A characterization of strategy-proof probabilistic assignment rules

Abstract: We study the classical probabilistic assignment problem, where finitely many indivisible objects are to be probabilistically or proportionally assigned among an equal number of agents. Each agent has an initial deterministic endowment and a strict preference over the objects. While the deterministic version of this problem is well understood, most notably through the characterization of the Top Trading Cycles (TTC) rule by Ma (1994), much less is known in the probabilistic setting. Motivated by practical considerations, we introduce a weakened incentive requirement, namely SD-top-strategy-proofness, which precludes only those manipulations that increase the probability of an agent's top-ranked object. Our first main result shows that, on any free pair at the top (FPT) domain (Sen, 2011), the TTC rule is the unique probabilistic assignment rule satisfying SD-Pareto efficiency, SD-individual rationality, and SD-top-strategy-proofness. We further show that this characterization remains valid when Pareto efficiency is replaced by the weaker notion of SD-pair efficiency, provided the domain satisfies the slightly stronger free triple at the top (FTT) condition (Sen, 2011). Finally, we extend these results to the ex post notions of efficiency and individual rationality. Together, our findings generalize the classical deterministic results of Ma (1994) and Ekici (2024) along three dimensions: extending them from deterministic to probabilistic settings, from full strategy-proofness to top-strategy-proofness, and from the unrestricted domain to the more general FPT and FTT domains.