Representations

. 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 utility functions u : Z → R such that p % q ⇐⇒ E p [ u ] ≥ E q [ u ] for all u ∈ U for all simple lotteries p, q . In such case, the set U is essentially unique. Then, we show that the analogue 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 utility functions are bounded), providing a wide family of expected multi-utility representations.