Nonstandard numbers for qualitative decision making

Abstract

The consideration of nonstandard models of the real numbers and the definition of a qualitative ordering on those models provides a generalization of the principle of maximization of expected utility. It enables the decider to assign probabilities of different orders of magnitude to different events or to assign utilities of different orders of magnitude to different outcomes. The properties of this generalized notion of rationality are studied in the frameworks proposed by von Neumann and Morgenstern and later by Anscombe and Aumann. It is characterized by an original weakening of their postulates in two different situations: nonstandard probabilities and standard utilities on one hand and standard probabilities and nonstandard utilities on the other hand. This weakening concerns both Independence and Continuity. It is orthogonal with the weakening proposed by lexicographic orderings.