Mathematical Social Sciences 38 1999 139–156
Non-Archimedean subjective probabilities in decision theory and games
Peter J. Hammond
Department of Economics , Stanford University, Stanford, CA 94305-6072, USA
Abstract
To allow conditioning on counterfactual events, zero probabilities can be replaced by infinitesimal probabilities that range over a non-Archimedean ordered field. This paper considers a
suitable minimal field that is a complete 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., Brandenburger, A., Dekel, E., 1991. Lexicographic probabilities and choice under uncertainty, Econometrica 59, 61–79.] are
used to characterize preferences which: i reveal unique non-Archimedean subjective probabilities within the field; and ii can be represented by the non-Archimedean subjective expected value of
any real-valued von Neumann–Morgenstern utility function in a unique cardinal equivalence class, using the natural ordering of the field.
1999 Elsevier Science B.V. All rights reserved.
1. Introduction and outline
Recent game theory relies on players having conditional probabilistic beliefs after a counterfactual event, like a player deviating from a presumed equilibrium strategy.
Suggested formal frameworks for dealing with such counterfactuals include complete conditional probability systems, lexicographic hierarchies of probabilities, and logarith-
mic likelihood ratio functions. For a survey of these approaches and their relationship, see Hammond 1994.
Section 2 of this paper begins by pointing out a fundamental inadequacy of these previous suggestions. This arises because they do not permit a compound lottery to be
reduced unambiguously. Following some early steps taken in Hammond 1994, 1999c, this paper presents an
alternative approach, using the smallest possible extension of the real number field that allows for infinitesimal probabilities. Such extensions require consideration of a non-
E-mail address : hammondleland.stanford.edu. P.J. Hammond
0165-4896 99 – see front matter
1999 Elsevier Science B.V. All rights reserved. P I I : S 0 1 6 5 - 4 8 9 6 9 9 0 0 0 1 1 - 6
140 P
.J. Hammond Mathematical Social Sciences 38 1999 139 –156
Archimedean ordered algebraic field of possible probabilities, of the kind discussed in Section 3.
The key part of the paper begins with Section 4. First, for a general ordered field of possible probabilities, expected utility is derived from three axioms – ordering,
independence, and an ‘‘algebraic continuity’’ axiom. Next, Section 5 introduces additional axioms, similar to those devised by Anscombe and Aumann 1963 for
real-valued probabilities, and by Blume et al. 1991 henceforth BBD for lexicographic probabilities. These are used to prove existence of a unique cardinal equivalence class of
real-valued von Neumann–Morgenstern utility functions and also, except in the case of universal indifference, unique possibly infinitesimal subjective probabilities.
Section 6 considers the special case when probabilities belong to Re, the smallest
ordered field including both the real line and at least one positive ‘‘basic’’ infinitesimal.
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This is extended to the special field R e of all power series in e, as the smallest
complete metric space that includes Re when the latter is given a suitable metric. In
Section 7 this extension allows the algebraic continuity condition to be replaced by a more intuitive ‘‘extended continuity’’ condition. In this setting, moreover, applying the
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natural ordering of R e to expected utilities induces a preference ordering which
corresponds to a familiar lexicographic expected utility criterion, with a real-valued von Neumann–Morgenstern utility function and non-Archimedean subjective probabilities.
2. Non-reduction of compound lotteries
