Theory of decision under uncertainty / Itzhak Gilboa.

Format:

Book

Author/Creator:

Gilboa, Itzhak.

Contributor:

Cambridge University Press.

Rosengarten Family Fund.

Series:

Econometric Society monographs ; no. 45.

Econometric Society monographs ; 45

Language:

English

Subjects (All):

Decision making.

Uncertainty (Information theory).

Physical Description:

1 online resource (xiv, 214 pages).

Place of Publication:

Cambridge ; New York : Cambridge University Press, 2009.

System Details:

text file

Contents:

I Intuitive Definitions 1

1 Motivating Examples 3

2 Free Will and Determinism 5

2.1 Can Free Choice Be Predicted? 5

2.2 Is the World Deterministic? 6

2.3 Is Free Will Observable? 6

2.4 The Problem of Free Will 7

2.5 A Rational Illusion 10

2.6 Free Will and the Decision Matrix 12

3 The Principle of Indifference 14

3.1 Will a Canonical Space Help? 14

3.1.1 The Canonical State Space 14

3.1.2 Difficulties with a Uniform Prior on [0, 1] 15

3.1.3 Conclusion 16

3.2 What's Special about the Coin? 17

3.2.1 Symmetry 17

3.2.2 Smooth Beliefs 18

4 Relative Frequencies 20

4.1 The Law of Large Numbers 20

4.2 The Problem of Induction 21

4.2.1 Hume's Critique 21

4.2.2 Goodman's Grue-Bleen Paradox 22

4.2.3 Kolmogorov Complexity and Its Dependence of Language 23

4.2.4 Grue-Bleen Again 26

4.2.5 Evolutionary Explanations 31

4.3 Problems with the Frequentist Approach 34

5 Subjective Probabilities 37

5.1 Linda the Bank Teller 37

5.2 Pascal's Wager 38

5.3 Classical versus Bayesian Statistics 40

5.3.1 Basic Definitions 40

5.3.2 The Gambler Fallacy 41

5.3.3 Exchangeability 42

5.3.4 Confidence Is Not Probability 44

5.3.5 Classical Statistics Can Be Ridiculous 45

5.3.6 Different Methods for Different Goals 46

II Behavioral Definitions 49

6 A Case Study 51

6.1 A Characterization Theorem for Maximization of Utility 51

6.2 Proof 53

6.3 Interpretations 55

6.3.1 A Few Definitions 56

6.3.2 A Meta-Scientific Interpretation 59

6.3.3 A Normative Interpretation 62

6.3.4 A Descriptive Interpretation 63

6.4 Limitations 64

6.4.1 Semiorders 65

6.4.2 Other Ways to Measure Utility 71

7 The Role of Theories 72

7.1 Theories Are Always Wrong 72

7.2 Theories and Conceptual Frameworks 74

7.3 Logical Positivism as a Metaphor 76

8 Von Neumann-Morgenstern's Theorem 78

8.1 Background 78

8.2 The Theorem 79

8.3 Proofs 83

8.3.1 The Algebraic Approach 83

8.3.2 A Geometric Approach 84

8.3.3 A Separation Argument 85

8.4 The Three Interpretations 86

9 De Finetti's Theorem 89

9.1 Motivation 89

9.2 The Theorem 90

9.3 A Proof 91

9.4 The Three Interpretations 92

10 Savage's Theorem 94

10.1 Background 94

10.2 States, Outcomes, and Acts 96

10.3 Axioms 97

10.3.1 P1 97

10.3.2 P2 97

10.3.3 Notation 99

10.3.4 Null Events 99

10.3.5 P3 100

10.3.6 P4 102

10.3.7 P5 102

10.3.8 P6 103

10.3.9 P7 104

10.4 The Result for a Finite Outcome Set 105

10.4.1 Finitely Additive Measures 105

10.4.2 Nonatomic Measures 107

10.4.3 The Theorem 108

10.5 The Case of a General Outcome Set 108

10.6 Interpretations 109

10.7 The Proof and Qualitative Probabilities 110

11 The Definition of States 113

11.1 Causality 113

11.1.1 Newcomb's Paradox 113

11.1.2 States as Functions from Acts to Outcomes 114

11.1.3 A Problem 115

11.2 Hempel's Paradox of Confirmation 116

11.2.1 Are All Ravens Black? 116

11.2.2 A State-Space Formulation 117

11.2.3 What Is a Confirmation? 117

11.2.4 A Resolution 118

11.2.5 Good's Variation 118

11.2.6 What Do We Learn from This? 119

11.3 Monty Hall Three-Door Game 120

12 A Critique of Savage 123

12.1 Criticizing Critiques 123

12.1.1 An Example 123

12.1.2 The General Lesson 124

12.2 Critique of P3 and P4 125

12.2.1 Example 125

12.2.2 Defense 126

12.2.3 State-Dependent Utility 127

12.2.4 The Definition of Subjective Probability 128

12.2.5 When Is State Dependence Necessary? 129

12.3 Critique of P1 and P2 130

12.3.1 The Basic Problem 130

12.3.2 Reasoned Choice versus Raw Preferences 131

12.3.3 Schmeidler's Critique and Ellsberg's Paradox 132

12.3.4 Observability of States 136

12.3.5 Problems of Complexity 137

13 Objectivity and Rationality 138

13.1 Subjectivity and Objectivity 138

13.2 Objective and Subjective Rationality 139

14 Anscombe-Aumann's Theorem 142

III Alternative Behavioral Theories 145

15 Choquet Expected Utility 147

15.1 Schmeidler's Intuition 147

15.2 Choquet Integral 149

15.3 Comonotonicity 150

15.4 Axioms and Result 151

16 Prospect Theory 154

16.1 Background 154

16.2 Gain-Loss Asymmetry 154

16.3 Distortion of Probabilities 156

16.4 Rank-Dependent Probabilities and Choquet Integration 158

17 Maxmin Expected Utility 160

17.1 Convex Games 160

17.2 A Cognitive Interpretation of CEU 161

17.3 Axioms and Result 163

17.4 Interpretation of MMEU 163

17.5 Generalizations and Variations 165

17.6 Bewley's Alternative Approach 165

17.7 Combining Objective and Subjective Rationality 166

17.8 Applications 168

IV Cognitive Origins 171

18 Case-Based Qualitative Beliefs 173

18.1 Axioms and Result 173

18.2 Four Known Techniques 175

18.3 The Combination Axiom in General Analogical Reasoning 177

18.4 Violations of the Combination Axiom 179

19 Frequentism Revisited 180

19.1 Similarity-Weighted Empirical Frequencies 180

19.2 Intuition 181

19.3 Axiomatization 182

19.4 Empirical Similarity and Objective Probabilities 184

20 Future Research 188.

Notes:

Includes bibliographical references (pages [191]-198) and index.

Electronic reproduction. Cambridge Available via World Wide Web.

Description based on print version record.

Local Notes:

Acquired for the Penn Libraries with assistance from the Rosengarten Family Fund.

ISBN:

9780511840203

0511840209

Publisher Number:

99966742045

Access Restriction:

Restricted for use by site license.