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

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. Elinor now found the difference between the expectation of an unpleasant event, however certain the mind may be told to consider it, and certainty itself. — Jane Austen, Sense and Sensibility, ch. 48. . . . a more attractive and manageable theory may result from a non-Archimedean representation. . . . One must keep in mind the fact that the refutability of axioms depends both on their mathematical form and their empirical interpretation. — Krantz, Luce, Suppes and Tversky (1971, p. 29). Non-Archimedean Probabilities