Theory of Decision under Uncertainty

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  • Edition: 1st
  • Format: Paperback
  • Copyright: 2009-03-16
  • Publisher: Cambridge University Press
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This book describes the classical axiomatic theories of decision under uncertainty, as well as critiques thereof and alternative theories. It focuses on the meaning of probability, discussing some definitions and surveying their scope of applicability. The behavioral definition of subjective probability serves as a way to present the classical theories, culminating in Savage's theorem. The limitations of this result as a definition of probability lead to two directions first, similar behavioral definitions of more general theories, such as non-additive probabilities and multiple priors, and second, cognitive derivations based on case-based techniques.

Author Biography

"This is a fantastic book. It presents an intelligent, rigorous, and thought-provoking treatment of the theory of choice under uncertaintly. The combination of philosophical and mathematical approaches is a treat. Graduate students and professional economists alike have much to learn from this book." -Daron Acemoglu, Massachusetts Institute of Technology

Table of Contents

Prefacep. xiii
Intuitive Definitionsp. 1
Motivating Examplesp. 3
Free Will and Determinismp. 5
Can Free Choice Be Predicted?p. 5
Is the World Deterministic?p. 6
Is Free Will Observable?p. 6
The Problem of Free Willp. 7
A Rational Illusionp. 10
Free Will and the Decision Matrixp. 12
The Principle of Indifferencep. 14
Will a Canonical Space Help?p. 14
The Canonical State Spacep. 14
Difficulties with a Uniform Prior on [0, 1]p. 15
Conclusionp. 16
What's Special about the Coin?p. 17
Symmetryp. 17
Smooth Beliefsp. 18
Relative Frequenciesp. 20
The Law of Large Numbersp. 20
The Problem of Inductionp. 21
Hume's Critiquep. 21
Goodman's Grue-Bleen Paradoxp. 22
Kolmogorov Complexity and Its Dependence of Languagep. 23
Grue-Bleen Againp. 26
Evolutionary Explanationsp. 31
Problems with the Frequentist Approachp. 34
Subjective Probabilitiesp. 37
Linda the Bank Tellerp. 37
Pascal's Wagerp. 38
Classical versus Bayesian Statisticsp. 40
Basic Definitionsp. 40
The Gambler Fallacyp. 41
Exchangeabilityp. 42
Confidence Is Not Probabilityp. 44
Classical Statistics Can Be Ridiculousp. 45
Different Methods for Different Goalsp. 46
Behavioral Definitionsp. 49
A Case Studyp. 51
A Characterization Theorem for Maximization of Utilityp. 51
Proofp. 53
Interpretationsp. 55
A Few Definitionsp. 56
A Meta-Scientific Interpretationp. 59
A Normative Interpretationp. 62
A Descriptive Interpretationp. 63
Limitationsp. 64
Semiordersp. 65
Other Ways to Measure Utilityp. 71
The Role of Theoriesp. 72
Theories Are Always Wrongp. 72
Theories and Conceptual Frameworksp. 74
Logical Positivism as a Metaphorp. 76
Von Neumann-Morgenstern's Theoremp. 78
Backgroundp. 78
The Theoremp. 79
Proofsp. 83
The Algebraic Approachp. 83
A Geometric Approachp. 84
A Separation Argumentp. 85
The Three Interpretationsp. 86
De Finetti's Theoremp. 89
Motivationp. 89
The Theoremp. 90
A Proofp. 91
The Three Interpretationsp. 92
Savage's Theoremp. 94
Backgroundp. 94
States, Outcomes, and Actsp. 96
Axiomsp. 97
P1p. 97
P2p. 97
Notationp. 99
Null Eventsp. 99
P3p. 100
P4p. 102
P5p. 102
P6p. 103
P7p. 104
The Result for a Finite Outcome Setp. 105
Finitely Additive Measuresp. 105
Nonatomic Measuresp. 107
The Theoremp. 108
The Case of a General Outcome Setp. 108
Interpretationsp. 109
The Proof and Qualitative Probabilitiesp. 110
The Definition of Statesp. 113
Causalityp. 113
Newcomb's Paradoxp. 113
States as Functions from Acts to Outcomesp. 114
A Problemp. 115
Hempel's Paradox of Confirmationp. 116
Are All Ravens Black?p. 116
A State-Space Formulationp. 117
What Is a Confirmation?p. 117
A Resolutionp. 118
Good's Variationp. 118
What Do We Learn from This?p. 119
Monty Hall Three-Door Gamep. 120
A Critique of Savagep. 123
Criticizing Critiquesp. 123
An Examplep. 123
The General Lessonp. 124
Critique of P3 and P4p. 125
Examplep. 125
Defensep. 126
State-Dependent Utilityp. 127
The Definition of Subjective Probabilityp. 128
When Is State Dependence Necessary?p. 129
Critique of P1 and P2p. 130
The Basic Problemp. 130
Reasoned Choice versus Raw Preferencesp. 131
Schmeidler's Critique and Ellsberg's Paradoxp. 132
Observability of Statesp. 136
Problems of Complexityp. 137
Objectivity and Rationalityp. 138
Subjectivity and Objectivityp. 138
Objective and Subjective Rationalityp. 139
Anscombe-Aumann's Theoremp. 142
Alternative Behavioral Theoriesp. 145
Choquet Expected Utilityp. 147
Schmeidler's Intuitionp. 147
Choquet Integralp. 149
Comonotonicityp. 150
Axioms and Resultp. 151
Prospect Theoryp. 154
Backgroundp. 154
Gain-Loss Asymmetryp. 154
Distortion of Probabilitiesp. 156
Rank-Dependent Probabilities and Choquet Integrationp. 158
Maxmin Expected Utilityp. 160
Convex Gamesp. 160
A Cognitive Interpretation of CEUp. 161
Axioms and Resultp. 163
Interpretation of MMEUp. 163
Generalizations and Variationsp. 165
Bewley's Alternative Approachp. 165
Combining Objective and Subjective Rationalityp. 166
Applicationsp. 168
Cognitive Originsp. 171
Case-Based Qualitative Beliefsp. 173
Axioms and Resultp. 173
Four Known Techniquesp. 175
The Combination Axiom in General Analogical Reasoningp. 177
Violations of the Combination Axiomp. 179
Frequentism Revisitedp. 180
Similarity-Weighted Empirical Frequenciesp. 180
Intuitionp. 181
Axiomatizationp. 182
Empirical Similarity and Objective Probabilitiesp. 184
Future Researchp. 188
Referencesp. 191
Indexp. 199
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