Red Herrings: Some Thoughts on the Meaning of Zero-Probability Events - PDF document

Red Herrings: Some Thoughts on the Meaning of Zero-Probability Events and Mathematical Modeling Edi Karni* Kicking off the discussion following Savage's presentation at the 1952 Paris colloquium, Arrow raised what he considered to be a difficulty with the intuitive interpretation of Savage's theorem. It suggests that decision makersng on an event of measure of that event. Within the realm of the revealed-preference methodology and limited verifiability, Arrow's difficuthe problem he poses has it origin in technical aspects of the Savage'sbstantive aspect of it. Keywords: Zero-probability events, Null events, revealed preference * Department of Economics, Johns Hopkins University. E-mail address benefited from discussions and comments with Itzhak Gilboa, Joseph Greenberg, Dov Samet, Marco Scarsini, and David Schmeidler. None of themnecessarily shares the andthecornersofthedie.Theeventthatthedielandsononeofitsedgesisobservable,hence,veriable.Incontrasttothesecases,theexperimentdescribedbyArrowisnotim-plementable.Drawingapairofrealnumbersintheunitsquareaccordingtoanonatomicmeasureisamathematicalidealization.Anyreal-lifechancedevicemayonlyapproximate(e.g.,byanitedecimalexpansion)thesenum-bers.Yetconsiderationofbettingoneventssuchasrequiresthattheexactpairofrealnumbersbedetermined.Accordingtorevealed-preferencemethodology,achoicebetweenactsthatagreeoutsidesucheventsisstrictlyhypotheticaland,consequently,inadmissible.Putdierently,sincestatesrepresentresolutionsofuncertainty,itimpossibletoattainthecompleteres-olutionoftheuncertaintyasrequiredbytheformalmodel.Realityadmitsonlypartialresolutionofuncertainty,whichrenderssomemathematically-distinctbetsobservationallyequivalent.Thefollowingexamplemayhelpclarifytheissueofnonveriabilityofconceivableevents.Supposethedecisionmakerisaskedtochoosebetweenbettingontheeventthatthevelocityofaparticleinagivenpositiontakesoneofanitenumbersofvaluesandbettingonitbeingoneofthesevalues.Thisquestionisinadmissibleintheframeworkoftherevealedpreferencemethodology.3ConclusionSupposethatadecisionmakerfacedwiththechoiceofbettingonchooses,asArrowpredicted,tobeton,evenifthepayoisslightlysmaller.Furtheryet,supposethatheiswillingtopaysomethingfortherighttoparticipateinthebeton.Whatconclusionsmustbedrawnfromthesechoices?Theobviousconclusionisthatthedecisionmakerdoesnotunderstandtheproblem.Thisisaclear-cutcaseinwhichthemathematicalstructureofthe Inthecontextofgames,thisobservationsmeansthatifthesetofpurestrategieshasthepowerofthecontinuum,itisimpossibletoimplementamixedstrategyaccordingtowhichthepurestrategyisselectedrandomlyaccordingtoadensityfunction.AccordingtotheHeisenberguncertaintyprinciple,itisimpossibletomeasuresimulta-neouslythepositionandthevelocityofaparticlewithanydegreeofaccuracyorcertainty.Consequently,itisimpossibletodeterminethepayoofthebet. modelallowstheformulationofquestionsthathavenoempiricalsubstance.Posingsuchaquestionintherstplaceismisleading.Takingthedecisionmaker’sresponsetosuchquestionsseriouslyasatestofSavage’stheoryismisguided.Thechoicequestionshouldberephrasedasfollows.Howmuchareyoubewillingtopaytoparticipateinabetongiventhat,becauseitisimpossiblendoutifobtained,youcanneverwin?Iconjecturethattheanswerwillbe“nothing”,whichisconsistentwithtreatingasanullevent.[1]ColloquesInternationauxduCentreNationaldelaResearchSceintiXL,Econometrie,(1953).[2]Blume,L.,Brandenburger,A.,andE.Dekel(1991)”LexicographicProb-abilitiesandChoiceUnderUncertainty,”Econometrica.