Theory of Decision under Uncertainty

by Itzhak Gilboa

• ISBN13: 9780521741231
• ISBN10: 0521741238
• Edition: 1st
• Format: Paperback
• Copyright: 2009-03-16
• Publisher: Cambridge University Press

Summary

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

Preface	p. xiii
Intuitive Definitions	p. 1
Motivating Examples	p. 3
Free Will and Determinism	p. 5
Can Free Choice Be Predicted?	p. 5
Is the World Deterministic?	p. 6
Is Free Will Observable?	p. 6
The Problem of Free Will	p. 7
A Rational Illusion	p. 10
Free Will and the Decision Matrix	p. 12
The Principle of Indifference	p. 14
Will a Canonical Space Help?	p. 14
The Canonical State Space	p. 14
Difficulties with a Uniform Prior on [0, 1]	p. 15
Conclusion	p. 16
What's Special about the Coin?	p. 17
Symmetry	p. 17
Smooth Beliefs	p. 18
Relative Frequencies	p. 20
The Law of Large Numbers	p. 20
The Problem of Induction	p. 21
Hume's Critique	p. 21
Goodman's Grue-Bleen Paradox	p. 22
Kolmogorov Complexity and Its Dependence of Language	p. 23
Grue-Bleen Again	p. 26
Evolutionary Explanations	p. 31
Problems with the Frequentist Approach	p. 34
Subjective Probabilities	p. 37
Linda the Bank Teller	p. 37
Pascal's Wager	p. 38
Classical versus Bayesian Statistics	p. 40
Basic Definitions	p. 40
The Gambler Fallacy	p. 41
Exchangeability	p. 42
Confidence Is Not Probability	p. 44
Classical Statistics Can Be Ridiculous	p. 45
Different Methods for Different Goals	p. 46
Behavioral Definitions	p. 49
A Case Study	p. 51
A Characterization Theorem for Maximization of Utility	p. 51
Proof	p. 53
Interpretations	p. 55
A Few Definitions	p. 56
A Meta-Scientific Interpretation	p. 59
A Normative Interpretation	p. 62
A Descriptive Interpretation	p. 63
Limitations	p. 64
Semiorders	p. 65
Other Ways to Measure Utility	p. 71
The Role of Theories	p. 72
Theories Are Always Wrong	p. 72
Theories and Conceptual Frameworks	p. 74
Logical Positivism as a Metaphor	p. 76
Von Neumann-Morgenstern's Theorem	p. 78
Background	p. 78
The Theorem	p. 79
Proofs	p. 83
The Algebraic Approach	p. 83
A Geometric Approach	p. 84
A Separation Argument	p. 85
The Three Interpretations	p. 86
De Finetti's Theorem	p. 89
Motivation	p. 89
The Theorem	p. 90
A Proof	p. 91
The Three Interpretations	p. 92
Savage's Theorem	p. 94
Background	p. 94
States, Outcomes, and Acts	p. 96
Axioms	p. 97
P1	p. 97
P2	p. 97
Notation	p. 99
Null Events	p. 99
P3	p. 100
P4	p. 102
P5	p. 102
P6	p. 103
P7	p. 104
The Result for a Finite Outcome Set	p. 105
Finitely Additive Measures	p. 105
Nonatomic Measures	p. 107
The Theorem	p. 108
The Case of a General Outcome Set	p. 108
Interpretations	p. 109
The Proof and Qualitative Probabilities	p. 110
The Definition of States	p. 113
Causality	p. 113
Newcomb's Paradox	p. 113
States as Functions from Acts to Outcomes	p. 114
A Problem	p. 115
Hempel's Paradox of Confirmation	p. 116
Are All Ravens Black?	p. 116
A State-Space Formulation	p. 117
What Is a Confirmation?	p. 117
A Resolution	p. 118
Good's Variation	p. 118
What Do We Learn from This?	p. 119
Monty Hall Three-Door Game	p. 120
A Critique of Savage	p. 123
Criticizing Critiques	p. 123
An Example	p. 123
The General Lesson	p. 124
Critique of P3 and P4	p. 125
Example	p. 125
Defense	p. 126
State-Dependent Utility	p. 127
The Definition of Subjective Probability	p. 128
When Is State Dependence Necessary?	p. 129
Critique of P1 and P2	p. 130
The Basic Problem	p. 130
Reasoned Choice versus Raw Preferences	p. 131
Schmeidler's Critique and Ellsberg's Paradox	p. 132
Observability of States	p. 136
Problems of Complexity	p. 137
Objectivity and Rationality	p. 138
Subjectivity and Objectivity	p. 138
Objective and Subjective Rationality	p. 139
Anscombe-Aumann's Theorem	p. 142
Alternative Behavioral Theories	p. 145
Choquet Expected Utility	p. 147
Schmeidler's Intuition	p. 147
Choquet Integral	p. 149
Comonotonicity	p. 150
Axioms and Result	p. 151
Prospect Theory	p. 154
Background	p. 154
Gain-Loss Asymmetry	p. 154
Distortion of Probabilities	p. 156
Rank-Dependent Probabilities and Choquet Integration	p. 158
Maxmin Expected Utility	p. 160
Convex Games	p. 160
A Cognitive Interpretation of CEU	p. 161
Axioms and Result	p. 163
Interpretation of MMEU	p. 163
Generalizations and Variations	p. 165
Bewley's Alternative Approach	p. 165
Combining Objective and Subjective Rationality	p. 166
Applications	p. 168
Cognitive Origins	p. 171
Case-Based Qualitative Beliefs	p. 173
Axioms and Result	p. 173
Four Known Techniques	p. 175
The Combination Axiom in General Analogical Reasoning	p. 177
Violations of the Combination Axiom	p. 179
Frequentism Revisited	p. 180
Similarity-Weighted Empirical Frequencies	p. 180
Intuition	p. 181
Axiomatization	p. 182
Empirical Similarity and Objective Probabilities	p. 184
Future Research	p. 188
References	p. 191
Index	p. 199
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