Consistent Probabilistic Social Choice

Three fundamental axioms in social choice theory are consistency with respect to a variable electorate, consistency with respect to a variable agenda, and consistency with respect to composed preference profiles. In the context of traditional non-probabilistic social choice, these axioms are known to be highly incompatible. We show that in the context of probabilistic social choice, the axioms uniquely characterize a function proposed by Fishburn (Rev. Econ. Stud., 51(4), 683--692, 1984). The function returns so-called maximal lotteries, i.e., lotteries that correspond to optimal mixed strategies of the underlying plurality game. Maximal lotteries are guaranteed to exist due to von Neumann's Minimax Theorem and are almost always unique.