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Non-Archimedean Subjective Probabilities in Decision Theory and Games

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Peter J. Hammond

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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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Paper provided by Stanford University, Department of Economics in its series Working Papers with number 97038.

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Date of creation: 07 Dec 1997
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Handle: RePEc:wop:stanec:97038

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Related research
Keywords: Non-Archimedean probabilities; subjective expected utility; Anscombe--Aumann axioms; lexicographic expected utility; conditional probability systems; reduction of compound lotteries;

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References listed on IDEAS
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.:
  1. McLennan, Andrew, 1989. "The Space of Conditional Systems is a Ball," International Journal of Game Theory, Springer, vol. 18(2), pages 125-39.
  2. David M Kreps & Robert Wilson, 2003. "Sequential Equilibrium," Levine's Working Paper Archive 618897000000000813, David K. Levine. [Downloadable!]
    Other versions:
  3. 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. [Downloadable!] (restricted)
  4. McLennan, Andrew, 1989. "Consistent Conditional Systems in Noncooperative Game Theory," International Journal of Game Theory, Springer, vol. 18(2), pages 141-74.
  5. Myerson, Roger B, 1986. "Multistage Games with Communication," Econometrica, Econometric Society, vol. 54(2), pages 323-58, March. [Downloadable!] (restricted)
    Other versions:
  6. Blume, Lawrence & Brandenburger, Adam & Dekel, Eddie, 1991. "Lexicographic Probabilities and Choice under Uncertainty," Econometrica, Econometric Society, vol. 59(1), pages 61-79, January. [Downloadable!] (restricted)
  7. Hammond, P.J. & , ., 1987. "Consequentialist foundations for expected utility," CORE Discussion Papers 1987016, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  8. Blume, Lawrence & Brandenburger, Adam & Dekel, Eddie, 1991. "Lexicographic Probabilities and Equilibrium Refinements," Econometrica, Econometric Society, vol. 59(1), pages 81-98, January. [Downloadable!] (restricted)
  9. 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. [Downloadable!] (restricted)
  10. Roger B. Myerson, 1977. "Refinements of the Nash Equilibrium Concept," Discussion Papers 295, Northwestern University, Center for Mathematical Studies in Economics and Management Science. [Downloadable!]
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Cited by:
(explanations, 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.)

  1. Antonio Quesada, 2003. "Negative results in the theory of games with lexicographic utilities," Economics Bulletin, Economics Bulletin, vol. 3(20), pages 1-7. [Downloadable!]
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