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Elementary Non-Archimedean Representations of Probability for Decision Theory and Games

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Patrick Suppes: Scientific Philosopher

Part of the book series: Synthese Library ((SYLI,volume 234))

Abstract

In an extensive form game, whether a player has a better strategy than in a presumed equilibrium depends on the other players’ equilibrium reactions to a counterfactual deviation. To allow conditioning on counterfactual events with prior probability zero, extended probabilities are proposed and given the four equivalent characterizations: (i) complete conditional probability systems; (ii) lexicographic hierarchies of probabilities; (iii) extended logarithmic likelihood ratios; and (iv) certain ‘canonical rational probability functions’ representing ‘trembles’ directly as infinitesimal probabilities. However, having joint probability distributions be uniquely determined by independent marginal probability distributions requires general probabilities taking values in a space no smaller than the non-Archimedean ordered field whose members are rational functions of a particular infinitesimal.

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Authors and Affiliations

  1. Department of Economics, Stanford University, CA, 94305-6072, USA

    Peter J. Hammond

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  1. Peter J. Hammond
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  1. Corcoran Department of Philosophy, University of Virginia, Charlottesville, VA, USA

    Paul Humphreys

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Hammond, P.J. (1994). Elementary Non-Archimedean Representations of Probability for Decision Theory and Games. In: Humphreys, P. (eds) Patrick Suppes: Scientific Philosopher. Synthese Library, vol 234. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0774-7_2

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  • DOI: https://doi.org/10.1007/978-94-011-0774-7_2

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