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1MoralImpossibilityinthePetersburgParadox:ALiteratureSurveyandExperimentalEvidenceTiborNeugebauerUniversityofLuxembourgAbstract:ThePetersburggambleconstitutesanimportantparadoxinthehistoryofideas.
Ithasbeenthought-provokingandledtoimportantdevelopmentsinthenaturalandbehavioralsciences.
Theproposedresolutionsoftheparadoxhaveinvolveddeepreflectionsaboutthehumanmindbysomeofthemostcelebratedscientistsofthepastthreecenturies.
Thispaperdescribestheparadox,andfocusesontheresolutionsthathavebeenadvancedintheliteraturewhilealludingtothehistoricalcontext.
Inparticular,Bernoulli'smoralimpossibilityconceptisrevisitedanddiscussed.
Thestudycontributesexperimentaldatatothediscussionoftheparadox.
Thedecision-makingofsubjectsisinlinewiththenotionofmoralimpossibility;invaluingthegamble,peopleneglecttheeventsthatoccuronlywithsmallprobability.
Thestudyelicitsthesizeoftheprobabilitythatexperimentalsubjectsseemtoneglectwhentheyformulatetheirwillingness-to-payforthePetersburggamble.
Itisarguedthatthisbehaviorisboundedlyrational,astheindividuallevelofmoralimpossibilitycanbeinterpretedasanindividualaspirationlevelintheartofconjecturing.
Keywords:Petersburggamble;experimentJEL-code:B3;C44;C9;D8;G1;N0Author'saddress:UniversitédeLuxembourgFacultédeDroit,d'EconomieetdeFinanceCampusKirchberg,K24,rueAlbertBorschetteL-1246Luxembourg+352466644-6285(telephone);-6811(fax)Tibor.
Neugebauer@uni.
luIthankStephanLengsfeld,FrancisLagos,JuanAntonioLacomba,HikmetArslanoluandzgürToparlakfortheirsupportinthedatacollectionprocessandJohnHeyforrevisinganddiscussingtheinstructions.
HelpfulcommentsbyOttwinBecker,MichaelBirnbaum,JimCox,RachelandDavidCroson,ErnanHaruvy,JohnHey,GuillaumeHollard,AstridHopfensitz,RudolfKerschbamer,PatrickKinsch,MartinKocher,LouisLevy-Garboua,UlrichSchmidt,KlausSchredelseker,ReinhardSelten,MatthiasSutterandseminarparticipantsatLUISS,UCFullerton,UniversityofInnsbruck,UniversityofLuxembourg,UniversityIofParis,UniversityofDallas,2008ESAmeetingTucsonandthe2008GEW-TagungMannheimareacknowledged.
PartofthisworkwasdoneduringmyresearchstayatLUISSinFebruary2007.
IthankLUISSandespeciallyJohnHeyforthehospitalityreceived.
2[Since]itisonlyrarelypossibletoobtaincompletecertaintythatiscompleteineveryrespect,necessityanduseordainthatwhatisonlymorallycertainbetakenasabsolutelycertain.
Itwouldbeuseful,accordingly,ifdefinitelimitsformoralcertaintywereestablishedbytheauthorityofthemagistracy.
Forinstance,itmightbedeterminedwhether99/100ofcertaintysufficesorwhether999/1000isrequired.
Thenajudgewouldnotbeabletofavoroneside,butwouldhaveareferencepointtokeepinmindinpronouncingajudgment.
(JamesBernoulli1713;quotedfromthecommentedtranslation2006,p.
321)1Intheartofconjecturing(JamesBernoulli1713),i.
e.
themeasuringofanevent'sprobability,theapplicabilityofprobabilitytheorytoempiricaldatacruciallyhingesontheconceptofmoralcertainty.
Somethingismorallycertainifitsprobabilitycomessoclosetocompletecertaintythatthedifferencecannotbeperceived.
Incontrast,somethingismorallyimpossibleifithasonlyasmuchprobabilityasthemorallycertainfallsshortofcompletecertainty(Bernoulli2006,p.
316).
Moralimpossibilityisthusdefinedasthesubjectiveprobabilitylevelwhichisnegligiblysmallintheconjecturingonthelikelycasesofevents.
Onlybyestablishmentofsuchaprobabilitylevelcanweachievethecertaintythatweneedforscientificknowledgeaccumulation.
Withoutacceptingsuchaprobabilitylevel,conjecturingaboutthelikelycasesofeventsisonlypossibleinextremecasessincecompletecertaintycannotusuallybeattainedthroughexperimentationandobservation.
Onehistoricallyimportantexampleofthegamesofchanceinwhichmoralcertaintyandcompletecertaintyleadtoextremelydifferentvaluationsistheso-calledPetersburgparadoxinwhichafaircoinisrepeatedlytosseduntilheadsshowsupforthefirsttime.
Whileanon-zeroprobabilityexiststhatthecoinistossedforever,afactthatposesanon-finitemathematicalexpectedpayoffforthegamble,theoccurrenceofaninfiniterunoftailsismorallyimpossible.
2Alongfiniterunisalsomorallyimpossible,however,inasingletrialofthegamble.
Longsimulationsinexperimentalstatisticshardlyeverreporttheoccurrenceofmorethantwentysubsequenttosses.
Arunoftwentytailsapproximatelycorrespondstotheoddsofoneinamillion.
Inthehistoryofthought,ithasbeenproposedthatinasingletrialofthePetersburggambleevenarelativelysmallnumberoftossescanbeconsideredasmorallyimpossible.
Inthispaper,thispropositionisexperimentallytestedthroughtheelicitationofsubjects'willingness-to-payforvarioustruncatedversionsofthePetersburggamblethatdifferinthemaximumpayoff.
TheexperimentaldatashowthatallversionsofthePetersburggamblewhichallowformorethansixrepeatedtossesoftailselicitthesamewillingness-to-pay.
Fromthisevidenceitisconcluded1Throughoutthepaperthe"ArsConjectandi"ofJamesBernoulli(1713)isquotedbyreferringtoDudleySylla'sEnglishtranslationandcommentary,"TheArtofConjecturing…,"whichwaspublishedin2006.
2Anevenmorestrikingexamplewheremoralimpossibilityispittedagainstmathematicalprobabilityistheeventthatamonkeyrandomlyhittingkeysonatypewriterwillwriteagiventext,suchastheBible(EmileBorel1914).
Again,whilethiseventhasamathematicalprobabilitygreaterthanzero,itismorallyimpossiblethatitoccurs.
Orimaginethepossibilitythatwithinagenerationofmankindonlyboysandnogirlsarebornorviceversa;althoughthiseventismorelikelythananinfiniterunoftailswhenafaircoinistossedrepeatedlyitisstillmorallyimpossible.
3thatsubjectsneglectthoseoutcomesinthePetersburggamblewhichoccurwithaprobabilitysmallerthanorequaltooneinsixty-four.
Althoughthenotionofmoralimpossibilityisolderthanexpectedutilitytheory,ithasbeenlargelyneglectedineconomicsand,inparticular,inthemoderndiscussionofthePetersburgparadox.
ExpectedutilityhasbeentheonlystandardsolutiontothePetersburgparadoxineconomicstextbooks.
Nevertheless,thenotionthatthereexistsasmallestprobability-unitofinterestfordecision-makingunderuncertaintyhasapplicationstoeconomicsandthesocialsciencesfarbeyondthePetersburggamble.
3Ineverydaylife,peopleneglectcaseswhoseeventstheyperceiveasbeingunlikely.
Ultimatelytheneglectofsmallprobabilitiesmighthaveincreasedthelikelihoodoftheoccurrenceofgreatdisasters(NassimNicholasTaleb2007;seealsoKlausSpremann2008).
4Thepaperisorganizedasfollows.
ThefollowingsectiondescribesthePetersburggambleandalludestothehistoricalperspective.
5Inadditiontoearliersurveys,thepresentpaperincludestheexperimentalcontributionstothePetersburgparadox(insection3),highlightstheapplicationofmoralimpossibilityanduncoversinterestingdetailsthathavebeenneglectedinthehistoryoftheparadox.
Theseunnoticeddetailsinclude,forinstance,thetensionbetweentheoutcomeofthelawoflargenumbersandthetheoremoninfiniteserieswhichmusthavepuzzledthefounderofthegamble,NicholasBernoulli,andwhichheproposedtoresolvebyinvokingthemoralimpossibilityconcept;theparadoxicalriskpreferencethatarisesforasellerofthePetersburggamble;andthedilutionconcernthatcanarisewhenaninstitutionoffersthegambleforsale.
Thereviewoftheliteraturecontinuesinsection2withafocusonthesolutionconcepts3Amongotheradvantages,weobtainawell-definedunboundedutilityfunction(PeterWakker1993).
4ApamphletbyColeenRowleyarguesthattheFBIhadpriorinformationabouttheterroristattackof9/11,andifthetracehadbeenseriouslyfolloweduptheattackcouldhavebeenprevented(http://www.
time.
com/time/covers/1101020603/memo.
html).
Evidently,therisksofdeterioratingrealestatemarketswereneglectedinwhatledtothesubprimecrisis(http://knowledge.
wharton.
upenn.
edu/article.
cfmarticleid=1998).
Inrecentmonths,thecolliderexperimenthasstartedinGeneva.
Doubtshavebeenraisedthat,intheexperiment,blackholescanbegeneratedthatmaysubsumetheworld.
Thesefearshavebeenplayeddownbyspecialistsas'baloney',althoughwehavenoexperiencewithblackholesandnomathematicalproofhasbeenprovidedthatrendersthesefearsinvalid.
JohnHuth,aprofessorofphysicsatHarvard,wascited(http://edition.
cnn.
com/2008/TECH/09/08/lhc.
collider/)assayingthat"thegravitationalforceissoweakthatyou'dhavetowaitmany,many,many,many,manylifetimesoftheuniversebeforeoneofthesethingscould[get]bigenoughtoevengetclosetobeingaproblem.
"Thisstatementcouldbetakenasaconfirmationthataterribledisastercanhappenwithavery,very,very,very,verysmallprobabilitygreaterthanzero.
Theworldcommunityseemstoacceptrunningthisrisk.
5ThePetersburgparadoxhasseendedicatedsurveysbyEmanuelCzuber(1882),PaulA.
Samuelson(1977),GérardJorland(1983;1987),GlennShaffer(1988),andJacquesDutka(1988).
Theliteraturehasbeenpartlysurveyedandhistoricallydiscussed,furthermore,inIsaacTodhunter(1865),JohnMaynardKeynes(1921),KarlMenger(1934),GeorgeStigler(1950),KennethArrow(1952),LeonardSavage(1954),OttoSpiess(1975),MauriceAllais(1979),GilbertBassett(1987),andinLorraineDaston(1988).
Furthermore,thestoryiscoveredinmanyinterdisciplinarybooks,includingeconomics(e.
g.
Hans-WernerSinn1980;KlausSchredelseker2002),historyofscience(PeterBernstein1996),mathematicsandstatistics(RichardEpstein1977;WarrenWeaver1982),philosophy(RichardJeffrey1983;MichaelResnik1987;IanHacking2001),sociology(RussellHardin1982),psychology(ScottPlous1993)andappearsonseveralInternetpages(e.
g.
http://www.
wikipedia.
org;http://plato.
stanford.
edu/archives/win1999/entries/paradox-stpetersburg/).
4thatconstitutethetestablehypothesesfortheexperiment,theresultsofwhicharedetailedinsection4.
Inlinewiththepresentedevidence,moralimpossibility(i.
e.
theneglectofsmallprobabilities)seemstobethemostconvincinghypothesis.
Thisconclusionisbackedupwiththepresentationofthreeexperimentalstudies.
ThefirststudyofthePetersburggambleshowsthatunlikelypayoffshavenoimpactonsubjects'revealedwillingness-to-payasthesamevaluationsareelicitedfordifferentlengthofthegamble.
Theothertwostudiesfollowuponthisresultandelicitthethresholdlevelbeyondwhichthesevaluationsdonotchange.
Allgamblesthatinvolvedprobabilitylevelssmallerthan1/16andmaximumpayoffsgreaterthan16Euroelicitedthesamedistributionofvaluations.
Fromthisobservationitisconcludedthatthesmallprobabilitiesofhigherpayoffsareneglected,assumingthatthesmallamountofmaximumpayoffcannotrepresentalevelofmaximumutility.
Section5summarizesanddiscussestheresultsofthepaper.
1ThebirthoftheproblemOnSeptember9,1713,sothestorygoes,NicholasBernoulliproposedthefollowingprobleminthetheoryofgamesofchance,after1768knownastheStPetersburgparadox(SandorCsrg2001,p.
62),inalettermailedfromBaseltothemathematicianPierreReymonddeMontmort(DanielBernoulli1954,p.
33).
6Petertossesacoinandcontinuestodosountilitshouldlandheadswhenitcomestotheground.
HeagreestogivePauloneducatifhegetsheadsontheveryfirstthrow,twoducatsifhegetsitonthesecond,fourifonthethird,eightifonthefourth,andsoon,sothatwitheachadditionalthrowthenumberofducatshemustpayisdoubled.
SupposeweseektodeterminethevalueofPaul'sexpectation.
Theauthoroftheproblem,NicholasBernoulli(1687–1759),wastheleadingfigureinstochasticsintheseconddecadeoftheeighteenthcentury(Csrg2001,p.
60;seealsoAndersHald1990).
In1711,heprovidedageneralsolutiontothemostdifficultprobleminstochasticsofthattime,thedurationofplay(KarlKohli1975),whichwasageneralizationofBlaisePascal's(1623–1662)6ThegamblereceiveditsnamefromthefactthattheseminalpaperofDanielBernoulliwaspresentedandpublishedinthejournaloftheImperialAcademyofSciencesinSaintPetersburg,whichwasfoundedbyCatherinetheGreatin1725.
IhavefoundafirstreferencetothePetersburgprobleminthememoir23ofJeanleRondd'Alembert(1768,p.
78);inhiscontribution,d'Alembertrefersfirsttothe"probleminthememoirsofPetersburg"beforeheswitchestotheterm"Petersburgproblem"whichheusesthereafterinalllatercontributions(seealsoJorland1987,p.
165).
TheoriginalproblembyNicholasBernoulliinvolvedtherollofthedie(Montmort1713,p.
402).
TheflipofthecoinwasintroducedtotheproblembyGabrielCramer,whothusreformulatedtheproblemofNicholasBernoulliintoitsdefiniteforminaletterfromLondondatedMay21,1728(seeSpiess1975).
Intheoriginalproposal,NicholasBernoullipostedfiveproblemstoMontmort,ofwhichthelasttwowerepredecessorsofthePetersburggamble(seeequations(1)and(2)below).
Thefourthprobleminvolvedalotterythatpaidone"ECU"foreachrollofanordinarydieuntilsixpointswereachievedforthefirsttime,i.
e.
thelotterypaysk+1ifkisthenumberofrollsofthediebeforesixpointsshowupfirst.
Thefifthprobleminvolvedthepowerseriesofpayoffs2k,3k,k2,andk3,substitutingforkinthelotteryofthefourthproblem.
Thedifferencebetweenthefourthandthefifthproblemisthatthefourthproblemyieldsanexpectedpayoffofsix,whiletheexpectationsinproblem5donotallexist.
5problemofpointsandHuygens'sgambler'sruinproblem(seealsoKeithDevlin2008).
ItshouldbenotedthattheproblemofthedurationofplaywasintroducedbyMontmort(1708)andthatitismethodologicallysimilartothePetersburggamble.
7Nicholaswasthestudentandnephewofthefounderofprobabilitytheory,JamesBernoulli,andwrotetheforewordtotheposthumouslypublishedArsConjectandi,i.
e.
theunfinishedmasterpieceofhisuncle,in1713,whichcontainedthefirstproofofthelawoflargenumbers.
8NicholasalsoprovidedaproofofthelawoflargenumbersinalettertoMontmortwhereheindirectlyintroducedthenormaldistributionthusprecedingAbrahamdeMoivre(OscarSheynin1970,p.
232).
James,themostfamousrepresentativeofthewholeBernoullihadthegreatvisiontoprovide,viathelawoflargenumbers,ajustificationforthemeasurementofempiricalprobabilitiesfromobservedfrequencies.
Withincreasingsamplesize,heargued,onecanlearnfromexperiment,aposteriori(i.
e.
aftertheeventhashappened),thehiddenprobabilitiesofcasesinwhichaneventcanoccur.
Owingtohisearlydeath,however,hecouldnotfinishtheimportantfourthpartoftheArsConjectandi,whosetitleindicatesapplicationsofprobabilitytheorytocivil,moralandeconomicaffairsbutwhosecontentlackssuchapplications.
9NicholasBernoulli(1709)carriedonthisprojectandappliedhisuncle'stheorytomoralsandthesocialsciencesinhisthesisDeusuartisconjectandiiniure.
10Healso7Theproblemofthedurationofplay,alsoknownastheruinproblem(seebelowfortheformulationoftheproblemofruinbyMauriceAllais1979),maybeformulatedasfollows:twoplayers,PeterandPaul,areendowedwithmandnducatsrespectively.
TheyrepeatedlyplayagameinwhichPaulhasprobabilityofwinningpandPeterhasprobabilityq=1–p.
Thewinnerinagamegetsaducatfromtheloser.
Thegameisrepeateduntiloneoftheplayershaslostallhisducats.
WhatistheprobabilitythatthegameendsatthekthgambleorbeforeMontmort(1708)solvedthisgameforthespecialcaseofm=n=3andp=q=.
8ThelawoflargenumbersshowsthatthesequenceofindependentBernoullitrialsconvergestoitsexpectationwithanincreasingnumberofobservations.
Beforethename"thelawoflargenumbers"wasintroducedtotheliteraturebySimeonDennisPoisson(1781–1840),itwascalledBernoulli'stheorem(seeTodhunter1865).
9JamesBernoullihadalreadystartedworkingonhismasterpiecetwentyyearsbeforehisdeath.
Fromthereadingofhisscientificdiary,Mediationes,onecanconcludethattheproofforthelawoflargenumberswaswrittenearlier,sometimeduringtheyears1689to1692(ibid.
,p.
32).
Jamessaidthedelayinpublicationwasbecauseofhisbadhealthandlazinessatwriting(ibid.
,p.
36).
Healsosaidthatthemostimportantpart,theapplicationtocivil,moralandeconomicmatters,wasmissing.
Thusithasbeensuggestedthathedidnotpublishthebookbecausehelackedbothdataandknowledgeofeconomicissuestowhichhecouldapplyhistheory(ibid.
,p.
49).
HeaskedGottfriedWilhelmLeibniz(1646–1716)forbothdata(intheformofabookonlifeannuitiesbyJandeWitt(1675),aformerstudentofRenéDescartes(1596–1650))andproposalsforapplications,butdidnotreceiveeitherbeforehisdeath(ibid.
,p.
49).
AsNicholasBernoulliwritesintheforewordtotheArsConjectandi,thepublishersmighthavehopedthatJames'sbrother,John(=Johann)Bernoulli(1667-1748),whowasJames'ssuccessortothechairofmathematicsatBaselandthesupervisorofNicholas'sthesisafterJames'sdeath,wouldsupplythemissingpart.
Nicholasreiteratesthattheyalsotriedtogivethejobtohim,buthedeclinedbecausehefelthimselfunequaltoit(ibid.
,p.
129).
WhileNicholaswrotetheforewordandsuppliedapageoferratatothepublication,hewasnotthepublisheroftheArsConjectandiashasrepeatedlybeenstatedintheliterature.
Asrecentresearchshowed,therehasbeenhistoricalconfusion(possiblyresultingfromtheentrytotheMathematischesLexikonbyChristianvonWolffin1716),asthesonofJames,apainternamedNicholas"theyounger"andborninthesameyearasNicholas,tookthebooktothepublisher(Bernoulli2006,p.
60).
10ThetitleofhisthesistranslatesintoEnglishas"theusageoftheartofconjecturinginjurisprudence.
"NicholassubmittedthethesistothelawfacultyatBaselin1709tobecomeadoctoroflaw.
Intheforewordtohisthesisheacknowledgesthegreatinfluenceofhisuncle'sunpublished6offeredtosupplytheunfinishedpartstotheArsConjectandi(ashementionedinalettertoLeibniz)but,intheend,thefamilydecidedtopublishthemasterpiece"asis"inordertounderlineJames'spriorityinthefoundationofprobabilitytheory(Bernoulli2006,p.
61).
Intheforewordtohisuncle'smasterpiece,Nicholasinvitesthereaderand,namely,MontmortanddeMoivre,toapplythecalculusofprobabilitiestomorals,economicsandpolitics.
Accordingtothetheoryofthesummationofinfiniteseries,whichwealsoowetoJamesBernoulli,oneobtainsthemathematicalexpectationofthePetersburggamblebysummingtheseriesoftheprobabilityweightedpayoffs.
Eachproductofprobabilityandoutcomeinthisseriesyieldsonehalf;thefirsttossofthecoinendsthegameyieldingoneducatwiththeprobabilityonehalf,thesecondtossofthecoinendsthegameyieldingtwoducatswithaprobabilityofonequarter,etc.
LettheexpectationoperatorbedenotedbyE,andXistherandomvariablethatdescribesthepossibleoutcomes;equation(1)givesPaul'sexpectation.
∑=∞→*=+++=niiinXE021221lim.
.
.
844221][(1)Theseriesisdivergent;ithasnofiniteexpectation(foradiscussionseeJohnBroome1995).
Thefactthattherighthandsideisinfinitesuggestsariskneutralgamblershouldbewillingtopayanyfixedamounttopurchasetherighttoplaythegamble.
ThePetersburggamblewasthelastoffivemathematicalproblemswhichNicholaspresentedinthelettertoMontmort.
Thepreceding,fourth,problemwasidenticaltothefifthproblemintermsofprobabilitiesbutinvolvedapayoffstreamthatincreasedbyonlyoneducatpertossofthecoinratherthanbydoublingthestakesoneachtoss,asinequation(1).
Whilethefourthproblemallowsalsoanunboundedpayoff,thepayoff-probabilityproductsareconverging,andthereforetheexpectationisfinite,asacknowledgedinequation(2).
∑=∞→=*=+++=niiniXE0221lim.
.
.
834221][(2)Acomparisonofthesumsin(1)and(2)revealsthatthepayoffpowerfunctionofiproducesthedivergenceoftheexpectationinthefirstequation.
NicholasBernoullistatedthatthediscrepancybetweentheseproblemswas"mostcurious"(Montmort1713,p.
402).
11WithhisexpertiseonmanuscriptArsConjectandionhischoiceofsubject.
Inthethesis,healsoaddressedproblemsthatwerediscussedinJames'sscientificdiary,Meditationes,whichwasnotintendedforpublication.
Beforehisthesis,Nicholasrespondedtohisuncle'sworkonthesummationofinfiniteseriesinhisdefenseforthemaster-of-artsdegreein1704(Bernoulli2006,p.
55).
HetaughtmathematicsattheUniversityofPaduabetween1716and1719whereheworkedondifferentialequationsandgeometry.
AttheUniversityofBaselhebecameaprofessoroflogicsin1722andaprofessoroflawin1731.
HecorrespondedwithMontmort(1678-1719)anddeMoivre(1667-1754)andalsowithLeibnizonconverginganddivergingseriesin1712and1713(JacquesDutka1988,p.
20).
BiographiesofNicholasBernoullihavebeensuppliedbyJoachimOttoFleckenstein(1968),Kohli(1975),AdolpheYouschkevitch(1987),Hald(1990),Csrg(2001),NorbertMeusnier(2006)andBernoulli(2006).
11Itisimaginablethatthisremarkandtheeye-catchingexpositionoftheproblemmayhavehadaninfluenceonsomeofthehistoricalsolutionproposals.
Forinstance,DanielBernoulli(1738)proposedtosumtheprobabilityweightedlogarithmofpayoffsin(1)insteadoftheprobabilityweightedpayoffs(seethefollowingsection).
7infiniteseriesandthelawoflargenumber,theexpectationsinequations(1)and(2)musthaveappearedextremelypuzzlingtoNicholas.
Ononehand,wehavetheapproximationduetotheinfiniteseriestheorem;ifafaircoinyieldsaninfiniterunofheads,theoutcomeexceedsanyboundinbothproblems.
Yet,theprobabilityweightedaverageisassumedtobeasmallfinitenumberinonecasebutinfiniteintheother.
Thequestionseemstobepermittediftherearedegreesofinfinity,andthatiswhatNicholasasked.
12Ontheotherhandandimpliedbythelawoflargenumbers,ifafaircoinistossedinfinitelyoftentherelativefrequencyofheadsandtailsmustbethesamewithprobabilityone.
Asaconsequence,aninfiniterunofheadsisimpossible.
ThisobvioustensionbetweentheoutcomesoftheinfiniteseriestheoremandthelawoflargenumberscanberesolvedinthetheoryofJamesBernoullibyrecurringtothemoralimpossibilityconceptwhichalsomakesitpossibletonarrowthefairvaluesofthetwogambles(1)and(2)whoseintuitivevaluesseemnottodifferbymuch.
Montmortwasnotinterestedintheapplicationofprobabilitytheorytomoralsandethics(Csrg2001,p.
61)andapparentlydidnotcontributetothediscussionoftheproblem.
13AfterMontmort'sdeath,theyoungmathematicianGabrielCramerproposedtoNicholasBernoullitworesolutionstotheproblemin1728.
InaletterwhichisreproducedinandappendedtothepaperofDanielBernoulli(1954,p.
33),Crameralsostatestheparadox;accordingtothecalculation,PaulmustgivetoPeteran"infinitesum"asanequivalent,"whichseemsabsurd,sincenopersonofgoodsense,wouldwishtogive20ducats.
"Thus,thePetersburgparadoxrepresentedacounter-exampletoPascal'swager.
Pascalhadarguedandlivedhisadultlifeinaccordancewiththetheorythatitisworthtoabandonallrichesforthesmallchanceofwinninganinfinitepleasure.
TheresolutionstothePetersburgparadoxwhichhavebeenpresentedintheliterature,includingthatofCramer,arediscussedinthefollowingsection.
2.
ProposedresolutionsoftheparadoxTherehavebeenseveralproposalsfortheresolutionofthePetersburgparadoxintheliteraturethroughoutthecenturies.
Themostfamousconcept,expectedutilitytheory,whichhasbeenthestandardapproachreferredtoineconomicstextbooks,wasproposedbyNicholasBernoulli'scousin,DanielBernoulli.
DanielBernoullisubmittedacopyofhisunpublishedpapertoNicholasBernoullionApril4,1732,anditwaspublishedin1738.
2.
1Expectedutility1412IntheoriginalcorrespondencetoMontmort(1713),NicholasBernoullipointedoutthatthe"expectedinfinitesum"(orevengreatersum"ifitispermitted")cannotbethevalueofthelottery,"sinceitismorallyimpossiblethat[Paul]doesnotachieve[heads]inafinitenumberofthrows"(Spiess1975,p.
558).
AtranslationtoEnglishofSpiess'scollectionofNicholasBernoulli'scorrespondencesonthePetersburggamblehasbeenpublishedbyPulskamp(1999).
13Atfirst,Montmortseemeduninterestedintheproblem.
Laterhepromisedtoprepareamanuscriptontheissue,whichhasneverbeenfoundandtowhichnofurtherreferencewasmadeinanyknowncorrespondenceofNicholasBernoulli(Spiess1975).
Montmortdiedfromsmallpoxin1719.
14NicholasBernoullisenthisproblemsettoDanielBernoulli,whowasaprofessorofmathematicsinPetersburgatthattime.
InhisletterheindicatedtheparadoxaspointedoutbyGabrielCramerandsaidthatitwasirrationaltovaluethegambleabove20ducats.
DanielBernoullirepliedtohiscousininNovember1728thattheparadoxisfoundinthesmallprobabilitythatthegamblewilllastformore8InDanielBernoulli'sexpectedutilitytheory,prospectivewealthisweightedbytheprobabilityofoccurrence.
TheexpectedutilityforthePetersburggambleisthesumoftheprobabilityweightedutilitylevelsofwealthandthusdependsalsoontheinitialwealthαofthedecision-maker.
∑=∞→+=++++++=+niiinuuuuXEu02)2(21lim.
.
.
8)4(4)2(2)1()(ααααα(3)Equation(3)representsPaul'sexpectedutilityfromthePetersburggambleinthegeneralformulation.
15DanielBernoulliusedthelogarithmicutilityfunction,u(X)=logX.
Hearguedthat,atthemargin,utilityofadditionalwealthisinverselyproportionaltothepossessedwealth.
16Apersonwhosewealthamountsto100ducatsappreciatesanothercentapproximatelyasmuchasapersonwhohasaninitialwealthof1,000ducatsappreciatesanothertencents.
Inotherwords,themarginalutilitydecreasesininverseproportiontothepossessedwealth.
ThecertaintyequivalentforthePetersburggambleisthereforeanincreasingfunctionofpossessedwealth;theopportunityofplayingthePetersburggamblewouldbeworthtwoducatsifPaulpossessednothing,aboutthreeducatsifhisinitialwealthweretenducats,aboutfourifhisinitialwealthwere100ducats,andaboutsixifhisinitialwealthwere1,000ducats.
17Averyrichpersonwouldhavethecertaintyequivalentof20ducats(Bernoulli1954,p.
32).
GabrielCramerhadalreadysuggestedin1728that"menofgoodsense"valueaprospectbyits"moralexpectation"ratherthanbyits"mathematicalexpectation"(Bernoulli1954,p.
34).
Heassumed,inaseeminglyarbitrarymanner,thattheutilityfunctiontakestheformofthesquareroot,ashearguedthatonemayreceivedoublethepleasurefrom40millionthanfrom10million.
Hecomputedthecertaintyequivalentwithoutpayingattentiontoinitialwealthyieldinganapproximateequivalentof2.
9.
Hecanbecreditedwithbeingthefirsttoproposetheideaofdiminishingmarginalutility.
Hissolution,however,didnottakeintoaccounttheinitialwealththan20or30throws,andinafollow-upletterayearandahalflaterhestatedthatapersonwouldnotwageraninfinitesumwhentherewasonlyaninfinitesimallysmallprobabilityofwinning.
OnlyinJuly1731didhecomposeadraftofhisfamousexpectedutility.
TherehasbeenspeculationthatthedraftbenefitedfromdiscussionswithhisfriendandcolleagueLeonhardEuler(1707-1783),whopreparedbutdidnotfinishapaperonthesameissue,(itwasonlypublished79yearsafterEuler'sdeath)(Euler1862;seealsothediscussionofEuler'scontributionbyEdSandifer2004).
15ThegeneralformulationofexpectedutilityisowedtoJohnvonNeumannandOscarMorgenstern(1947),whoestablishedtheaxiomaticapproach(inthistheywereanticipated,however,byFrankRamsey1931).
Thestandardterm"expectedutility"isamoderntranslationoftheterm"emolumentummedium"usedbyDanielBernoulli(1738).
IntheEconometricatranslationofBernoulli'sarticle,theterm"moralexpectation"or"meanutility"isused(seeBernoulli1954,p.
24,footnote3).
Theterm"moralexpectation"or"moralvalue",however,wasintroducedbyGabrielCramer(1728),whenhediscussedthePetersburggamble.
ThetermwaslaterusedbyPierre-SimonLaplace(1820).
16Forhumanperceptionsofjustnoticeabledifferences,thepsychophysical"Weber-Fechner"lawsuggestsalogarithmicrelationshipbetweenstimulusandperception(forasurveyseeDuncanLuceandPatrickSuppes2002).
17Thecertaintyequivalentistheamounttopaythatmakesyouindifferentwhetheryoupurchasethegambleornot.
Forapersonwithinitialzerowealthandalogarithmicutilityfunction,DanielBernoulli(1954)computedtheexpectedutility,oflog2,andthecertaintyequivalent,of2ducats,astheinverseutilityfunction.
9positionandthusBernoulli'ssolutionmustbeconsideredassuperiorinbothsophisticationandreason.
18Laplace(1820)acceptedtheideaofdiminishingmarginalutilityandcalleditthetenthprincipleofprobability.
Heshowedthatmathematicalexpectationwasthelimitofmoralexpectationwhenthedivisionofrisksbecomesinfiniteandthususeditasthefoundationforhistheoryofinsurance(Jorland1987,p.
171).
Duringthefollowingtwohundredyears,DanielBernoulli'sutilitytheorywasdiscussed,appreciatedforthepossibilityitgaveofmakinginterpersonalcomparisons(KnutWicksell1900),andgeneralized.
FrancisEdgeworth(1881)rejectedthelogarithmicutilityfunctionforbeingasarbitraryasanyotherconcavefunction,AlfredMarshall(1890)replacedthewealthargumentinthefunctionbyincome,VilfredoPareto(1893)replaceditbyconsumption,andMaxWeber(1908)suggestedthattheutilityfunctioncanvaryforonegoodoranother.
KarlMenger(1934),finally,showedthatutilitymustbeboundedfromabove.
HebasedhisargumentonthePetersburggamble.
GivenBernoulli'slogarithmicutilityfunction,theparadoxisreinstatedbyreplacingthepower-series2iinequation(1)bytheexponentiallyincreasingpowerseriesie2sincethispayoffsseriesyieldsanon-finiteexpectedutilityandthusanon-finitecertaintyequivalent.
Sinceacorrespondingsuper-powerseriescanbefoundforeveryunboundedutilityfunction(Menger1934,p.
468f),theonlywaytocircumventtheparadoxintheexpectedutilityframeworkrequiresacut-offlevelforutilitywhereanyfurtherincreaseinpayoffleavesthedecision-makeratthesameutilitylevel.
Suchacut-offleveltoutilitywasalsosuggestedbyGabrielCramerin1728.
Hearguedthatthesumof2100or21,000ducatswouldgivehimnomorepleasureandattracthimmoretoacceptthegamblethanapayoffof224(≈16million)ducats;amaximumpayoffofsixteenmilliongivesanexpectedpayoffofthirteenducats.
MotivatedbyMenger'sdiscussion,vonNeumannandMorgenstern(1947)introducedtheaxiomaticapproachtoexpectedutilityinthesecondeditionoftheir"TheoryofGamesandEconomicBehavior"(seeMenger1967,p.
211).
KennethArrow(1971)showedthattoavoidthesuper-Petersburgparadoxandforacompleteorderingofallprobabilisticoutcomes,theutilityfunctionmustbeboundedfrombothaboveandbelow(seealsoStigler1950;Savage1954;D.
L.
Brito1976;RobertAumann1977).
Thedouble-sidedboundednessandotherimportantpropertiesoftheutilityfunctiontoavoidthesuper-PetersburggamblearealsodiscussedinSamuelson(1977).
Whilemathematicallyboundednessisanecessaryconditionfortheutilityfunctiontobewell-defined,theflatutilityfunctionbeyondacertainwealthasproposedbyGabrielCramerseemstomisrepresenthumanbehavior.
Althoughpeoplemightfeelindifferentbetweenapayoffof2nand2n+1forn≥24oranarbitraryinteger,asCramersuggested,itisdoubtfulthatpeoplewouldchoose2nover2n+1whenfacingthechoice.
Evenifthedecision-makerdoesnotchoosethe18George-LouisLeclercBuffon(1707–1788),wholearntthePetersburggamblefromCrameronatriptoGenevain1731(YvesDucelandThierryMartin2001),contributedseveralapproachestoaresolutionoftheparadoxwhichhepublishedinhisfamousessay(Buffon1777).
Hisfirstsolutioninvolveddiminishingmarginalutility,whichhecommunicatedtoCrameronOctober31730,thatis,priortoDanielBernoulli'spublication(Buffon1777,p.
75ff).
Buffon'sutilitytooktheinitialwealthpositionofPaulasapointofreferenceandweightedlossesmorethangains(seealsobelow).
10doubleamountfortheirownwellbeing,thedoubleamountwouldsimplybechosenforthesakeofpassingontheiradvantagetothechildrenandallchildren'schildren.
Moregenerallyspeaking,aboundonutilityseemstostandincontradictiontotheobservedcompetitivenatureofthehumanrace,theconqueroroftheearthandofoursunsystem.
Thereshouldbenodoubtthathumansprefergoverningtheentireuniversetogoverningtheuniverseexceptfortheearth.
ThePetersburggambleisultimatelyagambleontheearthandtheuniverse.
2.
2MoralimpossibilityandmoralcertaintyInreactiontotheproposalforusingtheexpectedutilityapproachtotheresolutionofthePetersburgparadox,NicholasBernoulli(1732)repliedtohiscousinonApril5,1732:Ihaveread[yourmanuscript]withpleasure,andIhavefoundyourtheorymostingenious,butpermitmetosaytoyouthatitdoesnotsolvetheknotoftheprobleminquestion.
Thereisnotagreedtomeasuretheuseorthepleasurethatonederivesfromasumthatonewins,northelackofuseorthesorrowthatonehasbythelossofasum;thereisagreednolongertoseekanequivalentbetweenthethingsthere;butthereisagreedtofindhowaplayerisobligedinjusticeorinequitytogivetoanotherfortheadvantagethatthereinaccordshiminthegameofchanceinquestion,orinothergamesingeneral,sothatthegameisabletobedeemedfair,asforexampleagameisconsideredfair,whenthetwoplayersbetanequalsumonagameunderequalconditions,althoughinyourtheory,andinpayingattentiontotheirwealth,thepleasureortheadvantageofgaininthefavorablecaseisnotequaltothesorroworthedisadvantagethatonesuffersinthecontrarycase.
19Mr.
Cramerhasalsotriedtoresolvetheproblembyreflectingonuseoronpleasurethatmenareabletoderivefrommoney,butwithoutpayingattentiontothesumofgoodsthatonealreadypossesses.
Hereisthatwhichhehaswrittentomein1728onthismatter:(ItfollowsaquoteoftheletterofGabrielCramer1728.
)Ihaveindicatedtohimnextthatitwouldseemtomethatinadmittingthisassumption,thatamanofgoodsenseisnotwillingtogive20ducats,becauseheestimatesallthecaseswhichgivehimalessersumthan20ducatspossible,andeachoftheothers,whichareabletogiveagreatersum,impossible;thatinadmitting,Isay,thisassumption,oneisabletoevaluatehisexpectation212.
.
.
0320163218161481241121=+++++++(4)Iclaimthatthisreasoningisnottooexact,butIbelievethatinmatchingtogetheryourideaandthatofMr.
Cramerandmyownonthatitisnecessarytoestimateasmall19NotethatNicholas'notionofafairgamblerequiresPeterandPaultobeindifferent,whileDaniel'ssolutionassignsafairvalueofthegambleonlytoPaul.
11probabilityasnull,oneisabletodetermineexactlythesoughtequivalent[forthePetersburggamble].
20(Spiess1975,p.
566f,emphasisadded)ThisquoteisanexcerptfromNicholasBernoulli'slastpreservedletteronthePetersburggamble.
Theknotinquestionisthemoralimpossibilityofobtainingmorethan20ducatsasapayoffinthePetersburggamblesincethelikelihoodofsuchaneventistoosmall;inhislettertoGabrielCramerheismoreexhaustiveonthisissue.
21Asstatedabove,thetermmoralimpossibilitywasintroducedbeforebyhisuncleJamesintheArsConjectandi.
Itwasdefinedasthereciprocalofmoralcertainty.
22MoralcertaintyisoneofthekeyconceptsinJamesBernoulli'sartof20DanielBernoulli(1954,p.
33)referredtotheletterofhiscousininhisfamousarticle.
"Inalettertome…,[NicholasBernoulli]declaredthathewasinnowaydissatisfiedwithmypropositionontheevaluationofriskypropositionswhenappliedtothecaseofamanwhoistoevaluatehisownprospects.
However,hethinksthatthecaseisdifferentifathirdperson,somewhatinthepositionofajudge,istoevaluatetheprospectsofanyparticipantinagameinaccordwithequityandjustice.
"21NicholasBernoullirepliedtoGabrielCrameronJuly3,1728(Spiess1975,p.
562f):TheresponsethatyougiveforthesolutionofthesingularcaseproposedtoMrdeMontmortpage402,Prob.
5satisfiesonlypartofit;itsuffices,asyousay,tomakeseethat[Paul]mustnotgiveto[Peter]aninfiniteequivalent;butitdoesnotdemonstratethetruereasonforthedifferencethatthereisbetweenthemathematicalexpectationandvulgarestimate;forexampleinthecaseofHeadsandTailsthereisnopersonofgoodsensewhowishedtogive20ECU,notforthisreasonthattheuseorthepleasurethatoneisabletodrawfromaninfinitesumisbarelygreaterthantheonewhichcanbetakenofasumof10,or20,or100millions,butbecauseingivingforexample20ECUonehasaverysmallprobabilitytowinsomething,andthatonebelievesthelossmorallycertain.
Thevulgarneitherstakeshereinthestorylinenorofmillions,norofhundredsofECUpayingnoattentionatalltothisthatthetermsofthegeometricprogression1,2,4,8,16,etc.
becomingfairlygreattheyareabletobeconsideredequal,heisenlistedthroughthisneithertoacceptnortorefusethegame,itisdeterminedsolelybythedegreeofprobabilitythathehastowinorlose;tohimaverysmallprobabilitytowinagreatsumdoesnotcounterbalanceaverygreatprobabilitytoloseasmallsum,heregardstheeventofthefirstcaseasimpossible,andtheeventofthesecondascertain.
Itisnecessarytherefore,inordertosettletheequivalentjustly,todetermineasfaraswherethequantityofoneprobabilitymustdiminish,sothatitbeabletobedeemednull;buthereisthatwhichisimpossibletodetermine,anyassumptionthatonemakes,oneencountersalwaysdifficulties;thelimitsofthesesmallprobabilitiesarenotprecise,buttheyhaveacertainlatitudewhatoneisnotabletofixeasily;aprobabilitywhichforexamplehas1/100certitudemustnotbereputednullinsteadthatwhichhas1/99certitude.
Itseemstomethereforeinadmittingthisassumptionthatamanofgoodsenseisnotwillingtogive20ECU,becauseheholdsforcertainthatthesumwhichwillfalltohimwillbelessthan20ECU,oneisabletofindtheequivalentsoughtbythefollowingreasoning:byhypothesisitismorallyimpossiblethatheobtains20ECU;itwillbethereforealsomorallyimpossiblethatheobtains32ECUorsomeothernumberofECUinthisprogression32,64,128,etc.
;ortheprobabilitytoobtainanumberofthisprogressionis1/64+1/128+1/256+…=1/32,thereforethismanofgoodsensereputesaprobabilitywhichdoesnotsurpassasnull,andaprobabilitywhichhasasatotalcertitude,consequentlyhisexpectationwillbeworthbytherule1/21+1/42+…+1/3216+032+0+…=2.
5.
[Theoriginalreads…1/168+016+…=2,butiscorrectedinthelaterlettersenttoDanielBernoulli].
ButIdonotknowifthisotherreasoningwillbemorejust:Amanwhodoesnotwanttogivemorethan20ECUestimatesallthecaseswhichgivetohimalessersumthan20ECUpossible,andeachoftheothers,whichareabletogiveagreatersum,impossible;heregardsthereforeonlytheprobabilitieslessthan1/32asnull,consequentlyhisexpectationwillbeworth1/21+1/42+…+1/3216+032+0+…=2.
5[Theoriginalreads=2].
Therewillbewellagainsomethingstosayonthismatter,butnothavingtheleisuretoarrangeinorderortodeveloptheideaswhicharepresentedtomyspirit,Ipassovertheminsilence.
22AccordingtoJamesBernoulli'sdefinition,"themorallycertainisthatwhoseprobabilityisalmostequaltocompletecertaintysothatthedifferenceisinsensible"(Bernoulli2006,p.
316).
Moral12conjecturing.
Accordingtothelawoflargenumberthelikelihoodincreasesthattheobservedrelativefrequencyfallsinsideagivenneighborhoodofthetrueprobabilitywithanincreasingnumberofindependentobservations.
However,completeconvergenceisonlyattainedinthelimitwherethenumberofrepetitionsincreasesbeyondanybound.
Tobaseaconclusiononafinitenumberofobservationswemustdiscardoutliersthatoccurwithsmallprobabilitiesbydefiningaminimumprobabilitylevel,i.
e.
thelevelofmoralimpossibilitybeyondwhichsmallprobabilityeventscanbetreatedaszero.
Indeedacertaindegreeofsubjectivearbitrarinessmaybeunavoidablewhenwefixtheaspirationlevelofmoralimpossibility.
Therefore,Jamesproposedthatthelevelofmoralcertaintyoritscounterpartmoralimpossibilitymustbeestablishedbythejudgeaccordingtothecircumstances,whether99/100ofmoralcertaintyissufficientorwhether999/1,000isneeded(Bernoulli2006,p.
321).
Giventhelevelofmoralcertainty,hecontinued,onecandetermineaposteriori(i.
e.
empirically)whatwecannotderiveapriori(i.
e.
therealodds)byextractingitfromarepeatedobservationoftheresultsofsimilarexamples.
Onlybyfixingalevelofmoralcertaintycanwemakeajudgmentontheoddsoftheeventsofcases.
Alevelofmoralimpossibilityisnecessaryfortheadvancementofknowledgeinthenaturalsciencesunderuncertainty.
Todetermineempiricalprobabilities,JamesBernoulliwhoalsowasaprofessorofexperimentalphysicsproposedexperimentssuchastheonesthathadbeenreportedearlierbyAntoineArnauldandPierreNicole(1662)inthe"ArtofThinking";hesaidthat"…thisempiricalmethodofdeterminingthenumberofcasesbyexperimentisnotneworuncommon,"(Bernoulli2006,p.
328).
TheArsConjectandiabruptlyfinishesafterJamescomputes,forhisfirstandonlyurnexample,therequirednumberofn>25,500experimentsatacertaintylevelof1/1,000.
23IthasbeenarguedthatthisnumbermighthaveappeareddisappointinglylargetoJamesBernoulli,ashishometownofBaselnumberedfewerinhabitantsinthosedays.
GiventhatBernoulliwasinterestedinapplyingprobabilitytheorytocivilproblemssuchalargenumberposedapracticaldatacollectionproblem.
StephenStigler(1986,p.
77)certaintyhasbeenintroducedbyJeanCharlierdeGerson(1363–1429),chancelloroftheUniversityofParisaround1400.
ItissaidthattheconceptgoesbacktoastatementinAristotle'sNicomacheanEthics"thatonemustbecontentwiththekindofcertaintyappropriatetodifferentsubjectmatters,sothatinpracticaldecisionsonecannotexpectthecertaintyofmathematics.
"Descartesputitincirculation;hedescribes"morallycertain"ashavingsufficientcertaintyforapplicationtoordinarylife"(thosewhohaveneverbeeninRomehavenodoubtthatitisatowninItaly,eventhoughitcouldbethecasethateveryonewhohastoldthemthishasbeendeceivingthem)"(ReneDescartes1985,p.
290).
Moralcertaintyhashaditsrelevanceinjurisprudence,whereitmeansbeyondanyreasonabledoubt(thisisthehighestlevelofproofwhichisusedmainlyincriminaltrials).
Leibnizdiscussedmoralcertaintyanddegreesofprobabilityinjurisprudenceandintroducedimpossibilityandpossibilityaseventswithzeroandunityprobabilityin1665(Keynes1921,p.
155).
In1699,JohnCraigdiscussedlevelsofcertaintyinhistheologiaechristianaeprincipiamathematica,abookNicholasBernoullicitedinhisthesiswhendiscussingwitnessesanddegreesofcertainty.
WhenGabrielCramerlecturedonlogicsinabout1745,hediscussedmoralcertaintyandmoralimpossibilityinlinewiththeArsConjectandiandtheUsuArtisConjectandiinIure(ThierryMartin2006).
23Actually,thisnumberislargerthantherequirednumberowingtotwocrudeapproximationsinJamesBernoulli'sproof;Hald(2007,p.
14)reportsn>12,243.
13suggestedthatBernoulliquittedhisworkinfrustrationwhenhesawthehugenumber.
24Onewayofrevisingtherequirednumberofobservationstoasmaller,moreavailablesamplesizewouldbe,infact,tolowertheaspirationlevelonmoralimpossibility.
NicholasBernoullibelievedthatthePetersburggamblerequirestheapplicationofthemoralimpossibilityconcepttomakeitfair.
25Itisconceivablethathehopedtoobtainaconsentlevelofmoralimpossibilityfortheapplicationinthegamesofchance.
InletterstoCramerandDanielBernoulli,Nicholassuggestedthat1/64andsmallerprobabilitylevelsshouldbetreatedaszero.
OtherauthorsproposeddifferentlevelsofmoralimpossibilityinthePetersburggamble.
D'Alembert(1764,p.
7)suggestedthatonewouldnotwanttoriskafairamountofmoneyonoutcomesthatoccurwithsuchasmallprobabilityof1/128orless,evenifthepotentialearningswereimmense.
Morerecentlyandwithoutfurtherjustificationorreference,SamuelGorovitz(1979)proposedthattheprobabilityof1/128wasnegligibleinthePetersburggamble.
Buffon(1777),whoprovidedseveralproposalsfortheresolutionofthePetersburggamble,arguedthataprobabilitysmallerorequalto1/10,000generallycannotbedistinguishedfromazeroprobability.
Inhisdays,theoddsthata56-year-oldmanwoulddieinthecourseofadaywere1:10,180.
Heclaimedthatsuchasmallprobabilityisnothingtobeworriedabout.
DanielBernoulliapprovedtheideaofnegligibleprobabilitiesinalettertoBuffondatedMarch19,1762butdemandedtheapplicationofthemoreconservativelevelofonein100,000(1777,p.
75);thatwasalsotheprobabilitylevelinphysicsthatHuygensregardedasbeingequivalenttoamathematicalproof(Dutka1988,p.
33).
Astheexpectedintensityofthefearofdeathinthecourseofthedaydisappearsifitslikelihoodissmallerthan1/10,000,andasthisfearismuchgreaterthantheintensityofallothersentimentssuchasfearorhope,Buffon(p.
90)believedthatamoralimpossibilitylevelof1/1,000shouldbeappliedtotheestimateofthemoralvalueofmoney.
HesuggestedthatallpayoffsofthePetersburggamblethatoccurwithaprobabilityoflessthan1/1,024canberegardedasalmostzero,sothattheyareirrelevantfordecision-making.
Buffon(1777)underlinedthisclaimbyexperimentaldata.
HeconductedthefirstrecordedexperimentinstatisticstodetermineempiricallythelikelyoutcomesinthePetersburggamble.
Achildplayedoutn=211=2,048trialsofthePetersburggamble(Buffon1777,p.
48f).
ThereportedgambleoutcomesarereproducedinthetableA1oftheappendix.
Alltrialsendedafteratleast24JamesBernoullisearchedover20yearsforquestionstowhichhecouldaddresstheartofconjecture.
HeeventuallylearnedabouttheexistenceofdeWitt'sworkonliferentsincludingmortalitytables(theworkisprintedinKohli1975b)andrepeatedlycalledforitssubmissiontohimbyLeibniz.
FromthecorrespondencewithLeibnizitisevidentthatJamesBernoulliwasdesperatelysearchingfordatatoapplyhistheory.
Itisalsopossiblethathelefttheapplicationofhistheoryunwritten,sinceheneverreceivedthecopyofdeWitt'sbookfromLeibniz.
25NicholasreferredtomoralcertaintyinhisreplytoMontmortin1713(Spiess1975,p.
558),whenhemotivatedhisfifthproblemwhosereformulationbecamelaterknownastheStPetersburgparadox.
NicholasBernoulli(1709)alsoacknowledgedtherelevanceofmoralcertaintyintheforewordtohisthesis;"theartofconjecturingconcernsuncertainanddoubtfulmatters,aboutwhich,althoughcompletecertaintyisimpossible,wecanneverthelessbyconjecturesdefinehowgreattheprobabilityisthatthisorthatwillbe,whatprobablywillhappen,whichoutcomeismoreprobablethananother,orhowmuchthisorthatconclusiondivergesfromcompletecertainty"(Bernoulli2006,p.
55).
14ninetossesandtheaveragepayoffwas4.
9ducats.
Buffonconcludedthataboutfiveducatsshouldbeafairentryfeetothegamble.
AugustdeMorgan(1912)reportedreplicationsofBuffon'sexperimentbythreeanonymouscorrespondentsandDutka(1988)ranelevenBuffonexperimentsonthecomputer.
ThesefourteenBuffonexperimentsgeneratedanaveragepayoffof7.
3ducatsperPetersburggamble(Dutka1988,p.
35ff).
DavidTolmanandJamesFoster(1981)ran1,000Buffonexperimentswiththecomputergeneratinganaveragepayoffof9.
8andamedianaverageof6.
8ducats.
AllanCesar(1984),whovariedthenumberoftrials,n,from100to20,000repetitions,confirmedthetheoreticalresultimpliedbythelawoflargenumbers(seethesectionbelow),inthatanincreaseinrepetitionsleadstoanincreaseoftheaveragepayoff,sincelongerseriesmakehigherpayoffsmorelikely.
ManuelRussonandSJChang(1992)reportedinconsistenciesoftheirdatawiththistheoryastheyfoundthatlongruns(involvingmorethan20tails)didnotoccurincomputertrials.
Contrarytothisobservation,RobertVivian(2004)reportedevidencethatwasconsistentwiththetheory.
Insummary,theresultsintheliteraturearemixed.
Thelongestruneverreportedinvolved28tails(LudgerHinners-Tobrgel2003).
26FollowinguponthediscussionofBuffon,d'Alembert(1761)raisedthequestionwhethersomeunlikelyevents,e.
g.
100successivetossesoftails,canoccuratall.
Heconcludedthatsomeprobabilitylevelsaresimplytoosmalltobephysicallyrelevant;somecasesheclaimedarepurelymetaphysicallypossibleandphysicallyimpossible.
27TheideaofphysicalimpossibilitywasmostprominentlyrepresentedlaterbyAugusteCournot(1843).
28Cournot(p.
78)statedthatitismathematicallypossiblethataheavyconestandsinbalanceonitsvertex,butitisphysicallyimpossibleastheprobabilityofthateventisvanishinglysmall.
Similarly,hesuggestedthatinalongsequenceoftrialsitisphysicallyimpossibleforthefrequencyofaneventtodiffersubstantiallyfromtheevent'sprobability(1843,p.
121f).
Fromthesestatements,theso-called"Cournot'slemma"hasbeenintroducedintotheliterature(Fréchet1948;GlennShafer2006).
26LolaLopes(1981)ran100simulations(representingbusinesses)withonemillionPetersburggambles(representingbuyers)forfourdifferententryfees.
Abusinesswouldstayafloataslongasitsbalancesheetafterfeesandprizeallocationsremainedabove-$10.
000,butitwouldclosedownotherwise.
ThestudydoesnotrevealthelongestrunoftailsasLopesfocusedonthesurvivalofbusinesses.
Witha$25feeonlynineteenbusinessesstayedopenafteronemillionbuyers;witha$100fee,90businessesstayedopenandtheaveragegainwas$55.
9million.
27SimilarlytoBuffon(1707-1788),d'Alembert(1717-1783)contributedseveralessaystothestudyofthePetersburggamble.
Inoneessay,hearguedthatarunof100tailsinarowmaybemetaphysicallypossiblebutitisphysicallyimpossible;suggestingthattherearelimitstotheapplicationofmathematicstotherealworld.
Inmemoir27,d'Alembert(1768,p.
299)statesthefollowing;given"2100playerswhocasteachonehundredtimesinsequenceasimilarpieceintotheair;Isaythatonecanwagerwithoutanyriskthatanyoftheseplayerswillbringforthneitherheadsnortailsonehundredtimesinsequence"(quotedfromthetranslationbyPulskamp2004).
28ForthePetersburggamble,Cournot(1843,p.
109)suggestedamarketsolution.
Heproposedtoselllotteryticketswhichyieldedanon-zeropayoutforoneparticularsequenceoftossesonly.
Heclaimedthattherewouldbeonecriticallengthbeyondwhicheveryticketwouldremainunsoldandcitedempiricalevidence.
IntheFrenchlottery,theadministrationhadtakenoutaprizewhichoccurredwithaprobabilityofonein44million,becauseitwastooseldomgambledon.
Cournot(1843,p.
106)saidthat"oneimagineswellthattheremustbealimittothesmallnessofchance"(seealsoJorland1987,p.
182).
15Theprinciplestatesinitsweakandstrongformsthatasmallprobabilityeventwillhappenrarelyonrepeatedtrialsanditwillnothappenatallinaparticulartrial,respectively.
Cournot'slemmahasbeenviewedasthefundamentallawwhichlinksprobabilitytheorytotherealworldbymanyfamoustheoreticians,includingPaulLevy(1925),AndreiMarkov(1900),AndreiKolmogorov(1933)and,mostdrastically,Borel(forasurveyseeGlennShafer2006).
Borel(1939,p.
6f)calledtheprinciplethataneventwithverysmallprobabilitywillnotoccuratanytime"thesinglelawofchance".
Hedistinguishedimpossibleeventsbymeasure;impossibilityonthehumanscale:p+∑=∞→εNnXPnmmn(6)whereXmdenotesthepayoffinthemthtrial,andε>0denotesanarbitrarilysmallnumber.
Inequation(6),boththeexpectationandthevariancearefiniteconstants.
Intheclassicalsense,theexpectedpayofffortherepeatedgamblewouldbeconsideredasthefairvalueofthegamble.
Lacroix(1802)alreadynoticedthattheexpectationshouldonlybeusedasafairvalueiftheexpectationexists.
IntheoriginalPetersburggamble,theexpectationdoesnotexistyetandnordoesitsvariance.
Forthiscaseandbymakinguseofatruncationargument,Feller(1945)provedthattheentryfeeenwhichmakesthePetersburggamblefairintheclassicalsensegrowswiththenumberofrepetitions.
Fellershowedthattheexpectationofthegambleisen=0.
5log2n.
400lim1→>∑=∞→εnnmmnenXP(7)Inthelimit,i.
e.
wherethegambleisrepeatedwithoutend,thefairentryfeeforthePetersburggambleconvergestoinfinity.
TheoriginalPetersburggambleissuchanextremecasesincethespreadaroundtheexpectationissoextremethatonlyanindefinitenumberofrepetitionsofthegambleleadstoalikewiseindefiniteaveragepayoff.
Notethatthedescribedentryfeecanonlybefairfortherepeatedgamble,butitisnotapplicabletotheone-shotgamble,sinceforn=1,en=0.
Allais(1979,p.
500ff)likeLaplace(1820)andothers(e.
g.
Lacroix1802;Czuber1882)beforehimconsideredthesinglegambleasthelimitingcaseformathematicalexpectations.
41Allaisagreesthatpsychologicalvaluessuchasthediminishingmarginalutilityargumentorthezeroweightingofsmallprobabilitiesofwinningmayplayarolefordecision-makinginthesingle39JakobFriedrichFries(1842)andCzuber(1882,p.
20)pointedoutthattheconceptofmathematicalexpectationisvalidonlyinthecaseofmanyrepetitions,i.
e.
ifthelawoflargenumberscanbeapplied.
Forinstance,ifyouplayn=180,000230gambleswithN=30,themathematicalprobabilitythattheaveragepayoffdeviatesbymorethan1%fromtheexpectationis"freefromtheinfluenceofchance.
"40AccordingtoDutka(1989,p.
36),en=0.
5log2(nlog2n)isalsoapossiblefairentryfeeasitisasymptoticallyequivalenttoFeller'sfunction.
FollowingFeller'sapproach,mathematicianshavediscussedlimittheoremsforthePetersburggamble(Steinhaus1949;Martin-Lf1985,2005;CsrgandDodunekova1990;CsrgandSimons1993-94,1996,2002,2005;Berkeretal.
1999;Csrg2003).
Inhispaper,Feller(1945)didnotmakeanyreferencetoBuffon(1777),whoderivedthesamelimitdistributionthroughhisexpectationheuristic,orCondorcet(1781),whofirstappliedthelawoflargenumberstothetruncatedgamble(bothhavebeenacknowledgedabove).
41FrankKnight(1921,p.
234)statedthatiftheexperimentcannotoftenberepeatedindefinitely,theprobabilitiesareirrelevanttotheindividual'sconduct.
Similarviewswereheldbytherepresentativesoffrequencytheory(Edgeworth1922,p.
277f);theprobabilityconceptcannotbeappliedtosingleeventsbutonlytoseriesinthesenseofVenn(1866).
21gamble.
NicholasBernoulli(1732)andBuffon(1777)concededthatbothcomponentsplayarole,andsodidMenger(1934)andKahnemanandTversky(1979).
Yettheoccurrenceofeventsthatareasunlikelyas1/10,000becomepracticallycertainifthenumberofrepetitionnbecomeslarge.
Allaisarguedthatthemathematicalexpectationcanbeconsideredtobetherationalmodelintherepeatedgamble,inparticularforacorporationthatmaximizesexpectedpayoffsratherthanpsychologicalvalue.
Hedistinguisheddifferentcasesregardingthenumberofgamblesandthecapitalrequirements,i.
e.
whethersettlementistobemadeaftereachgambleorwhetherthenetsumisdueonlyafterthelastgambleisover.
Neglectingfinitetimelimitationsandconsideringfinite,constantentryfees,AllaisstatesthatthegamblewillalmostcertainlyleadtoPeter'sruinifthesettlementisbeingmadeonlyafterthelastgamblewhennmovestoinfinity,whileifsettlementistobemadeaftereachgambletheruinofPaulismorelikely.
42Infact,iftheprobabilityofruinisleftoutoftheaccountthemathematicalexpectationoftheplayerremainsatPeter'sfortune(giventhatverylongrunsoftailsmaterializeinthelongrun).
Indeed,aclosely-relatedcasewasmadebySamuelson(1963)inhis"fallacyoflargenumbers.
"Heproposedthatahundred-foldrepetitionofafavorablegamblewithsettlementafterthelastrepetitionispreferabletotheone-shotgamble,asitdecreasestheriskofruin,i.
e.
theprobabilityofabiglossbecomesverysmall,althoughitisnoteliminatedcompletely.
43Toavoidtheriskofthegambler'sruinintherepeatedPetersburggambleforthecaseofimmediatesettlementofpayoffs,Paulshouldstakeaconstantproportionofhisportfolioratherthanafixedamount(WilliamWhitford1886;SydneyLupton1890).
44Thus,Whitford'swayofdiversifyingriskacrossrepetitionsofthegambleleadstothelogarithmicfunctionproposedbyDanielBernoulli(1738).
Indeed,DanielBernoulli(1954)alludedtoapossibleimplicationofhistheoryindiversification,buthiscousinNicholasBernoulli(1732)deniedsucharelationshipfortheone-shotgambleifnodiversificationopportunityexists.
4542Thisissuemayalsoexplainwhycasinosrequiresettlementineachgambleandlimitthestakesforbets(seealsoMartin-Lf1985;andLopes1981).
43Samuelsonofferedtohiscolleaguethelotteryinwhichonewins200orloses100withthesameprobability.
Hiscolleaguerejectedthegamble,butwaswillingtoplay100repetitionsofthegamble.
44SimilarlytoWhitworth,othercontributorsalsoproposedabettingruleinproportiontotheplayer'savailablefunds(JohnWilliams1936;JohnKelly1956;Durand1957;LeoBreiman1961;RobertBellandThomasCover1980).
Thefocusofthesebettingsystemsisonthegrowthoftheportfoliovaluebymaximizationofthegeometricmeanofthereturn(seealsoGaborSzékelyandDanielRichards2004,2005;andChristopherRump2007).
Samuelson(1969,p.
245)believesitisincorrect"thatifoneisinvestingformanyperiods,theproperbehavioristomaximizethegeometricmeanofreturnratherthanthearithmeticmean.
"45NicholasBernoulli(1732,p.
567)repliedtohiscousinthatexpectedutilitytheorydoesnotimplysuchadiversificationasit"…onlyshowsthatoneriskstopermitagreatersorrowinplacingagreatsumwithasingledebtorthaninplacingthesamesuminpartsamongmanydebtors;butitdoesnotshowthatonerisksalsotomakeagreaterloss….
Weknowwithoutpayingattentiontoyourprinciple…thatonedoes.
.
.
bettertoplace500coinsin2places,than1,000coinsinasingleplace,becauseoneisnotexposedtolosingaseasilyall1,000coins….
Onemustnotputtoomanyeggsinonebasket,saysourBlois.
Butwhatcanyoudo,ifyouneededtomakeworthofyourmoneyincreditingittoamerchant,andifyoudonothavetheoptiontoplaceitbysmallparts"222.
6RiskKeynes(1921,p.
315)believedthataresolutiontothePetersburgproblemmustaccountfortheincreasingriskrelativetoexpectedpayoffwhenthegamblelengthisincreased,N→∞.
Asapossibledefinitionofriskheproposedafunctionofprobabilityandthedeviationfromtheexpectation.
Hearguedthat,otherthingsbeingkeptequal,thevalueofagambledecreaseswithanincreaseinrisk.
Keynes'sapproachrepresentsaprecursorofthemean-risktheorybyHarryMarkowitz(1952),46wherethestandarddeviationisusedasariskmeasure(seealsoArrow1951).
JohnSennetti(1976)appliedthistheorytothePetersburgparadoxtoconcludethatthegamblewillnotbeplayedforalargeentrancefee.
ThomasEpps(1978,p.
1455)cameupwiththeparadoxicalimplicationthat,owingtotheunboundedrisk,themean-riskcriterionwouldrejectthePetersburggambleevenatazeroentryfee.
PaulWeirich(1985)studiedthesuper-Petersburggambleandshowedthatthemean-riskcriteriondoessolvetheparadoxunderspecialassumptionssimilartoboundedutilityornegligenceofsmallprobabilities.
WhilethisapproachmayhaveitsmeritsitisnotclearwhatconclusionscanbedrawnfromitsapplicationtothePetersburggamble.
IfwelookmorecarefullyintoKeynes'scontribution,Keynesexplicitlyacknowledgedtheriskofoverpayingthewager,butthisaspectmaybedescribedasakindoflossaversionbytoday'sstandardsratherthanriskaversion(Buffon1777;Camerer2005).
Withrespecttothestandardmeasurementofriskaversion,oneparadoxicalissuearisesintheoriginalPetersburgparadox.
Givenanyfinitecertaintyequivalentsofthegamble,Paulisperdefinitionrisk-averseevenifheiswillingtopayastaggeringlyhighnumberforthewager,e.
g.
everyamountupto$21,000,whilePeterisdefinedasriskseekingforanyfinitewillingnesstoaccept,e.
g.
evenifheisnotwillingtosellthewagerforanysmalleramountthan$21,000(guaranteeingthepayofffromthegamblewithhislife).
Inthiscase,theclassificationbytheArrow-PrattmeasureiscounterintuitiveasbothplayersmaybeadjudgedinsanealthoughinoppositiontotheArrow-Prattjudgment;Paulisimprudentlyrisk-lovingandPeterisridiculouslyrisk-averse.
2.
7ExpectancyheuristicAnalternativeresolutiontothePetersburgparadox,theso-calledexpectancyheuristic,hasbeenproposedbyMichelTreisman(1983).
Insteadofcomputingtheproductofprobabilitiesandpayoffs,theexpectancyheuristicsuggestscomputingtheoutcomeattheexpectedgamblelength.
Aspointedoutintheliterature(e.
g.
inSamuelson1977),theexpectedgamblelengthistwo.
Itiscomputedequivalenttoequation(2)whereeachtossisrewardedbyonemoreducat(insteadoftheamountsbeingdoubled).
AtagamblelengthoftwothepayoffistwoducatswhichisthereforethevalueofthePetersburggambleaccordingtotheexpectancyheuristic.
Notethatthemedianpayoffwillbeclosetothevalueoftheexpectancyheuristic,too.
46SeealsoArrow(1951,pp.
423-426)fordiscussionsofearliernotionsofthemean-standarddeviationcriterionandthequadraticutilityfunction.
233.
ExperimentalresearchBottomandcolleagues(1989)designedanexperimenttotestseveralhypothesesincludingtheexpectancyheuristicofTreismaninthePetersburggamble(thesamedataarealsopresentedinJCRiveroetal.
1990).
Theauthorsstatedtheywerealsoconsideringthepossibilitythatsmallprobabilitiesareneglected.
Theycitedthedeminimisliteratureaccordingtowhichprobabilitiesbelow1/10,000and1/1,000,000arecommonlyignored.
Fromtheselevelstheyconstructedhypotheseswhichvaluedthepayoffatthemathematicalexpectation,cuttingoffthegambleafter14and20tosses,respectively.
TheyalsocomputedGabrielCramer'sandDanielBernoulli'sutilityfunctionassumingzerowealthasanadditionalhypothesis.
Finally,afinitewealthhypothesisassumedthatPeterhadnomorefundsthan220ducats.
Thus,onlypayoffsbelowthatamountwereconsideredreasonable.
Fortheexperiment,Bottomandcolleagues(1989)recruitedfromtwosubjectpoolsofstudentsandprofessionals.
Thelatterwerespecialistsinstatistics,economicsandmanagementscience.
Bidswerecollectedinasealedbidauctionunderfourhypotheticalconditions,withnopayoffsorentryfeesinvolved.
Subjectswereaskedtowritedownasealedbidforeachofthefourconditionsandtoimaginethatthesebidswereactuallycompetinginanauction(Bottometal.
1989,p.
142).
ThefirstconditioninvolvedthestandardPetersburggambleasproposedinequation(1),butwithdoubledpayoffs;inthesecondconditionthepayoffsforeachpossibleoutcomewereincreasedbyfivedollars;inthethirdconditiontendollarsmorepayoffwereofferedoneachpossibleoutcome;andinthefourthcondition,payoffsweredoubledfromthefirstconditionforeachpossibleoutcome.
Theresearchersconcludedthattheirresultssupportedtheexpectancyheuristic,meaningthatthemedianbidswereapproximatelyequaltotheexpectedmedianpayoff.
DespitetheageandtheimportanceoftheproblemonlyafewexperimentsonthePetersburggamblehavebeendocumented.
47ItisprobablethatthesolvencyproblemwhichrenderstheexperimentalapproachtotheoriginalPetersburggamblemeaninglesshasbeenamajorcaseagainstit.
Tocircumventthisproblemtheexperimentaldesignmustinvolvethetruncatedgamblewhichlimitsthegambletoafinitenumberofpossibletosses(Fontaine1764).
SuchasettinghasbeenusedinthepresentstudyandinindependentresearchbyJamesCoxandcolleagues(2007),whoconductedthefirstexperimentswiththetruncatedPetersburggambleinFebruary2007.
TheresultsoftheexperimentshavealsobeenreportedinCoxandVjollcaSadiraj(2008)andCoxandcolleagues(2009).
Theauthorsalludetothecalibrationproblemsinthestandardtheoriesofdecision-makinginaccommodatingthePetersburgparadox.
Theirdesign47HaimLevyandMarshallSarnat(1984,p.
110)reportedintheirtextbook,withoutgivingfurtherdetailsontheirprocedure,thattheymadeinquiriesofagroupofstudents,ofwhom"mostwerepreparedtopayonlytwoorthreedollarsforachancetoplay.
Afewwerewillingtopayasmuchaseightdollarsbutnooneofferedmorethanthat.
"ThisstudyhasbeencitedinJürgenJerger(1992).
Vivian(2003,p.
342)statedinafootnotethathehasoftenaskedstudentstoindicatetheamounttheywouldbepreparedtorisktoplaythegame.
"Nostudentiseverpreparedtoriskmorethanafewdollarstoplaythegame…someevenindicatethatzeroisareasonableamounttoplaythegame.
"24involvedninepossibletruncationsofthegambleincluding{1,2,3,.
.
,9}tossesofthecoin.
48Thirtysubjectswereinvitedtoplaythegamblefor0.
25dollarsbelowtheexpectedvalueofthegamblewiththeirownmoney;onegamblewaschosenatrandomandplayedoutforreal.
Mostoftheirsubjectswereunwillingtoplaythegamble.
Thereforetheyconcluded"thatamajorityofsubjectsintheexperimentareriskaverse,notriskneutral"(Coxetal.
2009,p.
224).
Thisresultalsoseemstobesupportedbythedatareportedinthecurrentstudy.
AnotherrecentandindependentexperimentalstudyonthePetersburggamblewhichincludesbothhypotheticalandrealincentivesisreportedbythetwobiologistsBenjaminHaydenandMichaelPlatt(2009).
Theprimaryinterestoftheirstudyisontherepeatedgamble.
49Theirdatashow,inlinewiththelawoflargenumbers,thatthevaluationpergambleincreaseswiththenumberofrepeatedgambles.
Inparticular,subjects'valuationsseemtoconvergetothemedianoutcome.
Inlightoftheseobserveddecisions,wemustgiveaddedacknowledgementtotheproposalofTolmanandFoster(1981)thatthemedianvaluationisareasonablechoicefortherepeatedPetersburggamble.
504ThetruncatedPetersburg-gambleexperimentInthefollowingsubsections,severalexperimentaldatasetsarepresented.
IamextremelygratefulforthesupportofmycolleaguefriendsandstudentswhocollecteddataonmybehalfatdifferentlocationsinEurope.
ThepresentationwillincludedatagatheredinclassroomexperimentsattheLeibnizUniversityofHannover,GermanyandtheUniversityofGranada,SpainandafieldexperimentthatwasconductedinHannover.
Theexperimentalstudyisdedicatedtothestudyoftheone-shotPetersburggamble.
Theunderlyingassumptioninthestudyisthatthefollowinggeneralexpectedutilityfunctionalformrepresentspreferences.
∑∞==1)()()(iiixupfXU(8)MostofthevarioushypothesesthathavebeenputforthinthehistoryoftheparadoxtruncatethePetersburggambleincludingtheboundedutilityorthefinitewealthhypotheses,whichfordifferentreasonsputanupperboundontheutilityofthegamble,andthephysicalimpossibilityorthemoralimpossibilityhypotheses,whichassumethatasmallprobabilityissetequaltozero,f(p)=0,p2500},allamountsinEuro.
Intotal,352subjects,or98.
5percent,repliedtotheincomequestion.
Fromthesedata,thefollowingobservationcanbemade;subject'soffersincreasesignificantlywithincome(inthetreatmentsthatinvolveamaximumpayoffofatleast32Euro).
Forthesampleinvolvingthetreatmentswithamaximumpayoffofatleast32Euro,thepooledregressionresultofthestatedwillingness-to-payontheincomecategoriesisasfollows(theasteriskindicatessignificanceatthe5percentlevel,theparenthesesquotestandarddeviations).
statedwillingness-to-pay=2.
63*+0.
74*IncCat(1.
013)(0.
335)TheregressionresultisapparentlyinlinewiththeassumptionofDanielBernoulli(1954),accordingtowhichthewillingness-to-payforthePetersburggambleshouldbeanincreasingfunctionofwealth(orincome,inlinewithMarshall1890).
Theevidenceonthesmallprobabilityneglect,however,wasnotanticipatedinDanielBernoulli'stheory,aswaspointedoutbyNicholasBernoulli(1732).
Theothertwotreatmentsindicatenosignificantwageeffect,andgenderandage57Asubject-pooleffectisverylikely.
InFebruaryandMarch2007,IalsorantwoclassroomexperimentsinRomeatLUISS.
Theresultsindicatedsubject-pooldifferences;elevenfirst-yearstudentsofaMasterofArtscoursesubmittedamedianbidof1.
00Eurofortheoriginal(non-truncated)Petersburggamble,whilenineMBAstudentssubmittedamedianbidof6.
00EuroforthetruncatedPetersburggamble,wherethepayoffmaximumwas100Euro.
Thedifferenceissignificant;thep-valueofthetwo-tailedMann-Whitneytestis0.
002(thetwo-tailedtestisusedherebecauseofthedifferentsubjectpools).
Actually,theMBAsampleelicitedsignificantlyhigherbidsthananyotherreportedclassroomexperiment.
30arenosignificantdeterminantsofthestatedwillingness-to-pay,either.
Oneshouldalsopointoutthatthecorrelationbetweenthestatedageandthestatedincomecategoryissignificantinoursample.
TheSpearmanrankcorrelationcoefficientis0.
260,andthep-valueofsuchanextremeresultis0.
000.
Asindicatedinthetable3,thestatedwillingness-to-paywasextraordinarilyhighinthefieldexperiment.
58Ofthe99personswhostatedanamountatorabovefiveEuroastheirwillingness-to-pay,however,onlynineteen,or19percent,werewillingtopurchasethegambleatapriceoffiveEuro.
Conversely,fourpersons,or1.
5percent,statedanamountbelowfiveEuroandpurchasedthegambleforfiveEuro.
Eliminatingthedatapointsofthesesubjectsdoesnotleadtodifferentconclusions,buttoalowermedianwillingness-to-pay.
4.
3Isthesmallprobabilityeffectrobusttosubject-poolchanges-Yes,itseemstoberobust.
Anadditionalclassroomexperimentwithanincentivecompatibledesignwasconductedtochecktherobustnessoftheresultsfromthefield.
Atotalof232studentsofathird-yeareconomicscourse59attheUniversityofGranada,Spainparticipatedintheexperiment.
Inspring2008,thecoursewastaughttothreedifferentgroupsbyFrancisLagos.
AttheendofthefinallectureinMay2008,DrLagosinvitedthestudentstoparticipateintheexperimentandranonmybehalfasecond-priceauctionasdescribedinsection4.
1onthetruncatedPetersburgwithmaximumpayoffs16,32,and64Euro,respectively.
InApril2009,thesamecoursewastaughttotwodifferentgroupsbyDrJuanAntonioLacomba,whoinvitedthestudentstoparticipateandconductedtwoexperimentswithmaximumpayoffsof128and1024Euro.
IndifferencetotheGermanclassroomexperimentbutinlinewiththefieldstudy,allpossiblepayoffswerepresentedonthedecisionsheetandthetombola-sheetwasnotpresented.
Thebasicstatisticsoftheexperimentarereportedinthetable3andthepair-wisetestresultsarerecordedinthetableA5oftheappendix.
Theresultsreplicatetheobservationofthefieldexperiment.
60TheobservedbiddingbehaviorisinlinewiththehypothesisofNicholasBernoulli(1732),thatsubjectssetsmallprobabilitiesbelow1/32equaltozero.
Thebetween-countrycomparisonofthebiddataforallPetersburggambleswhichinvolveamaximumpayoffabove16indicatesnodifferencesfortheSpanishandGermanclassroomexperiments.
61Table3.
BidsinthetruncatedPetersburggamble(Granada)58Thesubjectswhostatedawillingness-to-payof50or100EuroasrecordedinTable3rejectedtheoffertopurchasethegambleatapriceoffiveEuro.
59Notethatthird-yeareconomicscourseswerealsoimplicatedinGermany.
Thus,cross-countrycomparisonofthebiddatamaybepossible.
60Theone-tailedMann-WhitneytestsuggeststhatthePetersburggambletruncatedat16leadstosignificantlylowerbidsthanthegambletruncatedat32,64,1024;thep-valuesare0.
041,0.
016,and0.
032,respectively.
Thesamplesizeofthe128cohortisapparentlytoosmalltoreflectthedifferencetothe16cohortata5percentlevel;thep-levelis0.
095.
Thepair-wisetestresultsforallthetreatmentsarenotsignificant,theyarerecordedinTableA5oftheappendix.
61Thetwo-sampletwo-tailedMann-Whitneytestdoesnotrejectthenullhypothesiswhichassumesnodifferentindividualbidsinbothlocations(thep-valueis.
548),andtheeight-sampletwo-tailedKruskal-Wallistestdoesnotrejectthenullofequalbids,either(thep-valueis1.
000).
31maximumpayoff:1632641281024Meanbid1.
442.
062.
092.
162.
12Medianbid1.
101.
581.
501.
942.
00Maximumbid5.
008.
0010.
007.
006.
05Minimumbid0.
010.
100.
500.
020.
00Standarddeviation1.
091.
701.
711.
851.
63Numberofparticipants40445928625ConcludingremarksThisarticlereportsexperimentalevidencethatpeople'selicitedwillingness-to-payforthePetersburggambleiscompatiblewiththeviewthatsubjectsneglectsmallprobabilitiesofwinning.
Morespecifically,therevealedwillingness-to-payforthetruncatedversionofthePetersburggambledidnotdifferfromtheonefortheoriginalversionofthegamble.
Thesmallest,stillnotable,probabilitylevelinthedatais1/32,maybeasurprisinglylowlevel.
ButthislevelofmoralimpossibilityhasbeenproposedbyBernoulli(1732)who,throughhiscontinuedinterestinthePetersburggamble,evidencedbypersistenceindiscussingthisproblemwithvariousresearchersovertheyears1713to1732,introduceditintotheliterature.
Thestatisticalequivalenceofthewillingness-to-payforthevariouslengthsofthegambleimpliesthatsubjectsconsiderthepayoffsupto32ducatsonly.
ItthusseemsreasonabletoacceptthattheneglectofsmallprobabilityeventsisamorerelevantdecisioncriterioninthePetersburggamblethanitsbestknownalternatives,boundedutility(whichequalizesthepleasuresofgaining32ducatswithunlimitedwealth)andthelimitationoftheexperimenter'swealthat32ducats.
Menger(1934)pointedoutthateitheranupperboundonutilityoralowerboundonprobabilitymustbeinstigatedtoresolvethePetersburgparadoxes.
Wakker(1993)showedthatwithdenumerableprobability,itispossibletoinstateawell-definedunboundedutilityfunction.
Intermsofprobabilitylevels,anequivalentsize,namelythe5percentlevel,hasbeenproposedbyappliedstatisticians(followingFisher1925)asaconservativelevelofmoralimpossibilityforthesocialsciences,sincetwostandarddeviationsofthedistributionareincludedandonlyveryextremeoutliersareexcluded.
Indeed,dependingonthesignificanceofthesubjectmatter,higherlevelsofmoralimpossibilitywillbeapplied.
Someconsentlevelshavebeenestablishedforscientificresearch,healthcare,orforensicproofofevidence.
Thisapproachisboundedlyrationalastheparticularcriticalprobabilitylevelseemssomewhatarbitrary.
Therefore,JamesBernoulli(1713)calledfortheauthoritiestofixthelevelsofmoralimpossibility,justasruleshavebeenestablishedforcollectiveorindividualsafetymeasures,personallibertyandmoralvalues.
AsNicholasBernoulliwasinterestedinapplyingstatisticstomorals,itisconceivablethatthePetersburggambleoccupiedhismindfortheexactpurposeoffixingthelevel.
AsthediscussiononthePetersburggambleremindsus,itisnotprudenttobetlargeamountsonextremeoutliersinaone-shotgame,butintherepeatedgambletheseextremeoutliersmayverywellmaterialize.
Inthisrespectthediscussioninthepaperreiteratedthattheone-shotgambleisdifferentfromtherepeatedgamble.
FollowingLaplace(1820),itisperfectlyreasonablefortheindividualtoinsureagainstapersonal(unlikely)misfortune,andforthe32insurancecompanytoinsureindividualsagainstsuchamisfortune.
Sincetheinsurancecompanyfrequentlytakesagambleonindividuals'misfortunes,theexpectedvalueismorerelevanttotheinsurancecompanythantotheindividualwhogamblesonlyonce.
Thesmallertheprobabilityofsuchmisfortunes,however,thelessfrequentlydoindividualsinsureagainstthem,astheseeventsseemtoounlikelytooccur.
62Wehavereachedthepointwhereweapplybothourriskorlossaversionandthecancelingofsmallprobabilities,andeventuallywefaceatrade-offbetweenthesedecisionrules(Bernoulli1732;Buffon1777;Menger1932;andKahnemanandTversky1979).
Comingbacktotheexperimentalresults,itisobservedthatmostsubjects'elicitedwillingness-to-payfallsshortoftheexpectedpayoffofthegambles.
63Thisobservationisinlinewithariskaversionoralossaversionargument(Camerer2007;seealsoSchmidtandTraub2002)towhicheconomistsgenerallywouldsubscribeandhasbeenexperimentallysupportedforthefinitePetersburggamblebyCoxandcolleagues(2007,2008,2009).
Indeed,asBuffonpointedout,theremustbeadifferencebetweenthepleasureofwinningandthepainoflosing,sothatthelevelsofmoralimpossibilityforlossesandgainswillvary.
Peopleweighsmallprobabilitiesofmostoptimisticoutcomesdifferentlyfromsmallprobabilitiesofmostpessimisticoutcomes.
AsSamuelson(1977,42f)putsit:"ItisreasonableformetoignorethesmallprobabilitythatIshallfindadollaronmywaytowork.
64Butarationalmanwouldnotwanttoignorethesmallprobabilityofagreatdisaster.
"Whilelowprobabilityhighimpacteventshaveoccurredinthepastandarelikelytooccurinthefuture,itmayberationaltoinsureagainstthemandprudenttowagernosignificantamountonthem.
62Thepsychologicalliteraturehassuggestedthattheperceptionofsmallprobabilityeventsmaydependontheexperienceoftheiroccurrence.
Recentlyexperiencedeventsinfluencedecisionmaking,astylizedfactwhichhasbeencalledtherecencyeffect(RobinHogartandHillelEinhorn1992).
Inrecentexperimentalworkithasbeenshownthattherecencyeffectplaysaroleinsubjects'decisionstobetonsmallprobabilityeventswithrepeatedgambles(RalphHertwigandcollaborators2004).
Experimentalsubjectswhodonotwitnessrareeventsdonotbetonthemandviceversa.
63Theexperimentalliteratureshowsthatthewillingness-to-payforalotteryisgenerallysmallerthanthewillingness-to-accept(forasurveyseeUlrichSchmidtandStefanTraub2009).
Giventhisdisparityitislikelythatasksarehigherundertheconditionsofawillingness-to-acceptprocedure.
AnticipatingthatasaleofthePetersburggamblemayleadtoconsiderablelossesfortheseller,itisalsoperceivablethatthenegligenceofsmallprobabilitiesisnotableonlyformuchsmallerprobabilitiesthan1/32.
ThisissuecouldberelevantintheexplanationoftheexperimentalresultsbyKrollandVogt(2009).
64Presumably,thesamestorycanbetoldaboutlargermoneyamounts,too.
33AppendixTableA1.
Buffon'sresultsNumberof"tails"Buffon'sobservationsPayoffBuffon'sexpectationheuristica)010611210=1024+11494229=5122232428=2563137827=1284561626=645293225=326256424=167812823=88625622=49-51221=210-102420=1a)Buffon'sexpectedoutcomesmatchwiththegeometricaldistribution.
34TableA2.
OverviewofexperimentalstudiesSessionLocationAnnouncedmaximumpayoffSubjectsNumberofparticipantsMedianbidSession1LU.
Hannover10Firstyear460.
10LU.
Hannover100Firstyear190.
99LU.
Hannover1000Firstyear220.
99LU.
Hannover∞Firstyear280.
50Session2LU.
Hannover1,000Fourthyear302.
01Session3LU.
Hannover10,000Fourthyear472.
00Session4LUISS∞FirstyearMA111.
00Session5LUISS100MBA96.
00Session6LU.
Hannover250Thirdyear221.
85LU.
Hannover∞Thirdyear281.
50FieldstudyLUHannover10501.
00LUHannover16502.
00LUHannover32513.
00LUHannover50503.
00LUHannover100853.
00LUHannover1000753.
00Session7UGranada16Thirdyear401.
10Session8UGranada32Thirdyear441.
58Session9UGranada64Thirdyear591.
50Session10UGranada128Thirdyear281.
94Session11UGranada1024Thirdyear622.
0035TableA3.
ClassroomexperimentHannover:One-tailedMann-Whitneytestresultsforsubjects'bidsmaximumpayoff1,00010,000unlimited250.
130.
433.
3291,000.
859.
90410,000.
375TableA4.
Fieldstudy:One-tailedMann-Whitneytestresultsforsubjects'revisedwillingness-to-paymaximumpayoff1632501001,00010.
000.
000.
000.
000.
00016.
018.
049.
041.
02732.
756.
570.
37050.
388.
165100.
244TableA5.
ClassroomexperimentGranada:Mann-Whitneytestresultsforsubjects'bidsmaximumpayoff32641281,02416.
041.
016.
095.
03232.
438.
503.
43264.
565.
467128.
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55,pp.
95–115,1987.
42INSTRUCTIONSYouareabouttoparticipateinaneconomicexperiment.
Duringtheexperiment,pleasedonottalktoanyone.
Pleasemakeyourdecisiononyourownandfollowtheinstructionscarefully.
Itisinyourownintereststhatyouunderstandtheinstructions.
GENERALINFORMATIONIntheexperiment,youareaskedtosubmitabidforagamblethathasamonetarypayoff.
Youwriteyourbidandyournameonyoursheetofpaperwhichwillbecollectedlater.
Theparticipantwhosubmitsthehighestbidwinsthegamble,andwillpaytousthesecondhighestbid;theotherparticipantswillnotwinanythingandwillnotpayanything.
Thewinnerplaysthegambleandearnsthemonetarypayoffofthegamble.
Themoneyamountwillbepaidoutimmediately.
Pleasenotethatthewinnerofthegamblepaysapriceforthegamblewhichmaybehigherthanthemoneythatheorshewinsinthegamble.
Thereforeweadviseyoutochooseyourbidcarefully.
Therewillbenorefund,ifyoumakealoss!
Youbidonthefollowinggamble.
THEGAMBLETherearetwoballsinabag.
Theballsareidenticalexceptforthelabel.
Oneballislabelledwith'+',theotherisunlabelled.
Youdrawoneballfromthebag.
Ifyoudrawtheballlabelled'+'itisreturnedintothebagandyoudrawagain.
Eachtimeyoudrawthe'+'labelledballitwillberecorded.
Thegamecontinuesuntilyoudrawtheunlabelledballforthefirsttime.
Thenthegamblestopsandyoureceiveyourpayoffaccordingtotherecordednumberof'+'drawsthatyouhavemade.
Letthisnumberbedenotedbyk.
Thenyourpayofffromthegambleis2kEuro.
Inwords:YourpayoffistwoEurotothepowerofthenumberoftimesyoudrawthe'+'labelledball.
Fork≤5,theresultingpayoffsarepresentedinthefollowingtable.
YoursequenceofdrawsYourpayoffin20=1+21=2++22=4+++23=8++++24=16+++++25=32….
Notice,ofcourse,thatkmaybemoreorlessthan5.
Indeed,kcouldbeverylarge.
Yourminimumpayoffinthegambleis1Euro.
Thereisnomaximumpayoffinthegamble.
Onlyoneparticipantwinsthegambleandplaysitout.
Thepricethisparticipanthastopayfortheparticipationinthegambleisdeterminedbythesecondhighestbidsubmittedintheauction.
Pleasewriteyourbidinthespaceattheendofthissheetofpaper.
Youwillprivatelysubmityourbidwithoutknowingthebidsoftheotherparticipants.
Yourbidshouldbeyourmaximumwillingness-to-pay(inEuro)forparticipationinthegamble;ifyourbidisthehighestbid,thepriceyoupaywillnotexceedyourbidbutitmightequalyourbid.
AllbidsshallbeinEuro,decimalsafterthecommaindicateEurocents.
Thesmallestunitonecanbidis0,01Euroand0,01Euroisalsotheminimumbidforthegamble.
Afterallbidshavebeencollected,thewinnerintheauctionwillbedetermined.
Thegamblewillbeassignedtotheparticipantwhosubmittedthehighestbid.
Incaseofadraw,thatis,ifseveralparticipantssubmitthehighestbid,thewinnerwillbedeterminedbyrandomlyselectingoneofthehighestbidders.
Aswehavealreadysaid,thepricethewinnerpaysforthegambleequalsthesecondhighestbid.
Unlessthereismorethanonehighbidder,thepricewillbesmallerthanthehighestbid.
Example:Assumeoneparticipantsubmitsabidof0,02Euroandthesecondparticipantbids0,03Eurowhileallotherparticipantssubmitabidequalto0,01Euro.
Thesecondbidderwinsthegambleandpaysthesecondhighestpriceof0,02Euro.
Ifthesecondparticipantsubmitsabidof0,02Euroinstead,her/hisbidisequaltothebidofthefirstparticipant.
Thewinner,eitherparticipant1or2,mustbedeterminedthrougharandomdraw.
Thepricethewinnerpays,i.
e.
,thesecondhighestbid,isthenequaltoher/hisownbidof0,02Euro.
Writeyournamehere:Writeyourbidhere:,EuroComment:FURTHERINSTRUCTIONSThewinnerofthegamblewillbedetermined,andthegamblewillbeplayedout.
Thewinnerwilldrawballsfromthebag(withreplacement)untiltheunlabeledballshowsup.
GENERALINFORMATIONOnthissheetyouareaskedtoreplytothreefurtherquestionsregardingtheoutcomesofthegamble.
Thefirstquestionwillaskyoutostatethreeoutcomes.
Thesecondandthethirdarequizquestions.
Thissheetofpaperrepresentsyourlotforatombola.
Ifyourlot(i.
e.
,thissheet)isdrawninthetombolaandyouransweriscorrectyouwin10Euro,otherwiseyouwinnothing.
Intotal,thetombolainvolves3draws(withreplacement)fromthelots(i.
e.
,sheets)ofallparticipants.
Inotherwords,aftereachdrawthedrawnlotisreturnedtotheotherlotstoparticipateinthenexttombola.
QUESTIONS431)Onelotwillbedrawn(withreplacement)afterthegamblehasbeenplayedout.
Thenumberoftimesthe'+'labelledballhasbeendrawninthegamblewillbecomparedwiththethreeoutcomesyoustateonthissheet(ifyoursheetisdrawn).
Ifthesequencelengththathasrealizedinthegamblecoincideswithoneofthethreenumbersyoustatebelowandyourlotisdrawninthetombola,youwin10Euro.
Pleasewritedownherethethreenon-negativenumbers:k1=k2=k3=2)Thesecondlotwillbedrawnnext.
Thenumberofdrawsofthe'+'labelledballthathasa(mathematical)probabilityof1/128willbecomparedtothenumberyoustatebelow.
Youwin10Euroonthisquestion,ifyourlotisdrawnandyourreplyiscorrect.
k=3)Finally,thethirdlotwillbedrawn.
Thenumberyoustatehereafterwillbecomparedtotheexpectedpayoffofthegamble.
Ifyouransweriscorrectandyourlotisdrawn,youwin10Euroonthisquestion.
Expectedpayoffinthegamble=Writeyournamehere: