Money-Pump Arguments

102 Pages·2022·1.164 MB·English

by  Johan E. Gustafsson

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trade from B to A. But now you have been turned into a money
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of Expected Utility Theory can be defended by money-pump
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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
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DOI:10.1017/9781108754750
©JohanE.Gustafsson2022
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10.1017/9781108754750
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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.