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Non-Archimedean Subjective Probabilities in Decision Theory and Games

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  • Peter J. Hammond

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.

Suggested Citation

  • Peter J. Hammond, 1997. "Non-Archimedean Subjective Probabilities in Decision Theory and Games," Working Papers 97038, Stanford University, Department of Economics.
    • Handle: RePEc:wop:stanec:97038
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    1. Kreps, David M & Wilson, Robert, 1982. "Sequential Equilibria," Econometrica, Econometric Society, vol. 50(4), pages 863-894, July.
    2. 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..
    3. 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.
    4. F J Anscombe & R J Aumann, 2000. "A Definition of Subjective Probability," Levine's Working Paper Archive 7591, David K. Levine.
    5. Myerson, Roger B, 1986. "Multistage Games with Communication," Econometrica, Econometric Society, vol. 54(2), pages 323-358, March.
    6. Hammond, P.J. & , ., 1987. "Consequentialist foundations for expected utility," LIDAM Discussion Papers CORE 1987016, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    7. 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.
    8. LaValle, Irving H. & Fishburn, Peter C., 1996. "On the varieties of matrix probabilities in nonarchimedean decision theory," Journal of Mathematical Economics, Elsevier, vol. 25(1), pages 33-54.
    9. Blume, Lawrence & Brandenburger, Adam & Dekel, Eddie, 1991. "Lexicographic Probabilities and Equilibrium Refinements," Econometrica, Econometric Society, vol. 59(1), pages 81-98, January.
    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.
    11. 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.
    12. Howard Raiffa, 1961. "Risk, Ambiguity, and the Savage Axioms: Comment," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 75(4), pages 690-694.
    13. Bicchieri, Cristina & Jeffrey, Richard & Skyrms, Brian (ed.), 1999. "The Logic of Strategy," OUP Catalogue, Oxford University Press, number 9780195117158.
    14. K. J. Arrow, 1964. "The Role of Securities in the Optimal Allocation of Risk-bearing," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 31(2), pages 91-96.
    15. 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.
    16. 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.
    17. 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.
    18. 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.
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    Cited by:

    1. David McCarthy & Kalle Mikkola & Teruji Thomas, 2019. "Aggregation for potentially infinite populations without continuity or completeness," Papers 1911.00872, arXiv.org.
    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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