Subjective expected utility maximization is derived from four axioms, using an argument based on the separating hyperplane theorem. It is also shown that the first three of these axioms imply a more general maximization formula, involving an evaluation function, which can still serve as a basis for decision analysis.
1. Introduction
Decisions under uncertainty are commonly described in two ways: using a probability model or a state-variable model. In each case, we speak of the decision-maker as choosing among lotteries, but the two models differ in how a lottery is defined. In a probability model, lotteries are probability distributions over a set of prizes (for example, see Section 2.4 in [8]). In a state-variable model, lotteries are functions from a set of possible states of nature into a set of prizes (for example, see Chapter 13 in [8]).
The distinction between a probability model and a state-variable model is not simply a matter of mathematical style. A probability model is appropriate to describe gambles in which the prize will depend on events which have obvious objective probabilities; we shall refer to such events as objective unknowns. These gambles correspondto the 'roulette lotteries' of [1], or the 'risks' of [7]. For example, gambles which depend on the toss of a fair coin, the spin of a roulette wheel, or the blind draw of a ball out an turn containing a known population of identically-shaped but different-colored balls, all could be adequately described in a probability model. (There is an implicit assumption here that two objective unknowns with the same probability are completely equivalent for all decision-making purposes. For example, if we describe a lottery by saying that it "offers a prize of $100 or $0, each with probability 1/2", we are assuming that it does not matter whether the prize is to be determined by flipping a fair coin, or by drawing a ball from an urn which contains 50 white and 50 black balls.)