This paper extends the subjective expected utility model of decision making under uncertainty to include incomplete beliefs and tastes. The main results are two axiomatizations of the multiprior expected multiutility representations of preference relations under uncertainty. The paper also introduces new axiomatizations of Knightian uncertainty and the expected multiutility model with complete beliefs.

a b s t r a c tThis paper axiomatizes expected multi-utility representations of incomplete preferences under risk and under uncertainty. The von Neumann-Morgenstern expected utility model with incomplete preferences is revisited using a ''constructive'' approach, as opposed to earlier treatments that use convex analysis.

In this paper, we show that the presence of the Archimedean and the mixturecontinuity properties of a binary relation, both empirically non-falsifiable in principle, foreclose the possibility of consistency (transitivity) without decisiveness (completeness), or decisiveness without consistency, or in the presence of a weak consistency condition, neither. The basic result can be sharpened when specialized from the context of a generalized mixture set to that of a mixture set in the sense of Herstein-Milnor (1953). We relate the results to the antecedent literature, and view them as part of an investigation into the interplay of the structure of the choice space and the behavioral assumptions on the binary relation defined on it; the ES research program due to Eilenberg (1941) and Sonnenschein (1965), and one to which Schmeidler (1971) is an especially influential contribution.

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