Introduction

Scope and example

In the theory of utility, as presented in modern terms by von Neumann and Morgenstern (1947) and their followers, one starts from a set of prizes, say the finite set B with elements bq, q = 1,…, t. One then defines probability mixtures of prizes, by means of numerical probability vectors β on B, and observes the preferences of a decision-maker among such probability mixtures. When these preferences satisfy three simple axioms (complete order, independence and continuity – see Section 3 below), there exists a real-valued function u on B, called utility, such that preferences among probability mixtures of prizes are isomorphic with expected utilities. That is, the mixture β is preferred to the mixture if and only if

In the theory of games against nature, as presented in modern terms by Savage (1954) and his followers, one starts from a set of alternative, mutually exclusive states of the world, say the finite set S with elements s1,…, sn. Using, as before, a set B of prizes (‘consequences’), acts f, f′,…, are defined as mappings of S into B – that is, as state distributions of prizes (instead of the probability distributions of prizes considered in utility theory). One then observes the preferences of a decision-maker among acts.