Decision Theory

Contents

Preface

Acknowledgments

1     Introduction

1.1     Controversies

1.2     A guided tour of decision theory

Part One   Foundations

2     Coherence

2.1     The “Dutch Book” theorem

2.1.1     Betting odds

2.1.2     Coherence and the axioms of probability

2.1.3     Coherent conditional probabilities

2.1.4     The implications of Dutch Book theorems

2.2     Temporal coherence

2.3     Scoring rules and the axioms of probabilities

2.4     Exercises

3     Utility

3.1     St. Petersburg paradox

3.2     Expected utility theory and the theory of means

3.2.1     Utility and means

3.2.2     Associative means

3.2.3     Functional means

3.3     The expected utility principle

3.4     The von Neumann–Morgenstern representation theorem

3.4.1     Axioms

3.4.2     Representation of preferences via expected utility

3.5     Allais’ criticism

3.6     Extensions

3.7     Exercises

4     Utility in action

4.1     The “standard gamble”

4.2     Utility of money

4.2.1     Certainty equivalents

4.2.2     Risk aversion

4.2.3     A measure of risk aversion

4.3     Utility functions for medical decisions

4.3.1     Length and quality of life

4.3.2     Standard gamble for health states

4.3.3     The time trade-off methods

4.3.4     Relation between QALYs and utilities

4.3.5     Utilities for time in ill health

4.3.6     Difficulties in assessing utility

4.4     Exercises

5     Ramsey and Savage

5.1     Ramsey’s theory

5.2     Savage’s theory

5.2.1     Notation and overview

5.2.2     The sure thing principle ...