G Suppose that you prefer A to B, B to C, and C to A. Your u s preferences violate Expected Utility Theory by being cyclic. t a Money-pump arguments offer a way to show that such f s violations are irrational. Suppose that you start with A. Then s o you should be willing to trade A for C and then C for B. But n then, once you have B, you are offered a trade back to A for a Decision theory small cost. Since you prefer A to B, you pay the small sum to and Philosophy trade from B to A. But now you have been turned into a money pump. You are back to the alternative you started with but with less money. This Element shows how each of the axioms of Expected Utility Theory can be defended by money-pump arguments of this kind. This title is also available as Open Access M Money-Pump on Cambridge Core. o n e y - P arguments u m p a r g u m e n about the series series Editor ts This Cambridge Elements series offers an Martin Peterson extensive overview of decision theory in Texas A&M its many and varied forms. Distinguished University Johan E. Gustafsson authors provide an up-to-date summary of the results of current research in their fields and give their own take on what they sse rP bineflliueevnec ainreg trhees emarocsht ,s digranwifiicnagn ot rdigeibnaatle s ytisrev in U conclusions. e g d irb m a C y b e n iln o d e h silb u P 0 5 7 4 5 7 8 0 1 1 8 7 9 /7 1 0 1 .0 1 /g ro .io d Cover image: Martin Barraud / Getty Images //:sp IISSSSNN 22551177--44882179 ((opnrilnint)e) tth sse rP y tisre v in U e g d irb m a C y b e n iln o d e h silb u P 0 5 7 4 5 7 8 0 1 1 8 7 9 /7 1 0 1 .0 1 /g ro .io d //:sp tth Elements in Decision Theory and Philosophy editedby MartinPeterson TexasA&MUniversity MONEY-PUMP ARGUMENTS Johan E. Gustafsson University of York University of Gothenburg Institute for Futures Studies ShaftesburyRoad,CambridgeCB28EA,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre, NewDelhi–110025,India 103PenangRoad,#05–06/07,VisioncrestCommercial,Singapore238467 CambridgeUniversityPressispartofCambridgeUniversityPress&Assessment,a departmentoftheUniversityofCambridge. 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Money-Pump Arguments ElementsinDecisionTheoryandPhilosophy DOI:10.1017/9781108754750 Firstpublishedonline:September2022 JohanE.Gustafsson UniversityofYork UniversityofGothenburg InstituteforFuturesStudies Authorforcorrespondence:JohanE.Gustafsson, [email protected] Abstract:SupposethatyoupreferAtoB,BtoC,andCtoA.Your preferencesviolateExpectedUtilityTheorybybeingcyclic.Money-pump argumentsofferawaytoshowthatsuchviolationsareirrational.Suppose thatyoustartwithA.ThenyoushouldbewillingtotradeAforCandthen CforB.Butthen,onceyouhaveB,youareofferedatradebacktoAfora smallcost.SinceyoupreferAtoB,youpaythesmallsumtotradefromB toA.Butnowyouhavebeenturnedintoamoneypump.Youarebackto thealternativeyoustartedwithbutwithlessmoney.ThisElementshows howeachoftheaxiomsofExpectedUtilityTheorycanbedefendedby money-pumpargumentsofthiskind.ThistitleisalsoavailableasOpen AccessonCambridgeCore. Keywords:moneypumps,dynamicchoice,rationality,cyclicity,Expected UtilityTheory ©JohanE.Gustafsson2022 ISBNs:9781108718950(PB),9781108754750(OC) ISSNs:2517-4827(online),2517-4819(print) Contents 1 Money-Pump Arguments 1 2 Acyclicity 3 3 Completeness 24 4 Transitivity 39 5 Independence 50 6 Continuity 62 7 Against Resolute Choice 66 8 Against Infinite Money Pumps 74 AppendixA: Notation 78 AppendixB: Principles 78 References 82 Money-PumpArguments 1 1 Money-Pump Arguments It’s1955.You’vebeenofferedafullprofessorshipwithasalaryof$5,000.The dean calls you to his office to go over the final details. On his desk lie three contracts–labelledA,B,C. “As you know, your current offer (contract A) is a full professorship at $5,000,” the dean says, handing you the contract. “Yet…a little birdie told me you prefer an assistant professorship at $6,000 (contract C) since it pays a lot more. Potentially, I could offer you the assistant professorship. Potentially.” Thedeanrubshisthumbsoverhisindexandmiddlefingers–thegesturefor money.Messagereceived.Yousliphim$20. “It’sapleasuretoofferyoutheassistantprofessorship,”thedeansays,hand- ingyoucontractCinexchangeforA.“Evenso,I’vebeeninformedyouprefer anassociateprofessorshipat$5,500(contractB)sinceit’smoreprestigiousfor justalittlebitlessmoney.Mightthatbeworthsomethingtoyou?” Thedeanrubshisfingers.Allright.Yousliphimanother$20. “It’s, ahem, a pleasure to offer the associate professorship,” the dean says, handing you contract B in exchange for C. “Still, I’ve heard you prefer your first offer (contract A) since it’s still more prestigious for only slightly less money.” Thedeanrubshisfingers:Yousliphimanother$20. “It is a real pleasure to reoffer you the full professorship,” the dean says, handingyoucontractAinexchangeforB.“Welldeserved.” Oncemore,yougotyourfirstoffer,butnowyou’velost$60tothedean– who, by the way, is only getting started. The dean nods towards contract C, rubbinghisfingers.1 You’ve been reduced to a money pump!2 You’ve become the dean’s private cash dispenser. A victim of your own mind, you were brought to ruin by the structure of your preferences. You preferred A to B, B to C, and C to A. Thiscycleofpreferencesleftyouopentoblatantexploitation.Sosuchcyclic preferencesmust,itseems,beirrational. Argumentsofthiskindletusdemonstratethatsomeallegedrequirementof rationalityreallyisarequirementofrationality.Amoney-pumpargument for someallegedrequirementofrationalityconsistsofanargumentthatotherwise rationalagentswhoviolatetherequirementwouldinsomepossiblesituation 1 Davidson,McKinsey,andSuppes(1955,pp.145–6)introducedthisfirstmoney-pumpex- ample,creditingNormanDalkey. 2 Theterm‘moneypump’datesbacktoEdwards,Lindman,andPhillips1965,p.273. 2 DecisionTheoryandPhilosophy end up paying for something they could have kept for free even though they knewinadvancewhatdecisionproblemtheywerefacing. We will investigate whether there are compelling money-pump arguments thatrationalpreferencesconformtoExpectedUtilityTheory,whichisastruc- tural requirement on preferences over prospects. Let a final outcome be a description of the world that captures everything that the agent cares about.3 Letaprospectbeaprobabilitydistributionoverallpotentialfinaloutcomes.4 (Andletasureprospectbeaprospectwithasinglepossiblefinaloutcome.5) Expected Utility Theory, then, is the theory that prospects are preferred in accordancewithanexpected-utilityfunction:6 ExpectedUtilityTheory LetΩbethesetofpossiblefinaloutcomesand pX(o)betheprobabilityofoutcomeoinprospectX.Thenthereisareal- valuedfunctionusuchthat,forallprospectsXandY,itholdsthatXisat leastaspreferredasYifandonlyif ∑ ∑ u(o)pX(o)≥ u(o)pY(o). o∈Ω o∈Ω Rather than this general form, we will be concerned with Expected Utility Theory restricted to prospects with finite support – that is, prospects with a finitenumberoffinaloutcomeswithpositiveprobability.Giventhisrestriction, ExpectedUtilityTheoryisentailedbythefollowingbasicaxioms:7 • Completeness(Section3) • Transitivity(Section4.1) 3 Arrow1965,p.12. 4 Marschak1950,p.114. 5 Marschak1950,p.114. 6 Bernoulli1738,p.177;1954,p.24,vonNeumannandMorgenstern1944,pp.24–5,andJensen 1967,pp.172–3.ForaprecursortoExpectedUtilityTheorywhichdealswithexpectationsof valueratherthanutility,seeArnauldandNicole1662,p.467;1996,pp.273–4andPascal 1670,pp.56–7;S680/L418;2004,pp.212–13. 7 For proof, see Jensen 1967, pp. 172–82, Fishburn 1970, pp. 111–5, and Hammond 1998, pp.152–64.Fortheproof,wealsoneedtoassumethestandardalgebraofcombiningprospects. SeevonNeumannandMorgenstern1944,pp.26–7andJensen1967,p.170.Toderiveamore generalformofExpectedUtilityTheorywhichalsoholdsforprospectswithcountablyinfinite support(thatis,prospectswithacountablyinfinitenumberoffinaloutcomeswithpositive probability),weneedtosupplementthefourbasicaxiomswiththefollowingrequirementof rationality: EquiprobableWeak-PreferenceDominance IfthereareprospectsX1,X2,... and Y1,Y2,...andprobabilitiesp1,p2,...suchthattheprobabilitiessumto1,and,forall i=1,2,...,itholdsthatXi≿Yiandpi>0,thentheprospectof,foralli=1,2,..., probabilitypi ofXi isatleastaspreferredastheprospectof,foralli = 1,2,..., probabilitypiofYi. SeeBlackwellandGirshick1954,p.105andHammond1998,pp.189–90. Money-PumpArguments 3 • Thestrongstrict-preferenceversionofIndependence(Section5.3) • Continuity(Section6) Ourmaintaskwillbetoshow,withthehelpofmoney-pumparguments,that theseaxiomsarerequirementsofrationality.Forthefirstthreeaxioms,wewill findcompellingmoney-pumparguments.8But,forContinuity,wewillonlybe abletofindanargumentthatisalmostamoney-pumpargument. Money-pump arguments are often dismissed due to a number of influ- ential objections – for example: (i) that you could rationally avoid being moneypumpedifyouuseforesight,(ii)thatyoucouldrationallyavoidbeing money pumped if you are resolute and stick to a plan, and (iii) that money- pump arguments prove too much, because, in some cases with infinite series of trade offers, even agents who conform to Expected Utility Theory are exploitable. Wewillrebuttheseandotherobjections.Whileforesightblocksthestand- ardversionofthemoney-pumpargument,thereareotherversionsthatworkfor agentswhouseforesight(Section2.1).Oncetheresoluteapproachisspelled out in detail, the problems with escaping money pumps by being resolute become apparent (Section 7). And agents who conform to Expected Utility Theoryavoidexploitationevenincaseswithinfiniteseriesoftradeoffers,as longastheyuseforesight(Section8). Wewon’tstartwiththeaxiomsofExpectedUtilityTheory,however.Rather, we’llstartwiththemostdiscussedmoney-pumpargument:theargumentthat rationalpreferencesareacyclic.9 2 Acyclicity 2.1 Three-StepAcyclicity Considerhavingacupofcoffeewithone,two,orthreelumpsofsugar.Suppose thatyoucan’ttastethedifferencebetweenacupwithonelumpandacupwith twolumps.Norcanyoutastethedifferencebetweenacupwithtwolumpsand acupwiththreelumps.And,whenyoucan’ttasteanydifference,youprefer 8 Myorderingofthebasicaxiomsisn’tarbitrary.Themoney-pumpargumentforCompleteness does not rely on the other basic axioms. The argument for Transitivity relies on Com- pleteness. The argument for the strong strict-preference version of Independence relies on CompletenessandTransitivity.AndtheargumentforContinuityreliesonalltheotherbasic axioms. 9 Therearealsomoney-pumpargumentsforotherrequirementsofrationality.Notably,thereare money-pumpargumentsthatrationalcredencessatisfythelawsofprobability.(SeeRamsey 1931,p.182.)TheseargumentsareknownasDutch-bookarguments.(SeeLehman1955, p.251.)Foranoverview,seePettigrew2020. 4 DecisionTheoryandPhilosophy havinglesssugar(tokeepyourintakedown).Still,youcantastethedifference between a cup with one lump and a cup with three lumps – and, due to your sweettooth,youpreferthelatter.10 LetA,B,andCbethesureprospectsofhavingacupwithone,two,andthree lumpsofsugarrespectively.YoupreferAtoB,BtoC,andCtoA.Let‘X≻Y’ denotethatXis(strictly)preferredtoY.11 Thenwecanstateyourpreferences asfollows:12 (1) A≻B≻C≻A. Your preferences are cyclic. More specifically, your preferences violate the followingrequirement:13 Three-StepAcyclicity IfX≻Y≻Z,thenitisnotthecasethatZ≻X. AllviolationsofThree-StepAcyclicityhavethesameformasthepreferences in (1). So, to show that Three-Step Acyclicity is a requirement of rationality, allweneedtoshowisthatpreferencesofthekindin(1)areirrational. Thestandardversionofthemoney-pumpargumentrunsasfollows.14 Sup- pose that you start off with A. An exploiter offers you a trade from A to C. SinceyoupreferCtoA,youacceptthisoffer.Then,afterthisfirsttrade,you 10 ThisisavariationofanexampleinDummett1984,p.34.Bothexamples,however,have thesamestructureofindiscernibilityasanexampleinArmstrong1939,p.457n1andonein Luce1956,p.179(whichalsoinvolvescoffeewithvaryingamountsofsugar).Ng(1977,p.52) presentsasimilarexamplebasedonindiscernibilityinthreedimensions,wherenodimensionis moreimportantthantheothers.Anothersourceofcyclicpreferencesismajorityrule.Suppose thatroughlyonethirdofthepeoplehavethepreferencesA ≻ B ≻ C,thatroughlyonethird ofthepeoplehavethepreferencesB ≻C ≻A,andthatroughlyonethirdofthepeoplehave thepreferencesC ≻ A ≻ B.Now,supposethatyouareadedicateddemocratwhoprefers anoptionoveranotherifitispreferredtotheotheroptionbyamajorityofthepeople.Then youhavethepreferencesA ≻ B ≻ C ≻ A.SeeCondorcet1785,p.lxi;1976,p.54;1994, p.124,Dodgson1876,pp.8–12,andBlack1948,pp.32–3.May(1954,p.6)presentsasimilar example,whereyouassessoptionsonthreecriteria,andyoupreferanoptionifitissuperior onatleasttwoofthesecriteria.Then,inthesamewayasinthemajority-ruleexample,you endupwithcyclicpreferences.AfurtherexampleofcyclicpreferencesistheMere-Addition Paradoxinpopulationethics(seenote66).Finally,youmightpreferonedieoveranotherifit ismorelikelytowinthantheother,whichleadstocyclicpreferences–givenaspecialkindof dice.SeeGardner1970,p.110. 11 Debreu1959,p.8. 12 Weadopttheconventionthatchainsofrelationswithoverlappingrelatacanbecontracted.For example,‘X≻YandY≻Z’canbewrittenas‘X≻Y≻Z’.SeeDeMorgan1851,p.104and Fishburn1991,p.116. 13 Samuelson1947,p.151andSen1977,p.62.Forthestatementofprinciples,weadoptthe conventionthatfreevariablesareimplicitlyuniversallyquantified. 14 Edwardsetal.1965,p.273andPrattetal.1965,ch.2,p.10.Davidsonetal.(1955,p.146) presentasimilarcase,butit’snotclearthattheymeanttoshowthattheagentendsuppaying forwhattheycouldhavekeptforfree.SeeGustafsson2013,p.462fortheinterpretationthat Davidsonetal.onlyintendedtoillustratethattheagentwiththepreferencesin(1)wouldregret anychoicefromtheoptionset{A,B,C},whichmaybeasignofirrationalityinitself.Forthis alternativeargument,seeTullock1964,p.403.