Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- Part I Expected utility
- Part II Nonexpected utility for Risk
- Part III Nonexpected utility for uncertainty
- 13 Conclusion
- Appendices
- Appendix A Models never hold perfectly: how to handle their deficiencies?
- Appendix B Choosing from multiple prospects and binary choice: the principles of revealed preference
- Appendix C Dynamic decisions
- Appendix D Applications other than decision under uncertainty
- Appendix E Bisymmetry-based preference conditions
- Appendix F Nonmonotonic rank-dependent models and the Fehr–Schmidt model of welfare evaluation
- Appendix G Extensions of finite-dimensional results to infinite-dimensional results: a meta-theorem
- Appendix H Measure theory
- Appendix I Related textbooks and surveys
- Appendix J Elaborations of exercises
- Appendix K Skipping parts and interdependencies between sections
- References
- Author index
- Subject index
Appendix G - Extensions of finite-dimensional results to infinite-dimensional results: a meta-theorem
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Introduction
- Part I Expected utility
- Part II Nonexpected utility for Risk
- Part III Nonexpected utility for uncertainty
- 13 Conclusion
- Appendices
- Appendix A Models never hold perfectly: how to handle their deficiencies?
- Appendix B Choosing from multiple prospects and binary choice: the principles of revealed preference
- Appendix C Dynamic decisions
- Appendix D Applications other than decision under uncertainty
- Appendix E Bisymmetry-based preference conditions
- Appendix F Nonmonotonic rank-dependent models and the Fehr–Schmidt model of welfare evaluation
- Appendix G Extensions of finite-dimensional results to infinite-dimensional results: a meta-theorem
- Appendix H Measure theory
- Appendix I Related textbooks and surveys
- Appendix J Elaborations of exercises
- Appendix K Skipping parts and interdependencies between sections
- References
- Author index
- Subject index
Summary
The general extension of finite-dimensional behavioral foundations to infinite-dimensional ones is discussed in detail by Wakker (1993a). This book does consider infinite state spaces, but confines its attention to prospects with finitely many outcomes. Then the extension of results from finite state spaces to infinite state spaces is not difficult. The procedure will now be explained. We give an explanation for the special case of Theorem 4.6.4 (uncertainty-EU), and an explanation in general for all theorems of this book. For Theorem 4.6.4, explanations between square brackets should be read. For the other results, these explanations should be skipped, or adapted to more general models.
Assume that we have established preference conditions that are necessary and sufficient for the existence of a representation [EU] for finite state spaces S, with a utility function U that is unique up to unit and level. Assume that the behavioral foundation also involves one or more set functions [only one, namely P] defined on the subsets of S – or, for mathematicians, on an algebra of events – and that these set functions are uniquely determined. Besides the probability P in EU such as in Theorem 4.6.4, we deal with a nonadditive measure in rank-dependent utility and with two nonadditive measures in prospect theory. Then this behavioral foundation [of EU] also holds under Structural Assumption 1.2.1 (decision under uncertainty) if S is infinite. The proof is as follows.
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- Information
- Prospect TheoryFor Risk and Ambiguity, pp. 393 - 394Publisher: Cambridge University PressPrint publication year: 2010