Expected multi-utility representations of preferences over lotteries

Let ≿ be a binary relation on the set of simple lotteries over a countable outcome set Z. We provide necessary and sufficient conditions on ≿ to guarantee the existence of a set U of von Neumann–Morgenstern utilities u:Z→R such that p≿q⟺Ep[u]≥Eq[u]for allu∈Ufor all simple lotteries p,q. In this case, the set U is essentially unique. Then, we show that the analogous characterization does not hold if Z is uncountable. This provides an answer to an open question posed by Dubra, Maccheroni, and Ok in [J. Econom. Theory 115 (2004), no. 1, 118–133]. Lastly, we show that different continuity requirements on ≿ allow for certain restrictions on the possible choices of the set U of utility functions (e.g., all u are bounded), providing a wide family of expected multi-utility representations. Some implications are proved in a much wider setting.