Decision Theory with a Human Face Richard Bradley April 2016 ii For Shura Contents Preface xi Introduction xiii I Rationality, Uncertainty and Choice 1 1 Decision Problems 3 1.1 Maximisation of Expected Bene…t . 3 1.2 Framing Decisions . 6 1.3 Savage’sTheory . 9 2 Rationality 13 2.1 Moderate Humeanism . 13 2.2 The Choice Principle . 16 2.3 Subjectivism . 19 3 Uncertainty 23 3.1 Evaluative Uncertainty . 24 3.2 Option uncertainty . 27 3.3 Modal Uncertainty . 30 4 Justifying Bayesianism 33 4.1 Pragmatism . 34 4.2 Interpretations of Preference . 36 4.3 Representation Theorems . 38 4.4 Savage’sRepresentation Theorem . 40 4.5 Evaluation of Savage’saxioms . 44 4.5.1 The Sure-thing Principle . 45 4.5.2 State Independence / Rectangular Field . 46 4.5.3 Probability Principle . 47 4.6 Evaluation of Savage’sargument . 49 II Prospective Rationality 53 5 Rational Belief and Desire 55 5.1 Agents . 55 5.2 Prospects . 56 iii iv CONTENTS 5.2.1 Vocabulary . 58 5.2.2 Boolean Algebras* . 58 5.2.3 Conditional Algebras* . 59 5.3 Attitudes . 60 5.3.1 Comparative Relations . 63 5.3.2 Numerical Representation of Comparative Relations* . 65 5.4 Rational Belief and Desire . 66 5.4.1 Numerical Degrees of Belief and Desire . 66 5.4.2 Probabilism . 68 5.4.3 Desirabilism . 69 5.5 Aspirations of Objectivity . 73 6 Conditional Attitudes 77 6.1 Suppositions . 78 6.2 Suppositional Probability and Desirability . 80 6.3 Evidential Supposition . 82 6.3.1 Evidential Conditional Belief . 82 6.3.2 Evidential Conditional Desire . 84 6.4 Independence . 85 6.5 Foundations* . 86 6.5.1 Representation of Conditional Attitudes . 87 6.5.2 Evidential Supposition . 88 6.5.3 Independence and Separability . 89 6.6 Hypothetical Reasoning . 90 6.6.1 Dominance Arguments . 90 6.6.2 Averaging Arguments . 92 6.6.3 Summary . 94 7 Conditionals and the Ramsey Test 95 7.1 The Ramsey Test for Belief . 95 7.2 The Ramsey Test for Desire . 97 7.3 Chances and Counterfactuals . 98 7.4 Foundations . 101 7.5 Facing the Triviality Results . 103 7.6 Partitioning Conditionals . 106 7.7 Summary . 109 III Facing the World 111 8 Multidimensional Possible World Semantics 113 8.1 Prospects and Possible Worlds . 113 8.1.1 Possible Worlds Models . 113 8.1.2 Triviality Strikes Again . 117 8.2 Multidimensional Semantics . 119 8.2.1 Centering . 121 8.2.2 Multidimensional Possible World Semantics . 123 8.3 Probability . 124 8.3.1 Independence . 126 8.4 Desirability and Counterfactual Value . 128 CONTENTS v 8.5 Value Actualism . 130 8.5.1 World Actualism . 130 8.5.2 Prospect Actualism . 131 8.5.3 Restricted Actualism and Evidential Desirability . 132 8.6 Partitioning Conditionals . 133 8.7 Concluding Remarks . 135 9 Taking Action 139 9.1 Decision Principles . 139 9.2 Preference-based Choice . 141 9.3 Decisions Under Risk . 144 9.4 Decisions Making Under Uncertainty . 146 9.4.1 Savage Acts . 148 9.5 Bayesian Decision Theory . 150 9.5.1 Allais’Paradox . 152 9.5.2 Ellsberg’sParadox . 155 9.6 Causal Decision Theory . 157 10 The Learning Agent 163 10.1 Introduction . 163 10.2 Experience and Learning . 165 10.3 Classical Bayesian Conditioning . 168 10.4 Je¤rey Conditioning . 172 10.5 Adams Conditioning . 175 10.6 Preference Change . 178 10.6.1 Generalised Conditioning . 180 10.6.2 Belief-Driven Preference Change . 184 10.6.3 Desire-Driven Preference Change . 187 IV Rationality Within Bounds 189 11 Imprecise Bayesianism 191 11.1 Introduction . 191 11.2 Bounded Minds . 192 11.2.1 Unawareness and Incompleteness . 192 11.2.2 Types of Inconsistency . 194 11.2.3 Coherent Extendability . 196 11.3 Imprecise Bayesianism . 197 11.4 Severe Uncertainty . 202 11.5 Rational Incomplete Preference . 206 11.5.1 Coherently Extendable Preferences . 207 11.5.2 Representation Theorems . 208 11.6 Challenges to Imprecise Bayesianism . 210 12 Changing Your Mind 215 12.1 Attitude Change . 215 12.2 Changing An Incomplete State of Mind . 217 12.2.1 Attitude Revision . 220 12.2.2 Attitude Formation . 221 vi CONTENTS 12.2.3 Attitude Withdrawal . 222 12.2.4 Connections* . 223 12.3 Unawareness . 224 12.3.1 Rational Unawareness . 225 12.3.2 Representations of Unawareness* . 227 12.4 Changes in States of Awareness . 228 12.4.1 Domain Extension* . 229 12.4.2 Domain Restriction* . 230 12.4.3 The Problem of New Hypotheses . 231 13 Decision Making under Ambiguity 233 13.1 Introduction . 233 13.2 Reaching a Judgement . 234 13.2.1 Picking the ‘Best’ . 236 13.2.2 Aggregating . 237 13.3 Alternative Decision Rules . 240 13.3.1 Caution . 241 13.3.2 Con…dence . 243 13.3.3 Robustness . 244 13.3.4 Flexibility . 245 13.4 Cautious Decision Making . 246 13.5 Models of Ambiguity Aversion* . 248 13.6 The Rationality of Ambiguity Aversion . 251 13.7 Concluding Remarks . 254 14 Con…dence 257 14.1 The Paradox of Ideal Evidence . 257 14.2 Con…dence-Based Learning . 260 14.3 A Model of Con…dence . 263 14.3.1 Con…dence Relations* . 265 14.3.2 Deciding with Su¢ cient Con…dence . 267 14.4 Concluding Remarks . 269 15 Appendix: Proofs 271 List of Figures 1 Flow of Chapters . xvi 6.1 Causal Graph for Breathing Di¢ culties . 79 8.1 A Small-World Model . 118 10.1 Evidence-based Constraints on Belief . 164 10.2 Je¤rey Conditioning . ..