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# Theory of Decision under Uncertainty - …

Theory of Decision under Uncertainty Itzhak Gilboa July 2008. Contents 1 Preface 6. 2 Motivating Examples 9. 3 Free Will and Determinism 11. Can free choice be predicted? .. 11. Is the world deterministic? .. 12. Is free will observable? .. 13. The problem of free will .. 14. A rational illusion .. 19. Free will and the Decision matrix .. 21. 4 The Principle of Indi erence 23. Will a canonical space help? .. 24. The canonical state space .. 24. Di culties with a uniform prior on [0, 1] .. 25. Conclusion .. 26. What's special about the coin? .. 27. Symmetry .. 27. Smooth beliefs .. 28. 5 Relative Frequencies 31. The law of large numbers .. 31. The problem of induction .. 33. Hume's critique .. 33. 1. Goodman's grue-bleen paradox.

1 Preface The following are lecture notes for a graduate level class in decision under uncertainty. After teaching such classes for many years, it became more di ﬃ-

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1 Theory of Decision under Uncertainty Itzhak Gilboa July 2008. Contents 1 Preface 6. 2 Motivating Examples 9. 3 Free Will and Determinism 11. Can free choice be predicted? .. 11. Is the world deterministic? .. 12. Is free will observable? .. 13. The problem of free will .. 14. A rational illusion .. 19. Free will and the Decision matrix .. 21. 4 The Principle of Indi erence 23. Will a canonical space help? .. 24. The canonical state space .. 24. Di culties with a uniform prior on [0, 1] .. 25. Conclusion .. 26. What's special about the coin? .. 27. Symmetry .. 27. Smooth beliefs .. 28. 5 Relative Frequencies 31. The law of large numbers .. 31. The problem of induction .. 33. Hume's critique .. 33. 1. Goodman's grue-bleen paradox.

2 34. Kolmogorov complexity and its dependence of language 36. Grue-bleen again .. 40. Evolutionary explanations .. 47. Problems with the frequentist approach .. 51. 6 Subjective Probabilities 56. Linda the bank teller .. 56. Pascal's wager .. 58. Classical vs. Bayesian statistics .. 60. Basic definitions .. 60. The gambler fallacy .. 62. Exchangeability .. 63. Confidence is not probability .. 66. Classical statistics can be ridiculous .. 68. Di erent methods for di erent goals .. 69. 7 A Case Study 74. A characterization theorem for maximization of utility .. 74. Proof .. 77. Interpretations .. 80. A few definitions .. 80. A meta-scientific interpretation .. 86. A normative interpretation .. 89. A descriptive interpretation.

3 91. Limitations .. 93. Semi-orders .. 93. Other ways to measure utility .. 102. 8 The Role of Theories 104. Theories are always wrong .. 104. Theories and conceptual frameworks .. 107. Logical positivism as a metaphor .. 109. 9 von Neumann and Morgenstern's Theorem 112. Background .. 112. The theorem .. 114. Proofs .. 119. 2. The algebraic approach .. 119. A geometric approach .. 121. A separation argument .. 122. The three interpretations .. 124. 10 de Finetti's Theorem 127. Motivation .. 127. The theorem .. 128. A proof .. 130. The three interpretations .. 132. 11 Savage's Theorem 134. Background .. 134. States, outcomes, and acts .. 137. Axioms .. 139. P1 .. 139. P2 .. 139. Notation .. 141. Null events.

4 142. P3 .. 143. P4 .. 145. P5 .. 146. P6 .. 147. P7 .. 150. The result for a finite outcome set .. 151. Finitely additive measures .. 151. Non-atomic measures .. 154. The Theorem .. 154. The case of a general outcome set .. 156. Interpretations .. 156. The proof and qualitative probabilities .. 158. 12 The Definition of States 162. Causality .. 162. Newcomb's Paradox .. 162. States as functions from acts to outcomes .. 163. A problem .. 165. Hempel's paradox of confirmation .. 166. 3. Are all ravens black? .. 166. A state space formulation .. 167. What is a confirmation? .. 168. A resolution .. 168. Good's variation .. 169. What do we learn from this? .. 171. Monty Hall three-door game .. 172. 13 A Critique of Savage 176.

5 Criticizing critiques .. 176. An example .. 176. The general lesson .. 178. Critique of P3 and P4 .. 179. Example .. 179. Defense .. 181. State dependent utility .. 182. The definition of subjective probability .. 183. When is state dependence necessary? .. 185. Critique of P1 and P2 .. 187. The basic problem .. 187. Reasoned choice vs. raw preferences .. 188. Schmeidler's critique and Ellsberg's paradox .. 190. Observability of states .. 195. Problems of complexity .. 196. 14 Objectivity and Rationality 198. Subjectivity and objectivity .. 198. Objective and subjective rationality .. 199. 15 Anscombe-Aumann's Theorem 203. 16 Choquet Expected Utility 208. Schmeidler's intuition .. 208. Choquet Integral .. 210.

6 Comonotonicity .. 212. Axioms and result .. 214. 4. 17 Prospect Theory 217. Background .. 217. Gain-loss asymmetry .. 218. Distortion of probabilities .. 219. Rank-dependent probabilities and Choquet integration .. 222. 18 Maxmin Expected Utility 225. Convex games .. 225. A cognitive interpretation of CEU .. 226. Axioms and result .. 228. Interpretation of MMEU .. 230. Generalizations and variations .. 231. Bewley's alternative approach .. 232. Combining objective and subjective rationality .. 233. Applications .. 236. 19 Case-Based Qualitative Beliefs 241. Axioms and result .. 241. Four known techniques .. 244. The Combination axiom in general analogical reasoning .. 247. Violations of the Combination axiom.

7 249. 20 Frequentism Revisited 251. Similarity-weighted empirical frequencies .. 251. Intuition .. 252. Axiomatization .. 253. Empirical similarity and objective probabilities .. 257. 21 Future Research 262. 22 References 264. 5. 1 Preface The following are lecture notes for a graduate level class in Decision under Uncertainty . After teaching such classes for many years, it became more di - cult to squeeze all the material into class discussions, and I was not quite sure about the appropriate order in which the issues should be presented. Trying to put everything in writing, and thereby finding at least one permutation of topics that makes sense, is supposed to help me forget less and confuse the students less.

8 As a by-product, the notes may be useful to students who miss classes, or who wish to know what the course is about without having to show up every week. The graduate classes I normally teach are geared towards students who want to do research in economics and in related fields. It is assumed that they have taken a basic microeconomic sequence at the graduate level. In teaching the class, I hope that I can convince some students that Decision Theory is a beautiful topic and that this is what they want to do when they grow up. But I'm fully aware of the fact that this is not going to be the majority of the students. I therefore attempt to teach the course with a much more general audience in mind, focusing on the way that Decision Theory may a ect the research done in economics at large, as well as in finance, political science, and other fellow disciplines.

9 The present notes contain relatively few proofs. Interested students are referred to books and papers that contain more details. The focus here is on conceptual issues, where I find that each teacher has his or her own interpretation and style. The mathematical results are a bit like musical notes they are written, saved, and can be consulted according to need. By contrast, our presentation and interpretation of the results are akin to the performance of a musical piece. My focus here is mostly on the interpretation I like, on how I would like to perform the Theory , as it were. I comment on proofs mostly when I feel that some intuition may be gained from them, and that it's important to highlight it. 6. These notes are not presented as original thoughts or a research paper.

10 Many of the ideas presented here appeared in print. Some are very old, some are newer, and some appeared in works that I have co-authored. I try to give credit where credit is due, but I must have missed many references that I had and had not been aware of. There are also other thoughts that have not appeared in print, and originated in discussions with various people, in particular with David Schmeidler. When I recall precisely that a certain idea was suggested to me by someone, I do mention it, but there are many ideas for which I can recall no obvious source. This doesn't mean that they are original, and I apologize for any unintentional plagiarism. At the same time, there are also many ideas here that I know to be peculiar to me (and some to David and to me).