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 is unique in a well-deﬁned sense.
Dubra, Juan; Maccheroni, Fabio; and Ok, Efe A., "Expected Utility Theory without the Completeness Axiom" (2001). Cowles Foundation Discussion Papers. 1548.