tullock irrationality of intransativety 2662577

8/7/2019 Tullock Irrationality of Intransativety 2662577

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The Irrationality of IntransitivityAuthor(s): Gordon TullockSource: Oxford Economic Papers, New Series, Vol. 16, No. 3 (Nov., 1964), pp. 401-406Published by: Oxford University PressStable URL: http://www.jstor.org/stable/2662577 .

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THE IRRATIONALITY OF INTRANSITIVITYBy GORDON TULLOCK

A RECENT discussion of welfare economics 'summarizes the question oftransitivityas follows. There is not now conclusive reason to believe thatthe assumption of transitivityof preferencein individual choice is deeplysuspect. There are, however,grounds forbeing willingto make deep em-pirical investigationsinto the validity of the assumption'.' At anotherpoint the same author says: 'My conclusions indicate that we currentlypossess no strongevidence to warrantdropping(the assumption of transi-tivity).'2 Given the role that transitivityplays in much of the modernliterature,and in Rothenberg's book, his attitude towards it seems verydefensive.We do not have strongevidence against it; thereis no conclusivereason to suspect deeply the assumption. In sum, the evidence againsttransitivityis not sufficientto condemn it, but Rothenberg is clearlydoubtful. In this he is more or less in accord with the bulk of currentopinion on the matter.

It is the purpose of this article to present a contraryview. It will beargued that the assumption of transitivityis not particularlydoubtfulordubious. In view of the apparent suspicion of the assumption by the bulkofmodernwelfare economiststhis mightseem to be a revolutionaryposi-tion, but the practice of the same economistshas indicated relativelylittlereal doubt of the transitivenature of preferenceorderings. Articleafterarticlehas appeared in which this assumption is made, somethingwhichwouldseem to indicatethat the writershave at least someconfidencein itsreliability. Further,the chains ofreasoning based upon this assumptiondo not appear to have led to demonstrablyfalse conclusionsabout thereal world. This fact,in itself,is powerfulalthoughnot conclusiveevidenceforthe truth of the transitivityassumption.The debate about the assumption of transitivityturnsupon the inter-pretationof certain real world experiences. If an individual is asked totake his choice among a large numberofpairs of alternatives,with eachalternative appearing as one memberofquite a numberof thepairs,3or alarge number of people are asked to judge between three alternatives

1 The Measurement of Social Welfare,Jerome Rothenberg (Prentice-Hall, Inc., Engle-wood Cliffs,N.J.) 1961, p. 231.2 Ibid., p. 19.3 See Arnold M. Rose, 'Astudy of irrationaljudgements',The JournalofPoliticalEconomy,Oct. 1957, pp. 394-402 for an example. In this case the choice among the alternativesinvolved judgementsofthe severityof a crimeratherthan preference,but the transitivityproblem is the same.


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402 THE IRRATIONALITY OF INTRANSITIVITYgroupedin a set ofpairs,1it is highlylikelythat at least one case where anindividual has chosenA overB, and B overC, then C overA will appear.There are two possible explanations for this phenomenon. The first,offeredby May,2 is that the preferenceschedules are actually intransitive.The individual at one and the same time does preferA to B, B to C, and Cto A. The second, offeredby Rose,3 holds that the effectis an artifactarisingin the course ofthe experiment. This latter explanation is rathercomplicated,since manythingscan go wrong. In the firstplace, theexperi-mentnecessarily will take time, and the subject may change his mind.Whenhe was asked to compareA and B he not only preferredA to B, healso preferredA to C. The questions,however,forcedhimto thinkaboutthe generalsubject, and bythe time the experimentergot around to askinghimto compare A with C he had changedhismind; he now prefersC toA and to B. The apparent intransitivitymerelyreflectsthe change.Anotherproblemwhichmay arise involves the fact that in manyexperi-mentsofthis sortthe subject is requiredto choose betweenthealternativeseven though he may be actually indifferent.This could easily lead toapparent intransitivities.The subject may also be bored and simplyfillin the blanks on the experimentformhelter-skelterwithoutreally con-sulting his preferences.4They miayalso make a numberofother'errors'in recordingtheirjudgements. The result of all of these factorsis that itis quite possible foran experimentalsubject who has completelytransitivepreferencesto select A overB, B over C, and C over A.Thuswe have a well-establishedempiricalphenomenonand twopossibletheoreticalexplanations forit. Rose in his article produced considerableevidence that the second explanation was the trueone, but the doubt onthe point expressed by Rothenberg would appear to justifyanotherat-temptto demonstratethat intransitivityis not the explanation. For thispurposewe need only considerthree-elementintransitiveloops since anylargerorderloop will necessarilycontain at least one three-elementloopwithinit. A demonstrationthatthree-elementloops do not occur will alsoserveto eliminatethe possibilityoflargerintransitivecycles.As a firststep I should like to presenta basic statementabout the realworld:A man confrontedwith a small (say 5) collectionofalternativeswilleither (a) preferone of them,(b) be indifferentamong a subset oftwo ormoreofthe collectionbutpreferany memberof that subsetto the remain-der of the collection,or (c) be indifferentamong all the alternatives. Itseemsunlikelythat anyonewillseriouslyquestion this proposition,5but if

1 See Kenneth 0. May, 'Intransitivity, utility,and the aggregationof preferencepat-terns',Econometrica,Jan. 1954, pp. 1-13. 2 Op. cit. pp. 7-13. 3 Op. cit.pp. 395-6.4 Rose foundthat the number of apparent intransitivitiesin the judgementsof each ofhis subjects was inverselycorrelatedwith the timethey took to complete the form(p. 399).5 An ingenuouscolleague has offeredthe criticismthat the individual in question might


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G. TULLOCK 403someone does, it is easily tested empirically. Simple observation of acafeterialine would probably be the easiest 'experiment'. Anyone whoaccepts this statementmust,if he is rational, agree that preferencesaretransitive.The proof of transitivityis somewhat easier if we initially confineour-selves to a world in which indifferenceis impossible. In this restrictedworld our basic statementshrinksto the simple formulathat a man willpreferone of a small group of alternatives. The proof of transitivityis asimple example of reductioad absvurdur. If the individual is alleged topreferA to B, B to C, and C to A, we can inquire which he would preferfromthe collectionof A, B, and C. Ex-hypothesi he must preferone, sayhe prefersA to B or C. This,however,contradictsthe statementthat heprefersC to A, and hence thealleged intransitivitymust be false.There does not seem to have been any discussion of what a man withintransitivepreferenceswould do if presented with a choice among allelementsof the intransitiveloop simultaneously.He clearly wouldnot beable to choose,and he equally clearlycould not be called indifferentamongthe alternatives. The only patternof behaviour consistentwith initransi-tive preferenceswould be a closed loop in which the subject rapidly wentroundand roundhispreferentialcycle. Kenneth 0. TMay'sarticle,'Intransi-tivity,utility,and aggregationin preferencepatterns',' is surelythe localsclassicus forthe intransitivityhypothesis. In the course ofthe experimentreported there he firstfound apparent intransitivitiesin the preferenceorderingof 17 college studentsby paired comparisons. lie then presentedthem with all three alternativessimultaneouslyand asked them to rankthem. They had no difficultyin doing so.2 This result is, of course,completelyinconsistent with the hypothesis that they had intransitivepreferenceschedules.

May, however,did not point out that the ability of his experimentalsubjects to rank the alternatives proved that they had, at least at theperiodwithinthe experimentat which they made the ranking,transitivepreferenceschedules. He does not even discuss the point,but merelygoeson to presenta mechanismto explain the intransitivitywhichhe believeshis experiments showed. The simple explanation forMay's results, ofcourse, is that the subjects changed their minds duringthe courseof theexperimentand that the 'intransitivity'merelyreflectssuch changes. Inorder to accept the intransitivityexplanation of the phenomenonit isnecessary to assume quite complex patterns. One would be that thebe totally ignorant of the alternatives and hence neitherhave preferencesnor be trulyin-different.This, of course,involves the question of exactly what we mean by 'indifference'.Fortunately this semantic problem can be skipped. If the man in question hasno preferenceswith regard to the alternatives it is at least clear that he does not have intransitivepre-ferences. 1 Econornetrica,Jan. 1954, pp. 1-13. 2 pp. 6-7.


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404 THE IRRATIONALITY OF INTRANSITIVITYsubjects did not change theirmindsduringthe periodin which they wereasked the firstthreequestions,comparingA withB, B with C, and C withA, but that they then all changed their minds whentheywereasked thefourthquestion whichcalled forrankingall threealternatives.There is an alternativeexplanation forMay's experimentalresultwhichpreservesintransitivityas conceivable. If we assume that his subjects'preferenceswere not only transitivebut also depended upon irrelevantalternativesthen his resultsare explicable. Theoreticallyitwouldbe poss-ible fora man to preferA toB, B to C, and C to A while at the same timepreferringA ifpresentedwitha collectionofA, B, and C, ifhispreferenceswithrespectto A and C weredependentupon whetherornot B was pre-sent. If he prefersC to A whenB is not presentbut prefersA to C whenB is present,thenMay's experimentalresultscould be explainedwithoutassuming that his subjects changed their mindsduring the course of theexperiment. Since the timeofOccam, however,we have normallychosenthe simpler theory, and surelyin this case the hypothesisthat the sub-jects had transitivepreferenceschedules but changed theirminds is thesimplerone.So far, however,we have been discussinga simplifiedworld in whichindifferenceis absent. If we admit indifferencethe situation is morecom-plicated, although the same basic result will follow. Let us considerthesituationwhereit is allegedthat a man prefersA toB, B to C, and C to A,and that indifferenceis possible. If confrontedwith thecollectionofA, B,and C, then the individualmight,in keepingwithourbasic statement,beindifferentbetween A and B but preferboth to C. This would contradicttwo linksin the alleged intransitivitychain,the preferenceof A to B andthe preferenceof C to A. Hence, once again, the allegation of intransi-tivity must be untrueif our basic statement is true. Finally, the indi-vidual mightbe indifferentamongthe threealternatives. Spelled out,thiswould involvenot preferringA to B or C, notpreferringB to A or C andnot preferringC to A orB. Clearlythiscontradictsall threelinksin thealleged intransitivityloop.If indifferenceis permitted,however,morecomplicatedtypesofintran-sitivityare possible. An individualmightbe alleged to preferA to B andB to C but to be indifferentbetween A and C. It would be tedious to gothroughthevarious possibilities,but this would also be inconsistentwithourbasic statement. A moreinterestingcase would involve an individualwho was indifferentbetweenA and B and also between B and C but whopreferredA to C. This is, ofcourse,perfectlypossible.1 When confronted

I The intransitivityof the indifferencerelation has been analysed by W. E. Armstrongina long series of articlesbeginningwith 'The determinatenessof the utility function', TheEconomic Journal, Sept. 1939, pp. 453-67.


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G. TULLOCK 405withthe choice betweenA, B, and C, the individual will select A. Sincethis would appear to contradicthis indifferencebetweenA and B, somediscussion seemsin order.Assumean individual wvhoprefersblue to green and will always choosethe 'bluest' colourwhen given an opportunity.1He is, however,unableto make infinitelyfinejudgements. The minimumchange of colour whichhe is able to perceiveis some small amountwhichwe will denominate Y.We presenthim withthree colours,A, B, and C, in which A is the mostblue and C the mostgreen,but with the differencebetween A and B lessthan Y. The differencebetween B and C is also less than Y, but the dif-ferencebetweenA and C is greaterthan Y. If asked to choose which liepreferswhen confrontedby these alternativesin pairs,our subject shouldchoose A to C, but be indifferentbetween A and B and B and C. If he ispresentedwitha choicebetween A, B, and C, he should have no difficultyin selectingA because the presenceofC makes it possible forhimto dis-tinguishbetween A and B. He can see that A is bluerthan C, but cannotdistinguishbetween B and eitherA or C. The realizationthat B must liebetweenA and C easily follows. The same generalprocesscould operateinall cases whereindifferenceis foundto be intransitive.The addition ofthethird alternative, by changing the informationavailable to the chooser,increasesthe finenessof his discriminatorypowers. Hence his expressedorderis, in a sense,actually changed.So farwe have beenverylargelydiscussingthelogicof thesituation.Canwe design experimentswhich would serve to confirmor refuteour conclu-sions ? Theanswertothis questionisyes,andhenceour conclusionsarefully' operational'. Speakingformyself,however,theresultstobe expectedfromthe experimentsdiscussedbelow have been so obvious that they have re-mained Gedankenexperimentsinstead of experimentsto be actually per-formed.Perhapsone ofthereaderswillbe rathermorescepticalofmy logic.The firststep in any experimentwould be to give transitivitytests ofthe sort developed by May and Rose to a number of subjects in ordertofindapparent intransitivityloops. When such loops werefoundit wouldbe easy to distinguishtrue intransitivityfromapparent intransitivityarisingfromexperimentaldifficultiesby simply confrontingthe subjectwithall threeof the alternativeson the loop. Ifhe is able to rankthem,asMay's subjects were,it shouldbe clear that the 'intransitivity'is onlyanexperimental artifact. This procedure would only be repeating May'sexperiment,but his sample was rather small (only 17 subjects with in-transitiveloops) and it mightbe argued that witha larger sample a caseofgenuine intransitivitymighthave turned up.

I Dr. James M. Buchanan suggestedthe followinganalysis. I am also indebted to him formuch otherhelpfulcomment.


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406 THE IRRATIONALITY OF INTRANSITIVITYMay's experiment,however,did not cover another possibility. His sub-jects were required to choose between the alternatives, and could notsimplysay that they were indifferent.It is possible that, with less restric-tive conditions a differentresultmighthave turned up. Perhaps some ofhis subjects who had chosen A overB, B over C, and C over Al mighthaveexpressed indifferencewhen confrontedwith all three alternatives as agroup. This result could be explained either in terms of experimentalproblems,particularlythe factthatthe subject may have changedhis mind,orby a complicatedcombinationof true intransitivityand a most peculiarformof interdependenceamong the alternatives. Experimental determi-nation of which of these was the correctexplanation would be easy. Thereare a number of routines,but the simplestwould be to reduce the choiceonce again. The subject could be told that since he was indifferent,theexperimenterwould make the choice. The experimentercould then saythat personally he disliked C and was indifferentbetween A and B andhence would flipa coin to decide between them. The subject, if he has thepostulated intransitiveloop, should object and suggestthat A be markedas the preferredalternative. If he does not, he clearlyno longer prefersA to B, and hence the apparent intransitivitymust be put down to achange of mind rather than true intransitivity.This procedure can befurtherelaborated to obtain even greaterassurance.Note that thediscussionhas involvedno directassumptionofrationalityon thepart of thepersonswhohave preferenceschedules,onlyon thepartof the student consideringthem. If our basic statementabout the realworldis accepted, thenany intransitivepreferenceorderingmust involvethe simultaneous holding of directlycontrary preferences. The subjectmust preferA to B and B to A. Even a schizoid must split these twopreferencesbetween his two personalities. Our proof does not involvethe basic questionof human rationality,but only bars one particulartypeofirrationality.

Universityof Virginia1 Or chosen A over B, B over C, but been indifferentbetweenA and C, if indifferencewere permitted.

tullock irrationality of intransativety 2662577

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