ECONOMICS ANDLANGUAGE

ARIEL RUBINSTEIN

Nancy L. Schwartz Lecture

Delivered May 27, 1998, at theKellogg Graduate School of Management

Northwestern University2001 Sheridan Road

Evanston, Illinois

NANCY L. SCHWARTZ

A dedicated scholar and teacher, Nancy Schwartzwas the Morrison Professor of Decision Sciences,Kellogg’s first female faculty member appointed toan endowed chair. She joined Kellogg in 1970,chaired the Department of Managerial Economicsand Decision Sciences and served as director of theschool’s doctoral program until her death in 1981.Unwavering in her dedication to academic excel-lence, she published more than 40 papers and co-authored two books. At the time of her deathshe was associate editor of Econometrica, on theboard of editors of the American Economic Reviewand on the governing councils of the AmericanEconomic Association and the Institute ofManagement Sciences.

The Nancy L. Schwartz Memorial Lecture serieswas established by her family, colleagues andfriends in tribute to her memory. The lectures pres-ent issues of fundamental importance in currenteconomic theory.

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ARIEL RUBINSTEIN

Ariel Rubinstein is the Salzberg Chair professor ofeconomics at Tel-Aviv University, where he hasbeen teaching since 1990, and a professor of eco-nomics at Princeton University, where he has beenteaching since 1991. He received his Ph.D. in eco-nomics from the Hebrew University in 1979, withearlier degrees in mathematics and economics fromthe same institution. His past teaching and researchaffiliations include The Hebrew University,Nuffield College (Oxford), Bell Laboratories, theLondon School of Economics, the University ofChicago, the University of Pennsylvania, ColumbiaUniversity, New York University, and the Russell-Sage Foundation.

Professor Rubinstein is a Fellow of the IsraeliAcademy of Sciences and of the EconometricSociety, and a foreign honorary member of theAmerican Academy of Arts and Sciences and of theAmerican Economic Association. He has deliveredmajor invited lectures at universities and congressesaround the world including the Walras-BowleyLecture of the Econometric Society, and has servedas editor for a large number of publications includ-ing Econometrica, Journal of Economic Theory,Review of Economic Studies, and Games andEconomic Behavior.

An author of three books and more than 60research papers, Professor Rubinstein is one of theworld’s leading researchers in the areas of econom-ics and game theory. He has done path-breakingwork on dynamic strategic interaction, bargaining,interactive epistemology, and bounded rationality,and his papers are published in the leading journalsof economics, decision theory and applied mathe-matics.

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ACKNOWLEDGEMENTS

Parts of this lecture are based on my ChurchillLectures delivered at Cambridge, England, in May1996, and which will eventually be published byCambridge University Press. Chapter 2 of this lec-ture is based on Rubinstein (1996) and Chapter 3is based on Glazer and Rubinstein (1997). I wouldlike to thank Kobi Glazer for permitting me to useour joint paper for this lecture and to Ehud Kalaiand Bart Lipman for their comments.

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TABLE OF CONTENTS

Chapter One: Economics and Language . . . . .41.1 A Personal Note 1.2 Economics and Language

Chapter Two: The Functionality of Properties of Binary Relations . . . . . . . . . . . . . . . . . . .72.1 Binary Relations 2.2 Indication-friendliness 2.3 Informativeness

Chapter Three: Strategic Considerations in Pragmatics . . . . . . . . . . . . . . . . . . . . . .143.1 Grice’s Principles and Game Theory 3.2 Debates 3.3 A Model

References . . . . . . . . . . . . . . . . . . . . . . . . . .24

ECONOMICS AND LANGUAGE

Chapter One: Economics and Language

1.1 A Personal NoteThe psychologist Joel Davitz once wrote: “I suspectthat most research in the social sciences has rootssomewhere in the personal life of the researcher,though these roots are rarely reported in publishedpapers.” (Davitz, 1976). The first part of this state-ment is definitely true about this lecture. AlthoughI work in several fields in economics and game the-ory, all my academic research has been motivatedby my childhood desire to understand the way thatpeople argue. In high school, I wanted to studylogic. The main reason was that I thought that itwould be useful for the political debates, which Iwas planning to be involved in or in the legal bat-tles against evil, which I hoped to conduct afterbecoming a solicitor.

Unfortunately, I became neither a lawyer nor apolitician. In the meantime, I came to understandthat logic is not a very useful tool in those areas inany case…. But I continued to explore formalmodels of game theory and economic theory, neversustaining any desire to predict human behavior,never wishing to anticipate the stock prices, andnever having illusions about capturing all of realityin one tiny model. I simply kept being interested inthe arguments people bring in debates and in thereasons motivating decisions. And, I am still puz-zled, even fascinated, by the magic of the linksbetween the formal language of the mathematicalmodels and natural language. This brings me to thesubject of this lecture, “Economics and Language.”

1.2 Economics and LanguageThe title of the lecture may be misleading. Thecaption “economics and language” is somewhat

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“attractive” but it is also vague. As a heading, it cov-ers many different subjects that I will not raise here.For example, I will not talk about the language ofdecision-makers. When decision-makers makedeliberate choices, they often formulate their judg-ments and decisions in words. Were I to speak onthis domain, I would present an investigation ofthe assumption that decision-makers formulatetheir considerations by using some language. Thisis a particularly attractive assumption when the“decision maker” is a collective of individuals but itis also appealing when we refer to a decision-makeras an individual. The formalization of this assump-tion requires the tools of mathematical logic. Theanalytical task would be to identify the constraintson the set of preferences induced from naturalrestrictions on the language used by the decision-maker to define his preferences. For example, suchconstraints make the lexicographic preferencesmuch more appealing when compared to a stan-dard textbook consumer’s utility function, such aslog(x1+1)x2. But today, I leave this subject aside.

And I will not talk about the language of econom-ics. Much can be said about the rhetoric of eco-nomic theory and game theory. An economicmodel is not just a mathematical model. It is a com-bination of a mathematical model and an interpre-tation. The investigation of the language we use forattaching interpretations to economic models is, inmy opinion, of much interest. If I were to discussthis subject here, I would try to argue that the rhet-oric of game theory is actually misleading in creat-ing an impression that it is much more “useful”than it actually is. In particular, I would try to per-suade you that basic game theoretical notion of“strategy” is modeled in a way quite different fromwhat is suggested by its meaning in natural languageand that the term “solution” carries a deterministicflavor which is not appropriate.

But, I will not discuss here the “language of eco-nomic agents” and nor the “language of econom-ics.” I will make do with topics within a research

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domain, which is probably far away from tradi-tional economics: “economics about language.”

Why would economic-type thinking be relevant tosome linguistic issues? Economic theory is anattempt to explain regularities in human interac-tion and the most fundamental nonphysical regu-larity in human interaction is our natural language.In economic theory we have studied, quite care-fully, issues concerning the design of social systems;language is partially a mechanism of communica-tion. Economics tries to explain social institutionsas regularities derived from optimizing certainfunctions which they serve; one can try to use thismethod regarding language as well. In this lecture Iwill try to demonstrate what we, economists, cando in this area by presenting two short investiga-tions in which we use “economic” reasoning toaddress linguistic issues.

And before starting, I owe my audience an apology.Browsing through the literature while preparingthese lectures, I came across a short article writtenby Jacob Marshack called, “Economics ofLanguage” (Marschak, 1965). The article startswith a description of a discussion between engi-neers and psychologists regarding the design of thecommunication system of a small fighter plane.After the presentation, Marschak states: “The pres-ent writer…apologizes to those of his fellow econ-omists who might prefer to define their field morenarrowly, and who would object to…identificationof economics with the search of optimality in fieldsextending beyond, though including, the produc-tion and distribution of marketable goods.” Hethen continues: “Being ignorant of linguistics, heapologizes even more humbly to those linguistswho would scorn the designation of a simple dial-and-buttons systems a language.” I believe that likeMarschak, I am not really apologizing to econo-mists…but like Marschak, I do feel apologetictoward linguists and philosophers of languagebecause my knowledge of this territory is much toolimited.

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Chapter Two: The Functionality ofProperties of Binary Relations

2.1 Binary RelationsA binary relation on a set Ω specifies a connectionbetween elements within Ω. Such binary relationsare common in natural language. For example,“person x knows person y,” “tree x is to the right oftree y,” “picture x is similar to picture y,” “chair xand chair y have the same color” and so on. I willavoid binary relations like “Professor x works foruniversity y” or “the social security number of x isy,” which specify “relationships” between elementswhich naturally belong to two distinct sets.Actually, I will also restrict the term a “binary rela-tion” to a binary relation which is irreflexive: noelement relates to itself. The reason is that usually,the term “x relates to y” when x=y is very differentthan the meaning of the term “x relates to y” whenx≠y. For example, the statement “a loves b” is verydifferent than the content of the statement that “aloves himself.”

The nature of many binary relations requires thatthey satisfy certain properties. For example, therelation “x is a neighbor of y” must, in any accept-able use of this relation, satisfy the symmetry prop-erty (if x is a neighbor of y then y is a neighbor ofx). The relation “x is to the right of y” must be alinear ordering (that is satisfying the properties ofcompleteness, asymmetry and transitivity). On theother hand, the nature of many other binary rela-tions, such as the relation “x loves y,” does notimply any specific properties that the relation mustsatisfy a priori. It may be true that in a particulargroup of people, whenever “x loves y” then “y lovesx” as well. However, there is nothing in our under-standing of the relation “x loves y” which makesthat symmetry necessary.

In fact, the objects of the investigation here areproperties of those binary relations which appear innatural language. (Formally, a property of the rela-

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tion R is a sentence in the language of the calculusof predicates which uses a name for the binary rela-tion R, variable names, connectives and qualifiers,but does not include any individual names of theset of objects Ω.) I will refer to a combination ofproperties as a structure of a binary relation.

I am curious about the structures of binary rela-tions which appear in natural language and I lookfor explanations of why, out of the infinite numberof potential properties, we find that only a fewproperties are common in natural languages. Forexample, it is difficult to find natural properties ofbinary relations such as:

A1: For every x and y, the number of z forwhich xRz is equal to the number of z forwhich yRz.

A2: If xRy and xRz (y≠z), and both yRa andzRa, then also xRa.

Or, it is difficult to conceive examples of naturalstructures of binary relations which require com-pleteness and asymmetry (being a tournament) butdo not require transitivity. One exception whichcomes to mind is the structure of the relation “x isin a clockwise direction relative to y (in the short-est arch connecting x and y).” Is it just a coinci-dence that only a few structures are familiar innatural language?

The starting point for the following presentationwill be that binary relations fulfill some functionsin life. One can think about many criteria by whichto examine the functionality of binary relations.Here, I will examine only two. I will argue that cer-tain properties, all shared by linear orderings, per-form better according to each of these criteria. Ofcourse, other criteria are likely to provide alterna-tive explanations for the frequent use of differentcommon structures such as equivalence and simi-larity relations.

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2.2 Indication-friendlinessTwo parties observe a group of trees; the speakerwants to refer to a certain tree. If the tree, was theonly olive tree in the grove, the speaker would justuse the term “the olive tree.” If there is no mutuallyrecognized name for the tree and the two partieshave a certain binary relation defined on the set ofthe trees in their mutual vocabulary, the user mayuse this relation to define the element. For example,the phrase “the third tree on the right” is a way ofindicating one tree out of many by using the linearordering “x stands to the left of y” when the groupof trees is well-defined and the relation “being to theleft of” is a linear ordering. Similarly, the phrase“the seventh floor” is a way of indicating a locationin a building given the linear ordering “floor x isabove floor y.” There would be no need to use thephrase if the floor is “the presidential floor.” On theother hand, the relation “line a on the clock isclockwise to line b (at the smallest angle possible)”does not enable the user to indicate a certain line ona numberless clock: any formula which is satisfiedby 3 o’clock is satisfied by 4 o’clock as well.

Thus, binary relations are viewed here as tools usedfor indicating elements out of a set whose objectsdo not have names. We are looking for a structureof binary relations which guarantees that a binaryrelation which satisfies it also enables the user tounambiguously single out any element out of anysubset which contains it. We are led to the followingdefinition:

Definition: A binary relation R on a set Ω isindication-friendly if for every A Ω, and everyelement a∈A, there is a formula fa,A(x) (in thelanguage of the calculus of predicates with onebinary relation and without individual con-stants) so that a is the only element in A satis-fying the formula (when substituting a in placeof the free variable x).

Any linear ordering is indication-friendly. If R is alinear ordering, for A Ω, the formula⊇

⊇

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P1(x)=∀y(x≠y→xRy) defines the “maximal” element in the set A. The formula P2(x)=∀y(x≠yΛ–P1(y)→xRy) defines the “secondto the maximal” element, and so on. Note thatthere are “short cuts” for describing the differentelements. For example, the “short cuts” for P1(x)and P2(x) are “the first” and “the second.”

On the other hand, consider the set Ω=a,b,c,dand the nonlinear binary relation R, called “beat,”depicted in the diagram (satisfying: aRb, aRc, dRa,bRd, bRc and cRd):

Referring to the grand set Ω, the element “a” isdefined by “it beats two elements, one of whichalso beats two elements.” The element “b” isdefined by “it beats two elements, which beat oneelement each.” And so on. However, whereas therelation R allows the user to define any element inthe set Ω, the relation is not effective in definingelements in the subset a,b,d, where the inducedrelation is cyclical.

We will now demonstrate that if Ω is a finite setand R is a binary relation then R is indication-friendly if and only if R is a linear ordering. Wehave already noted that if R is a linear ordering onΩ, then for every A Ω and every a∈A, there is aformula indicating a. Assume that a binary relationR is indication-friendly. For any two elementsa,b∈Ω, in order to indicate the two elements in thetwo-member set A=a,b, it must be that either aRbor bRa but not both; thus, R must be complete andasymmetric. R must also be transitive because forevery three elements a,b,c∈Ω, in order to indicateeach of the elements in the set A=a,b,c, it must bethat there is no cycle.

⊇

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a b

c d

Conclusion 1: A binary relation enables the user toindicate any element in any subset of the grand setif and only if it is a linear ordering.

2.3 InformativenessAnother use of binary relations on a set Ω is that ofa means to transfer or store information concerninga specific relationship between the elements of Ω.Consider a case in which the grand set is the set ofarticle authors in some field of research and thespeaker is interested in describing the relation “xquotes y in his article.” The speaker may describethe relation by listing the pairs of authors who sat-isfy the relation. Alternatively, he may use thosebinary relations which are available in his vocabu-lary to describe the “x quotes y” relation. If he findshis vocabulary insufficient to describe the relation,he will use a binary relation which best approxi-mates the relation he wants to describe. For exam-ple, if the relation “x is younger than y” is welldefined, the speaker may use the sentence “anywriter who is younger than another writer quoteshim” to describe who quotes whom. As this maynot be entirely correct, he may add a qualifyingstatement such as “the exceptions are a who did notmention b (though he is younger), and c who didmention d (though he is not younger).” Thosequalified statements are the “loss” incurring fromthe use of an imprecise relation in order to indicatethe “who quotes whom” relation.

Our investigation is on the level of an imaginary“planner” who is able to design only one binaryrelation at the “initial stage of the world.” Ofcourse, real-life language includes many relations,and the effectiveness of each relation depends onthe entire fabric of the language. The assumptionthat the designer is planning only one binary rela-tion is made here solely for analytical convenience.

The design of one binary relation allows thespeaker to select one of four binary relations. Forinstance, he can say: “everyone is quoted by allauthors who are younger,” or “everyone is quoted

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by all who are not younger,” and of course he cansay also “everyone quotes everyone” and “no onequotes anyone,” two statements which do notrequire familiarity with any binary relation. Givena relation R, we will refer to those four relations asthe vocabulary spanned by R, and denote it byV(R). (Note that in defining the vocabularyspanned by R, we ignore other possibilities fordefining a binary relation using R, such as the useof statements of the type “xSy if there is a z suchthat xRz and zRy”).

Now, the speaker, who wants to refer to a binaryrelation, S, will use (so we assume) a relation inV(R) which is the “closest” to the one he reallywants to refer to. The loss incurred is measured bythe number of differences between the relation,which the speaker wants to describe, and the onehe finds in his available vocabulary. The distancebetween any two binary relations R' and R'' istaken to be the number of pairs (a,b) for which itis not true that aR'b iff aR"b. Note that by thismeasure, any pair for which R' and R" disagreereceives the same weight. Regarding the initialstate, it seems proper to put equal weights on allpossible “imprecisions.”

The designer’s problem is to minimize the expectedloss from optimal use of his vocabulary. It is assumedthat from his point of view, all possible binary rela-tions are equally likely to be needed by the speaker.Thus, the designer’s problem is minRΣSδ(S,V(R)),where δ(S,V(R))=minT∈V(R)δ(S,T) and δ(S,T) is thedistance between the relations S and T. One canshow (see Rubinstein, 1996) that choosing R so thatit will include half of the pairs (a,b) where a≠b is“nearly optimal” for the designer if he wishes toreduce the expected number of imprecisions.

We reach the final point of this section: Whenplanning a binary relation on Ω, the designer, sowe assume, also considers the possibility that therelation will eventually be used in reference to asubset of Ω (this is analogous to the condition

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made in the previous section that the speaker willwish to indicate a from any subset of the grand set).Completeness and asymmetry guarantees that if Rhas this structure then for every subset Ω' Ω, theinduced relation R|Ω' includes exactly half the pairsin Ω'–(ω,ω)|ω∈Ω'.

Conclusion 2: For the task of expressing binaryrelations as accurately as possible using a vocabu-lary spanned by a single binary relation, it is nearlyoptimal that the binary relation will be completeand asymmetric.

⊇

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Chapter Three: StrategicConsiderations in Pragmatics

3.1 Grice’s Principles and Game TheoryThe study of language is traditionally (seeLevinson, 1983) divided into three domains: syn-tax, semantics, and pragmatics. Syntax is the studyof language as a collection of symbols detachedfrom their interpretation. Semantics studies therules by which an interpretation is assigned to asentence independently of the context in which thesentence is uttered. This part touches the thirddomain, pragmatics, which is the study of the waythat the context in which an utterance is madeaffects the way it is interpreted. It views an utter-ance as a signal which conveys information withina context: the speaker, the hearer, the place, thetime, and so forth. What the hearer thinks aboutthe intentions of the speaker and what the speakerthinks about the presuppositions of the hearer arefacts which are relevant for understanding theutterance.

Take, for example, a conversation between A, whoconverses from home, and his friend, calling froma telephone booth:

A: B, I am about to go for a walk. How’sthe weather outside?

B: It’s not raining heavily.

Normally, A will conclude from B’s statement thatit is raining but not heavily. This conclusion doesnot follow from the semantic interpretation of thesentence “it’s not raining heavily,” which allows forthe possibility that it is “raining but not rainingheavily” as well as “it’s not raining at all.”Furthermore, there are circumstances under whichthe utterance “it’s not raining heavily” will indeedbe interpreted as not excluding the possibility thatit is not raining at all. For example, imagine that Bis in a hut and has told A that it is dark outside and

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that he is deep under his blanket. In such a case, Bwould only realize that it is raining if it is “pouringrain” and the rain pounds his roof. In this case, Awill indeed infer from B’s statement that it may notbe raining at B’s place. Or,

A: “What do you see?”

B: “Well…it’s not a rose.”

Ordinarily, A will understand this statement that Bis seeing a flower which is not a rose, though B hasnot said that he is looking at a flower.

In these examples, the fact that a statementappeared within a conversation between two peo-ple made the meaning of each of B’s statements dif-ferent from that we would have given it had webeen asked to interpret the sentence in isolation. Atheory which aims to describe the rules for inter-preting daily conversational utterances was sug-gested in the 1960s by the philosopher Paul Grice(see especially Grice, 1989). Grice’s initial point is:

“…while it is no doubt true that the formaldevices are especially amenable to systematictreatment by the logician, it remains the casethat there are very many inferences and argu-ments, expressed in natural language and notin terms of these devices, which are neverthe-less recognizably valid. So there must be a placefor unsimplified…logic of the natural counter-parts of these devices;….” (p. 23–4)

I will not enter into a discussion of Grice’s theory.Let me just say that I was amazed how much gametheoretical considerations are at the heart of thattheory. I will now attempt, by employing workconducted jointly by Kobi Glazer and myself, todemonstrate the possible applications of game the-oretical methods in explaining a pragmatics’ phe-nomenon in the special context of a debate.

3.2 DebatesBy a “debate,” I am referring here to a situation in

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which two or more parties who disagree regardingsome issue raise arguments in an attempt to per-suade a third party of their position. This is, ofcourse, not the only form of debate taking place inlife. Sometimes the purpose of the debaters is toargue just for the sake of arguing, and sometimesthey aim to influence one another rather than athird party. Debates can be thought of as a specialform of conversation in which the parties have dif-ferent interests. Although debates are very commonin real life, debates have rarely been investigatedwithin the economic and game theoretical litera-ture (some exceptions are Lipman and Seppi, 1995;Piketty and Spector, 1996; and Shin, 1994).

The aim of this presentation is to provide an expla-nation to one phenomenon often observed indebates. At the outset, we should note that theGricean theory does not apply to debates becausehis logic of conversation is based on the principle ofcooperation, which does not hold in the situationof interest conflict of interests characterizing adebate.

To motivate the discussion, let us start with theresults of a survey conducted among several groupsof students at Tel Aviv University. The followingquestion was presented to one group of students:

Question 1: You are participating in a publicdebate about the level of education in theworld’s capitals. You are trying to convince theaudience that in most capital cities, the level ofeducation has risen of late. Someone is chal-lenging you by bringing up indisputable evi-dence showing that the level of education inBangkok has deteriorated. Now it’s your turnto respond. You have similar, indisputable evi-dence to show that the level of education inMexico City, Manila, Cairo, and Brussels hasgone up. However, because of time constraints,you can argue and present evidence only aboutone of the four cities mentioned above. Whichcity would you choose for making the

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strongest counter-argument against theBangkok results?

Another group of subjects was presented withQuestion 1 but with the sole modification thatBangkok was replaced by Amsterdam. A third, sim-ilar group of students was asked to answerQuestion 2, in which the subjects were asked toselect one opening argument by choosing fromamong the four cities Mexico City, Manila, Cairoand Brussels.

The following table presents the results of theirresponses to Questions 1 and 2. Manila is thefavorite counter-argument to Bangkok; Brussels isthe favorite counter-argument to Amsterdam. Incontrast, the subjects split quite evenly between thefour arguments when answering Question 2.

A puzzling element appears in the survey results. Iftwo arguments contain the same quality of infor-mation, why is it that one is considered to be astronger counter-argument than the other? Thefact that Manila is closer to Bangkok than it is toMexico City seems irrelevant to the substance ofthe debate, and yet it appears to affect dramaticallythe choice of the better counter argument.

We believe that this phenomenon is connected toconsiderations of pragmatics. Within a debate, aresponder’s counter-argument to “Bangkok,” ifusing anything but Manila, is interpreted as anadmission that Manila is also an argument in favorof the opponent’s position.

In the rest of the discussion we do not pretend to

QUESTION 1 QUESTION 1 QUESTION 2BANGKOK AMSTERDAM

n 38 62 24Mexico City 19 % 15 % 21 %Manila 50 % 3 % 21 %Cairo 11 % 5 % 25 %Brussels 21 % 78 % 33 %

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fully explain the rules by which people comparearguments. We just would like to point out that thelogic of debate should not necessarily include anaxiom stating that if an argument x beats an argu-ment y (in the sense of being a successful counter-argument against it) then argument y should notbeat argument x.

3.3 A ModelWe view a debate as a mechanism designed toextract information from debaters. We assume thatthe aim of the mechanism’s design is to increase theprobability that the right conclusion will be drawnby the listener subject to the constraints imposedon the debaters and the listener in terms of the timeand cognitive abilities they can invest in theprocess. It will be shown that fulfilling this aimmay lead to debating rules in which arguments andcounter-arguments are not treated symmetrically.

The setup is very simple: An uninformed listener hasto choose between two outcomes, O1 and O2. The“correct” outcome, from his point of view, is deter-mined by five aspects, numbered 1,…,5. An aspect imay be realized to be either 1 or 2, with the inter-pretation that if an aspect i gets the value j, aspect iis evidence supporting the outcome Oj. A stateω=(ωj)j=1,..,5 is a five-tuple of 1s and 2s whichdescribes the realizations of the five aspects. The lis-tener assigns equal weights to all five aspects, and thecorrect outcome at state ω, C(ω), is the outcomethat is supported by the majority of the arguments.

The listener is ignorant of the state but the twodebaters, named debater 1 and debater 2, have fullinformation about the state. The “problem” is thatthere is a conflict of interests between the twodebaters and the listener: Each debater i wishes thatoutcome Oi be chosen, whatever the state, whereasthe listener wants the correct outcome to be chosen.

A debate is taken to be a mechanism in which eachdebater reveals pieces of information in order topersuade the listener to choose the debater’s

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favorite outcome. A debate consists of two ele-ments: The procedural rules which specify the orderand what sorts of arguments each debater isallowed to raise; the persuasion rule which specifiesthe relationship between the arguments presentedand the listener’s conclusion.

We need to specify the “language” that the debaterscan use: It is assumed that debaters cannot makeany moves other than raising arguments of the type“argument j supports me.” Thus, one debater can-not raise arguments that support the outcome pre-ferred by the other debater. We will assume furtherthat debaters have to prove their claims, namely,debater i cannot claim that the value of aspect j is iunless it is indeed i.

We want to characterize the optimal debate givenan effective constraint about the length of thedebate. Of course, if it is possible for three argu-ments to be raised during the debate, the listenercan obtain the correct outcome with certainty. Hedoes so by the request that one of the debaters pres-ents three arguments; that debater will win thedebate if and only if he fulfills this task. Thus, tomake the length constraint effective, let us assumethat the number of arguments, which can be raised,is two.

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

1) The set of feasible moves of each debateris a subset of 1,…,5.

2) There can be at most two moves: Eitheronly one of the debaters is allowed to make atmost two arguments (the one-speaker debate);or the two debaters move simultaneously, eachone making at most one argument (the simul-taneous debate); or the debate is a two-stageprocess where, at each stage, one debater moveswhile making at most one argument (thesequential debate).

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3) One of the outcomes, O1 or O2, isattached to each terminal history.

A debate Γ and a state ω determine a game Γ(ω),which will be played by the two debaters. The two-player game form Γ(ω) is obtained from Γ bydeleting, for each debater i, all aspects which do notsupport his position in ω. (If player i has to moveafter a history h and if at ω none of the argumentshe is allowed to make at h support his position,then the history h in the game Γ(ω) will become aterminal history and the outcome Oj is attached toh.) As to the preferences in Γ(ω), debater i strictlyprefers outcome Oi to the other outcome, Oj.

The game Γ(ω) is a two-person zero-sum game(the listener is not a player in the game). The gamehas a value, v(Γ,ω), which is a lottery over the set ofoutcomes. Let m(Γ,ω) be the probability thatv(Γ,ω) assigns to the incorrect outcome. Whenm(Γ,ω)=1, we say that debate Γ induces a mistakein state ω. In debates which are not simultaneous,m(Γ,ω) is either 0 or 1. In simultaneous debates,v(Γ,ω) may be a non-degenerate lottery(1>m(Γ,ω)>0). All mistakes are weighted equallyand the optimal debate is taken to be the onewhich minimizes m(Γ)=Σωm(Γ,ω).

To demonstrate the calculation of m(Γ), let usreview some examples:

1) Let Γ be a debate in which only debater1 is asked to present two arguments; he wins ifhe raises two arguments either from the set1,2,3 or from the set 4,5. M(Γ)=4: the fourmistakes in favor of debater 1 occur in stateswhen only aspects 4 and 5 or two aspects outof the set 1,2,3 support debater 1. Actually,this debate has the least number of mistakes inthe set of one speaker’s debates.

2) Let Γ be a simultaneous debate in whicheach debater can present at most one argu-ment. Debater 2 wins the debate if debater 1

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argues regarding aspect x and debater 2’s argu-ment is regarding aspect x+1(mod5) or aspectx–1(mod5). Here, debater 1 rightly wins inany state ω where there are three successiveaspects (ordered on a circle) in his favor, and herightly loses in any state where only zero, one,or exactly two nonconsecutive arguments arein his favor. There are ten other states: five ofthem 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, thevalue of the induced game is the lottery thatselects the two outcomes equally. Thus,m(Γ)=10(1/2.)=5. Actually, one can see that theminimal number of mistakes in the family ofsimultaneous debates is 5.

In Glazer and Rubinstein (1997), we found thatany optimal debate procedure is sequential and theminimal m(Γ) over all debates is three. The follow-ing is a debate with the minimal number of mis-takes. First, debater 1 and then debater 2 are askedto present an argument. Debater 2 wins if and onlyif he counter-argues argument i with an argumentregarding an aspect which is listed in the secondcolumn in row i. This debate induces three mis-takes, two in favor of debater 1 (in states(1,1,2,2,2) and (2,2,1,1,2)) and one in favor ofdebater 2 (in state (1,2,1,2,1)).

What does the listener understand from anexchange of arguments where debater 1 arguesregarding aspect i and debater 2 responds withaspect j? Is it merely that aspects i supports debater1 and aspects j supports debater 2? We can identifythe listener’s thoughts by considering the sequential

IF DEBATER 1 DEBATER 2 WINS IF AND ONLY IF HEARGUES FOR… COUNTER ARGUES WITH…

1 22 4,53 44 1,55 2,3

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equilibria of the three-player game constructedfrom the debate game by adding the listener as athird player who has to choose the outcome of thedebate at any terminal history. The above optimalpersuasion rule is supported by a sequential equi-librium in which Debater 1’s strategy is to raise thefirst argument i for which debater 2 does not havea proper counter-argument. If debater 2 has aproper counter-argument for each of his feasiblearguments, debater 1 chooses the first argumentwhich supports him. Debater 2’s strategy is torespond with a successful counter-argument when-ever it is possible. The listener chooses the outcomeaccording to the above persuasion rule. Note, forexample, that if debater 1 raises argument 3 anddebater 2 raises argument 4, it is optimal for the lis-tener to rule in favor of debater 2, since he con-cludes that in addition to aspect 4, aspects 1 and 2are in favor of debater 2.

Note that the three-player sequential debate gamehas other sequential equilibria as well. One of thoseis particularly natural: In any state ω, debater 1raises the first argument i which is in his favor anddebater 2 responds with the argument j, which isthe smallest j>i in his favor. The listener’s strategywill be guided by the following logic: Debater 1, inequilibrium, is supposed to raise the first argumentin his favor. If he raises argument i, the listenerbelieves that arguments 1,2,…,i–1 are in favor ofdebater 2. Debater 2, in equilibrium, is supposedto raise the first argument in his favor followingargument i. Hence, if debater 2 raises argument j,the listener believes that arguments i+1,..,j–1 are infavor of debater 1. The listener chooses O1 if thenumber of aspects he (the listener) believes are infavor of 1 is larger than those which he believes tosupport 2. This equilibrium induces six mistakes.To conclude, both equilibria demonstrate the pointthat the interpretation of the arguments is a matterof equilibrium and not a matter of just the seman-tic content of the arguments.

An interesting fact about the above optimal persua-

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sion rule is that aspect 5 is a persuasive counter-argument against aspect 2 and aspect 2 is a persua-sive counter-argument against debater 1’sargument regarding aspect 5. This is not coinci-dental. Actually we showed:

Conclusion 3: Any optimal debate is sequentialand has a persuasion rule, which does not treat theplayers symmetrically. That is, there is a pair ofaspects, i and j, so that when presented in sequence,i is a persuasive counter-argument against j and j isa persuasive counter-argument against i.

To conclude, in this lecture the rules of debate weretreated as tools which were designed to enable thebest elicitation of information, given the con-straints on the length of the debate and the inter-ests of the debaters. It was shown that in the courseof debates, the listener concludes from argumentsmore than what is said. The content of an argu-ment is an equilibrium phenomenon. In particularit was concluded that it is not necessarily a fallacythat in debates argument i defeats argument j and,at the same time, argument j defeats argument i. Inother words, the “logic of debate” does not “have”to contain a rule that if “p defeats q,” “q should notdefeat p.”

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References

Davitz, J. (1976), The Communication of EmotionalMeaning, Greenwood Publishing Group.

Glazer, J. and A. Rubinstein (1997), “Debates andDecisions: On a Rationale of ArgumentationRules”, Tel-Aviv University, Discussion Paper.

Grice, P. (1989), “Logic and Conversation” inStudies in the Way of Words, Cambridge: HarvardUniversity Press.

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

Lipman, B.L. and D.J.Seppi (1995), “RobustInference in Communication Games with PartialProvability”, Journal of Economic Theory, 66, 370-405.

Marschak, J. (1965), “The Economics ofLanguage”, Behavioral Science, 10, 135-140.

Piattelli-Palmarini, M.(ed.) (1970), Language andLearning: The Debate between Jean Piaget and NoamChomsky, Cambridge: Harvard University Press.

Piketty, T. and D. Spector (1996), “Rational DebateLeads to One-Dimensional Conflict”, mimeo.

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

Rubisnetin, A. (1996), “Why are Certain Propertiesof Binary Relations Relatively More Common inNatural Language?” Econometrica 64, 343-356.

Shin, H.S. (1994), “The Burden of Proof in aGame of Persuasion”, Journal of Economic Theory,64, 253-264.

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

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PREVIOUS NANCY L. SCHWARTZMEMORIAL LECTURES

1983 Hugo SonnenscheinThe Economics of Incentives: An IntroductoryAccount

1984 Andreu Mas-ColellOn the Theory of Perfect Competition

1985 Menachem E. YaariOn the Role of ‘Dutch Books’ in the Theory ofChoice Under Risk

1986 Robert J. AumannCooperation, Rationality, and BoundedRationality

1987 Robert E. Lucas Jr.On the Mechanics of Economic Development

1988 Truman F. BewleyKnightian Uncertainty

1989 Reinhard SeltenEvolution, Learning, and Economic Behavior

1990 Vernon L. SmithExperimental Economics: Behavioral Lessons forTheory and Microeconomic Policy

1991 Gary S. BeckerOn Habits, Addictions, and Traditions

1992 Kenneth J. ArrowInformation and Returns to Scale

1993 Peter A. DiamondIssues in Social Insurance

1994 Robert B. WilsonNegotiation with Private Information:Litigation and Strikes

1995 Roy RadnerEconomic Survival

1996 Nancy L. StokeyShirtsleeves to Shirtsleeves: The Economics ofSocial Mobility

1997 David M. KrepsAnticipated Utility and Dynamic Choice

1.3m/5-99/KGSM-ThD