Rational decisions

214 Pages·2011·1.123 MB·English

by  Binmore, K. G

Rational Decisions
The Gorman Lectures in Economics
SeriesEditor,RichardBlundell
LawlessnessandEconomics:AlternativeModesofGovernance,
AvinashK.Dixit
RationalDecisions,
KenBinmore
Aseriesstatementappearsatthebackofthebook
Rational Decisions
Ken Binmore
PrincetonUniversityPress
PrincetonandOxford
Copyright©2009byPrincetonUniversityPress
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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.
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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