Theory of Decision under Uncertainty Thisbookdescribestheclassicalaxiomatictheoriesofdecisionunderuncertainty, aswellascritiquesthereofandalternativetheories.Itfocusesonthemeaningof probability,discussingsomedefinitionsandsurveyingtheirscopeofapplicability. Thebehavioraldefinitionofsubjectiveprobabilityservesasawaytopresentthe classicaltheories,culminatinginSavage’stheorem.Thelimitationsofthisresultas adefinitionofprobabilityleadtotwodirections–first,similarbehavioraldefinitions ofmoregeneraltheories,suchasnonadditiveprobabilitiesandmultiplepriors,and second,cognitivederivationsbasedoncase-basedtechniques. Itzhak Gilboa is a professor in the Berglas School of Economics, Tel-Aviv Uni- versity,Israel,andaprofessorinthenewlyestablishedDepartmentofEconomics andDecisionScience,HEC,Paris.Earlier,hebecameachairedprofessoratNorth- westernUniversityin1992,visitedattheDepartmentofEconomics,Universityof Pennsylvania(1995–1997),andwasaprofessoratBostonUniversity(1997–1999). ProfessorGilboaalsoservedasaFellowattheCowlesFoundationatYaleUniver- sityfrom2001to2007.TherecipientofaSloanFellowship,amongotherawards, he has published articles in the leading economic theory journals, primarily on decisionunderuncertainty.ProfessorGilboacoauthoredATheoryofCase-Based DecisionswithDavidSchmeidler(CambridgeUniversityPress,2001).Hereceived hisPh.D.fromTel-AvivUniversityin1987. EconometricSocietyMonographs Editors: AndrewChesher,UniversityCollegeLondon GeorgeJ.Mailath,UniversityofPennsylvania TheEconometricSocietyisaninternationalsocietyfortheadvancementofeconomictheoryin relationtostatisticsandmathematics.TheEconometricSocietyMonographSeriesisdesigned to promote the publication of original research contributions of high quality in mathematical economicsandtheoreticalandappliedeconometrics. Othertitlesintheseries: G.S.MaddalaLimiteddependentandqualitativevariablesineconometrics,0521338255 GerardDebreuMathematicaleconomics:TwentypapersofGerardDebreu,0521335612 Jean-MichelGrandmontMoneyandvalue:Areconsiderationofclassicalandneoclassical monetaryeconomics,0521313643 FranklinM.FisherDisequilibriumfoundationsofequilibriumeconomics,0521378567 AndreuMas-ColellThetheoryofgeneralequilibrium:Adifferentiableapproach, 0521265142,0521388708 TrumanF.Bewley,Editor,Advancesineconometrics–FifthWorldCongress(VolumeI), 0521467268 TrumanF.Bewley,Editor,Advancesineconometrics–FifthWorldCongress(VolumeII), 052146725X HerveMoulinAxiomsofcooperativedecisionmaking,0521360552,0521424585 L.G.GodfreyMisspecificationtestsineconometrics:TheLagrangemultiplierprincipleand otherapproaches,0521424593 TonyLancasterTheeconometricanalysisoftransitiondata,052143789X AlvinE.RothandMarildaA.OlivieraSotomayor,Editors,Two-sidedmatching:Astudyin game-theoreticmodelingandanalysis,0521437881 WolfgangHa¨rdleAppliednonparametricregression,0521429501 Jean-JacquesLaffont,Editor,Advancesineconomictheory–SixthWorldCongress(VolumeI), 052156610X Jean-JacquesLaffont,Editor,Advancesineconomictheory–SixthWorldCongress(VolumeII), 052148460X HalbertWhiteEstimation,inferenceandspecification,0521252806,0521574463 ChristopherSims,Editor,Advancesineconometrics–SixthWorldCongress(VolumeI), 052156610X ChristopherSims,Editor,Advancesineconometrics–SixthWorldCongress(VolumeII), 0521566096 RogerGuesnerieAcontributiontothepuretheoryoftaxation,0521236894,052162956X DavidM.KrepsandKennethF.Wallis,Editors,Advancesineconomicsandeconometrics– SeventhWorldCongress(VolumeI),0521580110,0521589835 DavidM.KrepsandKennethF.Wallis,Editors,Advancesineconomicsandeconometrics– SeventhWorldCongress(VolumeII),0521580129,0521589829 DavidM.KrepsandKennethF.Wallis,Editors,Advancesineconomicsandeconometrics– SeventhWorldCongress(VolumeIII),0521580137,0521589819 DonaldP.Jacobs,EhudKalai,andMortonI.Kamien,Editors,Frontiersofresearchineconomic theory:TheNancyL.SchwartzMemorialLectures,1983–1997,0521632226,0521635381 A.ColinCameronandPravinK.TrivediRegressionanalysisofcountdata,0521632013, 0521635675 SteinarStrom,Editor,Econometricsandeconomictheoryinthe20thcentury:TheRagnarFrisch CentennialSymposium,0521633230,0521633656 EricGhysels,NormanR.Swanson,andMarkWatson,Editors,Essaysineconometrics:Collected papersofCliveW.J.Granger(VolumeI),0521772974,0521804078,0521774969, 0521796970 Continuedonpagefollowingtheindex Tomyfamily (Davidincluded) Theory of Decision under Uncertainty Itzhak Gilboa TelAvivUniversity,Israel,andHEC,Paris CAMBRIDGEUNIVERSITYPRESS Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,Sa˜oPaulo,Delhi CambridgeUniversityPress 32AvenueoftheAmericas,NewYork,NY10013-2473,USA www.cambridge.org Informationonthistitle:www.cambridge.org/9780521517324 ©ItzhakGilboa2009 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2009 PrintedintheUnitedStatesofAmerica AcatalogrecordforthispublicationisavailablefromtheBritishLibrary LibraryofCongressCataloginginPublicationdata Gilboa,Itzhak. Theoryofdecisionunderuncertainty/ItzhakGilboa. p. cm. Includesindex. ISBN978-0-521-51732-4(hardback) 1.Decisionmaking. 2.Uncertainty(Informationtheory) I.Title. QA279.4.G556 2009 003(cid:1).54–dc22 2008042408 ISBN 978-0-521-51732-4hardback ISBN 978-0-521-74123-1paperback CambridgeUniversityPresshasnoresponsibilityforthepersistenceor accuracyofURLsforexternalorthird-partyInternetWebsitesreferredtoin thispublicationanddoesnotguaranteethatanycontentonsuchWebsitesis, orwillremain,accurateorappropriate.Informationregardingprices,travel timetables,andotherfactualinformationgiveninthisworkarecorrectat thetimeoffirstprinting,butCambridgeUniversityPressdoesnotguarantee theaccuracyofsuchinformationthereafter. Contents Preface pagexiii I IntuitiveDefinitions 1 1 MotivatingExamples 3 2 FreeWillandDeterminism 5 2.1 CanFreeChoiceBePredicted? 5 2.2 IstheWorldDeterministic? 6 2.3 IsFreeWillObservable? 6 2.4 TheProblemofFreeWill 7 2.5 ARationalIllusion 10 2.6 FreeWillandtheDecisionMatrix 12 3 ThePrincipleofIndifference 14 3.1 WillaCanonicalSpaceHelp? 14 3.1.1 TheCanonicalStateSpace 14 3.1.2 DifficultieswithaUniformPrioron[0,1] 15 3.1.3 Conclusion 16 3.2 What’sSpecialabouttheCoin? 17 3.2.1 Symmetry 17 3.2.2 SmoothBeliefs 18 4 RelativeFrequencies 20 4.1 TheLawofLargeNumbers 20 4.2 TheProblemofInduction 21 4.2.1 Hume’sCritique 21 4.2.2 Goodman’sGrue-BleenParadox 22 4.2.3 KolmogorovComplexityandItsDependence ofLanguage 23 4.2.4 Grue-BleenAgain 26 4.2.5 EvolutionaryExplanations 31 4.3 ProblemswiththeFrequentistApproach 34 vii viii Contents 5 SubjectiveProbabilities 37 5.1 LindatheBankTeller 37 5.2 Pascal’sWager 38 5.3 ClassicalversusBayesianStatistics 40 5.3.1 BasicDefinitions 40 5.3.2 TheGamblerFallacy 41 5.3.3 Exchangeability 42 5.3.4 ConfidenceIsNotProbability 44 5.3.5 ClassicalStatisticsCanBeRidiculous 45 5.3.6 DifferentMethodsforDifferentGoals 46 II BehavioralDefinitions 49 6 ACaseStudy 51 6.1 ACharacterizationTheoremforMaximization ofUtility 51 6.2 Proof 53 6.3 Interpretations 55 6.3.1 AFewDefinitions 56 6.3.2 AMeta-ScientificInterpretation 59 6.3.3 ANormativeInterpretation 62 6.3.4 ADescriptiveInterpretation 63 6.4 Limitations 64 6.4.1 Semiorders 65 6.4.2 OtherWaystoMeasureUtility 71 7 TheRoleofTheories 72 7.1 TheoriesAreAlwaysWrong 72 7.2 TheoriesandConceptualFrameworks 74 7.3 LogicalPositivismasaMetaphor 76 8 VonNeumann–Morgenstern’sTheorem 78 8.1 Background 78 8.2 TheTheorem 79 8.3 Proofs 83 8.3.1 TheAlgebraicApproach 83 8.3.2 AGeometricApproach 84 8.3.3 ASeparationArgument 85 8.4 TheThreeInterpretations 86 9 DeFinetti’sTheorem 89 9.1 Motivation 89 9.2 TheTheorem 90 9.3 AProof 91 9.4 TheThreeInterpretations 92