Rational Decisions The Gorman Lectures in Economics SeriesEditor,RichardBlundell LawlessnessandEconomics:AlternativeModesofGovernance, AvinashK.Dixit RationalDecisions, KenBinmore Aseriesstatementappearsatthebackofthebook Rational Decisions Ken Binmore PrincetonUniversityPress PrincetonandOxford Copyright©2009byPrincetonUniversityPress PublishedbyPrincetonUniversityPress, 41WilliamStreet,Princeton,NewJersey08540 IntheUnitedKingdom:PrincetonUniversityPress, 6OxfordStreet,Woodstock,OxfordshireOX201TW AllRightsReserved ISBN:978-0-691-13074-3(alk.paper) LibraryofCongressControlNumber:2008938513 BritishLibraryCataloging-in-PublicationDataisavailable ThisbookhasbeencomposedinLucidaBright TypesetbyT&TProductionsLtd,London Printedonacid-freepaper ∞ press.princeton.edu PrintedintheUnitedStatesofAmerica 10 9 8 7 6 5 4 3 2 1 This book began with two lectures I gave in 2006 at University College London in memory of Terence Gorman who was a great economist and a good friend. He would have wished me to dedicate this book to the fondly remembered teacher at his grammar school in Northern Ireland who first inspired him with a love of mathematics. This page intentionally left blank Contents Preface ix 1 RevealedPreference 1 1.1 Rationality? 1 1.2 ModelingaDecisionProblem 2 1.3 ReasonIstheSlaveofthePassions 3 1.4 LessonsfromAesop 5 1.5 RevealedPreference 7 1.6 RationalityandEvolution 12 1.7 Utility 14 1.8 ChallengingTransitivity 17 1.9 CausalUtilityFallacy 19 1.10 PositiveandNormative 22 2 GameTheory 25 2.1 Introduction 25 2.2 WhatIsaGame? 25 2.3 ParadoxofRationality? 26 2.4 Newcomb’sProblem 30 2.5 ExtensiveFormofaGame 31 3 Risk 35 3.1 RiskandUncertainty 35 3.2 VonNeumannandMorgenstern 36 3.3 TheStPetersburgParadox 37 3.4 ExpectedUtilityTheory 39 3.5 ParadoxesfromAtoZ 43 3.6 UtilityScales 46 3.7 AttitudestoRisk 50 3.8 UnboundedUtility? 55 3.9 PositiveApplications? 58 4 Utilitarianism 60 4.1 RevealedPreferenceinSocialChoice 60 4.2 TraditionalApproachestoUtilitarianism 63 4.3 IntensityofPreference 66 4.4 InterpersonalComparisonofUtility 67 viii Contents 5 ClassicalProbability 75 5.1 Origins 75 5.2 MeasurableSets 75 5.3 Kolmogorov’sAxioms 79 5.4 ProbabilityontheNaturalNumbers 82 5.5 ConditionalProbability 83 5.6 UpperandLowerProbabilities 88 6 Frequency 94 6.1 InterpretingClassicalProbability 94 6.2 RandomizingDevices 96 6.3 RichardvonMises 100 6.4 RefiningvonMises’Theory 104 6.5 TotallyMuddlingBoxes 113 7 BayesianDecisionTheory 116 7.1 SubjectiveProbability 116 7.2 Savage’sTheory 117 7.3 DutchBooks 123 7.4 BayesianUpdating 126 7.5 ConstructingPriors 129 7.6 BayesianReasoninginGames 134 8 Epistemology 137 8.1 Knowledge 137 8.2 BayesianEpistemology 137 8.3 InformationSets 139 8.4 KnowledgeinaLargeWorld 145 8.5 RevealedKnowledge? 149 9 LargeWorlds 154 9.1 CompleteIgnorance 154 9.2 ExtendingBayesianDecisionTheory 163 9.3 MuddledStrategiesinGameTheory 169 9.4 Conclusion 174 10 MathematicalNotes 175 10.1 CompatiblePreferences 175 10.2 Hausdorff’sParadoxoftheSphere 177 10.3 ConditioningonZero-ProbabilityEvents 177 10.4 ApplyingtheHahn–BanachTheorem 179 10.5 MuddlingBoxes 180 10.6 SolvingaFunctionalEquation 181 10.7 Additivity 182 10.8 MuddledEquilibriainGameTheory 182 References 189 Index 197 Preface What is rationality? What is the solution to the problem of scien- tific induction? I don’t think it reasonable to expect sharp answers to such questions. One might as well ask for precise definitions of life or consciousness. But we can still try to push forward the frontier of rational decision theory beyond the Bayesian paradigm that represents thecurrentorthodoxy. Manypeopleseenoneedforsuchaneffort.TheythinkthatBayesian- ismalreadyprovidestheanswerstoallquestionsthatmightbeasked.I believethatBayesiansofthisstampfailtounderstandthattheirtheory applies only in what Jimmie Savage (1951) called a small world in his famous Foundations of Statistics. But the world of scientific inquiry is large—so much so that scientists of the future will look back with incredulity at a period in intellectual history when it was possible be takenseriouslywhenclaimingthatBayesianupdatingisthesolutionto theproblemofscientificinduction. JackGoodonceclaimedtoidentify46,656differentkindsofBayesians. My first priority is therefore to clarify what I think should be regarded as the orthodoxy on Bayesian decision theory—the set of foundational assumptions that offer the fewest hostages to fortune. This takes up mostofthebook,sinceItaketimeouttoreviewvariousaspectsofprob- abilitytheoryalongtheway.Myreasonforspendingsomuchtimeoffer- inganultra-orthodoxreviewofstandarddecisiontheoryisthatIfeelthe need to deny numerous misapprehensions (both positive and negative) aboutwhatthetheoryreallysays—orwhatIthinkitoughttosay—before getting on to my own attempt to extend a version of Bayesian decision theorytoworldslargerthanthoseconsideredbySavage(chapter9). Idon’tforonemomentimaginethatmyextensionofBayesiandecision theorycomesanywherenearsolvingtheproblemofscientificinduction, but I do think my approach will sometimes be found useful in applica- tions.Forexample,mytheoryallowsthemixedstrategiesofgametheory tobeextendedtowhatIcallmuddledstrategies(muchaspurestrategies wereextendedtomixedstrategiesbythecreatorsofthetheory). What is the audience for this book? I hope that it will be read not just by the economics community from which I come myself, but also by statisticians and philosophers. If it only succeeds in bridging some