Non-Archimedean subjective probabilities in decision theory and games
December 7, 1997 To allow conditioning on counterfactual events, zero probabilities can be replaced by infinitesimal probabilities that range over a non-Archimedean ordered field. This paper considers a suitable minimal field that is a complete metric space. Axioms similar to those in Anscombe and Aumann (1963) and in Blume, Brandenburger and Dekel (1991) are used to characterize preferences which: (i) reveal unique non-Archimedean subjective probabilities within the field; and (ii) can be represented by the non-Archimedean subjective expected value of any real-valued von Neumann--Morgenstern utility function in a unique cardinal equivalence class, using the natural ordering of the field. Keywords: Non-Archimedean probabilities, subjective expected utility, Anscombe--Aumann axioms, lexicographic expected utility, conditional probability systems, reduction of compound lotteries.
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Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- David M Kreps & Robert Wilson, 2003.
Levine's Working Paper Archive
618897000000000813, David K. Levine.
- McLennan, Andrew, 1989. "The Space of Conditional Systems is a Ball," International Journal of Game Theory, Springer;Game Theory Society, vol. 18(2), pages 125-139.
- Lavalle, Irving H & Fishburn, Peter C, 1992. "State-Independent Subjective Expected Lexicographic Utility," Journal of Risk and Uncertainty, Springer, vol. 5(3), pages 217-240, July.
- Lawrence Blume & Adam Brandenburger & Eddie Dekel, 2014.
"Lexicographic Probabilities and Choice Under Uncertainty,"
World Scientific Book Chapters,
in: The Language of Game Theory Putting Epistemics into the Mathematics of Games, chapter 6, pages 137-160
World Scientific Publishing Co. Pte. Ltd..
- Blume, Lawrence & Brandenburger, Adam & Dekel, Eddie, 1991. "Lexicographic Probabilities and Choice under Uncertainty," Econometrica, Econometric Society, vol. 59(1), pages 61-79, January.
- Hammond, P.J. & , ., 1987. "Consequentialist foundations for expected utility," CORE Discussion Papers 1987016, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
- R. Myerson, 2010.
"Refinement of the Nash Equilibrium Concept,"
Levine's Working Paper Archive
537, David K. Levine.
- Anderson, Robert M., 1991. "Non-standard analysis with applications to economics," Handbook of Mathematical Economics, in: W. Hildenbrand & H. Sonnenschein (ed.), Handbook of Mathematical Economics, edition 1, volume 4, chapter 39, pages 2145-2208 Elsevier.
- Myerson, Roger B, 1986.
"Multistage Games with Communication,"
Econometric Society, vol. 54(2), pages 323-358, March.
- Blume, Lawrence & Brandenburger, Adam & Dekel, Eddie, 1991. "Lexicographic Probabilities and Equilibrium Refinements," Econometrica, Econometric Society, vol. 59(1), pages 81-98, January.
- Rajan, Uday, 1998. "Trembles in the Bayesian Foundations of Solution Concepts of Games," Journal of Economic Theory, Elsevier, vol. 82(1), pages 248-266, September.
- K. J. Arrow, 1964. "The Role of Securities in the Optimal Allocation of Risk-bearing," Review of Economic Studies, Oxford University Press, vol. 31(2), pages 91-96.
- Karni, Edi & Schmeidler, David, 1991. "Utility theory with uncertainty," Handbook of Mathematical Economics, in: W. Hildenbrand & H. Sonnenschein (ed.), Handbook of Mathematical Economics, edition 1, volume 4, chapter 33, pages 1763-1831 Elsevier.
- F J Anscombe & R J Aumann, 2000. "A Definition of Subjective Probability," Levine's Working Paper Archive 7591, David K. Levine.
- McLennan, Andrew, 1989. "Consistent Conditional Systems in Noncooperative Game Theory," International Journal of Game Theory, Springer;Game Theory Society, vol. 18(2), pages 141-174.
- Reinhard Selten, 1973. "A Simple Model of Imperfect Competition, where 4 are Few and 6 are Many," Center for Mathematical Economics Working Papers 008, Center for Mathematical Economics, Bielefeld University.
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