Non-Archimedean subjective probabilities in decision theory and games

To allow conditioning on counterfactual events, zero probabilities can be r eplaced by infinitesimal probabilities that range over a non-Archimedean or dered field. This paper considers a suitable minimal field that is a comple te metric space. Axioms similar to those in Anscombe and Aumann [Anscombe, F.J., Aumann, R.J., 1963. A definition of subjective probability, Annals of Mathematical Statistics 34, 199-205.] and in Blume et al. [Blume, L., Bran denburger, A., Dekel, E., 1991. Lexicographic probabilities and choice unde r uncertainty, Econometrica 59, 61-79.] are used to characterize preference s which: (i) reveal unique non-Archimedean subjective probabilities within the field; and (ii) can be represented by the non-Archimedean subjective ex pected value of any real-valued von Neumann-Morgenstern utility function in a unique cardinal equivalence class, using the natural ordering of the fie ld. (C) 1999 Elsevier Science B.V. All rights reserved.