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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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Bibliographic Info

Article provided by Elsevier in its journal Mathematical Social Sciences.

Volume (Year): 38 (1999)
Issue (Month): 2 (September)
Pages: 139-156

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Handle: RePEc:eee:matsoc:v:38:y:1999:i:2:p:139-156

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Web page: http://www.elsevier.com/locate/inca/505565

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References

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  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. Lavalle, Irving H & Fishburn, Peter C, 1992. " State-Independent Subjective Expected Lexicographic Utility," Journal of Risk and Uncertainty, Springer, vol. 5(3), pages 217-40, July.
  3. David Kreps & Robert Wilson, 1998. "Sequential Equilibria," Levine's Working Paper Archive 237, David K. Levine.
  4. 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.
  5. Roger B. Myerson, 1984. "Multistage Games with Communication," Discussion Papers 590, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  6. F J Anscombe & R J Aumann, 2000. "A Definition of Subjective Probability," Levine's Working Paper Archive 7591, David K. Levine.
  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. 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.
  9. R. Myerson, 2010. "Refinement of the Nash Equilibrium Concept," Levine's Working Paper Archive 537, David K. Levine.
  10. Blume, Lawrence & Brandenburger, Adam & Dekel, Eddie, 1991. "Lexicographic Probabilities and Choice under Uncertainty," Econometrica, Econometric Society, vol. 59(1), pages 61-79, January.
  11. Blume, Lawrence & Brandenburger, Adam & Dekel, Eddie, 1991. "Lexicographic Probabilities and Equilibrium Refinements," Econometrica, Econometric Society, vol. 59(1), pages 81-98, January.
  12. Reinhard Selten, 1973. "A Simple Model of Imperfect Competition, where 4 are Few and 6 are Many," Working Papers 008, Bielefeld University, Center for Mathematical Economics.
  13. McLennan, Andrew, 1989. "Consistent Conditional Systems in Noncooperative Game Theory," International Journal of Game Theory, Springer, vol. 18(2), pages 141-74.
  14. 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.
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Cited by:
  1. repec:ebl:ecbull:v:3:y:2003:i:20:p:1-7 is not listed on IDEAS
  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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