Abstract
We discuss several features of coherent choice functions—where the admissible options in a decision problem are exactly those that maximize expected utility for some probability/utility pair in fixed set S of probability/utility pairs. In this paper we consider, primarily, normal form decision problems under uncertainty—where only the probability component of S is indeterminate and utility for two privileged outcomes is determinate. Coherent choice distinguishes between each pair of sets of probabilities regardless the “shape” or “connectedness” of the sets of probabilities. We axiomatize the theory of choice functions and show these axioms are necessary for coherence. The axioms are sufficient for coherence using a set of probability/almost-state-independent utility pairs. We give sufficient conditions when a choice function satisfying our axioms is represented by a set of probability/state-independent utility pairs with a common utility.
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References
Aizerman M.A. (1985) New problems in general choice theory. Social Choice and Welfare 2: 235–282
Anscombe F.J., Aumann R.J. (1963) A definition of subjective probability. The Annals of Mathematical Statistics 34: 199–205
Cozman F.G., Walley P. (2005) Graphoid properties of epistemic irrelevance and independence. Annals of Mathematics and Artificial Intelligence 45: 173–195
de Finetti B. (1974) Theory of probability (Vol. 1). Wiley, London
Gilboa I., Schmeidler D. (1989) Maxmin expected utility with non-unique prior. Journal of Mathematical Economics 18: 141–153
Kadane J.B., Schervish M.J., Seidenfeld T. (2004) A Rubinesque theory of decision. IMS Lecture Notes Monograph 45: 1–11
Kindler J. (1983) A general solution concept for two person, zero sum games. Journal of Optimization Theory and Applications 40: 105–119
Levi I. (1974) On indeterminate probabilities. Journal of Philosophy 71: 391–418
Levi I. (1986) Hard choices. Cambridge University Press, Cambridge
Pearce D. (1984) Rationalizable strategic behavior and the problem of perfection. Econometrica 52: 1029–1050
Savage L.J. (1954) The foundations of statistics. Wiley, New York
Schervish, M. J., Seidenfeld, T., Kadane, J. B., & Levi, I. (2003). Extensions of expected utility theory and some limitations of pairwise comparisons. In J.-M. Bernard, T. Seidenfeld, & M. Zaffalon (Eds.), Proceedings of the third international symposium on imprecise probabilities and their applications (pp. 496–510). Carleton Scientific.
Schervish, M. J., Seidenfeld, T., & Kadane, J. B. (2008). Proper scoring rules, dominated forecasts, and coherence. T.R. #865, Department of Statistics, Carnegie Mellon University.
Seidenfeld T., Schervish M.J., Kadane J.B. (1995) A representation of partially ordered preferences. Annals of Statistics 23: 2168–2217
Sen A. (1977) Social choice theory: A re-examination. Econometrica 45: 53–89
Wald A. (1950) Statistical decision functions. Wiley, New York
Walley P. (1990) Statistical reasoning with imprecise probabilities. Chapman and Hall, London
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Seidenfeld, T., Schervish, M.J. & Kadane, J.B. Coherent choice functions under uncertainty. Synthese 172, 157–176 (2010). https://doi.org/10.1007/s11229-009-9470-7
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DOI: https://doi.org/10.1007/s11229-009-9470-7