Strategy-Proof Probabilistic Social Choice Correspondences under Conditional Expected Utility

We study unanimous and strategy-proof probabilistic social choice correspondences (PSCCs), where the selected set of alternatives is interpreted as an interim outcome, and agents evaluate sets using conditional expected utility. We analyze two preference domains introduced by Barbera et al. (2001): the conditionally expected utility consistent (CEUC) domain and the conditionally expected utility consistent with equal probabilities (CEUCEP) domain. Our results characterize all unanimous and strategy-proof PSCCs on these domains and identify cases when randomization enlarges the class of admissible rules. On the CEUC domain, every unanimous and strategy-proof PSCC is a random dictatorship, showing that randomization over sets yields no additional flexibility. In contrast, the CEUCEP domain admits a richer family of unanimous and strategy-proof PSCCs. For at most three agents, these rules are precisely the random bi-dictatorial rules, which are convex combinations of bi-dictatorial rules introduced in Feldman (1980). For four or more agents, the characterization depends on the number of alternatives. When there are exactly three alternatives, the class expands to the larger family of coalition-weighted rules. Thus, randomization enlarges the class of strategy-proof correspondences in the three-alternative case, producing rules that are not convex combinations of deterministic strategy-proof correspondences. However, for four or more alternatives, the class of unanimous and strategy-proof probabilistic correspondences again collapse to random bi-dictatorships.