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Non-Archimedean subjective probabilities in decision theory and games

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  • Hammond, Peter J.

Abstract

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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  • Hammond, Peter J., 1999. "Non-Archimedean subjective probabilities in decision theory and games," Mathematical Social Sciences, Elsevier, vol. 38(2), pages 139-156, September.
  • Handle: RePEc:eee:matsoc:v:38:y:1999:i:2:p:139-156
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    2. Antonio Quesada, 2003. "Negative results in the theory of games with lexicographic utilities," Economics Bulletin, AccessEcon, vol. 3(20), pages 1-7.
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