It is widely held that Bayesian decision theory is the final word on how a rational person should make decisions. However, Leonard Savage--the inventor of Bayesian decision theory--argued that it would be ridiculous to use his theory outside the kind of small world in which it is always possible to "look before you leap." If taken seriously, this view makes Bayesian decision theory inappropriate for the large worlds of scientific discovery and macroeconomic enterprise. When is it correct to use Bayesian decision theory--and when does it need to be modified? Using a minimum of mathematics, Rational Decisions clearly explains the foundations of Bayesian decision theory and shows why Savage restricted the theory's application to small worlds. The book is a wide-ranging exploration of standard theories of choice and belief under risk and uncertainty. Ken Binmore discusses the various philosophical attitudes related to the nature of probability and offers resolutions to paradoxes believed to hinder further progress. In arguing that the Bayesian approach to knowledge is inadequate in a large world, Binmore proposes an extension to Bayesian decision theory--allowing the idea of a mixed strategy in game theory to be expanded to a larger set of what Binmore refers to as "muddled" strategies. Written by one of the world's leading game theorists, Rational Decisions is the touchstone for anyone needing a concise, accessible, and expert view on Bayesian decision making.

Preface ix

Chapter 1: Revealed Preference 1
1.1 Rationality? 1
1.2 Modeling a Decision Problem 2
1.3 Reason Is the Slave of the Passions 3
1.4 Lessons from Aesop 5
1.5 Revealed Preference 7
1.6 Rationality and Evolution 12
1.7 Utility 14
1.8 Challenging Transitivity 17
1.9 Causal Utility Fallacy 19
1.10 Positive and Normative 22


Chapter 2: Game Theory 25
2.1 Introduction 25
2.2 What Is a Game? 25
2.3 Paradox of Rationality? 26
2.4 Newcomb's Problem 30
2.5 Extensive Form of a Game 31


Chapter 3: Risk 35
3.1 Risk and Uncertainty 35
3.2 Von Neumann and Morgenstern 36
3.3 The St Petersburg Paradox 37
3.4 Expected Utility Theory 39
3.5 Paradoxes from A to Z 43
3.6 Utility Scales 46
3.7 Attitudes to Risk 50
3.8 Unbounded Utility? 55
3.9 Positive Applications? 58


Chapter 4: Utilitarianism 60
4.1 Revealed Preference in Social Choice 60
4.2 Traditional Approaches to Utilitarianism 63
4.3 Intensity of Preference 66
4.4 Interpersonal Comparison of Utility 67


Chapter 5: Classical Probability 75
5.1 Origins 75
5.2 Measurable Sets 75
5.3 Kolmogorov's Axioms 79
5.4 Probability on the Natural Numbers 82
5.5 Conditional Probability 83
5.6 Upper and Lower Probabilities 88


Chapter 6: Frequency 94
6.1 Interpreting Classical Probability 94
6.2 Randomizing Devices 96
6.3 Richard von Mises 100
6.4 Refining von Mises' Theory 104
6.5 Totally Muddling Boxes 113


Chapter 7: Bayesian Decision Theory 116
7.1 Subjective Probability 116
7.2 Savage's Theory 117
7.3 Dutch Books 123
7.4 Bayesian Updating 126
7.5 Constructing Priors 129
7.6 Bayesian Reasoning in Games 134


Chapter 8: Epistemology 137
8.1 Knowledge 137
8.2 Bayesian Epistemology 137
8.3 Information Sets 139
8.4 Knowledge in a Large World 145
8.5 Revealed Knowledge? 149


Chapter 9: Large Worlds 154
9.1 Complete Ignorance 154
9.2 Extending Bayesian Decision Theory 163
9.3 Muddled Strategies in Game Theory 169
9.4 Conclusion 174


Chapter 10: Mathematical Notes 175
10.1 Compatible Preferences 175
10.2 Hausdorff's Paradox of the Sphere 177
10.3 Conditioning on Zero-Probability Events 177
10.4 Applying the Hahn-Banach Theorem 179
10.5 Muddling Boxes 180
10.6 Solving a Functional Equation 181
10.7 Additivity 182
10.8 Muddled Equilibria in Game Theory 182


References 189
Index 197