Expected utility theory without the completeness axiom.

We study axiomatically the problem of obtaining an expected utility representation for a potentially incomplete preference relation over lotteries by means of a set of von Neumann-Morgenstern utility functions. It is shown that, when the prize space is a compact metric space, a preference relation admits such a Multi-utility representation provided that it satisfies the standard axioms of expected utility theory. Moreover, the representing set of utilities in unique in a well-defined sense.