 Aumann (1962) shows that if we have a partial but not total preference ordering, then we can represent it by a utility function (not unique-up-to-positive-affine-transformation) such that \(A \succ B\) implies \(\EU(A) \gt \EU(B),\) but not vice versa. Aumann shows that there will be at least one utility function that represents the preference ordering according to (objective) EU maximization. Thus, we can represent a preference ordering as the set of all utility functions that “one-way” represent the decision-maker’s preferences. Letting \(\EU_u(A)\) be the expected utility of \(A\) given utility function \(u\):