Elementary Non-Archimedean Representations of Probability for Decision Theory and Games (1994)
| Venue: | Suppes: Scientific Philosopher, Vol. I: Probability and Probabilistic Causality |
| Citations: | 16 - 5 self |
BibTeX
@INPROCEEDINGS{Hammond94elementarynon-archimedean,
author = {Peter J. Hammond},
title = {Elementary Non-Archimedean Representations of Probability for Decision Theory and Games},
booktitle = {Suppes: Scientific Philosopher, Vol. I: Probability and Probabilistic Causality},
year = {1994},
pages = {25--59},
publisher = {Kluwer}
}
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Abstract
1992 version is intended as a contribution to a two volume collection honouring Patrick Suppes, to be edited by Paul Humphreys and published by Kluwer Academic Publishers. 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 allowconditioning on counterfactual events with prior probability zero, extended probabilities are proposed and given the four equivalent characterizations: (i) complete conditional probability sys-tems; (ii) lexicographic hierarchies of probabilities; (iii) extended logarithmic likelihood ratios; and (iv) certain ‘canonical rational probability functions ’ representing ‘trembles ’ directly as in-finitesimal 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).
