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Decision making under uncertainty using imprecise probabilities. (English) Zbl 1119.91028

Summary: Various ways for decision making with imprecise probabilities-admissibility, maximal expected utility, maximality, E-admissibility, \(\Gamma\)-maximax, \(\Gamma\)-maximin, all of which are well known from the literature-are discussed and compared. We generalise a well-known sufficient condition for existence of optimal decisions. A simple numerical example shows how these criteria can work in practice, and demonstrates their differences. Finally, we suggest an efficient approach to calculate optimal decisions under these decision criteria.

MSC:

91B06 Decision theory
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[1] von Neumann, J.; Morgenstern, O., Theory of Games and Economic Behavior (1944), Princeton University Press · Zbl 0063.05930
[2] Herstein, I. N.; Milnor, J., An axiomatic approach to measurable utility, Econometrica, 21, 2, 291-297 (1953) · Zbl 0050.36705
[3] Berger, J. O., Statistical Decision Theory and Bayesian Analysis (1985), Springer · Zbl 0572.62008
[4] Raiffa, H.; Schlaifer, R., Applied Statistical Decision Theory (1961), MIT Press · Zbl 0181.21801
[5] Levi, I., The Enterprise of Knowledge. An Essay on Knowledge, Credal Probability, and Chance (1983), MIT Press: MIT Press Cambridge
[6] Dunford, N., Integration in general analysis, Transactions of the American Mathematical Society, 37, 3, 441-453 (1935) · JFM 61.0235.01
[7] Dunford, N.; Schwartz, J. T., Linear Operators (1957), John Wiley & Sons: John Wiley & Sons New York
[8] Kallenberg, O., Foundations of Modern Probability. Foundations of Modern Probability, Probability and Its Applications (2002), Springer · Zbl 0996.60001
[9] Walley, P., Statistical Reasoning with Imprecise Probabilities (1991), Chapman and Hall: Chapman and Hall London · Zbl 0732.62004
[10] Giron, F. J.; Rios, S., Quasi-Bayesian behaviour: a more realistic approach to decision making?, (Bernardo, J. M.; DeGroot, J. H.; Lindley, D. V.; Smith, A. F.M., Bayesian Statistics (1980), University Press: University Press Valencia), 17-38 · Zbl 0459.62006
[11] De Finetti, B., Theory of Probability: A Critical Introductory Treatment (1974-1975), Wiley: Wiley New York, (two volumes) · Zbl 0328.60003
[12] Gilboa, I.; Schmeidler, D., Maxmin expected utility with non-unique prior, Journal of Mathematical Economics, 18, 2, 141-153 (1989) · Zbl 0675.90012
[13] Satia, J. K.; Roy, J.; Lave, E., Markovian decision processes with uncertain transition probabilities, Operations Research, 21, 3, 728-740 (1973) · Zbl 0286.60038
[14] Zaffalon, M.; Wesnes, K.; Petrini, O., Reliable diagnoses of dementia by the naive credal classifier inferred from incomplete cognitive data, Artificial Intelligence in Medicine, 29, 1-2, 61-79 (2003)
[15] Good, I. J., Rational decisions, Journal of the Royal Statistical Society, Series B, 14, 1, 107-114 (1952)
[16] L.V. Utkin, T. Augustin, Powerful algorithms for decision making under partial prior information and general ambiguity attitudes, in: F.G. Cozman, R. Nau, T. Seidenfeld (Eds.), Proceedings of the Fourth International Symposium on Imprecise Probabilities and Their Applications, 2005, pp. 349-358.; L.V. Utkin, T. Augustin, Powerful algorithms for decision making under partial prior information and general ambiguity attitudes, in: F.G. Cozman, R. Nau, T. Seidenfeld (Eds.), Proceedings of the Fourth International Symposium on Imprecise Probabilities and Their Applications, 2005, pp. 349-358.
[17] Seidenfeld, T., A contrast between two decision rules for use with (convex) sets of probabilities: Gamma-maximin versus E-admissibility, Synthese, 140, 1-2, 69-88 (2004)
[18] de Cooman, G.; Troffaes, M. C.M., Dynamic programming for deterministic discrete-time systems with uncertain gain, International Journal of Approximate Reasoning, 39, 2-3, 257-278 (2004) · Zbl 1090.90194
[19] D. Kikuti, F.G. Cozman, C.P. de Campos, Partially ordered preferences in decision trees: computing strategies with imprecision in probabilities, in: R. Brafman, U. Junker (Eds.), Multidisciplinary IJCAI-05 Workshop on Advances in Preference Handling, 2005, pp. 118-123.; D. Kikuti, F.G. Cozman, C.P. de Campos, Partially ordered preferences in decision trees: computing strategies with imprecision in probabilities, in: R. Brafman, U. Junker (Eds.), Multidisciplinary IJCAI-05 Workshop on Advances in Preference Handling, 2005, pp. 118-123.
[20] Schechter, E., Handbook of Analysis and Its Foundations (1997), Academic Press: Academic Press San Diego · Zbl 0943.26001
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