ABSTRACT

We review the axiomatic foundations of subjective utility theory with a view toward understanding the implications of each axiom. We consider three different approaches, namely, the construction of utilities in the presence of canonical probabilities, the construction of probabilities in the presence of utilities, and the simultaneous construction of both probabilities and utilities. We focus attention on the axioms of independence and weak ordering. The independence axiom is seen to be necessary to prevent a form of Dutch Book in sequential problems.

Our main focus is to examine the implications of not requiring the weak order axiom. We assume that gambles are partially ordered. We consider both the construction of probabilities when utilities are given and the construction of utilities in the presence of canonical probabilities. In the first case we find that a partially ordered set of gambles leads to a set of probabilities with respect to which the expected utility of a preferred gamble is higher than that of a dispreferred gamble. We illustrate some comparisons with theories of upper and lower probabilities. In the second case, we find that a partially ordered set of gambles leads to a set of lexicographic utilities, each of which ranks preferred gambles higher than dispreferred gambles.

I. INTRODUCTION: SUBJECTIVE EXPECTED UTILITY [SEU] THEORY

The theory of (subjective) expected utility is a normative account of rational decision making under uncertainty.