price (1973),Maynard Smith (1982), and especially the more contempo-rary literature of evolutionary game theory, which formu-lated a coherent argument for the emergence of words. Thebasic definition is:

Definition 1: A strategy s* is an Evolutionary StableStrategy (ESS) if for any strategy s either u(s,s*),u(s*,s*)or, u(s,s*)5u(s*,s*) and u(s,s),u(s*,s).


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The rationale for this concept is by now familiar to usersof game theory. A mode of behavior s* will not remain astable unique mode of behavior in a community if, follow-ing a mutation of an arbitrarily small fraction of the popu-lation to strategy s, the mutants will not, on average, doworse than the members of the community who have notmutated. The above conditions guarantee that whateverthe extent of the mutation, the mutants will do worse thanthe nonmutants (for every «.0 small enough,

u(s,«s1(1 2 «)s*),u(s*,«s1(12«)s*)).

This standard form of ESS does not eliminate the poolingequilibria. In fact, if W includes more than one elementthen no ESS exists. If s* is an ESS, then any mutationwhich changes the response to words which are not usedwill be a “resistible mutation.” Thus, for ESS to be a usefulconcept, we must modify its definition. At least two pos-sible directions have been proposed in the literature.

The first method is to utilize a multi-valued solutionconcept. The following is one such alternative definition(for others of this type see, for example, Weibull, 1995):

Definition 2: A set of strategies S* is an evolutionarystable set if there is a probability measure p on S* with fullsupport, such that:

(1) u(s,s9);u* for every s,s9[S*(2) For any sÓS*, u(s, p),u* or u(s, p)5u* and u(s,s),

u(s*,s) for any s*[S*.

Thus, there is one payoff u* such that, in an evolutionarystable set, all members of the community receive the sameexpected payoff u*. The fact that all of them have the samepayoff maintains the stability of the set. The criteria forthe success of a mutation are derived from the same ratio-nale as that of the ESS. Mutants will do worse than allstrategies which exist in the stable population of strate-gies. It is easy to verify that the set of all strategies whichinstructs the player as an observer to remain silent is anevolutionary stable set. Essentially, the placing of a highprobability on the strategy which responds to any word by


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panicking will make any mutant worse off than with anyof the S* strategies.

Note a difficulty with the above definition of evolution-ary stable set. Following the appearance of a mutation, notall strategies in S* will do equally well and thus the distri-bution of strategies in S* may be unstable. My impressionis that other modifications of the ESS, which appear in theliterature, suffer from similar difficulties.

The second approach to modifying the ESS is to weakenit by demanding that a successful mutation must yieldstrictly higher payoff. This “modified” version (as Binmoreand Samuelson, 1992, call it) is appropriate when themutation is largely intentional and members are interestednot in maximizing payoffs but in having the highest payoffin the group:

Definition 3: A strategy s* is a modified evolutionarystable strategy if for any strategy s either u(s,s*),u(s*,s*)or u(s,s*)5u(s*,s*) and u(s,s)#u(s*,s).

The strategy dictating the use of the word w* to warn ofa hazard and the response to any message with the action Ais a modified evolutionary stable strategy. However, so isthe strategy to maintain silence (choose shhh) and torespond to any message with panic. Thus, the modifiedESS does not provide an explanation of the emergence of a“warning.”

2.5 Complexity and evolutionary forces

This final section of the presentation introduces a newelement into the model: the complexity of the strategies. Itis assumed that evolutionary forces not only affect thebasic payoffs but also the complexity of strategies. In par-ticular, evolutionary forces eliminate strategies whichdemand resources to respond to never-used messages. Thecosts of complexity put pressure on strategies not toinclude a “threat” to respond to messages with theharmful action P. This clears the way for mutations thattransfer and utilize observed information.

At this point we should formally define what we mean


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by the “simplicity of a strategy.” A strategy is evaluated bythe number of different actions, besides “shhh” and “D,”which it plans to use under certain circumstances eventhough they will probably not be realized. Denote the com-plexity of a strategy s by comp(s). The simplest strategy,with complexity 0, dictates that the player chooses “shhh”as an observer and takes the action D as a listener whateverthe message he receives. It is the simplest strategy in thesense that its execution does not require any resources tomonitor the situation. The most complex strategy is ofcomplexity 3. Such a strategy specifies a message to be sentin case of a hazard, a nonempty set of messages to whichthe response is the action A and a nonempty set of mes-sages to which the response is P.

As mentioned previously, the basic assumption of thissection is that evolutionary forces work in favor of strate-gies that yield either higher payoffs or lower complexity. Itis assumed that the costs of complexity are lexicographi-cally secondary to the payoffs of the original game. Thus,there are two possible modifications of the ESS notion,both of which support the equilibria in which informationis transmitted and used.

Definition 4 is analogous to definition 2 with the addi-tion of the complexity considerations:

Definition 4: A set of strategies S* is an evolutionarystable set (with complexity considerations) if there is aprobability measure p, with the whole set S* as support,such that:

(1) u(s,s9);u* for every s,s9[S* and all strategies in S* areequally complex.

(2) For any sÓS* u(s,p),u* or u(s,p)5u* and for all s*[S*either u(s,s),u(s*,s) or [u(s,s)5u(s*,s) and comp(s*),comp(s)].

Thus, for a mutated strategy s to be successful, either (1) itdoes better on average than any of the strategies in S* or,(2) it is equally successful on average as any strategy in S*but does better than any strategy in S* when the followingtwo criteria are applied lexicographically: (a) payoff while


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playing against the mutation s and (b) the complexity ofthe strategy.

It is clear that there is only one evolutionary stable set(with complexity considerations) which includes any strat-egy which instructs the user to utter some word in W* whena hazard is seen and to respond with A to any word in W*.Thus, this concept “predicts” that information will betransmitted and used, though it has the unattractive featurethat any utterance in W* serves as a meaningful signal.

The final definition is analogous to definition 3:

Definition 5: A strategy s* is a modified evolutionarystable strategy with complexity considerations if for anystrategy s either

u(s,s*),u(s*,s*), or[u(s,s*)5u(s*,s*) and u(s,s),u(s*,s)] or[u(s,s*)5u(s*,s*), u(s,s)5u(s*,s) and comp(s*)$comp(s)].

Thus a modified evolutionary stable strategy (with com-plexity considerations) is a modified ESS by which mutantsurvival is compared using three criteria which are appliedlexicographically: the payoff while playing against s*, thepayoff while playing against s, and the complexity of thestrategy.

If s* is such a modified ESS (with complexity considera-tions) and if s* assigns “shhh” to the observer, then thecomplexity considerations dictate that the response of s*to any message is D. It follows that a mutation s whichuses the word w to point out a risk and responds to w withA, will be a successful mutation.

If a modified ESS (with complexity considerations), s*,dictates that w* be uttered when a hazard is observed thenit must be that s* responds to w* with A. According to thecomplexity considerations, there are no circumstancesunder which any word will trigger the action P and thus s*must dictate that an observer utters w* when a hazard isobserved and a listener responds with A to any message insome set W* which contains w*. It is easy to verify thatsuch a strategy is indeed a modified evolutionary stablestrategy (with complexity considerations).


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2.6 A final comment

This chapter was meant to provide a short overview of thebasic evolutionary game-theoretic explanation for theemergence of words (terms) in natural language. It is debat-able whether the model is persuasive. I do not have a firmopinion on this issue. However, I would like to raise threepoints of criticism:

First, the connection between biological powers andwhat we model as evolutionary forces is far from beingclear. The definitions are not motivated by biological evi-dence or by intuition.

Second, the scenario analyzed here is not sufficientlygeneral. The emergence of the term “Be careful” is ana-lyzed independently of the other elements of the language.In our analysis we classify the situation of “two membersgoing on a boat trip” as a well defined category. We assumethe existence of a well defined meaning for the notion“there is a hazard.” We assume that the listener hasexactly three possible responses. The only element to bedetermined in the model is whether the observer shouts awarning when a hazard is observed and whether the shoutis interpreted correctly. A persuasive explanation of theemergence of a linguistic concept requires a much moregeneral setting. One may claim that this criticism is appli-cable to any game-theoretic model. Indeed models of gametheory are usually as simplistic and specialized as the onepresented here. However, I believe that models of gametheory are meant to describe and analyze human reasoning– and do not directly deal with reality. Since people focuson a small number of factors in decision making, gametheory is able to use simplified models to explain behavior.In contrast, evolutionary theories are intended to explainreal-world phenomena and thus need to be constructed in amore general setting.

Finally, the explanation of such phenomena using evolu-tionary forces requires an explanation not only for the exis-tence of human language but also for the lack thereof amonganimals. It is interesting to note that traces of this criticismalready appeared in the writings of Adam Smith (see Levy,


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1992), for an extensive discussion of this issue. In chapter IIof book I, Smith discusses the fact that while human beingsconduct trade, “Nobody ever saw a dog make a fair anddeliberate exchange of one bone for another with anotherdog.” Adam Smith points out that trade requires language.He attributes the existence of flexible language amonghuman beings to their “hardware”, which accounts for thedifference between humans and animals. While AdamSmith discusses the link between the evolution of trade andthe existence of language, the approach discussed in thischapter has attempted to explain the evolution of language.So what then explains the difference between humans andanimals with regard to the development of language?Haven’t we simply replaced one puzzle with another?

R E F E R E N C E S

Banerjee, A. and J.W. Weibull (1993) “Evolutionary Selectionwith Discriminating Players,” Stockholm: The IndustrialInstitute for Economic and Social Research

Binmore, K., M. Piccione and L. Samuelson (1998)“Evolutionary Stability in Alternating-offers BargainingGames,” Journal of Economic Theory, 80, 257–91

Binmore, K. and L. Samuelson (1992) “Evolutionary Stability inRepeated Games Played by Finite Automata,” Journal ofEconomic Theory, 57, 278–305

Blume, A., Y.-G., Kim and J. Sobel (1993), “EvolutionaryStability in Games of Communication,” Games andEconomic Behavior, 5, 547–75

Farrell, J. (1993) “Meaning and Credibility in Cheap-talkGames,” Games and Economic Behavior, 5, 514–31

Levy, D. (1992) The Economic Ideas of Ordinary People,London: Routledge

Marschak, J. (1965) “The Economics of Language,” BehavioralScience, 10, 135–40

Maynard Smith, J. (1982) Evolution and the Theory of Games,Cambridge: Cambridge University Press

Maynard Smith, J. and G.R. Price (1973) “The Logic of AnimalConflict,” Science, 246, 15–18

Rubinstein, A. (1986) “Finite Automata Play the RepeatedPrisoner’s Dilemma,” Journal of Economic Theory, 39,83–96


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Warneryd, K. (1993) “Cheap Talk, Coordination and EconomicStability,” Games and Economic Behavior, 5, 532–46

(1998) “Communication, Complexity and EvolutionaryStability,” International Journal of Game Theory, 27,599–609

Weibull, J. (1995) Evolutionary Game Theory, Cambridge, MA:MIT Press


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C H A P T E R 3

STRATEGIC CONSIDERATIONS INPRAGMATICS

3.1 Grice’s principles and game theory

The study of languages (see Levinson, 1983) is traditionallydivided into three domains: syntax, semantics, and prag-matics. Syntax is the study of language as a collection ofsymbols detached from their interpretation. Semanticsstudies the rules by which an interpretation is assigned toa sentence independently of the context in which the sen-tence is uttered. This chapter deals with the domain ofpragmatics.

Pragmatics examines the influence of context on theinterpretation of an utterance. An “utterance” is viewed asa signal which conveys information within a context. Thecontext comprises the speaker, the hearer, the place, thetime, and so forth. How the hearer views the intentions ofthe speaker and how the speaker views the presuppositionsof the hearer are relevant to the understanding of an utter-ance. Thus, in game-theoretic terms, the way in which anutterance is commonly understood may be thought of asan equilibrium outcome of a game between speakers of alanguage.

To illustrate the type of phenomena explained by prag-matics, consider the following natural language conversa-tions:

Example 1: A telephone conversation between A at homeand his friend B, calling from a telephone booth:

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Sections 3.2–3.4 are based on Glazer and Rubinstein (1997). For basic read-ings on pragmatics see Levinson (1983). Grice (1989) is a collection ofGrice’s papers on the subject.

A: “B, I am about to go for a walk. What is the weather likeoutside?”

B: “It is not raining heavily now.”

Normally, A concludes from B’s statement that it is rainingbut not heavily. This conclusion does not follow from thesemantic interpretation of the sentence “it’s not rainingheavily,” which allows for the possibility that it is “rainingbut not raining heavily” as well as “it’s not raining at all.”Furthermore, there are circumstances in which the utterance“it’s not raining heavily” will indeed be interpreted as notexcluding the possibility that it is not raining at all. Forexample, imagine that B is in a hut and has told A that it isdark outside and that he is buried under his blanket. In such acase, B would only be aware of rain that is “pouring down”and hitting the roof loudly. In this case A could infer only thatit is not raining heavily from B’s statement since B would notknow whether it was raining lightly or not raining at all.

Example 2:

A: “Where does C live? I need to reach him urgently.”B: “C is somewhere in New York.”

Unless A suspects that B is trying to prevent him fromreaching C, B will normally conclude that A really doesnot know C’s exact address, although he did not say soexplicitly.

Example 3:

A, a married man, to B: “I went out with a woman lastnight.”

A’s statement is normally understood to mean that hedated someone who is not his wife although he did notexplicitly say so.

Example 4:

A: “What do you see?”B: “It is not a rose.”

Ordinarily, A will understand that B is looking at a flowerwhich is not a rose, although B has not said that he islooking at a flower.


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In these examples, the fact that a statement appearedwithin a conversation between two people gives the sen-tence a different meaning than if it were interpreted in iso-lation. A theory which attempts to describe the rules forinterpreting conversational utterances was suggested bythe philosopher Paul Grice in the 1960s (see especially hisarticle “Logic and Conversation” in Grice, 1989) asfollows:

while it is no doubt true that the formal devices are especiallyamenable to systematic treatment by the logician, it remainsthe case that there are very many inferences and arguments,expressed in natural language and not in terms of these devices,which are nevertheless recognizably valid. So there must be aplace for unsimplified … logic of the natural counterparts ofthese devices. (pp. 23–4)

According to Grice (1989), speakers of a language areguided by “the cooperative principle”:

Make your conversational contribution such as is required, atthe stage at which it occurs, by the accepted purposes or direc-tion of the talk exchange in which you are engaged. (p. 26)

Grice suggests four “maxims” which “in general yieldresults in accordance with the cooperative principle”:

Quantity: “Make your contribution as informative as isrequired … Don’t make your contribution more informa-tive than is required.”

Quality: “Try to make your contribution one that istrue … 1. Do not say what you believe to be false. 2. Do notsay that for which you lack adequate evidence.”

Relation: “Be relevant.”

Manner: “Be perspicuous (avoid obscurity, avoid ambigu-ity, be brief and be orderly).”

A central concept in Grice’s theory is that of conversa-tional implicature:

Strategic considerations in pragmatics

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A man who, by saying that p has implicated q, may be said tohave conversationally implicated that q, provided that (1) he ispresumed to be observing the conversational maxims, or atleast the Cooperative Principle; (2) the supposition that he isaware that, or thinks that, q is required in order to make hissaying or making as if to say p, consistent with this presump-tion; and (3) the speaker thinks (and would expect the hearer tothink that the speaker thinks) that it is within the competenceof the hearer to work out, or grasp intuitively, that the supposi-tion mentioned in (2) is required. (1989, pp. 30–1)

The inference mechanism which is used in conversa-tional implicature is described by Grice as follows:

He had said that p; there is no reason to suppose that he is notobserving the maxims, or at least the Cooperative Principle; hecould not be doing this unless he thought that q; he knows (andknows that I know that he knows) that I can see that the suppo-sition that he thinks that q is required; he has done nothing tostop me from thinking that q; he intends me to think, or is atleast willing to allow me to think, that q; and so he has impli-cated that q. (1989, p. 31)

Following are some examples of how Grice’s theory isapplied:

Example 5: A is a driver who stops in a remote village andsees B who looks clearly a local.

A: “I am out of petrol.”B: “There is a garage around the corner.”

A’s line of thought might be as follows:

• There is no reason for me not to believe that B is co-operative.

• B’s utterance must not interfere with the maxim “be rel-evant.”

• It must be that B means that petrol is sold at thatgarage.

Let us return to example 2: Unless B wants to prevent Afrom reaching C, the maxim of quality seems to be vio-lated, unless B does not know C’s address. In example 3,


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the hearer excludes the possibility that the speaker doesnot know the name of the woman he was dating last night.The fact that A does not know who the woman is mustindicate that he does not want to give the name to B,which normally implies that she is not his wife.

Note that Grice’s inference does not apply to circum-stances in which it is clear that the relationship betweenthe speaker and the hearer is not cooperative.

Example 6: A and B play a zero-sum game: B hides a coinin one of his two hands. A will get the coin if he correctlyguesses the hand holding the coin.

A: “Hmmm …”B: “The coin is in my right hand.”

Here, the cooperative principle clearly does not apply. Awould normally conclude that B’s statement is an attemptto confuse him.

Grice’s explanation is not always persuasive. Let usreturn to example 1:

A: “B, I am about to go for a walk. What is the weather likeoutside?”

B: “It is not raining heavily now.”

For A to conclude that B means that it is raining but notheavily, A has to go through the following steps:

• B knows more than he has told me. He knows the stateof the world exactly, which may be “not raining” (NR),“light rain” (L), or “heavy rain” (H).

• He wants to be cooperative and is giving me true infor-mation.

• He wants to be relevant so he does not give me the exactinformation.

• If it was not raining at all, he would tell me so.• Since he says that it is not H, and if it were NR he would

tell me, I conclude that the state of the world is L.

The weak link in this chain is A’s assumption that if Bwere to observe that it was not raining, he would conveythis information to A. If B wants to be relevant and “it is


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not raining heavily” is the relevant information, whywould he not say so if it were not raining at all?

Students of game theory will find this discussion to bein the spirit of game-theoretic reasoning. Note, in particu-lar, the considerations which Grice imposes on the lis-tener: “he (the speaker) could not be doing this unless hethought that q; he knows (and knows that I know that heknows) that I can see that the supposition that he thinksthat q, is required …” In other words, Grice’s theory isessentially a description of one agent thinking about howanother is thinking. This is precisely the definition of stra-tegic reasoning and is the essence of game theory (for anearly paper which gives an account of Gricean communi-cation using the framework of game theory see Parikh,1991).

As Grice noted, the theory may also be applicable toother forms of interactions between informed and unin-formed agents, in which the informed agent takes anaction (not necessarily an utterance) with the intention ofinfluencing the hearer’s subsequent actions. The similar-ity to game theory is clear.

Furthermore, if game theory is to shed light on real-lifephenomena, linguistic phenomena are the most promisingcandidates. Game-theoretical solution concepts are mostsuited to stable real-life situations which are “played”often by large populations of players. Thus, game theoretictools may function most effectively when used to explainlinguistic phenomena.

3.2 Debates

To demonstrate the possible applications of game-theoretic methods in explaining Gricean phenomena, theremainder of this chapter will focus on a phenomenonrelated to debates. In this chapter, a “debate” will refer to asituation in which two parties who disagree regardingsome issue raise arguments in an attempt to persuade athird party of their positions. This is, of course, not theonly possible form of a debate. In some cases, the intentionof the debaters is to argue just for the sake of arguing. In


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other cases, their aim is to influence one another ratherthan a third party. Debates can be thought of as a specialform of conversation in which the parties have differentinterests. Debates differ from other mechanisms of “con-flict resolution,” such as bargaining and war, since poweris not used (or is used less) in debates to influence theoutcome. Although debates are common in real life, theyhave rarely been investigated within the economic andgame theoretic literature (notable exceptions are Shin,1994; Lipman and Seppi, 1995 and Spector, 2000).

The aim of this chapter is to provide an explanation ofthe following phenomenon, often observed in debates: Twoarguments may have a different degree of persuasiveness ascounter-arguments even though they would be viewed asequally persuasive when stated in isolation. At the outset,we should point out that the Gricean theory does not applyto debates since Grice’s logic of conversation is based onthe principle of cooperation, which does not apply to theconflict of interests characteristic of a debate.

To initiate the discussion, the results of a survey con-ducted among several groups of students at Tel-AvivUniversity will be presented. The following question waspresented to one group of students:

Question 1: You are participating in a public debate on thelevel of education in the world’s capitals. You are trying toconvince the audience that in most capital cities, the levelof education has recently increased. Your opponent is chal-lenging you with indisputable evidence showing that thelevel of education in Bangkok has deteriorated. Nowit is your turn to respond. You have similar indisputableevidence to show that the level of education in MexicoCity, Manila, Cairo, and Brussels has increased. However,owing to time constraints, you can present evidence foronly one of the four cities. Which city would you choose tomake the strongest counter-argument to the Bangkokresults?

Another group of subjects was presented with question 1with the modification that Bangkok was replaced byAmsterdam. To verify that the subjects regarded the four


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potential counter-arguments as equally effective, a thirdgroup of students was asked to answer question 2, in whichthe subjects are asked to select an opening argument ratherthan a counter-argument:

Question 2: You are participating in a public debate on thelevel of education in the world’s capitals. You are trying toconvince the audience that in most capital cities, the levelof education has recently increased. You have indisputableevidence showing that the level of education in MexicoCity, Manila, Cairo and Brussels has improved. Owing totime constraints, you can present evidence for only one ofthese cities. Which city would you choose in order to makeyour argument as convincing as possible?

The following table presents the responses to questions1 and 2. Manila is the favorite counter-argument toBangkok while Brussels is the favorite counter-argumentto Amsterdam. In contrast, the subjects split quite evenlybetween the four cities in question 2.

Question 1 Question 1 Question 2Bangkok Amsterdam

n 38 62 24

Mexico City (%) 19 15 21Manila (%) 50 3 21Cairo (%) 11 5 25Brussels (%) 21 78 33

A different group of students responded to question 3:

Question 3: Two TV channels have fixed weekly programschedules. You and a friend are debating which is the betterchannel in the presence of a third party. Your opponentargues in favor of channel A while you argue in favor ofchannel B. Both of you has access to the same five reportsprepared by a highly respected expert, each of which refersto a different day of the week and recommends onechannel over the other for that day. Your opponent begins


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the debate by quoting Tuesday’s report, which recom-mends channel A’s programs. The third party interruptshim and asks you to reply. Both Wednesday’s andThursday’s reports are in your favor – i.e. they recommendChannel B. However, you have time to present only one ofthese reports as a counter-argument, following which thethird party will make up his mind. Would you choose thereport for Wednesday or for Thursday as a better counter-argument to Tuesday?

The results were quite clear: About 70 percent of the 58subjects felt that Wednesday would be a better counter-argument than Thursday. Note that Thursday is the lastweekday in Israel and hence, one would expect it to bemore significant than Wednesday.

The survey results are puzzling. If two argumentscontain information of the same quality, why is one con-sidered to be a stronger counter-argument than the other?The fact that Manila is closer to Bangkok than it is toMexico City seems irrelevant to the substance of thedebate, yet it appears to dramatically affect the choiceof the better counter-argument. Similarly, althoughWednesday is less significant than Thursday with regard toTV programming, it is assessed as a better counter-argument against Tuesday.

We believe that this phenomenon is connected to prag-matics. A debater’s response to “Bangkok” with anycounter-argument other than Manila is interpreted as anadmission that Manila is also an argument in favor of theopponent’s position. In the context of question 3, if oneresponds to Tuesday with the counter-argument Thursday,rather than Wednesday, it is considered an admission thatWednesday’s report is not in his favor. The fact that thesentence “On Thursday Channel B is superior” is utteredin a debate as a counter-argument to “On Tuesday ChannelA is superior” gives the sentence a meaning different thanif the statement were made in isolation.

In what follows, I have no pretensions to fully explainthe rules by which people compare arguments. The soletarget is to point out that the logic of debate should not


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necessarily include an axiom stating that if an argument xis superior to argument y (in the sense of being a successfulcounter-argument against it) then argument y should notbe superior to argument x.

3.3 The model

We view a debate as a mechanism designed to extractinformation from debaters. We assume that the aim of thedesigner is to increase the probability that the right con-clusion will be drawn by the listener subject to the con-straints imposed on the debaters and the listener in termsof time and cognitive abilities. It will be shown that fulfill-ing this aim may lead to debating rules which do not treatarguments and counter-arguments symmetrically.

The setup is very simple: An uninformed listener has tochoose between two outcomes, O1 and O2. The “correct”outcome, from his point of view, is determined by fiveaspects, numbered 1, …, 5. An aspect i may be realized aseither 1 or 2. When an aspect i receives the value j itimplies that aspect i possibly supports the outcome Oj

(j51,2). A state v5(vj)j51,…,5 is a five-tuple of 1s and 2swhich describes the realizations of the five aspects. Thelistener assigns equal weights to all five aspects, and thecorrect outcome in state v, C(v), is the one which is sup-ported by the majority of the arguments.

While the listener is ignorant of the state, the two debat-ers, named debater 1 and debater 2, have full informationabout the state. There exists a conflict of interests betweenthe two debaters and the listener: Each debater i wishesthat outcome Oi be chosen, whatever the state, whereasthe listener wants the correct outcome to be chosen.

A debate is defined to be a mechanism in which eachdebater reveals information in order to persuade the lis-tener to choose his preferred outcome. A debate consists oftwo elements: The procedural rule which specifies theorder and types of arguments which are permitted and thepersuasion rule which specifies the relationship betweenthe arguments presented and the listener’s conclusion.

The “language” that the debaters are permitted to usemust be specified. It is assumed that debaters cannot make


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any moves other than raising arguments of the type “argu-ment j supports me.” Thus, a debater cannot raise argu-ments that support the outcome preferred by the otherdebater. We will further assume that the debaters mustprove their claims – i.e. debater i cannot claim that thevalue of aspect j is i unless it is indeed so.

The optimal debate should have an effective constrainton its length. Of course, if it is possible for three argumentsto be raised during the debate (that is, there is enough timefor raising three arguments and the listener has the cogni-tive ability to digest them all), then the listener can obtainthe correct outcome with certainty. He can accomplishthis by requesting that one of the debaters present threearguments – the debater will win the debate if and only ifhe fulfills this task. Thus, for the length constraint to beeffective, it is assumed that the number of argumentswhich can be raised during a debate is constrained to two.

Formally, we define a debate to be an extensive gameform in which:

(1) The set of feasible moves for each debater is a subset of{1, …, 5}.

(2) There can be at most two moves: Either only one of thedebaters is allowed to make at most two arguments (theone-speaker debate); or the two debaters move simulta-neously, each one making at most one argument (thesimultaneous debate); or the debate is a two-stageprocess in which one debater is allowed to make at mostone argument in each stage (the sequential debate).

(3) One of the outcomes, O1 or O2, is attached to each ter-minal history.

When debater j presents an argument to answer a previousargument of debater i, debater j’s argument is referred to asa counter-argument.

A debate G and a state v determine a game G(v) whichwill be played by the two debaters when the state is v. Thetwo-player game form G(v) is obtained from G by deleting,for each debater i, all aspects which do not support hisposition in v. If player i has to move following a history hand if at v none of the arguments he is allowed to make at


47

h support his position, then the history h in the game G(v)will become a terminal history and the outcome Oj

(debater i loses) is attached to h. As to the preferences inG(v), debater i strictly prefers outcome Oi to Oj.

The game G(v) is a two-person game; the listener is notconsidered a player. The game is a zero-sum game and hasa value, v(G,v), which is a lottery over the set of outcomessatisfying that each debater i has a strategy such thatwhatever the other debater’s strategy is, the distributionof outcomes is at least as good for i as this lottery. Letm(G,v) be the probability that v(G,v) assigns to the incor-rect outcome. When m(G,v)51, we say that debate Ginduces a mistake in state v. In debates which are notsimultaneous, m(G,v) is either 0 or 1. In simultaneousdebates, m(G,v) may be a nondegenerate lottery (1.m(G,v).0). For example, consider the simultaneous debate G withthe persuasion rule by which debater 1 wins if and only ifhe argues for some i and debater 2 does not argue for eitheri11(mod5) or i21(mod5). For the state v5(1, 1, 2, 2, 2), thevalue of the game G(v) is Oi with probability 1⁄2 for both i;thus m(G,v)51/2.

All mistakes are weighted equally and the optimaldebate is taken to be the one which minimizesm(G)5^vm(G,v). The following examples demonstrate thecalculation of m(G):

(1) Let G be the debate in which only debater 1 is allowedto speak and the listener is persuaded by any two argu-ments that support debater 1’s position. Then, amistake occurs in any state v where the number ofaspects which support debater 1 is exactly 2 – that is,m(G)510.

(2) Let G be a debate in which only debater 1 is asked topresent two arguments and he wins the debate if heraises two arguments in his favor from the set {1,2,3} orthe set {4,5}. In this case m(G)54: the four mistakes infavor of debater 1 occur in states where only aspects 4and 5 or two aspects out of the set {1,2,3} supportdebater 1. In fact, this debate has the least number ofmistakes in the set of one-speaker debates.


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(3) Let G be a simultaneous debate in which each debatercan present at most one argument. Debater 2 winsthe debate if debater 1 uses aspect x and he respondswith a counter-argument regarding aspect x11(mod5)or aspect x21(mod5). In this case, debater 1 rightlywins in any state v where there are three successiveaspects (ordered on a circle) in his favor and he loses inany state where only zero, one or two nonconsecutivearguments are in his favor. There are ten other states:five of them are the “shift permutation” of (1, 1, 2, 1, 2)and the other five are the “shift permutation” of(1, 1, 2, 2, 2). In each of these ten states, the value of theinduced game is the lottery that selects the two out-comes with equal probabilities. Thus, m(G)510(1⁄2)55.In fact, one can see that the minimal number of mis-takes in the family of simultaneous debates is 5.

(4) The following is a sequential debate which inducesthree mistakes. First, debater 1 and then debater 2 areasked to present an argument. Debater 2 wins if andonly if he counter-argues argument i with an argumentregarding an aspect which is listed in the secondcolumn in row i. This debate induces three mistakes,two in favor of debater 1 (in states (1,1,2,2,2) and(2,2,1,1,2)) and one in favor of debater 2 (in state(1,2,1,2,1)).

If debater 1 Debater 2 wins if and only ifargues for … he counter argues with …

1 {2}2 {4,5}3 {4}4 {1,5}5 {2,3}

Actually, one can show that this is an optimal debate (seeGlazer and Rubinstein, 1997). To conclude:

Claim: Any optimal debate procedure is sequential. Theminimal m(G) over all debates is three.


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Let us continue discussing the above optimal debate.What does the listener understand from an exchange ofarguments in which debater 1 argues regarding aspect i anddebater 2 responds with aspect j? Is it merely that aspect isupports debater 1 and aspect j supports debater 2? We cananalyze the listener’s thought process by considering thesequential equilibria of the three-player game constructedfrom the debate game by adding the listener as a thirdplayer who has to choose the outcome of the debate at anyterminal history. The above optimal persuasion rule issupported by a sequential equilibrium in which debater 1’sstrategy is to raise the first argument i for which debater 2does not have a proper counter-argument. If debater 2 has aproper counter-argument for each of his feasible argu-ments, debater 1 chooses the first argument which sup-ports him. Debater 2’s strategy is to respond with asuccessful counter-argument whenever possible. The lis-tener chooses the outcome according to the above persua-sion rule. Note, for example, that if debater 1 raisesargument 3 and debater 2 raises argument 4, it is optimalfor the listener to rule in favor of debater 2 since he con-cludes that in addition to aspect 4, aspects 1 and 2 are infavor of debater 2.

Note, however, that a three-player sequential debategame has other sequential equilibria. One of those is par-ticularly intuitive: In any state v, debater 1 raises the firstargument i which is in his favor and debater 2 respondswith the argument j, which is the smallest j.i in his favor.The listener’s strategy is guided by the following logic: Inequilibrium, debater 1 is supposed to raise the first argu-ment in his favor. If he raises argument i, the listener thenbelieves that arguments 1,2, …, i21 are in favor of debater2. In equilibrium, debater 2 is supposed to raise the firstargument in his favor following argument i. Hence, ifdebater 2 raises argument j, the listener believes that argu-ments i11, …, j21 are in favor of debater 1. The listenerchooses the outcome supported by the debater who hebelieves has more aspects in his favor; in the case of a tie,he rules in favor of debater 2 (stated differently, debater 2wins if (i21)11$(j2i)). This equilibrium induces six mis-


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takes, in the states (1, 1, 2, 2, 2), (1, 2, 2, 1, 1), (1, 2, 1, 2, 1),(1, 2, 1, 1, 2), (2, 1, 2, 1, 1), and (2, 1, 1, 2, 1).

It is interesting that according to the above optimal per-suasion rule, aspect 5 is a persuasive counter-argumentagainst aspect 2 and aspect 2 is a persuasive counter-argument against debater 1’s argument regarding aspect 5.This is not coincidental. In fact we can draw the followingconclusion (see Glazer and Rubinstein, 1997):

Conclusion: Any optimal debate is sequential and has apersuasion rule which does not treat the players symmetri-cally. In other words, there is a pair of aspects, i and j, suchthat when presented in sequence, i is a persuasive counter-argument against j and j is a persuasive counter-argumentagainst i.

3.4 Discussion

The discussion in this chapter is not intended to provide anexplanation for the precise manner in which statementswithin a debate are interpreted. It demonstrates a rationalefor a property which is often observed in real life: Thestrength of arguments when used as counter-argumentsmay be different than when they are used as arguments.The explanation is that persuasion rules are intended tocreate an environment enabling the best elicitation ofinformation, given the constraints on the length of thedebate and the interests of the debaters. Thus, the fact thatargument i defeats argument j and, at the same time, argu-ment j defeats argument i, is not necessarily a contradic-tion. In other words, the “logic of debate” does not include,and does not have to include, a rule that if “p defeats q,” “qshould not defeat p.”

Persuasion rules in natural language have to be stated innatural terms. The natural language terms in the “fivecities” example are the “relative geographical distance”and “geographical partition” of the group of cities into theset of Far Eastern cities {Bangkok, Manila} and non-FarEastern cities {Cairo, Brussels, Mexico City}. The optimalthree-mistake persuasion rule that was described in the


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previous section is not definable in these terms and thus itis not reasonable to expect that we would observe it in reallife. On the other hand, the following is an example of apersuasion rule for a sequential debate for the “five cities”example which can be described by the “Far East” and“non-Far East” terms: If the first speaker used a Far Easterncity in his argument, then the second must answer with aFar Eastern city, and similarly for a non-Far Eastern city.The sequential debate with this persuasion rule has sevenmistakes (one in the state where only Bangkok and Manilasupport the first speaker and the other six in each statewhere exactly two of the non-Far Eastern cities and exactlyone of the Far Eastern cities support the first speaker).This geographical partition also allows for a four-mistakeone-speaker debate in which the speaker is required topresent two arguments in order to win, either from the setof Far Eastern cities or from the set of non-Far Easterncities. Thus, the conclusion that there is a sequentialdebate which is superior to all other forms of debate is sen-sitive to the language in which the debate rules can bestated.

R E F E R E N C E S

Glazer, J. and A. Rubinstein (1997) “Debates and Decisions: Ona Rationale of Argumentation Rules,” Tel Aviv University,mimeo

Grice, P. (1989) Studies in the Way of Words, Cambridge, MA:Harvard University Press

Levinson, S.C. (1983) Pragmatics, Cambridge: CambridgeUniversity Press

Lipman, B.L. and D.J. Seppi (1995) “Robust Inference inCommunication Games with Partial Provability,” Journalof Economic Theory, 66, 370–405

Parikh, P. (1991) “Communication and Strategic Inference,”Linguistics and Philosophy, 14, 473–514

Spector, D. (2000) “Rational Debate and One-dimensional Con-flict,” Quarterly Journal of Economics, 115

Shin, H.S. (1994) “The Burden of Proof in a Game ofPersuasion,” Journal of Economic Theory, 64, 253–64


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P A R T 2

LANGUAGE OF ECONOMICS

C H A P T E R 4

DECISION MAKING AND LANGUAGE

4.1 Introduction

The starting point of this chapter is the view that decisionmakers deliberating before making a choice often verbalizetheir considerations. It follows that the “language” a deci-sion maker uses to verbalize his preferences restricts theset of preferences he may hold. Thus, interesting restric-tions on the richness of the decision maker’s language canyield interesting restrictions on the set of an economicagent’s admissible preferences.

Before we proceed to the main investigation, some back-ground is required. An economic agent in a standard eco-nomic model possesses a preference relation defined on aset of relevant consequences. The preferences provide thebasis for the systematic description of his behavior as wellas for welfare analysis. We usually assume that an eco-nomic agent is “rational” in the sense that his choicederives from an optimization given his preferences. Giventhat we adopt the rational man paradigm, the other con-straints imposed upon an economic agent’s preferences areoften weak. For example, in general equilibrium theory, weusually only impose conditions such as monotonicity,continuity, and quasi-convexity. On the other hand, werestrict the discussion in many economic studies to some

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The basic ideas in this chapter appeared in Rubinstein (1978), one of myearly papers which was never published. A related paper is Rubinstein(1984). Two parts of this chapter appeared as Rubinstein (1998b) andRubinstein (1999).

The mathematical tools I will be using are from standard MathematicalLogic (Boolos and Jeffrey, 1989, and Crossley et al., 1990, are good intro-ductory books).

family of preference relations which share a simple utilityrepresentation. Several questions thus arise: What is thereason that we do not further restrict the set of preferencesin the many cases where generality prevents us fromobtaining stronger results? Why do we often restrict ourattention to a small family of preferences even though theresults we obtain depend heavily on this restriction? Tophrase it more provocatively: Why does the utility func-tion (log(x1 11))x2 in a two-commodity world lie withinthe scope of classic textbooks whereas lexicographic pref-erences do not?

What are the general considerations that motivate us toinclude or exclude preferences from the scope of analysis?One consideration is that some preferences may betterexplain empirical data. Although I am not an empiricaleconomist, I am doubtful that this is a factor in the choiceof restrictions imposed on preferences in the economictheory literature. Another consideration is “analyticalconvenience.” This is a legitimate consideration, but onewhich must be treated with the necessary caution.Another consideration relates to “bounded rationality.” Itmay be argued that some preferences are more plausiblethan others since they can be derived from plausible proce-dures of choice. Finding such derivations is, of course, oneof the main objectives of bounded rationality models (for adiscussion of this point see chapter 2 in Rubinstein,1998a).

This analysis, however, will consider a different issue:the availability of a description of the preferences in a deci-sion maker’s language. The chapter assumes that when adecision maker is involved in an intentional choice, hedescribes his considerations, to himself or to other agentswho operate on his behalf, using his daily language. (For adiscussion of this point in the psychological literature see,for example, Shelly and Bryan, 1964.) Thus, “My first pri-ority is to obtain as many guns as possible and only afterthat do I worry about increasing the quantity of food” is anatural description of a preference relation. “I spend 35percent of my income on food and 65 percent on guns” is anatural description of a rule of behavior (one consistent

Language of economics

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with maximizing some Cobb–Douglas utility function).The function (log(x111))x2 , on the other hand, is a “text-book utility function” that is expressed by a relativelysimple mathematical formula, even though there is no ruleof behavior, stated in everyday language, which corre-sponds to this utility function.

Note that when a decision maker is a collective (recallthat decision makers in economics are often families,groups, or organizations) the assumption that preferencesmust be definable makes even more sense. In this case, adecision rule must be stated in words in order to be com-municated among the individuals in the collective, duringboth the deliberation and the implementation stages.

The assumption that the decision maker uses a limitedlanguage to express his preferences can have different inter-pretations. Most naively, it may be interpreted as a reflec-tion of the limited language that is available to the decisionmaker. But even if the language available to the decisionmaker is rich, he may not be able to use the richer languagebecause the relevant information he possesses is limited.And even if the language is richer and even if more informa-tion is available, the decision maker may choose, becauseof complexity considerations, to express his preferences bya formula, which must be stated in a limited language.

The aim of this chapter is to demonstrate that therequirement that preferences be definable can be fruitfullyanalyzed in formal terms. More specifically, we willanalyze two examples which illustrate the connectionbetween a decision maker’s language and the set of defin-able preferences. This analysis may serve as the first stepin a much more ambitious research program to study theinteraction between economic agents with “language” as aconstraint on agents’ behavior, institutions, communica-tion, etc. However, dreams aside, the goal of this chapter isquite modest.

4.2 Definable preferences on the basis of binary relations

Preferences are often defined on the basis of primitivebinary relations. For example, it is common that a decision

Decision making and language

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maker deliberating over the choice between two cars willstate his considerations explicitly and say something like: “Ihave three criteria in mind – price, safety, and the consumerreport ranking. I prefer one car over the other since accord-ing to most of these criteria, it is the superior car.” Whenthese binary relations are preferences, this aggregation is anoperation that we call a “social rule” in economics. Thestudy of this operation is at the core of social choice theoryin which basic preferences are interpreted as individualisticpreference relations and aggregate preferences are those ofthe collective as a whole. If one replaces the term “individ-ual i” with “property i,” social choice theory is transformedfrom a theory of social decisions into a theory of the forma-tion of individualistic preferences.

The assumption modeled and discussed in this section isthat in comparing two alternatives, the decision makeruses a formula that includes the names of the K binaryrelations, but not the names of the alternatives. The deci-sion maker is assumed to have a “rule” that determines hispreference between any pair of elements without referringto their names. His preference is expressed by a sentenceinvolving only phrases about the way that the K primitiverelations relate “one element” to the “other.”

In order to proceed with the analysis we must firstspecify the language formally. We will use the language ofpropositional calculus for this purpose. For every k, denoteby [xPky] the atomic proposition, which is interpreted as“the element x relates to the element y according to thekth binary relation.” Note that [xPky] is just a symbol rep-resenting a statement such as “one car is more expensivethan the other.”

Recall that a formula in the language of the calculus ofpropositions, with “atomic propositions” v1,…,vL, is astring of symbols constructed inductively by the followingrules: Any atomic proposition is a formula and, if w and care formulae, then 2w, w`c, w~c, w→c, and w↔c are alsoformulae. The truth value of such a formula w – i.e. whenvk receives the truth value tk (“True” or “False”) – isdenoted by w(t1,…, tL) and is defined inductively using thestandard truth tables of the connectives.


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Let A be a set of alternatives. We will refer to a functionthat assigns a binary relation on A to any profile ofK-tuples of binary relations on A as an aggregation rule. Anaggregation rule is attached to every formula with theatomic propositions [xP1y],…, [xPKy]:

Definition: An aggregation rule R is defined by w, aformula in the calculus of propositions with the atomicpropositions [xP1y],...,[xPKy], if, for any a, b[A and forevery profile of binary relations (R1,…,RK), aR(R1,…,RK)bif and only if w(aR1b,…,aRKb)5T.

If such a formula w exists, then the aggregation rule R isdefinable.

In other words, the determination whether aR(R1, …, RK)bis true or false is carried out according to the followingprocess: First, calculate the truth values of the K variables{aR1b, …, aRKb}; then calculate the truth value of w whenthe truth value of the atomic proposition [xPky] is set tothe truth value of aRkb. Note that the same formula w isused to determine the truth of aR(R1,…,RK)b for all pairsa,b[A and for all possible profiles of primitive binary rela-tions (R1,…,RK).

Two examples: The conjunction `k51,…,K[xPky] statesthe Pareto principle – one element relates to the other onlyif x relates to y by all K criteria. The majority rule can bewritten as the disjunction of all formulae of the type`k[MxPky, with M# {1,…,K} containing more than K/2elements.

Note that the requirement that in this case the aggrega-tion rule R be definable can be replaced by much morestandard requirements:

For any two profiles of binary relations {Rk9}k51,…,K and{Rk0}k51,…,K and for any two pairs of alternatives (a, b) and (c, d)of elements in A, if for all k [aRk9b if and only if cRk0d] thenaR(R19, …, RK9)b if and only if cR(R10, …, RK0)d.

This condition has three components which are familiarfrom social choice theory:


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(0) The rule which determines whether aRb or not aRbdepends only on {aRkb}k51,…,K; no other information isrelevant (such as whether bRka).

(1) Given any profile of binary relations, the same ruledetermines the “aggregated” relation between any pairof alternatives. This condition is in the spirit of theneutrality condition in social choice theory.

(2) For any pair of alternatives a and b, the same rule deter-mines how a relates to b for all profiles of primitiverelations. This condition is in the spirit of theIndependence of Irrelevant Alternatives in socialchoice theory.

It is not always this simple to identify a standard conditionthat is equivalent to a definability condition. In any case,the “definability” approach gives the standard conditions amore natural interpretation.

The binary relations in the domain of an aggregationrule are often naturally restricted to satisfy certain proper-ties which are systematic dependencies of the truth valueof aRkc on the truth values of aRkb and bRkc, for any a,b,and c. Such a requirement can be written as a formula, Tk,with the atomic propositions [xPky], [yPkz], and [xPkz], inthe form of a noncontradicting conjunction of some of theeight formulae d1[xPky]`d2[yPkz]→d3[xPkz], where each di

is either 21 or 11 (with the notation 21w;¬w and1 1w;w).

We say that a formula Tk is deterministic if Tk is aconjunction of exactly four formulae of the typed1[xPky]`d2[yPkz]→d3[xPkz], one for each of the possibleconfigurations of d1 and d2. If Tk is deterministic, the truthvalue of aRkc is fully determined by the truth values ofaRkb and bRkc (for any a,b, and c).

We say that Tk is degenerate if there is some d1 such thatd1[xPky] implies d3[xPkz] (or there is some d2 such thatd2[yPkz] implies d3[xPkz]) for some d3 – or, more precisely,there is some d1 such that both d1[xPky]`[yPkz]→d3[xPkz]and d1[xPky]`2[yPkz]→d3[xPkz] are conjuncts in Tk, orsome d2 such that both [xPky]`d2[yPkz]→d3[xPkz] andd1[xPky]`2d1[yPkz]→d3[xPkz] are conjuncts in Tk.


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For example, the formula ([xPky]`[yPkz]→[xPkz])`(¬[xPky]`¬[yPkz]→¬[xPkz]) corresponds to the require-ment that the kth primitive relation be a linear ordering.The formula ([xPky]`[yPkz]→[xPkz])`([xPky]`¬[yPkz]→¬

[xPkz])`¬[xPky]` [yPkz]→¬[xPkz]) corresponds to the case inwhich the primitive kth relation is an equivalence rela-tion. Neither formula is deterministic nor degenerate.

Since we are interested in the formation of preferences,we investigate the set of formulae that always define transi-tive relations. We require that the definition of the definedrelation be such that, when combined with the require-ments on the primitive relation as expressed in the formulaT5`k51,…,KTk, transitivity is a tautology.

Definition: The formula w (containing the variablesxP1y,…,xPKy) satisfies transitivity given (Tk)k51,…,K if theformula [w(x,y)`w(y,z)`T]→w(x,z) is a tautology.

(The formulae w(x,z) and w(y,z) are obtained fromw(x,y)5w after substituting each xPky with xPkz and yPkz,respectively.)

The requirement that the formula c5[w(x,y)`w(y,z)`T]→ w(x,z) be a tautology is at the core of the analysis. Thetransitivity of the defined relation is required for all truthconfigurations of {[xPky], [yPkz], [xPkz]}k as long as theysatisfy the properties required by (Tk)k.

Note that if we do not impose any restrictions on thebasic relations (Tk is “empty”), c is reduced to the formula[w(x,y)`w(y,z)]→w(x,z). However, the set of atomic proposi-tions {[xPky], [yPkz]}k, which appear in the conjunctionw(x,y)`w(y,z), is disjoint from the set of atomic propositions{[xPkz]}k appearing in w(x,z). This would make it impossiblefor w(x,y)`w(y,z)→w(x,z) to be a tautology unless w isalways true (a tautology) or always false (a contradiction).

Of course, for a given profile of basic relations satisfy-ing the requirements expressed by T, the formulac5[w(x,y)`w(y,z)`T]→w(x,z) may be satisfied for all triplesof elements even if c is not a tautology. This may occur ifthe profile of basic relations is not rich enough and config-urations of the basic relations that do not satisfy c are notrealized by any triple of alternatives.


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We can now present the main claim of this section:

Claim 4.1: Assume that each Tk is neither degenerate nordeterministic. If w is a formula that satisfies transitivitygiven T, then there is a set k*#{1,…,K} and a vector ofcoefficients {dk}kPk*, such that w↔`kPk*dk[xPky] is a tautol-ogy (and for every kPk*, dk[xPky]`dk[yPkz]→dk[xPkz] isimplied by Tk).

Claim 4.1 states that any definable aggregation rulesatisfying transitivity (given T), can be defined by a verysimple formula which is essentially a conjunction of prim-itive propositions or their negations.

Example: Assume that the basic binary relations are bothtransitive and negatively transitive (for all k,Tk5 ([xPky]`[yPkz]→[xPkz])`(¬[xPky]`¬[yPkz]→¬[xPkz]).This includes the case in which all primitive relationsare linear orderings. Then, any formula of the type`kPk*dk[xPky] for some set K* satisfies transitivity given T.By the above claim, any formula that satisfies transitivitygiven T is logically equivalent to a formula of the type`kPk*dk[xPky]. These formulae resemble the oligarchicsocial choice rules in the social choice theory literature.

Example: Consider the operation of constructing a clas-sification system for a set of objects (say, flowers) on thebasis of primitive equivalence relations (such as thenumber of leaves, color, and size). The formula Tk, whichembodies the assumption that the kth primitive rela-tion is an equivalence relation, is nondeterministic(¬[xPky]`¬[yPkz] does not imply either [xPkz] or ¬[xPkz])and is not degenerate. Any formula `kPk*[xPky] for someset k* satisfies transitivity and by claim 4.1, every formulain our language that satisfies transitivity given T is logi-cally equivalent to such a formula.

The proof of claim 4.1 (see appendix 1, p. 68) is similar toa standard result in the logic literature called the CraigLemma (see, for example, Boolos and Jeffrey, 1989):

Craig Lemma: If the formula w→c is a tautology, thenthere must be a formula, l, which uses only atomic propo-


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sitions appearing in both w and c such that both w→l andl→c are tautologies.

Note that in our analysis, the preference of a over brelies only on (aRkb)k51,…,K. This means that the aggrega-tion procedure cannot distinguish between the two cases“aRkb and bRka” and “aRkb and not bRka.” This would beacceptable if Rk was interpreted as a “strong preference of aover b” but not in the case of a weak preference.

To allow for “indifferences” the set of atomic proposi-tions should be expanded to include also the atomic propo-sitions (yPkx)k51,…,K. The formulae Tk would be required tobe a conjunction of ¬(xPky`yPkx) and eight formulae ofthe type xPky`yPkz→xPkz, xPky`yIkz→xPkz, where xIky isshorthand for ¬(xPky~yPkx). By a proof similar to that ofclaim 4.1 one can show that every definable aggregationrule (in the language of the calculus of propositions withthe proposition variables xP1y,…,xPKy, yP1x,…,yPKx) thatsatisfies transitivity and completeness given T is lexico-graphic: In other words, there is an enumeration of{1,…,K}, i(l),…, i(K), and a vector of coefficients {di(k)}k51,…,K

such that the aggregation rule is defined by the formula~k51,…,K[`t51,…,k21xIi(t)y`di(k)xPi(k)y]. Thus, the K criteriaare examined one by one, according to some order of prior-ity. One element, x, is preferred to another element, y, ifthe first k for which the i(k)th criterion is decisive deter-mines that x is preferred. Thus, although economistsalmost always exclude lexicographic preferences fromtheir models, from the point of view of this analysis theyare probably the most natural definable preferences.

4.3 A girl looks for a boyfriend in a foreign town

This section analyzes a second example illustrating thedefinability condition. Consider the “tale” of a girl whoarrives in a foreign town and wishes to find a boyfriend.She obtains information on each candidate in the form ofa list of his dates during the past T days. She has yet tomeet either the boys or the girls in town (i.e. they can be


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considered as “unknowns”). All she can glean from anytwo given names on the list is whether they are identicalor not. An alternative and probably better interpretationis of an entrepreneur who is comparing candidates for ajob, each of whom provides a list of his past employersabout whom the entrepreneur knows nothing.

Following are several principles which the girl may usewhen she compares two candidates, A and B (A and B arevariables, not individual names):

1. Prefer boy A if he dated Alice longer than B had.2. Prefer boy A if he has dated all the girls in town and B

has not.3. Prefer boy A if all the girls he dated, from the second

date on, had previously been dated by B.4. If the two boys dated the same girl on the first date,

prefer the one who dated her longer.5. Prefer the boy who made the most “switches.”6. Prefer the boy who has dated more girls.

Prior to the formal definitions, let us discuss the definabil-ity of these principles. Obviously, these principles aredefinable in some sense since I have just defined them.However, the question to be analyzed here is whether theyare definable in a limited language where: (i) the decisionmaker can not refer to a boy by his name, (ii) she can referto a girl only by the term “the girl he has dated at time t”and (iii) the only comparison the decision maker can makebetween “the girl he has dated at time t” and “the girl he(or someone else) has dated at time s” is whether they arethe same girl or not.

Principle (1) requires the decision maker to be able torefer explicitly to the name “Alice.” Principle (2) can bestated in the decision maker’s limited language using onlythe equality relation although it requires the use of a quan-tifier. Principles (3, 4, 5, 6) can be stated in this simple lan-guage. However principle (3) does not provide a definitionof a preference since (for the case of T52, for example) itimplies that the dating history (a,b) is preferred to (b,c),which is preferred to (c,a), which is preferred to (a,b). It iseasy to see that principle (4) does not define a preference


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either. Only principles (5,6) are definable in the limitedlanguage and induce preference relations.

We can now move on to the formal analysis. We charac-terize the binary relations that are definable in the “purelanguage with equality.” This is the language of the calcu-lus of predicates that includes symbols for equality only. Inthis language, an atomic formula is of the type z15z2,where z1 and z2 are variable names (such as x75x3 or x5x2).A formula is a string of symbols constructed according tothe following inductive rules: All atomic formulae are for-mulae; if w and c are formulae, then ¬w, w ~c and w`c,w→c and w↔c are also formulae; and if x is a free variable(does not appear with a quantifier) in the formula w, then'xw(x) and ;xw(x) are formulae as well.

A model in this simple language is simply a set G, inter-preted here as a set of girls. The validity of a formula w withthe free variables z1,…,zK, in a model G (when we substi-tute each zk with an element ak[G) is defined by the stan-dard truth tables and is denoted by G|5w(a1,…,aK).

We are interested in preferences on dating profiles oflength T that are defined by a formula w with free variablesx1,…,xT,y1,…,yT. Given a set G, the comparison betweentwo boys who have the dating profiles (a1,…,aT) and(b1,…,bT) is defined by G|5w(a1,…,aT,b1,…,bT). Notethat principle (2) can be expressed by the formula[;x~t51,…, T(xt5x)]`[¬;x~t51,…,T(yt5x)] while principle (3)can be stated by the formula `t52,…,T~m51,…, t21(xt5ym).

In the rest of the section we are interested in formulaethat induce binary relations that are transitive and asym-metric under all possible circumstances – i.e. in all models.

Definition: We say that a formula w with 2T free variablesinduces a transitive and asymmetric relation ordering ifthe formulae

;x1,…,xT,y1,…,yT,z1,…,zT[w(x1,…,xT,y1,…,yT)`w(y1,…,yT,z1,…,zT)→w(x1,…,xT,z1,…,zT)]

and

;x1,…,xT,y1,…,yT [w(x1,…,xT,y1,…,yT)→¬w(y1,…,yT,x1,…,xT)]

are tautologies – i.e. they are satisfied in all models.


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The analysis of this definability condition consists ofthree points: First, we can assume that the formula w iswritten without quantifiers. The “pure language withequality” has the property (see, for example, Robinson,1963) that for every model G (that is, for any set), everyformula has a logically equivalent formula containing thesame set of free variables with no quantifiers. To illustratethis point, consider the formula

w(x1,…,xT,y1,…,yT)5[;x~t51,…,T(xt5x)]`[¬;x~t51,…,T(yt5x)],

which states that a boy who has dated all the girls in townis preferred to one who has not. Although the quantifier“for all” is used here, the validity of w(a1,…,aT,b1,…,bT) ina model G depends only on the number of elements in Gand the number of elements in each of the two vectors. If Gincludes L#T elements then, when substituting (a1,…,aT)for (x1,…,xT), the validity of the formula [;x~t51,…,T(xt5x)]is the same as that of a formula stating that in the vector(x1,…,xT) there are L different elements, which can bewritten without quantifiers. (Take the disjunction of con-junctions, each of which corresponds to a partition of{1,…,T} into L nonempty sets {e1,…,eL} and state thatxi5xj for any i and j that are in the same ek and ¬xi5xj, if iand j are in distinct cells of the partition.)

The fact that for any model G, w has an equivalentformula with no quantifiers implies that for any givenmodel G, w is equivalent to a disjunction of configurationsof the variables {x1,…,xT,y1,…,yT}, where a configurationof {x1,…,xT,y1,…,yT} is the “description of which variablesare equal.” (In other words, it is a conjunction of formulaewhere, for each pair of variables z1 and z2 in{x1,…,xT,y1,…,yT}, either z15z2 or ¬z15z2 appears in theconjunction with the following constraint: If z15z2 andz25z3 appear in the conjunction, z15z3 is a conjunct aswell.)

Second, for any given cardinality of G, the only prefer-ences that are definable (in this simple language) are thosethat are induced by preferences on the “dating stabilityprofile” of each candidate. The term “dating stabilityprofile” refers to a partition of {1,…,T}, in which i and j are


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in the same element of the partition if and only if the boyhas dated the same girl at time i and time j. In other words,given a vector (a1,…,aT), define E(a1,…,aT) to be the parti-tion of {1,…,T} in which i and j are in the same partition ifand only if ai5aj. For example, if a boy is fickle and dates agirl for only one period, the corresponding dating structureis the partition {{1},…, {T}}; a “faithful” boy is characterizedby the dating structure {{1,2,…,T}}. One can show (seeappendix 2, p. 69) that if w defines an ordering in a modelG, then for any two dating profiles with E(a1,…,aT)5E(b1,…,bT), it is not true that

G|5w(a1,…,aT,b1,…,bT).

In other words, the decision maker is indifferent betweenthe two profiles (a1,…,aT) and (b1,…,bT).

We are still left with the possible dependency of thedefined preference on the cardinality of the set G: forany integer n there is a formula wn(x1,…,xT,y1,…,yT)with no quantifiers such that G|5;x1,…,xT,y1,…,yT

[wn(x1,…,xT,y1,…,yT)↔w(y1,…,yT,x1,…,xT)] for all modelsG with cardinality n. In other words, the decision maker’spreference relation on dating profiles may depend on thenumber of elements in G. This brings me to the thirdand last point: By the compactness argument there issome n* such that if G’s cardinality is at least n*,w(x1,…,xT,y1,…,yT) is equivalent to wn*(x1,…,xT,y1,…,yT).

Summarizing:

Claim 4.2: If w(x1,…,xT,y1,…,yT) is a formula whichinduces a transitive and asymmetric binary relation, thenfor any n, there is an ordering .n on the set of “datingstructures” of {1,…,T} such that if the size of G is n,

(a1,…,aT) relates to (b1,…,bT) if and only ifE(a1,…,aT).nE(b1,…,bT)

and there exists a number n* such that all .n are identicalfor n greater than some n*.

Thus, for any set G, the comparison of any two datingprofiles of a fixed length T (i) does not depend on any com-parison of the girls dated by the two boys (such as x35y7),


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(ii) for any given size of G, the comparison depends only onthe comparison between the dating stability structures ofthe two boys and (iii) is constant for sufficiently large sets.

4.4 Conclusion

This chapter has presented two examples of the applica-tion of a definability constraint on the set of admissiblepreference relations. It would also be interesting to investi-gate the definability-type constraints on the set of situa-tions that the decision makers may confront. It isappealing, in my opinion, to investigate models in whichthe emerging choice problems are limited to those whichare defined naturally. The language in which the choiceproblem is defined may affect the language of the decisionmaker. Imposing constraints both on the class of situa-tions as well as on the individuals’ preference relations,especially in interactive situations, may yield interestingresults. This would be a very challenging project, but theproof is, as always, in the pudding. The aim of this chapter,however, is much more modest – to persuade the readerthat the definability assumption on preferences is naturalas well as analytically tractable.

A P P E N D I X 1 P R O O F O F C L A I M 4.1

A basic proposition in the calculus of propositions states thatany formula w with the atomic propositions [xP1y],…, [xPKy] islogically equivalent to its disjunctive normal form, which is adisjunction ↔m[Mwm, where each wm is a truth configuration ofthe atomic propositions [xP1y],…, [xPKy] – that is, a formula ofthe form `k51,…,Kdk[xPky], where dk[{21,11}. We will say thata formula of the type `k[K*dk[xPky] is a decisive formula if`k[K*dk[xPky]→w(x,y) is a tautology. Let K* be a minimal (in theinclusion order) subset of {1,…,K} for which a decisiveformula `k[K*dk[xPky] exists. It is impossible that two dis-tinct sets K(1) and K(2) are such minimal subsets. If they were,then (`k[K(1)dk[xPky])`(`k[K(2)dk9[yPkz])`T→w(x,z) would be atautology. By the nondegeneracy condition the formula(`k[K(1)dk[xPky])`(`k[K(2)dk9[yPkz])`T implies a formula(`k[K«k[xPkz]) for which (`k[K«k[xPkz])→w(x,z) is a tautology,


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where for all k[K the formula dk[xPky]`dk9[yPkz]→«k[xPkz] is inT and K#K(1)>K(2). It follows that there is a unique minimalset K* for which `k[K*dk[xPky]→w(x,y) is a tautology.

Now, assume that there are two tautologies:`k[K*dk[xPky]→w(x,y) and `k[K*dk9[xPky]→w(x,y). Determinem[K* such that dm Þd9m . Since Tm is not deterministic, at leastone of the four conjunctions dm[xPmy]`dm[yPmz], dm[xPmy]`d9m[yPmz], d9m[xPmy]`dm[yPmz] and d9m[xPmy]`d9m[yPmz] doesnot imply the truth value of [xPmz]. If, for example, Tm does notimply any formula of the type dm[xPmy]`dm[yPkz]→d[xPkz]then consider the tautology [(`k[K*dk[xPky])`(`k[K*dk[yPkz])`(`kTk)]→w(x,z). Again, it follows that there is a tautology of thetype ` k[K*2{m}dk[xPkz]→w(x,z), in contradiction to the minimal-ity of the set K*.

A P P E N D I X 2 D AT I N G P R O F I L E S

We will show that for any dating structure, if two boys’ datingprofiles have this structure, the defined preference cannotprefer one to the other. For notational simplicity, we considerthe case in which each boy dates T distinct girls.

For any configuration c(x1,…,xT,y1,…,yT) in the disjunctivenormal form of w, where xiÞxj and yiÞyj for all iÞj, denoteN(c)5{t| there is an s such that xt5ys appears in c}. Let c* besuch a configuration, with the lowest number of elements inN(c). It is impossible that N(c*)50, since c* would then be theconjunction of all formulae of the type ¬z15z2 for all z1 Þz2[{x1,…,xK,y1,…,yK} and the asymmetry tautology would nothold.

It follows that

;x1,…,xT,y1,…,yT[c*(x1,…,xT,y1,…,yT)`c*(y1,…,yT,z1,…,zT)→w(x1,…,xT,z1,…,zT)]

is a tautology. Let c be a configuration which satisfies that forany t and r either xt5yr or ¬xt5yr appears within the configura-tion and xt5yr appears in the conjunction if and only if there isan s such that both xt5ys and xs5yr appear in c*. Then c mustbe a configuration in the disjunctive normal form ofw(x1,…,xT,y1,…,yT). N(c) is a subset of N(c*) and thus, by theminimality of N(c*) it must be the case that N(c*)5N(c).Denote by s the permutation on the set N(c*), defined by s(i)5jif xi5yj appears in c*. There must be an integer n such that sn isthe identical permutation. The formula w is the disjunction of


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conjunctions and one of these must be the conjunction of xi5yi

for i[N(c*) and the negation of all other equalities, contradict-ing the asymmetry tautology.

R E F E R E N C E S

Boolos, G.S. and R.C. Jeffrey (1989) Computability and Logic,Boston and London: Cambridge University Press

Crossley, J.N., J.C. Stillwell, C.J. Brickhill, N.H. Williams andC.J. Ash (1990), What Is Mathematical Logic? Oxford:Dover Publications

Robinson, A. (1963) Introduction to Model Theory and to theMeta Mathematics of Algebra, Amsterdam: North Holland

Rubinstein, A. (1978) “Definable Preference Relations – ThreeExamples,” Center of Research in MathematicalEconomics and Game Theory, RM 31, the HebrewUniversity of Jerusalem

(1984) “The Single Profile Analogues to Multi-profileTheorems: Mathematical Logic’s Approach,” InternationalEconomic Review, 25, 719–30

(1998a) Modeling Bounded Rationality, Cambridge, MA:MIT Press

(1998b) “Definable Preferences: An Example,” EuropeanEconomic Review, 42, 553–60

(1999) “Definable Preferences: Another Example (Searchingfor a Boyfriend in a Foreign Town),” Proceedings of the 11thInternational Congress of Logic, Methodology and Philoso-phy, forthcoming

Shelly, M.Y. and G.L. Bryan (1964) “Judgments and theLanguage of Decisions,” in Shelly, M.Y. and G.L. Bryan(eds.), Human Judgment and Optimality, New York: JohnWiley


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C H A P T E R 5

ON THE RHETORIC OF GAME THEORY

5.1 Introduction

This chapter contains some comments on the languageused by game theorists. As such, it is quite different fromthe rest of the book. Why am I interested in the study of therhetoric of Game Theory? Admittedly, “I am not thrilled”with the fact that Game Theory is viewed by many as“useful” in the sense of providing a guide for behavior instrategic situations. This view is reinforced by the lan-guage we game theorists use.

Consider, for example, John McMillan’s Games,Strategies and Managers (1992). On the cover of the paper-back edition is a quote from Fortune praising the book as“the most user friendly guide for business people,” andAkira Omori of Arthur Andersen Co. is quoted as sayingthat “[the book] will be helpful both for beginning manag-ers and for advanced strategic planners. In fact, I wouldrecommend that anyone engaged in the US–Japan negotia-tions read the book.”

Or consider Avinash Dixit and Barry Nalebuff’s bestseller,Thinking Strategically. The Financial Times (December 7,1991) says of the book: “Thinking Strategically … offersessential training in making choices and weighing possibil-ities not only in business but in daily life.” Schlomo Maitalsays in a review of the book published in Across the Board(June 1991): “Ever since 1926, when John von Neumann, abrilliant Hungarian mathematician and physicist, published

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Many of the ideas in this chapter have appeared in my previous writings.In particular, the discussion of “strategy” in section 5.3 is based onRubinstein (1991).

his path-breaking paper on Game Theory, it has been knownthat analytical models of games could be built. Thesemodels make it possible to find the best strategies and thebest solutions for games, including ones that embodycomplex business dilemmas.”

The public misconception of these books is reinforced bythe authors themselves. McMillan says: “Stripped of itsmathematics and jargon, it can be useful to people in man-agerial situations faced with important decisions and strat-egy,” and the opening sentence of his book asks: “What isGame Theory about, and how can it be used in makingdecisions?” Dixit and Nalebuff write: “It is better to be agood strategist than a bad one, and this book aims to helpyou improve your skills at discovering and using effectivestrategies,” “Our aim is to improve your strategy IQ,” and“Our premise in writing this book is that readers from avariety of backgrounds can become better strategists ifthey know these principles.”

The phenomenal success of Game Theory and the per-ception that it may “improve strategy IQ” are linked, inmy opinion, to the rhetoric of Game Theory. I doubt thatGame Theory would have attracted the same attention if itwere named “the theory of interactions between rationalagents.” The fact that one might conceivably find a book“about” Game Theory on the New York Times bestsellerlist and that students are fascinated by the “Prisoner’sDilemma” stems from the natural associations of gametheoretic jargon: Words like “game,” “strategy,” and “solu-tion” stimulate our imagination. Is the use of these wordsjustified?

Words are a crucial part of any economic model. An eco-nomic model differs substantially from a purely mathe-matical model in that it is a combination of mathematicalstructure and interpretation. The names of the mathemat-ical objects are an integral part of an economic model.When mathematicians use everyday concepts such as“group” or “ring,” it is only for the sake of convenience.When they name a collection of sets a “filter,” they do it inan associative manner and in principle they could call it an“ice cream cone.” When they use the term “good ordering”


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they are not assigning any positive ethical value. The situ-ation is somewhat different in mathematical logic inwhich, for example, the names of the connectives arestrongly related to their common meaning. In economictheory, and Game Theory in particular, interpretation is anessential ingredient of any model. A game varies accordingto whether the players are human beings, bees, or different“selves” of the same person. A strategic game changesentirely when payoffs are switched from utility numbersrepresenting von Neumann and Morgenstern preferencesto sums of money or to measures of evolutionary fitness.

The evaluation of Game Theory rhetoric is also impor-tant from the public’s point of view. The public’s interestin Game Theory is at least partially the result of efforts onthe part of academic economists to emphasize the practi-cal value of Game Theory as, for example, a guide to policymakers. Consultants use the professional language ofGame Theory in their arguments and it is in the publicinterest that these arguments be properly understood sothat the rhetoric does not distort their real content.Although the discussion of the usefulness of Game Theorycan take place on an abstract level without referring to thesubstance of game theoretic models, it does require anunderstanding of Game Theory beyond that required forthe “Prisoner’s Dilemma.”

I am doubtful about the practical applicability of GameTheory. However, I do not feel pessimistic since I don’tregard applicability as necessarily a virtue. I do not believethat the study of formal logic can help people become“more logical,” and I am not aware of any evidenceshowing that the study of probability theory significantlyimproves people’s ability to think in probabilistic terms.Game Theory is related to logic and probability theory andI doubt that it could prove useful to negotiators or othertypes of players. In fact, I suspect that Game Theory couldeven be misleading them since people often ignore the sub-tleties of game-theoretic arguments and treat solution con-cepts as instructions. (Nevertheless, I feel obliged tomention that I failed to demonstrate this in an experimentcarried out in collaboration with a group of Tel Aviv

On the rhetoric of game theory

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University students. The results of the experiment wereinconclusive.)

In fact, McMillan’s book is full of reservations on thedirect usefulness of Game Theory: “Game Theory does notpurport to tell managers how to run their business”;“Game Theory does not eliminate the need for the knowl-edge and intuition acquired through long experience”;“Game Theory offers a short cut to understanding the prin-ciples of strategic decision making. Skilled and experi-enced managers understand these principles intuitively,but not necessarily in such a way that they can communi-cate their understanding to others”; and “Game Theory,then, is a limited but powerful aid to understanding strate-gic interactions.” Dixit and Nalebuff (in spite of theirstatement “Don’t compete without it”) state that “in someways strategic thinking remains an art.”

And indeed when it comes to describing the usefulness ofGame Theory in actual cases, all authors adopt moremodest goals and become rather vague. Despite thepromise to be a user-friendly guide, McMillan concludeshis chapter on “Negotiating,” a topic which he considers tobe the “archetypical problem of Game Theory,” with themodest statement: “What advice for negotiators does GameTheory generate? The most important idea we have learnedin this chapter is the value of putting yourself in the otherperson’s shoes and looking several moves ahead.” Dixit andNalebuff’s “Rule 1” is to “Look ahead and reason back.”They state the general principle for sequential-move gamesto be “that each player should figure out the other players’future responses, and use them in calculating his own bestcurrent move,” and the final sentence of their book reads“You can bet that the most heavily played machines are notthe ones with the highest payback.”

If the most important lesson from reading a “user-friendly guide to Game Theory” is the value of puttingyourself in the other person’s shoes, and if improving yourstrategic IQ is boiled down to “anticipating your rival’sresponse,” I am not convinced that Game Theory is morevaluable than a detective novel, a romantic poem, or agame of chess in achieving these goals.


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Incidentally, is it clear that academic knowledge is neces-sarily valuable? Is it clear that improving strategic IQ is adesirable goal? David Walsh, a Washington Post writer, said“The problem is, of course, that if Dixit and Nalebuff canimprove your strategic IQ, they can improve your competi-tor’s as well.” This hints at an important point which isvalid for other social sciences as well: Improving strategicreasoning, even if it is possible (and I doubt that it is), shouldbe evaluated by social standards which take into accountnot only the success of a company’s strategic analyst butgeneral welfare as well. Is it desirable to improve the strate-gic IQ of a small sector of the population, especially onewhose position we might not wish to improve?

Incidentally, the discussion of the rhetoric of economicsis not a new subject. The best known research on thissubject was published in the mid-1980s (McCloskey, 1998,2nd edn.) and was followed by an extensive literature (see,for example, Henderson, Dudley-Evans and Backhouse,1993). My contribution in this chapter is quite limited. Iwill point out some of the rhetorical problems associatedwith the language of Game Theory. I will argue that therhetoric of Game Theory is actually misleading in that itcreates an impression that it has more practical applicationthan it actually does. In particular, I wish to show that:

(1) People look to Game Theory for advice on what strat-egy to adopt in game-like situations; however, thebasic notion of “strategy” in Game Theory can hardlybe interpreted as a course of action.

(2) The use of formulae in Game Theory creates an illusionof preciseness which does not have any basis in reality.

5.2 Strategy

“Strategy” in daily language

One of the central terms in Game Theory is “strategy.”What is a strategy in ordinary language? McMillan (1992)teaches us that the source for the word “strategy” is theGreek word for the leader of an army, which is rather dif-ferent from its usage today. Webster’s dictionary defines


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the word as “a method for making or doing something orattaining an end,” and the Oxford English Dictionarydefines the word as “a general plan of action.” We talkabout the strategy to win a war, the strategy of managers tomake profits or avoid being fired, and the strategy tosurvive a hurricane.

The informal definition of strategy according to gametheorists is not far from the one in everyday language.Martin Shubik refers to a strategy as “a complete descrip-tion of how a player intends to play a game, from beginningto end.” Jim Friedman defines it as “a set of instructions.”And John McMillan defines strategy as “a specification ofactions covering all possible eventualities.” Thus, it isappropriate to ask whether the formal definition of theterm “strategy” in Game Theory is similar to its everydaymeaning. If it is not, then Game Theory’s usefulness is putinto serious doubt; and even if it is, there is still a long wayto go before we can be persuaded that game-theoreticresults can teach something to managers.

A strategy in an extensive game

In an extensive game, a player’s strategy is required tospecify an action for each node in the game tree at whichthe player has to move. Accordingly, a player has to specifyan action for every sequence of events which is consistentwith the rules of the game. As an illustration, consider thetwo-player game form in figure 5.1:

According to the natural definition of strategy as a “com-plete plan of action,” player 1 is required to specify hisbehavior, “Continue” (C) or “Stop” (S), at the initial nodeand, if he plans to “Continue,” to make provisional plansfor his second decision node in the event that player 2chooses C. In contrast, the game-theoretic definition of


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1 C 2 C 1 C

S S S

Figure 5.1

strategy requires player 1 to specify his action at thesecond decision node, even if he plans to “Stop” the gameat the first node. Here, as in any game which requires aplayer to make at least two consecutive moves (and mostof the games which have been analyzed in economictheory fall into this category), a strategy must specify theplayer’s actions even after histories which are inconsistentwith his own strategy.

Why does the notion of strategy used by game theoristsdiffer from a “plan of action”? If we were investigating onlyNash equilibria of extensive games, then the game-theoreticdefinition would indeed be unnecessarily broad. The broaddefinition is, however, necessary for testing the rationalityof a player’s plan, both at the beginning of the game and atthe point where he must consider the possibility of respond-ing to an opponent’s potential deviation (the subgameperfect idea). Returning to the game form above, assumethat each player plans to choose “Stop” at his first decisionnode. Testing the optimality of player 2’s plan followingplayer 1’s deviation requires that player 2 specify his expec-tations regarding player 1’s plan at his second decision node.The specification of player 1’s action after both players havechosen C provides these expectations and has to be inter-preted as what would be player 2’s (as opposed to player 1’s)belief regarding player 1’s planned future play should player1 decide to deviate from what was believed to be his originalplan of action. Thus, a strategy encompasses not only theplayer’s plan but also his opponents’ expectations in theevent that he does not follow that plan. Hence, an equilib-rium strategy describes a player’s plan of action as well asthose considerations which support the optimality of hisplan (i.e. preconceived ideas concerning the other players’plans) rather than being merely a description of a “plan ofaction.” A profile of strategies provides a full analysis of thesituation and not merely a tuple of plans of actions.

A mixed strategy in a strategic game

The naive interpretation of “mixed strategy” requires thata player use a roulette wheel or some other random device


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to decide upon an action. Normally, we are reluctant tobelieve that our decisions are made at random; we prefer tobelieve that there is a reason for each action we take. Mostgame theorists never considered using random devices aspart of a strategy before being exposed to Game Theory.Indeed, the concept of mixed strategy has been a target forcriticism. To quote Aumann (1987): “Mixed strategy equi-libria have always been intuitively problematic” andRadner and Rosenthal (1982): “One of the reasons whygame theoretic ideas have not found more widespreadapplication is that randomization, which plays a major rolein Game Theory, seems to have limited appeal in manypractical situations.”

There are obviously cases in which players do chooserandom actions. When a principal (for example, anemployer or the government) monitors his agents (employ-ees or citizens), often only a few are monitored and they arechosen randomly. Of course, this is not a “mixed strategy.”The stochastic rule used for auditing is actually a purestrategy. The principal is not indifferent between monitor-ing no agents and monitoring all of them. His “pure strat-egy” is the parameters of the randomization to be used. Asan additional example, a parent who must allocate onepiece of candy to one of his two children may strictly preferto flip a coin. However, this flip of a coin is not a “mixedstrategy” either. Typically, the parent would not be indif-ferent between procedures for allocating the candy. Anallocation procedure should be regarded as a pure strategyeven if it involves flipping a coin.

The game-theoretic literature suggests a few more inter-pretations of this concept. The large population frequen-cies interpretation of a mixed strategy was used in chapter2. The game is played occasionally between players, eachdrawn from a population homogeneous in its interests butheterogeneous in its actions. The mixed strategy describesthe distribution of actions taken by the agents who aredrawn from the population attached to any given role inthe game. By a different interpretation, a mixed strategy isconceived of as a plan of action which is dependent onprivate information not specified in the model. According


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to this interpretation, a player’s behavior is actually deter-ministic although it appears to be random. If we add thisinformation structure to the model, the mixed strategybecomes a pure one in which actions depend on extraneousinformation. This interpretation is problematic since it isnot clear why people would base their behavior on factorswhich are clearly irrelevant to the situation (see Harsanyi,1973, for a more sophisticated interpretation whichresponds to this criticism).

There is an additional interpretation which has gainedpopularity during the past decade. According to this inter-pretation, a mixed strategy is the belief held by all otherplayers concerning a player’s actions (see Aumann, 1987).A mixed strategy equilibrium is then an n-tuple ofcommon knowledge expectations; it has the property that,given beliefs, all actions with a strictly positive probabilityare optimal. Thus, the uncertainties behind the mixedstrategy equilibrium are viewed as an expression of thelack of certainty on the part of the other players ratherthan an intentional plan of the individual player.

Does it matter?

It would appear that two central versions of the game-theoretic notion of a “strategy” – i.e. an extensive formstrategy and a mixed strategy in strategic games – are moreappropriately interpreted as beliefs rather than plans ofactions. However, if strategies are beliefs rather than plansof actions, the entire manner of speaking in Game Theoryhas to be reassessed. We can speak of player 1 choosing hisstrategy, but does he choose player 2’s belief about hisstrategy?

As a demonstration of the difficulties which follow fromthe beliefs interpretation, consider the sequential games lit-erature in which authors sometimes assume that strategiesare stationary in the sense that a player’s behavior is inde-pendent of the history of the game. This literature presentsstationarity as an assumption related to simplicity of behav-ior. For example, consider player 1’s strategy “always playa” in a repeated game. This strategy is simple in the sense


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that the player plans to make the same move independentlyof the other players’ actions. However, this strategy alsoimplies that player 2 believes that player 1 will play a even ifplayer 1 has played b in the first 17 periods of the game.Thus, stationarity in sequential games implies not onlysimplicity but also passivity of beliefs. This is counter-in-tuitive, especially if we assume simplicity of behavior. Ifplayer 2 believes that player 1 is constrained to choose a sta-tionary plan of action, then player 2 probably should believe(after 17 repetitions of the action b) that player 1 will con-tinue to play b. Thus, by assuming passivity of beliefs, weeliminate a great deal of what sequential games are intendedto model – i.e. the changing pattern of players’ behavior andbeliefs as players accumulate experience.

With regard to mixed strategies, the adoption of thebeliefs interpretation requires the reassessment of much ofapplied Game Theory. In particular, it implies that equilib-rium does not lead to a statistical prediction of players’behavior. Any action taken by player i as a best responsegiven his expectation about the other players’ strategies isconsistent as a prediction for i’s future action (this mighteven include actions which are outside the support of themixed strategy). This renders meaningless any compara-tive statics or welfare analysis of the mixed strategy equi-librium and brings into question the enormous economicliterature which utilizes mixed strategy equilibrium.

5.3 The “numbers” illusion

Although the question of whether we should represent anindividual by a preference relation or a utility functionmay seem immaterial (since a utility function simply rep-resents a preference relation numerically), I will argue thatthis is not the case and that the “numeration of GameTheory” sends a very misleading message with regard tothe precision and relevancy of Game Theory.

To demonstrate this point, let us briefly discuss theNash bargaining solution which is one of the most funda-mental models in modern economic theory. Nash’s theoryis designed to provide a “prediction” of the bargaining


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outcome based on two elements: (1) the bargainers’ prefer-ences which are defined on the set of possible agreements(including the event of disagreement), and (2) the bargain-ers’ attitude towards risk.

The primitives of Nash’s (two-person) bargainingproblem are the “feasible set” and a “disagreement point”,denoted by S and d, respectively. In Nash’s formalization ofthe bargaining problem each element of S corresponds tothe pair of numbers interpreted as the utility levelsobtained by the two bargainers at one (or more) of the pos-sible agreements. The utilities are understood to be vonNeumann–Morgenstern in that they are derived from pref-erence relations over lotteries that satisfy the expectedutility assumptions. The disagreement point is modeled asone of the points in S. The second basic concept of Nash’sbargaining theory is the bargaining solution, defined as afunction which assigns a unique pair of utility levels toeach bargaining problem (S,d). Thus, a bargaining solutionis supposed to provide a “unique prediction” of the bar-gaining outcome (in utility terms) for each of the problemsin its domain.

Nash (1950) showed that there is a unique bargainingsolution satisfying the following four axioms: Invarianceto Positive Affine Transformations (the re-scaling of eachbargainer’s utility values by a positive affine transforma-tion shifts the solution accordingly); Symmetry (a sym-metric problem has a symmetric solution point); ParetoOptimality (for each bargaining problem, the solution ison the Pareto frontier); and Independence of IrrelevantAlternatives (IIA) (the shrinking of the set of alternativeswithout eliminating the solution point and without elimi-nating the disagreement point does not alter the solutionpoint) which is the most problematic of the axioms interms of its interpretation. The unique solution satisfyingthese axioms is the Nash solution – i.e. the function

N(S,d)5argmax{(u12d1)(u22d2) | (u1,u2)[S and ui$di for both i}.

If the task of bargaining theory is to provide a “clear-cut”numerical prediction for a wide range of bargaining prob-lems, then Nash certainly achieved this goal. The fact that


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it is defined by a simple formula is a significant advantageof the theory, especially when we try to embed it in a largermodel that contains a bargaining component. But can thisprediction be tested as in the sciences? Does the formulapredict how people will share a pie with the accuracy thatwe can calculate the moment that a stone falling from atower will hit the ground?

I doubt it very much. The significance of Nash’s formuladerives from its abstract meaning regardless of its testabil-ity. The use of numbers, even if analytically convenient,obscures the meaning of the model and creates the illusionthat it can produce quantitative results. Does the Nashbargaining solution perhaps have a different, more abstractmeaning?

On the face of it, the answer is “no.” The formula of theNash bargaining solution lacks a clear meaning. What isthe interpretation of the product of two von Neumann–Morgenstern utility numbers? What is the meaning of themaximization of that product? Can we consider the max-imization of the product of utilities to be a “useful” princi-ple for resolving conflicts?

The use of numbers to specify the bargaining problemhas obscured the meaning of the Nash bargaining solution.Were game theorists to use a more natural language tospecify the model, the solution would become clearer andmore meaningful. In Rubinstein, Safra and Thomson(1992) the indirect language of utility is replaced by thedirect language of alternatives-preferences. The model’selements are interpreted directly using the words “alterna-tive,” “lottery,” “disagreement” and “preference.” TheNash problem, (S, d), is “replaced” by a four-tuple(X, D, s21, s22) where X is a set of feasible agreements, D is a“disagreement” event, and (s21, s22) is the bargainers’ pref-erence relations. As Nash’s theory aims to predict the bar-gaining outcome as a function of the players’ interests andtheir attitude to risk, preferences are taken to be definedover the set of lotteries whose “certain prizes” are theagreements in X and the disagreement event D.

Using this model, an alternative definition of the Nashbargaining solution can be formulated:


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An (ordinal) Nash solution for the problem (X,D, s21, s22)is an alternative y* such that for all p[[0,1] and all x[Xand i, if poxs2iy* then poy*s2jx (where pox is the lotterywhich gives x with probability p and D with probability12p).

The suggested interpretation of this solution is asfollows: A solution is a “convention” which attaches aunique agreement to any bargaining problem. The solutionembodies the assumption that players are aware that whenthey raise an objection to an alternative, they risk that thenegotiations will end in disagreement. The Nash solutionagreement is the only one which satisfies the followingproperty:

If–

• It is worthwhile for one of the players to demand animprovement in the convention, even at the risk of abreakdown in negotiations,

then–

• It is optimal for the other player to insist on followingthe convention even at the risk of a breakdown in negoti-ations.

In other words, the Nash bargaining solution is an agree-ment satisfying the condition that any argument of thetype “You should agree to my request x without delay,since this is preferable to you over the convention y* giventhe probability 12p of breakdown” is not profitable for theplayer making the request when he takes into account thesame probability of breakdown.

The Gulf War provides a concrete example of this defini-tion: The bargainers were Iraq and the USA. The set ofagreements contained the various possible partitions ofthe land in that region. The disagreement event was a war.When Saddam Hussein moved his troops he deliberatelytook the chance that the situation would deteriorate intowar before the USA capitulated. Apparently he preferredthe risk of war with a certain probability while hoping thatthe USA would agree to his request to annex Kuwait. Incontrast to his expectation, the USA preferred to take the


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risk of war and basically demanded a return to the statusquo rather than give in to Iraq’s demands. If the USA hadyielded to Iraq, it would have meant that the pre-invasionborders were not part of a Nash bargaining outcome.

When preferences satisfy the expected utility assump-tions, this definition and the standard one converge (seeRubinstein, Safra and Thomson, 1992). However, a byprod-uct of the ordinal definition is that Nash’s theory can beextended to a domain of preferences beyond those whichsatisfy the expected utility theory. The language of prefer-ences does not require the preferences to satisfy theexpected utility axioms; thus the definition can be appliedto a wider set of preferences on the set of lotteries over theagreements and the disagreement point. One can showthat for a wider class of preferences (which includes allpreferences satisfy the expected utility axioms), the Nashsolution is well defined and is the unique solution whichsatisfies a set of axioms which resemble the Pareto,Symmetry, and IIA axioms.

Indeed the switch to the alternatives-preferences lan-guage requires a restatement of the entire Nash theory.The details are beyond the scope of this book and I willcomment only on the change in the meaning of the IIAaxiom. The original IIA axiom requires that if u* is autility pair which is the solution of the problem (T,d), andif u* is a member of a set S which is a subset of T, then u* isalso the solution outcome of (S,d). As has often beenemphasized, this justification of the IIA axiom suits a nor-mative theory in which the solution concept is intended toreflect the social desirability of an alternative. When bar-gaining is viewed as a strategic interaction of self-interested bargainers, the validity of the IIA axiom isquestionable.

The following is an alternative statement of the IIAaxiom which does not require a comparison betweenproblems having different sets of alternatives. The“ordinal-IIA” states that a solution to a bargainingproblem y* remains invariant to a change in i’s preferencethat would make him less willing to raise objections to y*.Formally, assume that y* is the solution to the problem (X,


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D, s21, s22) and let s29i be a preference which agrees with s2i

on the set of deterministic agreements X such that:

(i)i for all x such that xs2i y*, if pox~i y* then poxa29i y* and(ii) for all x such that xa2i y*, if x~9i poy* then poxa29i y*.

Hence, y* is also the solution to the problem (X, D, s29i, s2j).The switch of player i’s preference from s2i to s29i reflects

his increased aversion to the risk of demanding alterna-tives which are better for him than the outcome y*. Thechange in player i’s preference makes player i “lesswilling” to object. Though bargainer i, with preference s29i,has the same ordinal preference over X as s2i, he is lesswilling to take the risk of demanding agreements whichare better than x*. The axiom states that this change doesnot affect the bargaining outcome. In other words, chang-ing player i’s preference to make him less willing toachieve agreements which are better than y* does notchange the bargaining outcome. With this interpretation ofIIA, the link between the axiomatization and the ordinaldefinition of the Nash solution seems more comprehen-sible. This axiom makes sense when the underlying bar-gaining procedure is such that there is a clear conventionand an objection can be “protected” from other objectionsonly by insisting on the retention of the convention. Itmakes less sense when counter-arguing can be carried outby other means, such as raising alternative demands, orwhen agreement is reached by the two parties graduallyreducing their demands.

Let us return to the main point of this section: The lan-guage of utility allows the use of geometrical presentationsand facilitates analysis; in contrast, the numerical presen-tation results in an unnatural statement of the axioms andthe solution. The switch to the language of alternatives-preferences allows a more natural statement and charac-terization of the Nash solution. One may argue thatthe equivalence between the terms “preferences” and“utility” in economic models implies equivalence in theiruse as well. However, as demonstrated here, the choice oflanguage affects the meaning of the results. The use of thelanguage of utility calls for arithmetic operations (such as


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multiplying utilities) which do not necessarily have mean-ingful interpretations. The use of the language of prefer-ences leads to the derivation of a solution which resemblesa consideration which could conceivably be used in reallife. Nevertheless, I am not convinced that Nash’s theoryhas done more than clarify the logic of one considerationwhich influences bargaining outcomes. I can not see howthis consideration will comprehensively explain real-lifebargaining results.

5.4 The term “solution”

In the previous two sections I discussed the use of the term“strategy” and the substitution of “utilities” for prefer-ences. In this section, I want to briefly discuss another keyterm in Game Theory – “solution.” The structure of agame-theoretic analysis is as follows: first, the modelerspells out the parameters of the game (identities of theplayers, rules, information available to the players, and theplayers’ interests). The analysis then moves on to “findingsolutions.” The result of applying a solution concept to agame is a set of profiles each of which assigns a strategy toeach player. A solution concept is a method which assignsto each game a set of profiles of strategies satisfying certainconditions of stability and rationality.

This form of investigation distinguishes between theassumptions embedded in the solution concept and thoseunderlying the description of the game. The former areregarded as being most solid, although they are in fact verystrong assumptions concerning the players’ behavior andprocess of deliberation. Since they are buried in the solu-tion concept, these assumptions are almost never discussedby users of Game Theory. Incidentally, these assumptionsare not all that clear to game theorists either, and it was along time before game theorists began examining the epis-temological assumptions underlying the various solutionconcepts.

Thus, whereas applied economists often debate theassumptions of a model they pay almost no attention tothe appropriateness of the assumptions which are incorpo-


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rated into the solution concept. A solution concept istreated as a machine which receives a situation as an inputand produces an output which is interpreted as a predic-tion of the way in which the game will be resolved.

In my opinion, the use of the particular word “solution”is central in creating the view that Game Theory is a “pre-diction machine.” The term “solution” implies the exis-tence of a problem. In the case of Game Theory theproblem is to figure out “what will happen” or “whatmove to make.” The word “solution” creates a sense ofbeing clearly defined and correct. The set of solutions for amathematical equation is certainly well defined; can wesay the same of the set of solutions to a game? Though thesolution of a game is often viewed as being analogous tothe solution of an equation or maximization problem, infact, a game-theoretic solution concept is no more than thestatement of a particular consideration which we havechosen to focus on.

Another term often used during discussions of themeaning of solution concepts is “recommendation.” It isoften said that a solution concept provides a profile ofrecommendations which are not “self-defeating.” Forexample, a Nash equilibrium is said to be a profile of rec-ommendations, one for each player, with the property thatif players believe that the recommendations will be fol-lowed, then no one has an interest in not following his rec-ommendation.

I think that the use of the term “recommendation” here ismisleading. The term implies “authority,” “an instruc-tion,” and the “right thing to do.” Can the term Nash equi-librium be considered as a synonym for recommendation? Arecommendation must be given by someone. Who ismaking the recommendation in this case? Is he just a gametheorist? What are his goals? Can a Pareto-inferior Nashequilibrium be considered a recommendation? And whyshould the source of the recommendation limit himself torecommendations which are not self-defeating? Would itnot be better to give each of the two players in a centipedegame a recommendation to never stop the game eventhough it is not a self-defeating recommendation?


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5.5 Final comments

The reader may have gotten the impression that thepurpose of this chapter is to diminish the importance ofGame Theory and to discourage potential users. On thecontrary, I think that Game Theory is moving into newareas of interest and we will soon be witness to excitingdevelopments. However, I have no expectations of GameTheory becoming “practical” as the term is understood bymost people. Since I have never understood these expecta-tions of Game Theory I am neither disappointed nor pessi-mistic.

I see analytical economic theory as a search for connec-tions between concepts, assumptions, and assertionswhich we use in understanding human interaction.Economic theory is an investigation of the different typesof considerations used by human beings to analyze humaninteraction. A strategic situation can be analyzed in manyways. Typically, there are many different arguments sup-porting one course of action or another. The most we cando analytically (as opposed to empirically) is to clarify theconnections between the different types of analysis.“Drawing links” and “understanding,” however, stop farshort of giving advice and being practical. In my opinion,achieving a better understanding of the arguments used inhuman interaction and reasoning is in itself a very worth-while and exciting project. For other views of the meaningof Game Theory, see Aumann (1987) and Binmore (1990).

At a conference where I presented this chapter, I wasasked by a listener whether I feel frustrated that my workand views have made no contribution to the world.Perhaps I am. However, I don’t think that this lecture isuseless in the sense of “influencing the world.” There arefew tasks in the academic world of the social sciencesmore important than fighting unjustified authority. Thischapter is my modest contribution to this importantbattle.


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CONCLUDING REMARKS

At the opening of this short series of lectures, I briefly pre-sented five issues which fall under the heading of “eco-nomics and language.” By now you have realized that theissues were indeed quite distinct. I found it difficult topoint out the common factor among the topics other thanthe title and the speaker.

I am deeply grateful to three people who have (surpris-ingly) agreed to comment on this manuscript. TilmanBörgers, who was also the original discussant inCambridge in 1996, Bart Lipman, and Johan van Benthem.Tilman and Bart bring some perspectives from economists,and Johan brings his perspective as a logician. It must behard to write such a review, especially when one wants tocriticize the author. Thus, I wish that the reader will pullout from the comments especially the critical commentsand the suggestions for further research.

The three discussants helped me out in finding thecommon features of the first four essays. As Bart Lipmanputs it, the first four chapters touch three related ques-tions: “What gives a statement its meaning?,” “Why is lan-guage the way it is?,” and “How does language affectaction?”

The discussants also persuaded me that the last chapteris very different in nature. If it belongs to this book beyondmy rhetorical use of the phrase “economics and language”,it is only because it presents my general approach to eco-nomic theory, which is a thread running through these lec-tures. By this approach economic theory

• speaks more about methods and modeling issues• talks less about economic substance

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• emphasizes the complexity issue as a major element inthe discussion

• treats economic theory as the study of the argumentsused by human beings

Illuminating the arguments which human beings use insocial interactions and investigating “the logic of the situ-ation,” a phrase Karl Popper uses to describe the aim of thesocial sciences, is indeed what I perceive to be the objec-tive of economic theory.

R E F E R E N C E S

Aumann, R. (1987) “What is Game Theory Trying toAccomplish?,” in K.J. Arrow and S. Honkapohja (eds.),Frontiers of Economics, Oxford: Blackwell

Binmore,K. (1990), “Aims and Scope of Game Theory,” in K.Binmore, Essays on the Foundations of Game Theory,Oxford: Blackwell

Dixit, A.K. and B.J. Nalebuff (1991) Thinking Strategically,New York: Norton

Harsanyi, J. (1973) “Games with Randomly Distributed Payoffs:A New Rationale for Mixed-strategy Equilibrium Points,”International Journal of Game Theory, 2, 486–502

Henderson, W., T. Dudley-Evans and R. Backhouse (eds.) (1993)Economics and Language, London: Routledge

McCloskey, D.N. (1998) The Rhetoric of Economics (Rhetoricof the Human Sciences), 2nd edn., Madison: University ofWisconsin Press

McMillan, J. (1992) Games, Strategies, and Managers, Oxford:Oxford University Press

Nash, J. (1950) “The Bargaining Problem,” Econometrica, 18,155–62.

Radner, R. and R. Rosenthal (1982) “Private Information andPure Strategy Equilibrium,” Mathematics of OperationResearch, 7, 401–9.

Rubinstein, A., (1991) “Comments on the Interpretation ofGame Theory,” Econometrica, 59, 909–24.

Rubinstein, A., Z. Safra and W. Thomson (1992) “On theInterpretation of the Nash Bargaining Solution,”Econometrica, 60, 1171–86.

Some concluding remarks

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P A R T 3

COMMENTS


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1 The ubiquity of language

Language is the air that we breathe when thinking andcommunicating, often without noticing. It makes humancognition possible, and at the same time constrains it, inoften invisible, but very real ways. Philosophy took a “lin-guistic turn” in this century, when this crucial mediumbecame a focus of attention, out in the open, especially inthe analytical tradition. The result were new insights intothe structure and functioning of language, which havegone into modern logic and linguistics, often under theheading of “semantics.” Ariel Rubinstein makes such alinguistic turn as an economist. He investigates “the eco-nomics of language” – i.e. optimization mechanisms thatmake language work, both as a structure and as an activity,but also “the language of economics,” linguistic mech-anisms underlying economic behavior and our theorizingabout it. More generally, his book is an attempt to positionGame Theory in a broad intellectual landscape of reason-ing and communication: which I find very congenial.Indeed, the author writes about Paul Grice’s views in“Logic and Conversation”: “[this] is essentially a descrip-tion of one agent thinking about how another is thinking.This is precisely the definition of strategic reasoning and isthe essence of game theory.” I found that quote interestingas a logician, because the analysis of many-agent com-munication is also a central topic on the modern agenda oflogic, computer science, and cognitive science. Are we allin the same boat? Let’s compare Rubinstein’s agenda withthat of people like me.

Attention to language is of the essence in logic. The

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logical forms that drive valid reasoning are linguistic innature, and hence issues of syntax and semantics havebeen central to the development of logic from the start.True, logicians have found it necessary to design special-purpose formal languages, as a supplement to natural lan-guages, that have started leading lives of their own. Forexample, when the author wants to survey possible formsof definition, he uses a first-order predicate logic, a nine-teenth-century formalism. But such formal languages arealso used in the study of various aspects of natural lan-guage. Grottlob Frege, the inventor of modern logic, com-pared its use to that of a microscope, versus naturallanguage, which is like the eye. As he says, formal lan-guages are more precise for specific purposes, but naturallanguage remains the much more versatile general-purposeinstrument. Logical theory since Frege has been describedas a “marriage of mathematics and linguistics.” Logicianswill typically study two sides of a coin: different syntactictypes of statement are connected with matching semantickinds of behavior (e.g. invariances across situations), oftendescribed mathematically.

2 From description to explanation

Much research in linguistics and logic is content withdescribing the facts of language and reasoning as they are –and not to explain how they came about, or continue func-tioning. A typical paper in, say, semantics of conditionalstatements in natural language might start with a descrip-tion of intuitively valid and invalid reasoning forms, andthen propose some underlying meaning for the key con-struction “if then” which shows how the valid inferencescome about, while avoiding the invalid ones. Here is a con-crete example. Consequents may be weakened freely: “if A,then B” implies “if A then B-or-C” – but antecedentscannot be strengthened freely: “if A, then B” does notalways imply “if A-and-C, then B.” (Let A be “I put sugar inmy coffee,” B “the coffee tastes fine,” C “I put Diesel oil inmy coffee”). Standard propositional logic does not explainthis fact, as the truth tables will validate both inferences.

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But the literature contains various proposals drawing anatural distinction. For example one often postulates apreference relation among situations (say of “relative plau-sibility”), and then calls a conditional “if A, then B” true ifB holds, not in all situations where A holds (which yieldsclassical entailment), but only in all most-preferred situa-tions where A holds. Other accounts of conditionals exist,too, providing “dynamic” alternative views of the samephenomena (including one by Grice). This is how far logicalanalysis usually goes. One systematizes the observablefacts of language use by postulating certain underlyingmechanisms, which may involve “theoretical terms” thathave no direct surface reflection. (Conditionals do not referto preferences explicitly.) The latter abstractions may thenbe used in other contexts too: e.g. preferences of variouskinds are gaining popularity in semantics.

Logical theories of this sort can have great sophistica-tion, and broad sweep. But what they do not attempt toexplain is how their key structures came about, or whythey function the way they do. These more ambitiousquestions are generally considered hard, having to do withunderlying mechanisms of cognition that we do not yetunderstand. Many people (including myself) believe thatnatural language strikes some kind of successful balancebetween expressive power for information content andresource complexity of cogitation and communication pro-cesses. But what do we mean by these terms in the absence(so far) of a truly fundamental theory of information andcognition? And another doubt: perhaps searching for deepexplanations may even be inappropriate. After all, humanlanguages are the result of a long evolution, with many his-torical accidents, where things could easily have goneanother way. We cannot “explain” human history. Whyshould we be able to “explain” language?

As opposed to this, Rubinstein offers a staunchly a prior-istic analysis of various linguistic phenomena, trying toshow how things function the way they do by being keptfirmly in place through the Invisible Hand of optimizationand strategic equilibrium. Economists in this mode areLeibnizian optimists: we live in “the best of all possible

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worlds,” or failing uniqueness, at least “in an optimal pos-sible world.” I myself love this kind of analysis (no matterits eventual prospects of success), which is in the samespirit as shortest-path principles in physics, or other grandideas driving science. Logic can profit from adding suchconcerns!

What does one want to explain about language? Let’s beambitious: (a) its structure (why it contains the expressionsthat it does), (b) the steady state of its competent use (therules that guide communication), (c) the mechanisms forlearning these skills (which take time), (d) the robustness oflinguistic practices, but also (e) the possibilities of failure,and then revision. The book offers a number of examples.One chapter explains certain lexical vocabulary, one ana-lyzes speech acts (warnings, and reactions to them), a thirdrules of dialogue (procedures for argumentation). The typesof explanation are different in each case, but I’ll come tothat. Of course, at this level of abstraction, we do not expectvery precise explanations for specific linguistic facts.Indeed, the book contains hardly any of the standard house-hold items for linguists or logicians. Nevertheless, what isoffered roughly follows the standard division into syntax,semantics, and pragmatics. And since there are no miracles,the simplicity of the game-theoretic analysis will necessar-ily be matched by the generality of the conclusions.

3 Explaining code

I will give my understanding of what Rubinstein does inhis case studies, and then compare with some logicalissues. The first batch of examples in chapter 1 of the bookconcerns linguistic code – namely, lexical meaningsexpressed by ordering predicates. Here analogies with stan-dard logical notions and approaches are clear, and the linksare obvious.

Why are certain orderings so ubiquitous?

According to the author, certain types of ordering relationare especially prevalent in natural language – transitive

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linear orderings and more generally, complete asymmetricorderings or tournaments. He explains the first phenome-non as follows. (Arguments always take place in somefixed finite model.) Linear orders allow one to describeeach individual in a domain uniquely via first-order logic,in terms of its position in the ordering. He then convertsthis observation (a typical mentality of a mathematician!),and shows that only linear orders have this unique identifi-cation property. In other words, given the importance ofunique reference, language must provide for it, and indeedit does. With tournaments, Rubinstein’s analysis is differ-ent, not in terms of maximal expressiveness, but minimalcomplexity. He considers definitions for binary relations Rthat we want to communicate by means of language toothers. Clearly, minimal complexity is a bonus here. Onenatural description format is this. Describe some generalproperties of R (again, in first-order formulas, such as thosedefining a tournament) and give some concrete examplesof ordered pairs that fall under it. Complexity can then bemeasured as the minimal number of concrete instancesone needs on top of some first-order description. He shows(modulo some technicalities) that the relations of lowestcomplexity in this sense are the tournaments – and, again,we are not surprised to see that language employs these inabundance. There is also fascinating stuff about approxi-mating relations in terms of hits-and-misses, but my aimis not to abstract the whole book.

Explanations for vocabulary

These analyses are very much in line with standard logicalanalysis, even though I have never seen quite these resultsin books on model theory. For example, consider the tour-nament example. Basically, a general first-order descrip-tion of a binary relation can only describe this up to its“isomorphism class”: all binary relations on the givenmodel sharing its first-order properties. To identify further,one needs to give some specific instances. Logicians haverather focused on the size of such isomorphism classes: theother side of the same coin, given that we are talking about

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(with n objects in our model) 2n2 binary relations alto-gether. As a minor quibble, I would certainly also want tocount the size of the general first-order formula as a param-eter of the complexity, not just the number of instances. Ifwe do that, we also get close to general principles in logicand computer science of so-called minimal descriptionlength. This is often implemented in terms of the“Kolmogorov complexity” of an object – i.e. the minimalsize of a program for some Turing machine that outputsthe object, in some effective representation. I think thatRubinstein’s analysis fits most naturally into this generalframework. Also, I think one can say more about thegeneral first-order part. Not all first-order descriptions areequal. One most effective device for saving on concreteinstances are “Horn clauses”: universal implications fromconjunctions of atoms to single atoms, such as transitivity:;xyz ((Rxy and Ryz)→Rxz). These automatically producefurther instances deterministically. Significantly, Hornclauses are the computational part of first-order logic,exploited in logic programming.

But logic has other methods than these for explainingthe occurrence of vocabulary – even though they areseldom presented together under that heading. I shallmention a few basic ones.

Type 1: natural invariances and their linguistic reflectionNature around us shows certain invariances – e.g. underEuclidean transformations of space (translations, rota-tions). It is an old idea of Helmholtz in the nineteenthcentury that such invariances will be naturally reflected inlanguage, which reserves basic lexical expressions for oper-ations or predicates that are invariant under these transfor-mations (such as geometric “betweenness” – incidentally,a basic ternary, not binary relation). There is a wholebunch of results in logic showing how logically definablepredicates in various languages coincide with invariantones of different kinds. For example, the basic logicalexpressions (Boolean set operations) are invariant for arbi-trary permutations of individuals in a domain. Furthercategories of expression in natural language, such as

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spatial prepositions (“inside,” “behind,” “toward”) will beinvariant for other classes of more geometrical transforma-tions. This type of invariance analysis occurs in thepresent book, too – but in connection with schemata ofdefinition for social choice.

Type 2: lowest complexity and its expressive reflectionEspecially in the logical literature on so-called generalizedquantifier expressions (“some,” “all,” “ten,” “most,”“few,” “enough”), there is a body of results of the follow-ing kind. Expressions of quantification may be associatedwith computation procedures that determine, given theobjects in a domain, whether the numerical truth condi-tion of the quantifier is satisfied. These procedures can beassociated with suitable automata. First-order quantifierscan be computed with finite-state automata, others, like“most,” rather require push-down store automata withunlimited memory. Now, it can be shown that natural lan-guage provides quantifier expressions for all “semanticautomata” of lowest complexity. Of course, quantifiers arejust one type of expression, close to logic. It would bemuch harder to explain the vocabulary in other categoriesof language, such as ordinary verbs (mathematicians onceattempted a classification of verbs via Catastrophe Theory)– but at some stage, we must also leave room for accidentsof our environment!

Note the two broad lines of thinking here. The firstexplains vocabulary as a reflection inside our cognitiveapparatus of invariances occurring outside of us in nature.The second explains vocabulary as a result of computa-tional limitations inside our cognitive apparatus. Bothseem attractive, and there need not be one uniform source.But there are still further approaches! The preceding wasabout individual agents, while we might also explainvocabulary through the necessities of social interaction(more congenial to game theory). Perhaps the oldest resultof this type is the Dutch Book Theorem from the 1950s,which explains the usual meanings of the propositionalconnectives in probabilistic reasoning as the only onesthat do not allow the making of unfair “Dutch books”

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against one’s opponents. This result has remained isolated,though (although Ariel Rubinstein informs me that therehas been some game-theoretic follow-up) – but I have oftenwondered about similar analyses for other logical notions.

Discussion

My conclusion is that Rubinstein’s analysis in chapter 1 isvery close to logic indeed – though certainly raising nicenew questions. But as we have seen, there are manyapproaches in this vein, and one should probably discusstheir comparative merits. On the critical side, I justmention one empirical point. Are the author’s observa-tions correct? Are linear orders or tournaments indeed thebasic stock of natural language? I doubt it. To me, the mostobvious linguistic category of binary relation are compara-tives. These are so basic that language even has a system-atic operation for building them: from “large” to “larg-er.”But the complete formal properties of comparatives areknown: they are asymmetric and “pseudo-connected”:;xyz (Rxy→(Rxz~Rzy)). This form of completeness isneither of Rubinstein’s two cases. So, I would say he stillowes us an equally clever account of comparatives.

4 Explaining speech acts

The next example (chapter 2), much more game-theoretic,is an account of the emergence of warning signals, utteredby speakers, and interpreted and then acted on appropri-ately by their hearers. This is closer to linguistic pragmat-ics, although it also involves semantics. The pattern ofexplanation here comes from evolutionary game theory,illustrated by a practical example. Basically, we canexplain the existence of warning speech acts as follows. Ifthe language did not have such a facility, a group of“mutants” could improve their communicative and practi-cal situation by endowing some expression with thismeaning, and utilizing the warnings amongst themselves.This phenomenon would then presumably spread eventu-ally to all users. Nevertheless, there is a difficulty. One

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cannot simply tamper with an existing language. If theaddition causes nonmutants to respond in harmful ways(e.g. panic as a response to the new warning signal), thenthe mutants may be worse off on average. Rubinstein thenexplains in progressively more technical terms how suchharmful reactions will not occur, provided we assume thatpreferences of language users do not just count utilities,but also the complexity of following a strategy. In particu-lar, panic is costly, so nonmutants will not bother.

I cannot pretend I got all the ins and outs of this, but thisgeneral pattern of explanation is very interesting. Languageeventually reaches some kind of maximal expressiveness,as measured by useful communicative functions. Never-theless, there are many arbitrary assumptions in theexample, and a more real challenge for me would be toprovide accounts involving similar assumptions for morecentral features of communication. In particular, althoughRubinstein discusses Grice’s Maxims of informativenessand relevance, he refrains from providing a game-theoreticunderpinning for them. Now that would be something!Another thing is this. We can distinguish between the per-formance of language users in “steady state” of maturecompetence versus their learning processes. How does thegame-theoretic account relate to this crucial distinction inunderstanding how language functions? Finally, I wouldlike to see some relation between the pattern of explana-tion given here and the more model-theoretic style of anal-ysis discussed in chapter 3. Take the issue of expressivecomplexity, which seems related to computational com-plexity for language users in getting the meanings right. Ithink complexity is over-riding: and language must do withwhat is feasible, rather than what is desirable. Thus, inmany cases, one would like to invert Rubinstein’s lexico-graphic preference for utility over complexity.

Finally, even though there are no direct analogues to thisevolutionary game analysis in logic, there are congenialthemes. For example, David Lewis’ famous account of theemergence of linguistic and nonlinguistic conventions hasa similar slant: what coordination moves will groups ofagents develop to achieve certain goals? In the subsequent

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literature, much emphasis has been placed on what theagents need to know about each other in order to com-municate effectively. I sense similar epistemic aspects tothe mutant thought experiment (e.g. mutants might cometo recognize mutants via signals, so as to exclude harm-ful confrontations with “old-timers”). But these remainimplicit in the analysis presented in the book.

5 Explaining rules

A third major example in the book are argumentation con-ventions (chapter 3). Rubinstein observes how argumenta-tion has a game-like structure, with moves and proceduralconventions. He then turns to specific empirical phenom-ena in “truth-finding.” In particular, one challenge is toexplain why certain arguments or counter-argumentsseem more persuasive than others to certain claims, evenwhen there is no obvious preference in terms of informa-tive value. Suppose someone claims that in most capitalsof the world, education levels are rising. Your opponentchallenges you and points out that it has sunk in Bangkokquite recently. You have positive evidence concerningManila, Cairo, and Brussels, but can present only one ofthese. Most people agree that you should mention Manila.But why? Rubinstein gives an intriguing analysis ofdebates and persuasion rules which shows that optimaldebates between players presenting conflicting evidence toa “listener” are sequential, and have a persuasion rule thatmust treat some arguments i, j symmetrically: i is persua-sive against j, but also vice versa.

This example is again very significant from a logicalpoint of view. Argumentation and dialogue games haveexisted in logic since the 1950s. Roughly speaking, validconclusions are those assertions C for which their propo-nent has a winning strategy in any debate against an oppo-nent granting the premises. In recent years, these gameshave been taken up by computer scientists who use theirprocedural, as well as compositional structure for model-ing interactive processes. This all concerns logical validity,and these games are very regimented. Little work has been

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done on games with more relaxed rules that would modelcommon sense reasoning or heuristics, let alone informalargumentation. Rubinstein’s new style of analysis maypoint in just the latter direction. Even so, I had sometrouble locating a crisp conclusion concerning the cities,because the author’s moral seems to depend on the debaterules in the end. But this may be just as well, because theobserved phenomenon is incomprehensible from a purelyinformational point of view – so there must be some addi-tional rationale based on implicit procedural conventions.

6 How does natural language function?

The three examples display very different explanatorystrategies. Indeed, they are very concrete, and specificassumptions about possible moves and utilities are cookedup ad hoc. This book does not attempt to provide a system-atic view of natural language. I do not find this an objec-tion, given its originality. But one would like to see ifaccounts like this would work more generally, for otherorderings, speech acts, or argumentation conventions.Also relevant is this. At least some empirical facts aboutlanguage use and complexity are known. Zipf’s Law, forexample, tells us that the highest-frequency expressionstend to be the lowest-complexity ones – as one wouldexpect of an optimal medium. This should surely berelevant to game-theoretic analysis. Also, linguists andlogicians have their own favorite Grand Challenges con-cerning natural language, clamoring for new break-through ideas, and it would be of interest to see if this kindof game theory has something to offer on those. Forexample, what is the game-theoretic explanation for theubiquity of the phenomenon of ambiguity in meanings?

I have two specific recommendations for broaderthemes. One concerns a convergence of interests. As I saidbefore, preferences are an obvious theme in the intersec-tion between game theory and modern logic and linguis-tics. One currently influential program in linguistics, forexample, is Optimality Theory, which attempts to explainsyntax and semantics through the selection of packages of

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constraints that are most-preferred in some suitable sense.(At lower unconscious levels of sentence comprehension,these preferences will be hard-wired – whereas they areconsciously manipulable at the higher level of discourse.)These analogies call for interaction. Another recommenda-tion is this. Since Rubinstein’s game-theoretic analysisoften concerns “what if” situations where people change alanguage, it may be important to analyze actual processesof new language formation, which occur around us allthe time – most notably, in the design of specializednatural, mathematical–logical, or computer programminglanguages.

7 Terms and formats in game theory

Part 2 of the book contains some analyses of the role of lan-guage in game theory. With some of this, I am not evensure that “language” is the right heading, because it seemsto concern general methodological issues, often similar towhat philosophers of science have long worried about. Forinstance, the status of terms in scientific theories has beendebated since the Vienna Circle. “Observational terms”are correlated with empirical observations, while “theoret-ical terms” serve the purposes of abstract organization, butdo not correspond directly to observation or commonsense. A good example are terms like “mass” or “energy”in physics, which bear only an oblique resemblance totheir original common-sense meanings. Very briefly,Rubinstein’s worries about the theoretical term “strategy”seem to dissolve then – the term is not the same as thecommon-sense notion, but is quite legitimate all the same.What remains is the warning that rhetorical uses of gametheory overlook these subtleties.

A more concrete and technical subject is the discussionof definable preferences. Rubinstein points to the impor-tance of linguistic formats for natural definitions of indi-vidual and collective preferences. He takes these fromvarious first-order formalisms, rather than the quantita-tive mathematical formulas often employed by gametheorists, which may lack reflection in any plausible

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description language. Results concern the use of con-straints narrowing down the admissible forms of defini-tion, such as desirable ordering properties, or suitableinvariances between ‘dating profiles’ of candidates for aperson seeking a partner. I will content myself with sayingthat chapter 4 is straight logical model theory, which couldalso have been published in a logic journal! Admissiblepreferences are defined in terms of their invariancebehavior (e.g. with respect to permuting individuals up tosameness of profiles), with an interesting use of the inter-polation theorem to ensure uniformity of definitionsacross domains of different sizes. What I would like toadvocate here is joining forces. Social choice theory, gametheory, logic and linguistics have obvious shared concernshere. Let me mention two current ones from my world.What is the dynamics of changing preferences? After all,informational preferences may change. I may consider allfour possibilities for two propositions A, B equally likely,without any preference at the start. But then, upon hearinga default conditional statement “if A, then B,” I upgradethis preference, making the A/not-B case less preferredthan the others. In the end, we want dynamics, not juststatics of preferences. Another point is the general move insemantics from one-agent to many-agent settings – e.g.moving from individual knowledge to common knowl-edge. This move is clear in game theory, where collectivescan form coalitions (cf. the “mutants” in an earlier-mentioned story). All these concerns seem to fit into onecoherent picture: but which?

8 Game theory and logic

I guess my conclusion is abundantly clear. I find this bookvery congenial, overlapping with logic all the way through,but very original precisely where it diverges. Of course, Iwould like to see more systematics in methods and aims,plus more grappling with the broader issues that drivelogic and the semantics of natural language. But thatseems feasible, and promising. Indeed, one can apply thesame game-theoretic style of analysis to many topics

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inside logic proper, it seems to me. (Why do we use theproof principles that we do?) As for the soul-searchingaspect, I also see similarities. Logic is about reasoning, butit never quite fits the observed phenomena. If the latterdiverge too much, logicians happily change their tune, andsay that logic serves to improve, rather than describe prac-tice. And indeed, in cognitive science, there is this veryreal phenomenon that theory can influence behavior –which makes the discipline much more subtle than thenatural sciences. So I’d say: “game theorists, relax!”

R E F E R E N C E S

van Benthem, J. (1986) Essays in Logical Semantics, Dordrecht:Reidel

(1996) Exploring Logical Dynamics, Stanford: CSLIPublications

(1998) “Logical Constants: The Variable Fortunes of anElusive Notion”, in R. Sommer and C. Talcott (eds.),Proceedings Feferfest, Stanford, forthcoming

(1999) Logic and Games, electronic lecture notes, Universityof Amsterdam, http://www.turing.uva.nl/~johan/teaching

van Benthem, J. and A. ter Meulen (eds.) (1997) Handbook ofLogic and Language, Amsterdam: Elsevier

Doets, H.C. (1998) Basic Model Theory, Stanford: CSLIPublications

Gärdenfors, P. (1988) Knowledge in Flux, Cambridge, MA: MITPress

Gärdenfors, P. and H. Rott (1995) “Belief Revision,” in D.Gabbay, C. Hogger and J. Robinson (eds.), Handbook ofLogic in Artificial Intelligence and Logic Programming, IV,Oxford: Oxford University Press, 35–132

Gilbers, D. and H. de Hoop (1998) “Conflicting Constraints: AnIntroduction to Optimality Theory” Lingua, 104, 1–12

Kemeny, J. (1955) “Fair Bets and Inductive Probabilities,”Journal of Symbolic Logic, 20, 263–73

Kuipers, Th. (1999) Structures in Science. Heuristic PatternsBased on Cognitive Procedures, Philosophical Institute,University of Groningen, Synthese Library, Dordrecht:Kluwer, forthcoming

Lewis, D. (1969) Convention, Cambridge, MA: HarvardUniversity Press

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Li, M. and P. Vitányi (1993) An Introduction to KolmogorovComplexity and its Applications, New York: Springer-Verlag

Prince, A. and P. Smolensky (1997) “Optimality Theory: FromNeural Networks to Universal Grammar,” Science, 275,1604–10

Shoham, Y. (1986) Reasoning about Change, Cambridge, M.A.:MIT Press

Veltman, F. (1996) “Defaults in Update Semantics,” Journal ofPhilosophical Logic, 25, 221–61

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Tilman Börgers

The quality of this book confirms that Ariel Rubinstein isone of the most important game theorists, and, more gen-erally, theoretical economists today. His importance liesin his originality. He urges researchers to break out ofestablished thought patterns. He opens up new lines ofthought. He is willing to disagree with predominant views.All these qualities are reflected in the current book.

As their author readily acknowledges, some chapters ofthis book are loosely, but not tightly connected. Chapters1, 2 and 3, however, have much in common. They are allbased on the claim that the ideas of game theory can befruitfully applied to the study of language. Reading andreflecting on this book has convinced me that this isindeed the case. The best evidence which I can offer is thefact that while reading the book I could think of manyadditional ideas in this area which appeared to me as prom-ising research subjects. Some of these ideas will be indi-cated below. Given the limitations of space for thiscomment, and because of the coherence of chapters 1–3 Ishall focus in my discussion on these three chapters.

The simplest game-theoretic model of language isdescribed in chapter 2. A parable can capture the essence ofthis model: Exactly one of several boxes contains a prizewhich individuals A and B can share if they find it.Individual A, but not individual B, observes the box intowhich the prize is placed. However, it is individual B whohas to open a box, and B has only one chance. Before Bchooses A can make one of several sounds, which can beheard and distinguished by A.

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I am grateful to V. Bhaskar for helpful discussions.

One equilibrium of this game is that a “language” devel-ops: There is a one-to-one relation between the soundsmade by A and the boxes, and A uses the language to truth-fully indicate which box contains the prize. B then opensthat box. Unfortunately, there are other equilibria of thisgame. For example, A may make a random sound, which Bthen ignores. This is an equilibrium because A has noreason not to make a random sound if he is ignored by B,and B has no reason to pay attention to A, given that A’sutterance is random. This equilibrium is, of course, lessattractive to A and B, but that doesn’t mean that it won’tbe played. Another bad equilibrium is that A always says“The prize is in box b,” independently of where it really is,and that B, when hearing this claim, opens some arbitrar-ily selected box, and otherwise, if A does not make thisclaim, refuses to open any box at all.

If we wish to provide a game-theoretic account of lan-guage, we need an explanation of why a communicationequilibrium rather than one of the bad equilibria is played.Chapter 2 describes an argument which enriches themodel by introducing complexity costs, and which refinesthe equilibrium notion involving evolutionary considera-tions. A key argument is that an evolutionary process willmove out of the bad equilibrium because a random muta-tion will appear where some individuals use and interpretsignals accidentally in the same way. Given the enormousnumber of sounds which humans can make, this looks likean extremely unlikely coincidence. Chapter 2 leaves meunconvinced that the multiplicity problem in languagegames has been resolved.

A variation of the simple model of chapter 2 has beenconsidered in the “cheap talk” literature: A and B both talkin stage 1, and take actions in stage 2. However, A and Bhave no factual information to communicate. They justtalk. The second-stage game is a coordination game with,say, two equilibria, one of which is better for both A and Bthan the other. Cheap talk games have equilibria in whichanything that is said in the first stage is ignored in thesecond stage. But they also have equilibria in which bothplayers say X in stage 1, and play the good equilibrium in

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stage 2 if and only if both agents said X in stage 1.Evolutionary arguments1 have been proposed to selectequilibria of the latter type (Kim and Sobel, 1995).

I mention cheap talk models because they contrast in aninteresting way with the model in chapter 2. Whereas inchapter 2 speech serves the purpose of communicatingobserved facts, in cheap talk games speech serves thepurpose of reinforcing some particular action. Cheap talkconsists thus of ethical (or other) imperatives. Cheap talkmodels might provide the beginning of an evolutionaryrationale for the large amount of ethical talk in which weall engage every day.

Chapter 1 is concerned with the more ambitious task ofexplaining the structure of language in detail, again usinggame-theoretic models. I would like to view the model inchapter 1 as an extension of that in chapter 2, although thisis not how Rubinstein presents chapter 1. Imagine that inour earlier parable the set of boxes in which the prize maybe can change. Sometimes there are boxes b, b9 and b0, etc.At other times, there are only boxes b9 and b0. Suppose thatbefore the communication game begins individuals A andB can see which boxes are there, and that they also can seehow these boxes are arranged. Specifically, assume that forany two boxes it is observable whether box b is to the leftof box b9 or vice versa. Finally, assume that another observ-able relation is whether all colors which appear on box b9also appear on box b or vice versa.

My interpretation of chapter 1 is that it is concernedwith an analysis of equilibria of this game, although thisgame is not explicit in chapter 2. As in chapter 1, the set ofequilibria of the communication game implicit in chapter2 is large. Rubinstein implicitly focuses on those equilibriaof this game in which words have meaning and communi-cation takes place. He also invokes a further exogenousrestriction regarding the equilibrium language. The lan-guage must identify objects only by use of a single binaryrelation. The question on which Rubinstein focuses iswhich of the available binary relations will be used.

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1 Unlike chapter 2, these arguments do not involve complexity costs.

The answer which Rubinstein proposes is that it will belinear orderings, such as the binary relation “to the left of.”An incomplete ordering, such as “all colors which appearon b9 also appear on b,” by contrast, will not be used.Rubinstein puts forward three reasons for this. One reasonis that only linear orderings allow unique identification ofobjects in any set of possible objects. Another argument isthat linear orders are particularly easy to teach, and henceeasy to transmit. Another argument fits less well into mycontext: Linear orders are well suited to describe otherbinary relations.

I want to emphasize two features of the argument inchapter 1. First, to understand the structure of language agame model is constructed in which some information iscommonly observed (in my parable: the set of boxes inwhich the prize might be, and their relations) and someinformation is privately observed (the particular boxwhich contains a prize). The language can use phrases(such as “the leftmost box”) which derive their specificmeaning from the commonly observed context. Almost allphrases of our languages have this property that they don’trefer to some specific individual, but can be understoodonly when some commonly observed context providesbackground information.

The second feature of chapter 1 that seems worth notingis that implicitly an equilibrium selection argument isused. This argument invokes two ideas. One is that lan-guage develops so that it is functional – i.e. useful tosociety. The second is that language develops so that it iseasy to transmit. The first half of this is even announced asan integral part of the “research agenda”: “features ofnatural language are consistent with the optimization ofcertain ‘reasonable’ target functions.” As indicated earlier,I have reservations about this argument. Ease of transmis-sion, though, seems to me a very natural criterion toinvoke when selecting among equilibria, although it mighteasily be in conflict with functionality.

Arguments similar to those in chapter 1 can be used toinvestigate other, possibly more basic issues. Suppose nobinary relations were observed. The equilibrium language

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could still make use of the fact that the set of boxes is com-monly observed. For example, it might be that, althoughmany of the boxes are made of cardboard, typically onlyone of the boxes placed in front of B is of cardboard, and allothers are of wood. Then a functional and easy to transmitlanguage might develop a single word to refer to cardboardboxes without differentiating between individual boxes.Thus one might start to construct a theory which explainswhy language so often uses category names to refer to indi-viduals.

Chapter 3 could also be fitted into a version of the frame-work that I have used to understand chapters 1 and 2.Unlike in chapters 1 and 2, however, in chapter 3 bothindividuals have some private information. Moreover,now their interests differ. The question which Rubinsteinasks is how a debate between A and B should be structuredif it needs to be short, and if we wish to elicit as muchinformation as possible.

The step from chapters 2 and 1 to chapter 3 is quite large.One might make a less ambitious step, and stay with amodel in which A’s and B’s preferences are identical.Consider, for example, the model of chapter 2, and supposewe enriched this model by just one ingredient, namely theassumption that A and B both have pieces of private infor-mation. Suppose it were exogenously given that A spokefirst, then B spoke, and then A spoke again.

One equilibrium of this game would involve A sayingeverything that he knows, followed by two rounds ofsilence in which neither agent says anything. This wouldbe efficient, in the sense that B could make a choice basedon all available information. But now suppose that speak-ing, and listening, is not without costs. Then both agentsmight be better off in another equilibrium in which A letsB speak first, and then establishes which of his own privateobservations are actually relevant, in the sense that theycould improve B’s choice, given what B already knows.

An analysis of equilibria of common interest games withinitial sequential talk might lead to the beginning of atheory of conversation. Conversation surely differs fromdebate in that the participants pursue common rather than

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opposed interests. Grice’s cooperative principles, quotedin Rubinstein’s chapter 3, seem more apt for conversationthan for debate. The above argument might provide aformal counterpart to Grice’s third maxim (“Relation: BeRelevant”). I find Grice’s four maxims of quantity, quality,relation, and manner highly appealing, and have secretlybeen toying with the idea of posting them in seminar andconference rooms. Among other things, they suggest,though, that I should conclude my contribution here.

R E F E R E N C E

Kim, Y.G. and J. Sobel (1995) “An Evolutionary Approach toPre-play Communication,” Econometrica, 63 1181–93

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Barton L. Lipman

“Is he – is he a tall man?’’

“Who shall answer that question?’’ cried Emma.“My father would say, ‘Yes’; Mr. Knightly, ‘No’;

and Miss Bates and I, that he is just thehappy medium.’’

(J A N E A U S T E N, E M M A . )

1 Introduction

While a reader of this book may be surprised to see a gametheorist writing about language, he should instead besurprised by how few game theorists have done so. AsRubinstein observes, language is a game: I make a statementbecause I believe you will interpret it in a particular way.You interpret my statement based on your beliefs about myintentions. Hence speaker and listener are engaged in agame which determines the meaning of the statement.

Furthermore, I think more “traditional’’ economistsshould be interested in models of language. The worldpeople live in is a world of words, not functions, and manyreal phenomena might be more easily analyzed if we takethis into account. For example, consider incomplete con-tracts. Our models treat contracts as mathematical func-tions and hence find it difficult to explain why agentsmight not fully specify the function. Of course, real con-tracts are written in a language and may not unambigu-ously define such a function – not its domain or range,much less the function itself.

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I thank Marie-Odile Yanelle for helpful conversations. No thanks are dueto the Dell Corporation.

Rubinstein gives an intriguing opening to this importanttopic. In what follows, I give a brief overview of what I seeas the important ideas in the book and use this as back-ground to comments on directions for future research. Inthe process, I will suggest that there are important and dif-ficult problems ahead.

2 Overview

While Rubinstein jokes that there is little to connect hislectures beyond the common title and speaker, I think it’sconceptually useful to consider the various ingredients of atheory of language the book gives. As I see it, chapters 1–4give us various ways to address three questions:1

1 What gives a statement its meaning?2 Why is language the way it is?3 How does language affect actions?

Rubinstein gives two distinct ways to answer the firstquestion. Chapters 1 and 4 give one of his approaches,modeling language as a logical system. In this approach,language is a set of building blocks and rules of construc-tion which implicitly define the meaning of statements.For example, the meaning of p~q is that either p or q is

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1 I have only one comment on chapter 5. Rubinstein criticizes the waygame theory is “sold’’ to business students, saying that the kind of con-ceptual understanding that game theory can provide “stop[s] far short ofgiving advice and claiming to be useful.’’ I agree up to a point. Havingtaught game theory to MBA students for a few years, I spent much timereflecting on what value, if any, the theory had for them. I think mostbusiness students want rules, not concepts, evidently believing that thebusiness world is a series of clearly framed problems, each with anappropriate rule to follow. This absurd notion influences the way text-books are written since these are supposed to appeal to business stu-dents. I disagree with Rubinstein, though, when he concludes that thereis nothing “useful’’ to such people in game theory. I think it is preciselythe concepts and understanding he alludes to which can benefitsomeone in business. Of course, their understanding will neverapproach Rubinstein’s! In particular, backward induction is indeed arevelation to business students and the concept is hard for them tolearn. Any reader who is skeptical on this point need only see how manybusiness students will err on a question involving sunk costs.

true – a meaning which, of course, depends on the under-lying meaning of p and q.

While this is a natural starting point, it is clear that lan-guage as spoken by real people is not logical in the formalsense. For example, as Rubinstein notes in chapter 3, a lis-tener generally makes inferences in response to a state-ment which go beyond the purely logical implications ofwhat the speaker said. Put differently, the fact that a givenstatement is made can itself signal some of the speaker’sprivate information. Later, I will discuss another way inwhich language is not purely logical.

Rubinstein’s second way to answer question 1 addressesthis drawback of the logical system approach, adoptinginstead what I’ll call the equilibrium approach. Thisapproach, used in chapters 2 and 3, treats language as a setof words and an equilibrium interpretation of their usageor meaning. For example, if a particular statement is madeonly in certain situations, then the meaning of the state-ment (in addition to any concrete evidence included, asstudied in chapter 3) is that one of these situations must betrue. While the equilibrium approach captures some ofwhat the logical system approach misses, it does so at acost: the language has no structure to it. Hence many ques-tions about language cannot be addressed in such a model.

Rubinstein’s approach to the second question, illus-trated nicely in chapters 1 and 3, is what I’ll call the struc-tural optimization hypothesis. In this approach, languageis that structure which maximizes the amount of informa-tion conveyed subject to constraints on the “complexity’’of the language. In chapter 1, the “complexity constraint’’is that the language can have only one relation. The ques-tion then is what relations are most useful for describingmany objects (or sets of objects) as precisely as possible.The complexity constraint in chapter 3 is that the listenercannot process more than two pieces of evidence. HereRubinstein considers the optimal way for the listener tointerpret statements by debaters in order to elicit informa-tion from them.

Finally, chapter 4 gives an intriguing approach toanswering the third question which I’ll refer to as the

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expressibility effect. The general perspective here is thatpeople perceive the world and form decisions in words.Because of this, the nature of the language people useaffects their actions. Chapter 4 looks at the expressibilityeffect through the requirement that decision rules or pref-erences be definable but, as Rubinstein notes, the principlecould be explored in many other ways and other aspects ofdecision making.

3 Further research

I now turn to three areas for future research in this areawhich strike me as particularly interesting. First, I amintrigued by the inherent circularity one gets when com-bining the structural optimization hypothesis and theexpressibility effect. By means of illustration, one responseto chapter 1 which I have heard is that it may be that linearorderings are common in our language not because of anyinherent usefulness they have but simply because linearorderings are common in the natural world. I think thiscriticism misses a crucial point: do we perceive linearorderings to be common in the world because they arecommon in our language? In other words, the structuraloptimization approach suggests that we structure languagein a way which seems useful to us given our perception ofthe world. On the other hand, because of the expressibilityeffect, once developed, language affects the way we see theworld. An analysis which includes such feedback effectscould be quite interesting.

Second, as noted earlier, both the logical systems andequilibrium approaches to meaning miss part of thepicture. It is clear that language does have structure even ifit is not fully logical. It is not obvious how to model thissimple notion. If the meaning of a sentence comes from itsequilibrium interpretation only, then there is no reason forstructure to relate to content. “I live in Wisconsin and it iscold’’ could be interpreted as the conjunction of “I live inWisconsin’’ and “It is cold’’ or the disjunction of “I amhungry’’ and “Rubinstein lives in Israel.’’ At the sametime, meaning cannot come purely from the content of the

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sentence in isolation from its context or other extra-logicalfactors. A model which combines the advantages of equi-librium models and logical systems would be more plausi-ble – and, perhaps, more useful.

It might be possible to develop such a model using thestructural optimization approach. In this case, a particularnatural complexity constraint is limited memory. It seemsintuitively obvious that it is easier to remember that “Ilive in Wisconsin and it is cold’’ means “I live inWisconsin’’ and “It is cold’’ instead of some notion unre-lated to Wisconsin or cold. Similarly, the structure ofwords might be derivable from memory limitations. Forexample, consider the use of prefixes and suffixes. Todeduce the meaning of a word beginning with “post,’’ it issufficient to remember the meaning of the prefix “post’’and the word which follows. To combine this with an equi-librium approach to meaning, one would presumably focuson “efficient’’ equilibria.

The last topic is another aspect of the issue of meaning:vagueness.2 Perhaps it is easier to understand what I meanby “vague’’ by contrasting it with “precise.” I will say aterm is precise if it describes a well defined set of objects.This is the way language works in most of the economicmodels where it appears: the set of objects is partitionedand a word is associated with each event of the partition.By contrast, a term is vague if it does not identify such aset.

To illustrate my meaning and to show why vague termsmake it difficult to model language as a logical system,consider the following version of the famous soritesparadox. Two facts seem clear regarding the way mostpeople use the word “tall.’’ First, anyone whose height is10 feet is tall. Second, if one person is tall and a secondperson’s height is within 1/1000 of an inch of the first, thenthe second person is tall as well. But then working back-ward from 10 feet, we eventually reach the absurd conclu-sion that a person whose height is 1 inch is tall. Of course,

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2 Keefe and Smith (1996) is a fascinating introduction to the philosophyliterature on the subject.

the source of the difficulty is clear: “tall’’ does not corre-spond to a clearly defined set. There is no fixed heightwhich defines the line between someone who is tall andsomeone who is not. Because of this, there is an inherentambiguity in the meaning of the word “tall’’ as the quote Ibegan with illustrates. Many other words have this prop-erty: consider “bald,’’ “red,’’ “thin,’’ “child,’’ “many,’’ and“probably.’’

The prevalence of vague terms in natural language posestwo intriguing and inter-related challenges. First, whatmeaning do such terms convey? Second, why are they soprevalent?

As to the first question, it seems clear that vague termsacquire their meaning from usage. That is, whatevermeaning “tall’’ has is due to the way people use the word,not any particular logical structure. Hence it seemsnatural to follow the equilibrium approach to studyingmeaning. The most obvious way to do so is to treat vagueterms as ones which are used probabilistically, so the prob-ability someone is described as “tall’’ is increasing inheight but is strictly between 0 and 1 for a certain range.This kind of uncertainty could correspond to mixed strate-gies or private information, either of which would give aclear notion of the meaning of a vague term.

However, this approach has a severe drawback: it cannotgive a good answer to the second question. In particular, ifthis is what vagueness is, we would be better off with a lan-guage which replaced vague terms with precise ones. Tosee the point, consider the following simple example.Player 2 must pick up Mr. X at the airport but has nevermet him. Player 1, who knows Mr. X, can describe him toplayer 2. Suppose that the only variable which distin-guishes people is height and that this is independently dis-tributed across people uniformly on [0,1]. 1 knows theexact height of Mr. X; player 2 does not. However, bothknow that when player 2 gets to the airport, there will bethree people there, Mr. X and two (randomly chosen)others. Player 2 has very little time, so he can ask only oneperson if he is Mr. X. If player 2 chooses correctly, 1 and 2both get a payoff of 1; otherwise, they both get 0. Clearly,

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there is a simple solution if 2 can observe the heights of thepeople at the airport and the set of possible descriptions is[0,1]: 1 can tell 2 Mr. X’s height. Because the probabilitythat two people are the same height is zero, this guaranteesthat Mr. X will be picked up.

However, this is a much bigger language than any in use.So let us take the opposite extreme: the only descriptions 1can give are “short’’ and “tall.’’ Further, player 2 cannotobserve the exact height of any of the people at the airport,only relative heights. That is, player 2 can tell who istallest, who is shortest, and who is in the middle.

In this case, it is not hard to show what the efficient lan-guage is: 1 should say “tall’’ if Mr. X’s height is greater than1/2 and “short’’ otherwise.3 Player 2 tries the tallest personat the airport when he is told that Mr. X is “tall’’ and theshortest when told that Mr. X is “short.’’ Note, in particu-lar, that there is no vagueness in the optimal language:“tall’’ corresponds to the set [1/2, 1].

What would “vagueness” mean in this context? Oneway to make “tall’’ vague would be if 1 randomizes. Forexample, suppose 1 says “short’’ if Mr. X’s height is below1/3, “tall’’ if it is greater than 2/3, and randomizes inbetween with the probability he says “tall’’ increasingwith Mr. X’s height. While this resembles the way “tall’’ isused in reality, there are no equilibria of this kind. If onetakes a nonequilibrium approach and assumes player 1 iscommitted to such a language, it is easy to show that bothplayer 1 and player 2 (and presumably Mr. X!) would bebetter off in the pure strategy equilibrium above.4

Alternatively, private information could give the neededrandomness. Suppose player 1 has observed some signal inaddition to height. In this case, the efficient language parti-tions not the set of heights but the set of height–signalpairs. Hence a word will correspond to a random statementabout height, the randomness being induced by the signal.

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3 Of course, the words themselves are not relevant to this equilibrium.An equally efficient language would reverse the roles of “tall’’ and“short,’’ or even replace them with “middle-aged’’ and “blond.’’

4 Lipman (1999) gives a generalization of this argument and some furtherdiscussion.

On the other hand, what is this other signal? First, supposeit is somehow intrinsically relevant. For example, player 1may know Mr. X’s weight and player 2 may be able toobserve relative weights. In this case, player 1 needs tocommunicate on two dimensions to player 2. The efficientlanguage will have terms which are precise in two dimen-sions even though this may make them imprecise in anyone dimension. This does not seem to be much of an expla-nation of an apparently unidimensional term like “tall.’’On the other hand, suppose the signal is not relevant. Thenit would be most efficient to ignore this signal and use thelanguage described above.

Why, then, are vague terms so prevalent? It seems ratherobvious that such words are useful – a moment’s reflec-tion will suggest that it would be difficult to say much ifone were not allowed to be vague!

I think the only way to formally understand the preva-lence of vague terms is in a model with a different kind ofbounded rationality than is considered in this book or inthe literature.5 There are at least three possibilities, whichI give in increasing order of ambitiousness. First, vague-ness may be easier than precision, for the speaker, listener,or both. For the speaker, deciding which precise term touse may be harder than being vague. For the listener, infor-mation which is too specific may require more effort toanalyze. With vague language, perhaps one can communi-cate the “big picture’’ more easily. This requires a differentmodel of information processing than any I know of.

A more difficult approach would be to derive vaguenessfrom unforeseen contingencies. If the speaker does notknow all the possible situations where the listener woulduse the conveyed information, it may be optimal to bevague. For example, contracts often use vague terms suchas “taking appropriate care’’ or “with all due speed’’instead of specifying precisely what each party should do.If agents fear that circumstances may arise that they havenot yet imagined, then they may avoid precision to retainflexibility. Hence the optimal contract may require the

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5 I give a fuller defense of this position in Lipman (1999).

parties to respond to unexpected circumstances “appropri-ately,’’ with the hope that the meaning of this word will besufficiently clear ex post.6 Given the difficulty of mod-eling unforeseen contingencies (see Dekel, Lipman andRustichini, 1998, for a survey), this approach is surely noteasy.

Finally, I turn to a still more ambitious approach. Tomotivate it, consider one seemingly obvious reason whyvague terms are useful: the speaker might not observe theheight of an individual well enough to be sure how to clas-sify him precisely. If we modify the example to includethis, however, 1 would have subjective beliefs about Mr.X’s height and the efficient language would partition theset of such probability distributions. Hence this objectionsimply shifts the issue: why don’t we have a precise lan-guage for describing such distributions?

An obvious reply is that real people do not form precisesubjective beliefs. In other words, it is not that peoplehave a precise view of the world but communicate itvaguely; instead, they have a vague view of the world.7 Iknow of no model which formalizes this. For example, itis not enough to replace probability distributions withnonadditive probabilities. If agents have nonadditive prob-abilities, then it is surely optimal to partition the set ofsuch beliefs precisely.

To sum up, I think this book gives some intriguing startson some important problems. As such, it opens the door tosome exciting and difficult research.

R E F E R E N C E S

Dekel, E., B. Lipman and A. Rustichini (1998) “Recent Develop-ments in Modeling Unforeseen Contingencies,’’ EuropeanEconomic Review, 42, 523–42

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6 This idea is very similar to the Grossman–Hart–Moore approach toincomplete contracts. See Hart (1995) on this approach and Dekel,Lipman and Rustichini (1998) for a discussion of the connectionbetween it and formal models of unforeseen contingencies.

7 The dividing line between unforeseen contingencies and this kind ofvague perception is itself quite vague!

Hart, O. (1995) Firms, Contracts, and Financial Structure,Oxford: Clarendon Press

Keefe, R. and P. Smith (1996) Vagueness: A Reader, Cambridge,MA: MIT Press

Lipman, B. (1999) “Why is Language Vague?,’’ Working Paper

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aggregation rule 59, 60alternatives–preferences 82, 85ambiguity 103arguments see debatesAumann, R. 78, 79, 88Austen, J. 114

Backhouse, R. 75Banerjee, A. 25, 27Bhaskar, V. 108binary relations 9, 97–8, 110–11

asymmetric and complete 18asymmetric and transitive 65–8comparatives 100cyclical 12–13and definable preferences 57–63describability 18–22functionality 22–3linear ordering 10–13natural kind 24properties 10pure language with equality 65,

66structure 10targets of 5to transfer or store information

15–18Binmore, K. 25, 31, 88Blume, A. 25Boolos, G. S. 55, 62Börgers, T. 89, 108–13bounded rationality models 56Bryan, G. L. 56

calculus of propositions 68cheap talk 26, 109–10communication

equilibria 109using game theory 25–7

compactness argument 67complexity 19, 97–101, 116

costs of 109and evolutionary forces 5, 31–3expressive 101natural 118resource 95

context 37conventions 101conversation 112conversational implicature 39–40cooperative principle 39, 40, 113Craig Lemma 62Crossley, J. N. 55

Davitz, J. 3debate 112

defined 47logic of 51model 46–51optimal 48–51rules 102–3: persuasion 46,

51–2; procedural 46sequential 49, 51valid conclusions 102

debates 6, 42–6decision-making

choice of boyfriend 63–8: datingprofiles 64, 69–70

language constraints 4, 6, 55–70Dekel, E. 122describability 18–22Dixit, A. 71–2, 74–5Dudley-Evans, T. 75Dutch Book Theorem 99

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INDEX

economic modelsexclude lexicographic

preferences 63include interpretation 6, 72–3

economic theory 4, 95–6equilibria 116–18

communication 109multiple 26Nash 29, 77, 87pooling 26, 29strategy 77

evolutionary approach 5, 109–10

evolutionary forces 22–3, 29–33evolutionary game theory 27–35,

100–2evolutionary stable set 30–1, 32–3Evolutionary Stable Strategy

(ESS) 29–31, 33expressibility effect 117

Farrell, J. 26Fishburn, P. 20Frege, G. 94Friedman, J. 76

game theory 5–6and communication 25–7criticism of use 34evolutionary 27–35, 100–2language in 104large population 28, 78and logic 105–6meanings of 88popular view of 71–2practical application 73–5, 88rhetoric 6, 71–8strategy in 76–80understanding an utterance

37–9see also equilibria; Nash

equilibriumGrice’s Maxims 101Glazer, J. 37, 49Goodman, N. 23Grand Challenges 103Grice, P. 6, 37–42, 93, 95, 113Grice’s principles 37–42, 113

Grin, F. 7Gulf War 83–4

Harsanyi, J. 79Helmholtz 98Henderson, W. 75Horn clauses 98

indication-friendliness 11–13, 17informativeness 15–18, 101Iraq 83–4

Jeffrey, R. C. 55, 62

Keefe, R. 118Kim, J. H. 20Kim, Y.-G. 25Kolmogorov complexity 98

languageexpressive power and resource

complexity 95formal and natural 94formation 104in game theory 104in logic 93–4as a logical system 115–16

language of propositionalcalculus 58

Levinson, S. C. 37Levy, D. 34Lewis, D. 101linear ordering 4–5, 10–13, 21–2,

24, 97, 100, 111, 117linguistic code 96–100Lipman, B. L. 43, 89, 114–22Lloyd, B. 23logic

and game theory 105–6language in 93–4

McCloskey, D. N. 75McMillan, J. 71, 72, 74, 75, 76Maital, S. 71Marschak, J. 7, 29Maynard Smith, J. 25, 27, 29meanings 5, 25–7minimal description length 98

Index

126

modified evolutionary stablestrategy 31, 33

Morgenstern, O. 73, 81, 82

Nalebuff, B. 71–2, 74–5Nash, J. 81Nash bargaining axioms 60, 81,

84–6Nash bargaining solution 6, 80

alternatives-preferences 82, 85Nash equilibrium 29, 77, 87

see also game theorynatural invariances 98–9

observational terms 104Omori, A. 71Optimality Theory 103–4

Parikh, P. 42Piatelli-Palmarini, M. 23Piccione, M. 25plan of action 76, 77, 79Popper, K. 90pragmatics 5–6, 37, 45, 100preference relations 80–1, 95

defined 55–6preferences 103

admissible 105changing 105definable 57–63, 64, 66, 68,

104–5lexicographic 63

Price, G. R. 29probability 119–20, 122

Quine, W. V. 23, 24

Radner, R. 78random actions 77–9, 120recommendation 87rhetoric 6, 71–8riddle of induction 23Robinson, A. 66Rosch, E. 23Rosenthal, R. 78Rubinstein, A. 9, 25, 37, 49, 55,

56, 71, 84Rustichini, A. 122

Saddam Hussein 83Safra, Z. 82, 84Samuelson, L. 25, 31semantics 37, 100sentence, nature of 21Seppi, D. J. 43sets

distance between 14–15splitting 13–15, 17

Shelly, M. Y. 56Shin, H. S. 43Shubik, M. 76Smith, A. 34–5Smith, P. 118Sobel, J. 25social choice theory 58, 59–60,

99, 105social rule 58solution 86–7sorites paradox 118Spector, D. 43strategic reasoning 42strategy

an extensive game 76–7and beliefs 79–80complexity costs 28in daily language 75–6equilibrium 77in game theory 76–80mixed 77–80simplicity of 32

strategy IQ 72–5structural optimization

hypothesis 116, 117–18syntax 37

Tel-Aviv student survey 43–5,102

Tetali, P. 20theoretical terms 104Thomson, W. 82, 84tournaments 10, 18, 21, 97, 100transitivity 10, 61–2, 65–8transmission ease 111

Ugly Duckling Theorem 23unary relations 13–15unforeseen contingencies 121–2

Index

127

United States 83–4utilities 82, 88–6utility function 56, 80

vagueness 118–22van Benthem, J. 89, 93–106vocabulary 97–100

“loss” 14–15, 16–17lowest complexity 99–100natural invariances 98–99

von Neumann, J. 71, 73, 81, 82

Walsh, D. 75

Warneryd, K. 25, 27Watanabe, S. 23–4Weibull, J. W. 25, 27, 30words

emergence of 29–31meaning 5, 25–7

Yanelle, M.-O. 114

zero-sum game 41Zipf’s Law 103


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