An Interview with Kenneth J. Arrow Author(s): J. S. Kelly and Kenneth J. Arrow Source: Social Choice and Welfare, Vol. 4, No. 1 (1987), pp. 43-62 Published by: Springer Stable URL: http://www.jstor.org/stable/41105852 . Accessed: 27/09/2013 14:15
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This content downloaded from 150.135.135.70 on Fri, 27 Sep 2013 14:15:11 PM All use subject to JSTOR Terms and Conditions " Soc Choice Welfare 4 : 43-62 Z (1987) Social# Choice ^Welfare © Springer-Verlag 1987
An Interviewwith Kenneth J. Arrow
J. S. Kelly Departmentof Economics,Syracuse University, Maxwell Hall, Syracuse,NY 13210, USA
The followingis an editedtranscript of an interviewconducted on March 4, 1986 withProfessor Arrow while he was visitingSyracuse University to deliverthe Frank W. Abrams LectureSeries to be publishedas The UncertainFuture and Present Action by Syracuse UniversityPress. This interviewwas to elaborate on his description,presented in Volume 1 of his CollectedPapers (Harvard University Press,1983), of the originsof his work in collectivechoice theory.
JK. You startedoff the story in the collectedpapers withremarks about studying relationallogic whileyou werein Townsend- Harris High School in New York City.
KA. Not in highschool in the senseof in myhigh school courses,but duringthis periodI was an omnivorousreader and gotinto all sortsof things. One of themwas Bertrand Russell's Introductionto MathematicalPhilosophy and it made a tremendousimpression on me.It was theidea oflogic that was inthere. I don't really recall,for example, if there was a formaldefinition of a relationas a set of ordered pairs, but I learned the ideas of mathematicallogic and its applications to mathematicsin Russell'sbook. It seemsto me thatI also read one or two otherlogic books around that time.
JK. Later,when you wentto CityCollege of New Yorkas a mathematicsmajor, you encounteredmore mathematical logic.
KA . Yes, butagain thelogic study was on myown, there were no coursesin it.I don't reallyremember exactly what I read. I rememberonce takingout the Principia Mathematicabut of course it's not thesort of thingone reallyreads from.I was lookingup some theoremsin it and thingslike that. I reallyam not preparedto tell you whatI read,but at somepoint things like the idea ofdefining rational numbers by orderedpairs and equivalenceclasses by orderedpairs was somethingI got to
This content downloaded from 150.135.135.70 on Fri, 27 Sep 2013 14:15:11 PM All use subject to JSTOR Terms and Conditions 44 J. S. Kelly know. I was fascinatedby thisand used to aggravatemy professors by writingout proofsin very strictly logical form, avoiding words as muchas possibleand thingsof that kind.
JK. You did takea formalcourse with Tar ski in thePhilosophy Department ; how did you happento take thatcourse?
KA. Yes. Well,I knewthat Alfred Tarski was a greatand famouslogician and there he was in mylast termin school and obviouslyI was going to take a course with AlfredTarski. It turnedout he had two courses.One was a kind of introductory courseand I feltI knewmore than that. The othercourse he gave was in thecalculus of relations.To say it was in the calculus of relationsmeant that he gave an axiomatictreatment of relations,although he motivatedit of course by motivating the axioms. You neverhad xRy; you onlyhad R and S and T. You see, he never mentionedindividuals in the formaltheory. He had an axiomatic theorylike an axiomatictreatment of set theory. Relations have somespecial aspects, in particular the idea of relativeproduct, RS. If thereis a z such thatxRz and zSy, thenxRSy. The relativesquare, R2=RR is especiallyinteresting; if the relative square is includedin R you have transitivity. So it was a fascinatingthing, although it was reallyvery elementary; really very easy. The concepts were not verysubtle compared withthe deep thingshe was workingon like the truthprinciple.
JK. At thispoint you wereinvolved in translatingsome of Tarskfs work.
KA. He wrotea textbookcalled Introductionto Logic [1] whichis one ofthe modern treatments,modern as of 1940. It had been publishedin German,may even have beenorginally published in Polish. I didn'ttranslate it. What happened was he had a translatorand I read theproofs. I wasjust finishingcollege and he asked me to read theproofs for him. He didn'tknow any English,you see. This was the interesting thing.He came to thiscountry in September,1939 forsome kind of congressor conferenceand was trappedhere by the outbreakof the war. He knew Polish, he knewGerman, but he didn'tknow any Englishso he spentthe Fall termlearning some Englishso he could teachus in theSpring. At firstwe couldn't understanda word he was sayingbut afterabout a week or so we began to catch on and we realizedit wasn't his rate of progress it was ourrate of progress that was relevant.His stresseswere all wrong. He was aware of this and thereforefelt he couldn't proofreadin English.It's ratherinteresting as a coincidencethat the translator was a Germanphilosopher named Olaf Helmerand Helmercomes back into mystory eightyears later. It's interesting. . . Tarski,although his Englishwas weak,had a verygood sense oflanguage and he kepton askingme "Is thatreally good English?"Not in thesense of being grammaticallycorrect, but, well forexample, Helmet was veryfond of usingthe word "tantamount"and Tarski got the feelingthat somehowit's not a word used veryoften. Actually his instinctsfor language were extremelygood. I suppose that was connected with his general work on formalizationsand metalanguages.Anyway, I was just a proofreader.
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JK. You writethat as a graduatestudent at Columbiayou spenttime as an exercise translatingconsumer theory in the logic of relationsand orderings.What got you startedon thatand whatdid you get out of it?
KA.I wentto Columbiabecause . . .wellthere were several problems. One was that wewere extremely poor and the question of going anywhere depended on resources. Columbiahad thegreat advantage, of course,that I could liveat home,which wasn'ttrue anywhere else. I didn'tget any financial support for my first year, none at all. Butanother of the things I had learned on myown at collegewas mathematical statisticsand I reallyhad become fascinated with it. There was a coursein statistics [atCCNY] ; theteacher, a man by the name of Robinson, had no realknowledge of itI wouldsay, basically - 1won't even say he had a goodreading list - buthe did list one book,J. F. Kenney[2] ifI remembercorrectly, which happened to havean excellentbibliography. It was not one of those cookbooks in statistics but actually didhave some attempts at mathematics.Kenney had references to R. A. Fisherand gaveyou enough to getyou interested. So I startedreading Fisher and oneof the firstthings was tryingto workout his derivationof the distributionof the correlationcoefficient under the null hypothesis,which was an integrationin ^-dimensionalspace. In Fisherit was done by intuition. I mean it's rigorous if you're sufficientlysophisticated ; to meit was gibberish. But I knewenough multivariate calculusto be able to translateit into rigorousform, at least a formthat I understood,and then I couldsee that he really was right. But I couldn'tsee it the way he wroteit. ThenI supposebecause of mylogical background what was really importantwas readingthe Neyman-Pearson papers which were then new and writtenin ratherobscure places, but they were available in the[CCNY] library. FromFisher alone, I thinkI would have been hopelessly confused about the logic of statisticaltests, although Fisher was greaton derivingdistributions. So, I knewI wantedto studymathematical statistics, which however was not a field,not a Departmentat Columbia.It was spreadout in otherDepartments. I knewthat Hotelling was one of themajor figures, but he was in theEconomics Department.I rather naively thought I would study mathematics and thenwould takethe statistics from Hotelling. I had no interestin Economics. I wasin the Mathematics Department, taking courses like Functions of a Real Variable,but I was goingto takecourses from Hotelling. In thefirst term he happenedto givea coursein MathematicalEconomics. So outof curiosity I took thisand gotcompletely transformed. The courseto an extentrevolved around Hotelling's own papers.But, as it happens,they were kind of central. He gavea rigorousderivation of supplyand demand.There was one paperon thetheory of the firm, one on thetheory of the consumer[3, 4]. And he gavea rigorousderivation of demandfunctions in the consumertheory paper and derived the Slutsky equations. I thinkhe knewabout Slutsky'swork, though I'm not sure he actually referred to Slutsky.So, anyway,this wasone of the best papers around at thetime. It's now a stapleof our literature but thenreally was novel.One of thethings, he was a very,very strong ordinalist, emphasizedthat all theseresults were invariant under monotone transformations, whichwas not a normalpractice in economics at thattime. Of course, all thosewho
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werecoming of age, like Paul Samuelson,would jump to thatposition ; it was the normal position of the avant garde. Well,the idea was thatit was an ordering.It was clearthat what they were saying was "jc is betterthan >>" and thatthis is a transitiverelationship. And I recognized thatthere were certain continuity axioms that had to be added to that.I was already familiarwith that because therewere certain similar things in the foundationsof probabilitytheory. In factI thinkI workedthat out for myself. I was playingaround once in collegetrying to workout an axiom systemfor probability theory, that was work on an Honors paper or something,and I ran across a set of axioms by Karl Popper. Research methods were prettyprimitive; I looked throughthe Union catalog and therewas a referenceto an article[5] in Mind by Popper. I realizedthat his axiom systemreally couldn't explain certain things that we take forgranted like the factthat cumulative distributions have a one-sidedcontinuity property. So I realizedthat you need some kind of extracontinuity axiom and I sortof invented countableadditivity all by myself.Later, of course,I foundthat Kolmogoroff and othershad done this,but I could see therehad to be an axiom. So I was kindof familiarfrom having worked it out therethat you neededthese continuityaxioms in orderto close yourpreference theory system. It was easy to provideand I suppose otherswere doing the same. I could also see thatwhile it was clarifyingfor me, it was hardlya contributionto knowledgebecause all I was doing was translatingto a language thatI knew.At least it got me thinking; whenever I saw a U for a utilityfunction I translatedto a preferenceordering. In factone thingthat struck me as an interestingproblem - thisis digressinga bit, but not entirely- why should there be a utilityfunction representing an ordering?Hotelling had neverreally asked thatquestion. Although he emphasized thatthe indifference map was theprimitive, and theutility function only represented it, he didn't really ask "Why should you have a representationin terms of numbers?"I was reallythinking about thisproblem when I happenedto runacross some papers [6] by Herman Wold who gave whathe called a "Synthesis"in some papers in SkandinaviskAktuarietidskrift which gave a long treatmentof demand analysiswhich did have essentiallyan axiomaticpoint of view. There he said you've got to provethere is a utilityfunction representation. He was thefirst person I know to realize,in print,that this was a problem.He gave an answer,extremely weak because he needed strongassumptions. Anyway,then I switchedto Economics fromMathematics. I had gone to Hotellingasking for a letterof recommendation for a fellowshipin theMathematics Department and he said, "Well, I'm sure I dont't have any influencein the MathematicsDepartment, but ifyou shouldenroll in Economics,I've foundin the past theyare willingto give one of my studentsa fellowship."I was bought. Incidentically,I impressedhim on about thesecond day of theclass because he was fascinatedby Edgeworth'staxation paradox ; in facthis paper on thetheory of thefirm was called "Edgeworth'sTaxation Paradox and theNature of Supplyand Demand Theory" [3]. Consider a case wherethere are firstclass and thirdclass railroadtickets as in theEnglish system. It turnsout thatif you imposea tax on one ticketthen, with suitable demand functions,you could lower the price of both commodities.At the time therewas a lot of excitementabout that; the public financepeople werepooh-poohing it, saying, "How can thisbe?" It had to do with
This content downloaded from 150.135.135.70 on Fri, 27 Sep 2013 14:15:11 PM All use subject to JSTOR Terms and Conditions An Interviewwith Kenneth J. Arrow 47 thenature of inter-relateddemand curves and thatwas thebig thingHotelling stressed,that demand functions depended on n variables,not one variable. But he saidhe was puzzled by the fact that he had never been able to produce an exampleof Edgeworth'sparadox with linear demand functions. So I satdown and wroteout theconditions for linear demand functions to yieldthe paradox ; theseconditions werecertain inequalities on thecoefficients and the inequalities were inconsistent. So I camein the next day and showedit to him.Really it was just a few.lines, but fromthat point on he was reallyimpressed with me. It was an extremelyeasy calculation,but thinking in inequalityterms was notcommon. Little pieces were quiteeasy to prove, but you couldn't do itin the mechanical fashion which you were doingwith, say, solving simultaneous equations or maximizations. Anyway,I enrolled in Economics and one of the things I read was a brandnew book,Hick's Value and Capital [7]. You know,after reading though the mish-mash likeMarshall and thingslike that, suddenly there was thisclear, well-organized view,you knew exactly what was happening. Just the sort ofthing to appealto me. Therewas a whole,messy, confused literature on capitaltheory ; all thosegreat debatesbetween Knight and von Hayek and all that. And now here was just the idea ofdated commodities and suddenlyscales fell from your eyes. A simpleidea like datedcommodities made whole issues transparent. Butas I readHicks, I couldsee there were things left out. I turnedto thisagain whenI returnedfrom the War, which was really pretty much of a hiatusin any work I wasdoing - 1was gone and very busy for about three and a halfyears. I haddone all myexaminations before I hadleft. So nowit was just a questionof my thesis. I decidedto take Valueand Capitaland redoit properly.I could see all kindsof specificpoints that were of concern.I wantedto combineit withSamuelson's stabilitytheory, which he haddeveloped in the meanwhile, the papers on dynamic stabilityin '41 and'42 [8].Maybe I wouldadd some stochastic elements to thestory becauseas a studentof probabilityand statisticstheory I couldsee noisein the system.Well, it was a lifetimeof work, really; it was a veryunrealistic thesis. Hotellingwas primarily interested instatistics atthis time and then he left for the UniversityofNorth Carolina. And Abraham Wald wasn't interested inEconomics anymore,either. The one I wasclosest to was Albert Hart, who was regarded then as a verypromising theorist, but somehow wasn't able to do whathe wascapable of. Nowpeople haven't even heard of Albert Gailord Hart. He hada goodanalysis of flexibilityina Festschriftfor Henry Schultz [9], another figure who has faded, but I was neverimpressed by HenrySchultz. Hart considered a problemwhere you're thinkingof buying a durablemachine and you're uncertain as tothe second period output; the trade-off isbetween a first machine that would be beautifully optimizing ifyou knew exactly the output but is not very good at slightly different outputs and a secondmachine that has coststhat are fairlyuniform along a widerange. The secondmachine might sometimes be preferred.He gavea sequentialanalysis; the ideathat your choice today can be dependent on your uncertainty about tomorrow. Elementaryas that point may seem, it just hadn't been expressed anywhere; it was veryrevelatory and came out in this Festschrift for Henry Schultz [10], published in '42, '43 or '44. Hartwas friendly and respectful but not very mathematical. One of the things he broughtto myattention was that firms are, after all, multiowner objects. It is true
This content downloaded from 150.135.135.70 on Fri, 27 Sep 2013 14:15:11 PM All use subject to JSTOR Terms and Conditions 48 J. S. Kelly thatall the ownersare interestedin the same thing,maximizing profit ; however, froma Hicksianpoint of view,the owners might have differentexpectations. Then theirrecommended investment policies today would be different.Each one, trying to maximizethe same thing,expected profit, would neverthelesshave a different choice. Now, froma modernpoint of view,we would probablytake a different position.In factthe whole idea on mypart was wrong;I didn'ttake accountof the verysimple point that owners could selltheir stock. I didn'tthink about untilI read Modigliani-Miller[11] years later and realized that my whole attack had been wrong.Those who weremost optimistic about thefirm would, in effect,buy it out fromeverybody else, who would do somethingelse withtheir money. So, in fact,in thatcontext, the votingparadox is irrelevant.But I did not thinkof it thatway, I thoughtof owners as glued to the firm,and just did not thinkabout the stock market.Within that context I thought,well, how would theydecide betweentwo actions. A reasonable thingis to assume it goes by majorityvote, a majorityof shares,of course. I startedwriting this down and it occurredto me that I have a preferenceof the firm defined by the statement that a majorityprefers investment A to investmentB. Then, frommy background, a naturalquestion is, is thisrelation transitive?Well, it didn'ttake morethan a couple of triesto see thatthis is not true. The minuteI saw it, I thought: This mustbe well known.In fact,I thoughtI mighthave seenit before. I have no idea fromthat day to thiswhat I could have seen. However,it is thesort ofthing which would appear maybeeven in a puzzle page ofa newspaper.It could have appeared in some quite triflingway. Anyway,I thought I'd seen itbefore and I didn'tthink it was major; all I thoughtwas itwas a nuisance because itwas spoilingmy theory. I endedup tryingto developa theoryon thebasis ofmaximizing profits weighted by share numbers, using that as a maximand.Later I gave up the whole thingbecause it seemed veryunwieldly and didn't cohere.
JK. Gave up thatsection or the wholedissertation ?
KA. The whole dissertation.Large partswere in no way novel. At best,I learned some ideas about myopiain investment,things that in lateryears I pursued.At one pointI had about thirtypages of outlineof ideas and resultsbut somehow it didn't sendme. I feltsomehow it was supposedto lead ultimatelyto empiricalwork. I was ratherdiscouraged because I was spendingquite a bitof time at this.It was a couple of years.
JK. Now comes the Hicks lecture. . .
KA. Intransitivitywas somethingI had discovered,but it was not on mymind; it was somethingI had dismissed.Hicks gave a lectureduring the winter of '46-'47. He had a veryinteresting idea. He was tryingto finda definitionof welfareinequality which was neverthelessconsistent with ordinalism.What was meant by saying "individualA is betteroff than individual B?" Hicks' statementwas thefollowing: Suppose individualA prefershis own bundle to B's and individualB also prefersA's bundle to his own. Then Hicks would say A is betteroff than B. Of course, this definitionsurfaced again about twentyyears later in thework of Duncan Foley [12],
This content downloaded from 150.135.135.70 on Fri, 27 Sep 2013 14:15:11 PM All use subject to JSTOR Terms and Conditions An Interviewwith Kenneth J. Arrow 49 but I'd neverheard of it before.I thinkI once foundsome indicationthat Trygve Haavelmo had had thatidea and I've triedto findthe reference again butI've never succeededin trackingit down. Hicks presented this lecture and wentthrough a lot of variationsof this point but he said JoanRobinson had criticizedit and he was a little worriedabout it.He said, ofcourse, that people might be non-comparable.A could preferhis bundleto B's and B could preferhis own to A's - he recognizedthat. I thoughtabout it sittingthere and finallysaid, "Would you want thisproperty of 'beingbetter off than' to be transitive?"I definedwhat I meantby transitive. "Well, in thecase, yourdefinition won't satisfythat, because thecomparison between A and B is based on A and B's orderings,the comparison between B and C is based on B and C's orderings,and thecomparison between A and C is based on A and C's orderings.And it's possibleto have A betteroff than B, B betteroff than C and C betteroff than A bythis." So I guesssomehow this idea of intransitivity fascinated me. Hicks said, "What? What? What?" I don't thinkhe quite got thepoint. Hart was veryquick ; he was chairingthe meeting, got thepoint and triedexplaining it. Interestingly,Hicks neverpublished this. Now all thismay be beside the point, because you may not want transitivityeven thoughit seems natural.In Foley's work,for example, this issue doesn't arise. Foley just said a pointis fairif nobody is betteroff than anybody else. But thisstory does show intransitivitywas bubbling aroundinside me eventhough I wasn'tconciously aware of pursuingthis line as a subjectof research. At this time,I receivedan invitationto the Cowles Commission.At firstI postponeda move because I was tryingto finishmy Hicksian dissertation before I wentthere, but I finallysettled on finishingit there.
JK. Wasn'tit unusual then to leave graduate school before finishing your dissertation ?
KA. You know,my knowledge of whatwas typicalwasn't very good. The people I knewwere at Columbia.I didn'tknow what was goingon at Harvardor Chicago. It was reallyvery provincial. In factfrom what I now know the Columbia situation was unusuallychaotic. One problemat Columbia,and I thinkit's true to thisday, is thatthe sense of communityamong studentsand facultyis veryweak; they'reall dispersed.In particularthe National Bureau of Research was a very strong organizationand some of the leading Departmentmembers went off there, especiallythe greatWesley Clair Mitchell.So theyweren't available. The Bureau was notnear the University and so theywere just simplyphysically somewhere else. One had the feeling,in fact,that they never talked to each other.
JK. Whatled to yourinvitation to Cowles?
KA. They came around and asked Wald and he recommendedme. While he was primarilytrained as a statistician,nevertheless he was interestedin economics. There weren'tmany in thatcategory and so theyasked him. Cowles was a funnykind of place because theywere kind of a persecutedsect ; the mathematicaland quantitativeemphasis was exceptionaland distrusted.My salarywas $ 3200 percalendar year - it was a calendaryear appointment. Even by the standardsof 1947, that wasn't verymuch.
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JK. What wereyou hiredto do ?
KA. Whatthey really wanted me to do was workon statisticalproblems but it was a free-wheelingplace. At the momentthe emphasiswas on the developmentof the econometricsof large-scale models. So called "large": three equations, five equations. Larry Klein ended up witha 20 equation model. Now Tinbergenhad even biggermodels in his League of Nations studybut this was simultaneous equations estimationwhich made thecomputational burden very much greater. I had some idea of using higherorder approximations.Others had gotten the asymptoticdistributions which were normal and I had been takinga course in Edgeworth-Cramérexpansions which you get fromhigher-order approximations. These ideas were originated by Edgeworthand quite ignored; interestingly, Edgeworth authored quite a fewnew ideas in statistics,most of which were ignored and then rediscovered.Edgeworth had this method of Cramer rediscoveredit. Actually, it was prettyhigh-powered mathematics and it really was probably beyondme. I knewhow to do it,but mathematical estimates of theerror term in the approximationwere a verysubtle and complexmatter. But I was thereto do anythingI pleased and I was veryobviously interested in theory.There was a feelingthat theoretical foundations were also an essentialpart. Finishingmy thesiscould fitinto this.
JK. Whileyou wereat Cowlesyou workedon thesingle-peakedness result.
KA. I reallyspent a yearthere not doing muchof anything,to tellyou thetruth. I wrote a few tiny papers, none of which amounted to anything.I was a great contributorto discussions: argumentative,finding exceptions, errors and counter- examples.But I reallyfelt very discouraged. Once, at lunch,we weretalking about politics,left parties and rightparties, and I rememberdrawing on a piece of paper theidea thata votermight have preferencesover the parties. It wasn'tso muchthat I saw theideas - it was the onlyway I could thinkabout it. It was not thatI thought - whydon't we representvoters as havingpreferences as soon as I thoughtabout the question,it couldn't occur to me therewas anyother way of doing it. So I wrotethis thingdown and startedlooking at the question of majorities.Its reallyhard to describeit. All I can say is,once you'veseen it, its obvious ; ittakes an houror two. If you ask thequestion, the answer is fairlyobvious. I spenta day or twoworking it up as a formalproof. And in myusual way,I sortof stalled about a monthon writingit up for publication. No, but that doesn't make any difference.I can't say a lost anything.It wouldjust establishthat I had theidea independently.But in a sense it didn't matterbecause withinabout a monthI picked up the JPE and there'sthe paper [13] by Duncan Black that had exactlythat idea. The coincidence I regard as an extremelyinteresting point in the historyof thought.It's an idea whichcould easily have occurredto Condorcet. It doesn't depend in any way on the developmentof mathematicsin the last 150 years.
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JK. Except to a sensitivityto thelogic of relations?
KA. I supposeso. Well,let me putit this way. The logicof relations was workedout at greatlength in thelatter part of the 19thCentury. Let's say it dependson Boole and theidea ofrelations as orderedpairs -even thatgoes back to 1910or 1911. But whilethe idea ofrelations as orderedpairs is a comfortingidea, in the sense that you have a logicalfoundation, it isn'tnecessary. The fellowwho developeda lot of the ideas about logical relations,rather sophisticatedly,was Charles Peirce, the philosopherand logician,founder of pragmatismaround 1880-1910. It was picked up bya Germannamed Schroder who in good Germanicfashion wrote three large volumes[14] around 1890. All the apparatus,all the sensitivitywas there.If you neededmore, the Principia surely supplied all thatwas needed.Nobody asked that question;that's all I can say. Black ofcourse, had been buildingup to it; he really did have the idea of votingas a mechanism.
JK. Had you read anythingof Black's beforethis paper ?
KA. I honestlycan't tellyou. The stuffbefore was awfullyformal and obvious. This was the onlypaper of his I seriouslyregard as havingsome excitementin it. So, anyway,there was mythird encounter with orderings. But thatreally developed out of amusement. Then thatsummer I wentto Rand Corporation- again throughsheer accident. My wife,who I metas a graduatestudent in Chicago,had previouslyworked in the AgricultureDepartment. She'd arrivedthere as a clerkand becamea professional,a statistician.Her boss was a verydistinguished mathematical statistician named M. A. Girshick.So I was friendlywith Girshick who had goneto theRand Corporation whenit was started.The Air Force neededsomeone to tellthem what was goingon inthe world, so theytook all thesewild characters and unleashedthem. Girshick was one of thoseinvited to go out thereand he commencedto spenda couple of years. He oftenvisited Chicago. He had been in contactwith the Cowles Commission anyway,because some of his work in multivariateanalysis really was veryclose mathematically,more than mathematically,close conceptually,to simultaneous equationestimation. He was givingadvice to them;in fact,he had some ingenious ideas. He contributeda good deal to the developmentof the limitedinformation methodand was neverreally given full credit for that. Anyway, Girshick had this connectionwith Cowles independentof us, butwhen he came to Chicago he visited us. One of thethings Rand was doingwas invitinglarge numbers of visitorsfor the Summerso Girshickurged me to come. Summerin Santa Monica didn'tseem like a bad idea to meand itturned out to be farmore intellectually exciting than anything I had planned because the halls were filledwith people workingon game theory. Everybodywas fooling with zero-sum games, how to calculate them, the fundamentaldefinition of theconcepts ; it was workat theconceptual level and at the technicallevel.
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JK. Was game theorysomething you'd studiedbefore you wentto Rand?
KA. Not really.I mean I knewabout thebook and had been vaguelypecking away at it,but I hadn'treally studied it at all carefully.It was not a bigtopic at theCowles Commission,although Marschak had writtena reviewof it. Anyway,Olaf Helmer was among those who had been broughtto Rand ; a philosopher.There were several people thereas a matterof factwho werebasically philosophers; Abraham Kaplan was another.Helmer said to me one day : "There's one thingthat disturbsme." They were taking game theoryand applying it especiallyto Soviet-U.S.relations : diplomaticconflict, potential tactical situations, war. However,the payofffunctions were defined in termsof utilityfunctions, as Von Neumannand Morgensternargued in theirappendix, and thesewere derived on thebasis of theindividual. The troublewas, theSoviet Union and theU. S. were not individuals.What is the meaningof this? You've got to give him creditfor proposingthe problem. Now I hadn'tspent a lot of timeor attentionon welfareeconomics. I had really been tryingto work on descriptivetheory and general equilibrium theory consideredas descriptive,rather than as normativetheory. But I did read. One of thethings I did whenI was supposed to be workingon mythesis was read and read and read.It was thetypical thing I was doing.I stilldo itto thisday. I'm supposedto be doingsomething and I findmyself picking up somethingalledgedly relevant and readingit. You pick up a lot of informationthat way at times.I had read Oscar Lange's expositoryarticle [15] on thefoundations of welfare economics. Lange was extremelyclear. It was onlyafterwards I began to feelhis claritywas purchasedat theprice of depth.He set forthvery clearly the conditions for a Pareto Optimum. But then he referredto the fact that one could consider maximizinga welfare function.You startedoff with Ul9 . . ., £/„,utility functions of individuals,and you want to maximize W(Ul9 . . ., £/„)but then thereare a whole subset of the maximizingconditions that don't involve W - thoseare essentiallythe conditions that define Pareto Optimality. If I recall correctly,he was quite clear on distinguishingthese concepts but he did have this W functionand he gave a rather casual referenceto Bergson. By this time, Samuelson's Foundationshad been publishedand he givesa veryfull account of welfareeconomics in Chapter8 which he bases on Bergson's paper. So I gave a quick replyto Helmer: "Economistshave thoughtabout thatand its reallyexplained by Bergson's social welfarefunction." "Oh, is that so," he said, "Whydon't you writeit up? I thinkit would be nicefor us all to have an exposition of how the Bergson social welfarefunction settles this."
JK. So youjust startedto writethis up.
KA. Well,of course,I dropped the C/'swhich I neverliked because I knewthe £/'s werejust disguisesfor R 's forpreference relations. I thought,while I was at it,I'd do an expositionstarting from just the orderings.Then I startedmusing about what informationis conveyed. Welfarecomparisons could be regardedas a series of pairwisevotes and I was obviouslyinterested in elections so thatjust seemedlike the
This content downloaded from 150.135.135.70 on Fri, 27 Sep 2013 14:15:11 PM All use subject to JSTOR Terms and Conditions An Interviewwith Kenneth J. Arrow 53 naturallanguage to use.One natural method of taking a bunchof R's andputting themtogether would be bypairwise comparisons by majority voting. And I already knewthat was going to leadto trouble!So I figured,well, majority voting was just oneof a verylarge number of possibilities, you just have to be moreingenious. I startedto writevarious possibilities down.
JK. For example ...
KA. I'm prettysure the Borda method came to mymind, because that was a very wellknown method. I didn'tknow it was Borda,you understand.
JK.It wassomething you'd encountered before?
KA.That was very well known ; it was a widelyused custom. A clubmight do that.It was somethingthat was donein practice.
JK.Did youask anypolitical scientists at Randabout voting procedures?
KA.I don'tthink at first they did have any political scientists. I don't know anybody I wouldhave regarded as a politicalscientist. I don't think Rand was interestedin traditionalpolitical science. Later I thinkthey had moretraditional political scientistsbut even when they did they had people interested in areastudies, Soviet experts.I don'tthink they ever had theoreticalpolitical scientists.
JK. So thisexperimentation was totallyisolated. You didn'task anybodyabout rules.
KA. Right,but that also reflectedme. It seemsto meI wastrying at somepoint to systematicallygothrough all possible rules. I tooksome examples and then consider relatedexamples. Beyond that, I can'ttell you which rules I explored.I didgrasp that,at somepoint in thisprocedure, that part of the point was I was onlyusing informationon thealternatives under consideration. But then that struck me as a verynatural thing to do.
JK. Verycrucial.
KA.That turned out to be verycrucial. In fact,if I hadrealized how crucial it was, I mighthave been moredisturbed. It seemedvery natural. Afterwards, when I formalizedit, I sawthe importance of it. Now it was obvious enough that if you let oneperson make the decision, there isn't any particular problem. So I wasassuming non-dictatorship.But, I thinkit was a luckything, I assumed non-dictatorship ina
This content downloaded from 150.135.135.70 on Fri, 27 Sep 2013 14:15:11 PM All use subject to JSTOR Terms and Conditions 54 J. S. Kelly verystrong form. You know at thestart I was onlylooking at triples,because that had theessential problem. I was assumingthere wasn't any dictator or anypairwise choice.And thenyou see, you need two or moreindividuals to be decisive.A little calculationshows that you can alwaysproduce orderings that violate this. It reallyis the Condorcetparadox restated.And thenwhen I triedto extendit to more than triples,I retainedthis postulate. So, in thisfirst version, which I did showabout a weeklater to AbrahamKaplan, therewas this idea of showingthat these conditionswere incompatible.But I thoughtI had reallyput a lot of emphasison the assumptionthat therewas no dictatoron any pair; so, I thought,well theremust be a solution if you allow differentdictators on differentpairs. This seemedto be absolutelycrucial to the argument,and I thoughtabout thisfor awhile. I thoughtit would be easyto produce an examplewhere you have a differentdictator on everypair. Now I didn't think thatwas a suitablesolution anyways, I thoughtan assumptionof non-dictatorship was a correctassumption, even on one pair,so I wasn'ttoo disturbed.But I thought, to clean out theexposition, to show exactlywhat was meant,I oughtto producean example whereyou have differentdictators on differentpairs. But no, if there's transitivitythat means dictatorship sort of propagatesitself. And finallyone night whenI wasn'tsleeping too well,I could see thewhole proof, you knowafter playing aroundwith it forawhile. And thatwas a couple of weekslater, that I had theidea thatthe non-dictatorship condition could be statedin thismuch weaker way, that the whole orderingcan't be determinedby one dictator.
JK. Whatdid youfeel at thispoint ?
KA. I feltthis was veryexciting. I thought"This is a dissertation."You know,its a funnything. One of myproblems had been feelingthat one has to be seriousand everytime I'd thoughtabout these voting questions theyseemed like amusing diversionsfrom the real grittyproblem of developinga good descriptivetheory. And in some senseI stillhave a littlebit ofthatfeeling. But whenI got theresult, I feltit was significant.I really did. It clearlydidn't conform to mypreordained ideas about what was significant.I would have said a prioriif somebody told me about this,my temptation would be to say,"Well, that'svery nice, but what importance is it?" But whenI did it,I felt,yes, this is something.This was at leastasking some very fundamentalquestions about the whole nature of social intercourseand parti- cularlyabout legitimationof collectiveaction. This wasn't just a technical issue in game theory.The technical and the philosophicalwere intimately merged. My whole workin general,not only in this fieldbut in others,has tendedto denythe idea we can takeoff the technique and put ithere and put thedeep issuesthere. Some ofthe so-called technical issues are really of theessence of theso-called deep issuesand you reallycan't separatethem at all. Each one illuminatesthe other. In factthey fuse together and in some cases they're identical.And nothingcan betterexemplify this than social choicetheory where the centralissues and the technicalissues were identical.
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JK. Let's go downthe list of some of thenames you've acknowledged and tellus what theycontributed. First the people at Rand: AbrahamKaplan.
KA. Well,he was one of thefirst persons to whomI showedthe results. He was the onlyone who combinedthe philosophicalside and at least some of the technical capacityto appreciatethis. I don't recallgetting anything specific from him, just an appreciationthat it reallywas important.
JK. Youngs.
KA. I don'tknow why I thankedYoungs. Youngs was a mathematicianwith whom I discussedsome issues of preferenceorderings and the like. I had been in close contactwith him and I discussedsome otheraspects about preferenceorderings, reallyabout individualpreference orderings. I felta kind of generalintellectual debt.
JK. David Blackwell.
KA. Again,well, Blackwell was a genius.He and I workedvery closely on other matters.He, Girshickand I wrote a paper on sequentialanalysis [16]. He did contributeone otherthing. There was thischapter [in Social Choiceand Individual Values]which has neverbeen followedup. I was raisingthe questionabout if the orderingswere restricted in some ways, when does theparadox exist. If you go to the extremeof single-peakedness we knowthe answer. So thequestion is, supposing you have somerestrictions on orderingsbut thereis a lot of freedomleft. I had a result, wherethe technicalpoint was when could you extenda quasi-orderingto a full orderingand it was Blackwellwho told me about Szpilrajn'sTheorem. However, unfortunately,my proof is notcorrect, because it suffers from the problem that Blau pointedout. I suspectthe theorem is corrector some theoremlike it is correctbut nobody'sever statedit and I've nevergone back to it.
JK. J. C. C. McKinsey.
KA. McKinseywas a veryinteresting fellow. He was theone who educatedall ofus to whatgame theory was all about. So theinfluence was indirect,but in a way itwas there.He was a beautifulexpositor. He was a logicianof considerablepower and had done some work earlieron the formalizationof logic; he was a disciple of Tarski's.The wholegame theory ambience, and thereforein particularMcKinsey - and Blackwellfor that matter on thetechnical side - wereinfluential in settingthe whole tone to this.
JK. The nextnames are from Chicago: TjallingKoopmans, Herbert Simon, Franco Modigliani,T. W. Anderson,Milton Friedman, David Easton.
KA. The expositionof thebook was developedin thenext year back in Chicago. I presentedthe materialover a numberof seminars.I was gratefulto thesepeople
This content downloaded from 150.135.135.70 on Fri, 27 Sep 2013 14:15:11 PM All use subject to JSTOR Terms and Conditions 56 J. S. Kelly because theythought it was a good idea, encouragedme and asked good questions; parts of the book are makingclear points theyfound obscure. Easton was a littledifferent. He was thefirst political scientist I talkedto about this.He gave me thereferences to theidealist position which was sortof the opposite idea. In a way theidealist position was theonly coherent defense that I could see in politicalphilosophy. It wasn't a veryacceptable position,but it was the only one thathad at least a coherentview of why thereought to be a social ordering.
JK. Whydid you call it a "Possibility"Theorem ?
KA. That was Tjalling'sidea. OriginallyI called it an impossibilitytheorem, but he thoughtthat was too pessimistic!He was my boss and a verysweet man, so I changed it forhim.
JK. Therewas a meetingof the Econometric Society where you presented these results.
KA. I guess I must have presentedit at the December, 1948 meeting.
JK Who was thereand whatwas thereaction ?
KA. I rememberLarry Klein was in the chair and Melvin Reder was reading anotherpaper at thesame session.My recollectionis thatthere were 30 or 40 people in the room. I distinctlyremember that in the audience was this contentious Canadian, David McCord Wright,who objectedbecause among the objectives,I hadn't mentionedfreedom as one of the essential values in social choice and - apparentlyhe wentout ofthe room saying that Klein and Arrowwere communists thiswas quoted to me at least by KennethMay who was also present. I thoughtunder the circumstances, I got a prettygood reception.I don't think anybodysaid "We've seen a revolutionbefore our eyes," but it was taken as a seriouscontribution. I wonderwhy it was accepted so well. There reallywas no resistance.It made my reputation. Therehad been,of course,a fairamount of controversyabout thefoundations of welfareeconomics, beginning with papers by Harrod,Hicks, then Kaldor, then the long chapter in Samuelson's Foundations,then Scitovsky in 1941 with intersectingcommunity indifference curves [17]. So uneaseabout thefoundations of - economic policy were there. So the debate was serious people were already concernedabout these things. Rightafter the summer I developedthis, on theway back to Chicago,I stopped at Stanfordto be interviewedfor a job. Girshickhad meanwhilemoved to Stanford to contributeto startinga StatisticsDepartment there. He was theirstar and he wantedme tojoin him.The EconomicsDepartment there had alreadyin factmade me an offera year earlier.
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JK Based on?
KA. Whathappened was due to Allen Wallis,who was recentlyChancellor of the Universityof Rochesterand is now, at the age of 75, Undersecretaryof State for EconomicAffairs. Wallis had been a Professorat Stanfordbefore the war, he was thefirst really major appointmentthey ever made. He didn'treturn after the war, but he was highlyregarded and apparentlythey asked himfor recommendations. He had workedwith Wald duringthe war and Wald had spokenabout me based on mywork as a studentat Columbia. It was verycommon at Stanfordto appoint AssistantProfessors who didn'thave a PhD ; theyassumed we would finish.I was appointedwithout a PhD. In fact,I got tenurewithout a PhD.
JK. Really?
KA. Well, I'm beinga littletechnical, but my statementis technicallycorrect. In thosedays you couldn't get your degree until your dissertation was printed.So I had thesetheorems and thensent my changed proposal into Albert Hart, who got kind of excitedabout it. I defendedthe thesisin Januaryof 1949. Stiglerwas on the examiningcommittee, Bergson had cometo Columbiain the meantime and theyput himon thecommitee. Of course,Bergson was askingsome searching questions but was veryfair and did havea highopinion of me. But I had no realinteraction - 1 sent in mydissertation and got it approved. Hart was immediatelyenthusiastic. He said, "I don't reallyunderstand it fully, but it soundslike you'redealing with very important issues," and I've heard later thataround Columbia it was held to be an excitingevent. Of course I had been regardeda kindof a starstudent. In fact,one ofthe things that had worriedme was whetherI was just an eternalstudent. Stanfordhad a customwhere all initialappointments were for one year.They keptthis idea sincethey frequently hired people withouteven interviewingthem - because of the geography. Even Moses Abramovitz who transformedthe Departmentwas hiredas a fullProfessor for just one yearjust beforeI came. Thejet has endedall this.So I came as an AssistantProfessor on a one yearappointment. My thesishad been approved,But I couldn'tget a degreeuntil it was printed.This wasjust about theend ofthatera. I musthave been one of thelast people to come underthat rule. In fact,while the thing was inthe process of being printed, I received a noticethat if I submitteda typedmanuscript, I could getmy degree immediately. But thatwas kind of expensive,to have somebodyretype that all up. I reallylost about a yearon mydegree by decidingto go ahead on theprinting. But thenthey gave me tenureon the basis of thisunpublished dissertation. You couldn't do it today. It would neverbe approved today. Thenit is interesting- the reception question. Hart, who didn't work in thisline, was veryenthusiastic; he had spoken, I gathervery well, around the Columbia faculty.The peopleat Stanfordwere very impressed ; essentially all I had forthem to see was thiswork - I hadn't done anythingelse excepttrivial stuff. They wereso impressedthat by theend of the interviewday - withina day theywere readyto makeme an offer.So it'sinteresting to getthis reception from all sortsof people not
This content downloaded from 150.135.135.70 on Fri, 27 Sep 2013 14:15:11 PM All use subject to JSTOR Terms and Conditions 58 J. S. Kelly logicallytrained, not mathematically trained. And whenthe book came out,it made a greatsuccess. It is a littlepuzzling - at thetime I took it as that'swhat happened. But in retrospectI sort of wonderwhy.
JK. Let's go to Blau's discoveryof a mistakein your proof [IS]. Thatwas eightyears later; was thata great surprise?
KA. Yes, it certainlywas. Blau was workingthat year at Stanfordand showedit to me. I was surprised,but I knewright away thata universalityassumption would correctthings. It had seemedobvious to me thatthe non-dictatorship property was hereditary,but it wasn't. I stillthink there is a bettercorrection than that one, but I've neverreally gone back to work on it. Blau's was a very,very nice result.It didn'tobviously change the basic impact, butit did showmy little attempts at generalizationdidn't work. It's interestingto see how easy it is to make a mistakeon thingsthat seem so airtight.
JK. Let's leave theorigins now. Over thesucceeding 40 years, whatwere the most importantdevelopments in social choice theory?
KA. Some of the work that has cohered around the original question is mathematicallyinteresting but not veryrelevant to theoriginal field. The literature that depends on small numbersof alternativesis in this category.I thinkthe alternativespace must be taken to be verylarge. I also have qualms about resultslike those of Kirman and Sondermann[19]. Whatdo we learnfrom Kirman-Sondermann exactly ? Whenyou havean infinityof voters,then the axioms as I wrotethem become consistentand you can produce votingsystems. But theyare consistentbecause in some sense the dictatorhas a differentmeaning; banninga dictatoris no longerenough. As it turnsout, if you have a sequence of decisive sets, each of which is properlycontained in its predecessor,the intersectionof thewhole sequence is empty,but everyonein that sequenceis thenless decisive.In some sense,the spirit of non-dictatorshipought to rule that out. Incidentally,there's a recentresult I haven'thad a chance to studyby a former studentof mine, Alain Lewis,who saysif you confine yourself to recursivefunctions thenthe voting paradox occurseven with an infinitenumber of voters; in thestrict sense,even withjust theordinary axioms. The examplesKirman and Sondermann use are non-constructive; something with cofinite sets is not somethingyou can actuallyconstruct - youjust show itexists. But sincethey are onlyexamples, that's not a proof that thereisn't a constructiveprocedure. But Lewis says he's givena proof and I have to studyit. If I can understandit.
JK. Whatabout the Gibbard-Satterthwaite [20] results?
KA. Gibbard's workwas a bombshell.That was veryexciting. I didn'tknow about Satterthwaite'swork for a couple of years,but it was verymuch the same thing.I
This content downloaded from 150.135.135.70 on Fri, 27 Sep 2013 14:15:11 PM All use subject to JSTOR Terms and Conditions An Interviewwith Kenneth J. Arrow 59 had takenthe liberty of abstracting from manipulability in my thesis and I never wentback to that issue. What's surprising is not really that there is an impossibility of non-manipulability,but thatthe issues should be essentiallythe same. That strikesone as a remarkablecoincidence. I stillfind it surprising and feel that we might not have the right proof. Somehow youfeel that if you had the right proof it would be obvious.But then I thoughtthat aboutmy work, too. My impossibilitytheorem ought to be totallyobvious when lookedat theright way. Yet everyproof involves a trick.Maybe not a bigtrick; I don'tthink it's a mathematicallyhard theorem. But somehow if you had the right wayof approaching it, it should be trivial. Yet, no matter how you present the proof, and they'reall prettyclose to equivalent,its not yet trivial. For example,when ultrafilterscame in, I thought,Aha !, this is a beautifulway of showing it. But it turns outthat to provethe decisive sets form an ultrafilterinvolves essentially all the originalcalculations.
JK.Still, it's a niceapproach conceptually.
KA.I don'tknow. I'm less convinced than when I firstsaw it. It has the advantage of referringto a knownbody of knowledge. But this is a bodyof knowledge which is somewhattechnical. You're bringing in a fairamount of technical apparatus and it oughtto payfor itself somewhere, ifI mayuse an economicapproach. It oughtto payfor itself in making the proof trivial. But in fact you need just about every step in theoriginal proof to showthat the issue is one ofa fixedultrafilter. So therefore, whybring in all thisapparatus? I was a littlesurprised by how little you get from all thatapparatus. I can'thelp feeling there's some way out of it. In thesame way, I alwaysfeel the Gibbard-Satterthwaite result should be moretransparent than it is. Butmaybe it can't be done.
JK. Whatabout Sen's Paretianliberal approach [21]; doesthat interest you?
KA. I thoughtthat was stunningand penetratingto a veryimportant issue. But. . .whydo wehave rights? What I amafter all is a kindof utilitarian manqué. Thatis tosay, I'd liketo beutilitarian but the only problem is I havenowhere those utilitiescome from.The problemI have withutilitarianism is not that it is excessivelyrational, but thatthe epistemologicalfoundations are weak. My problemis: Whatare those objects we are adding up? I haveno objectionto adding themup ifthere's something to add.But the one thing I retainfrom utilitarianism is that,basically, judgements are based on consequences.Certainly that's the sort of thingwe do inthe theory of the single individual under uncertainty ; you make sure utilityis definedonly over the consequences. I view rights as arrangementswhich mayhelp you in achieving a higher utility level. For example, if you are much better informedabout a certainchoice, because it's personal to you and not to me, I don't reallyknow anything about it, I shoulddelegate the choice to you.
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JK. You don't wantto allow preferencesover processes ?
KA. Well, one of the thingsI fearis emptiness.You put preferencesover enough things,then anythingthat happens can be defended. It destroysthe idea of discourse.Of course, it is a delicate issue, you can always say, of any particular processthat it is speciallyprivileged. You could take Nozick's point of view; you can have an absolute preferenceabout certainprocesses. For example,we have a propertysystem; if you and I make an agreementabout anythingwithin our propertyrights, that just fixesit, period. Now I've got to admitNozick's courageis good. Suppose somebodyinvented a curefor cancer and allowed it to be used only at an extremelyhigh price. Nozick says: No problem.Most everybodyelse would regard that as a fatal counterexample,but Nozick has the courage of his convictions.But that'sa strongexample of preferenceover processes.Most of the people who are advocatingrights are verydifferent, like Dworkin. They tend to supportso to speak left-wingrights rather than right-wingrights, but once you grantthat, who settleswhat rightsare legitimate?The consequentialistview - I won'tsay thatfully settles it either, but at leastyou have somethingto argueabout. So this is why I'm a littleunsympathetic to the rightsissue - everybodyjust multipliesthe rightsall over the place and you get total paralysis. Considerthe consenting adult example- say homosexuality- and thinkabout theconcept of externality. Now whydo we sayintercourse among consenting adults shouldbe allowed? One arguesbecause there'sno externality.But ifI care,there is an externality.I actually allude to thiseven in thefirst edition of my book. It'sjust a rhetoricalpassage and doesn't enterthe logic, but I mentionthat the concept of preferenceis just what everybodythinks their preferences are. Differentpeople mighthave differentideas ofexternalities. I took theview that all preferencescount. From the logical point of view,it doesn't matter;if you purifythe preferencesby rejectingthe nosy preferences,the theorem applies to whateveris left.There is, of course,a technicalproblem in systematicallycombing out inadmissiblepreferences. Transitivitysays you can't just look at separatepreference pairs, you have to look at thewhole system. That's what Gibbard's paper is reallydevoted to. Gibbard is not totallyconvincing, because there are some arbitrarychoices in his elimination procedure;he doesn't make it compellingthat his is the only way of doing it. It's just a way. It looked prettydevastating but it eliminatedmore than was really necessary. I'm quitepuzzled. People reallycare about consequencesas theysee them.If I'm reallyoffended because people are seeingobscene material, well, I'm hurt.I really am hurt.I'm hurtjust as muchas ifsomebody blew smoke in myeyes - or whatever yourfavorite form of pollution is. Indeeda lot ofpeople probably care muchmore. I reallyfind it difficultto decide. Unless somebodyproduces a logic of rightsin termsof whichwe can argue,I reallyfind the whole issue is unfocused.The reasonwhy it is compellingis thatthere are at least some cases wherewe do feelstrongly about therights. It's not clear you can always reduce those to utilitarianconsiderations like information.
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JK. One last question.What outstanding problem in social choicetheory would you mostlike to see solved?
KA. Well,if I had to pickjust one,it would be reformulatinga weakened form of the independenceof irrelevantalternatives which stops short of just dropping it completely.There are a lot of argumentsused today, extendedsympathy, for example,or therelevance of risk-bearingto social choiceas in Harsanyior Vickery [22], thatdo involve,if you look at themclosely, use of irrelevantalternatives. SupposeI'm makinga choicein Harsanyi'sstory among totally certain alternatives. I somehowuse preferencesamong risky alternatives as partof theprocess of social decision.We use a chain of reasoningthat goes throughirrelevant alternatives. It seemsquite open to acceptance,not at all unreasonable,that these are useful. I wouldnot want to ruleout in an argument,a lineof reasoning which goes through a chainof transitivity via an irrelevantalternative. And yetI don't wantto be in the positionof saying,well the whole thing depends on thewhole preference ordering. My currentfeeling is thatthat is themost central issue - themost likely way of really understandingissues.
JK. Are you anticipatingthat if you allow chains of transitivityover irrelevant alternativesyou willobtain a "'good'"social choiceprocedure or are you expectinga deeperimpossibility theorem ?
KA. I'm expecting- no, letme put it morecautiously - I'm hopingfor a possibility result.
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