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

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


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.


Nash Equilibrium Conditional Probability Sequential Equilibrium Extensive Form Game Bayesian Decision Theory 
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Copyright information

© Springer Science+Business Media Dordrecht 1994

Authors and Affiliations

  1. 1.Department of EconomicsStanford UniversityUSA

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