On Probabilistic Assignment Rules

We study the classical assignment problem with initial endowments in a probabilistic framework. In this setting, each agent initially owns an object and has strict preferences over the entire set of objects, and the goal is to reassign objects in a way that satisfies desirable properties such as strategy-proofness, Pareto efficiency, and individual rationality. While the celebrated result by Ma (1994) shows that the Top Trading Cycles (TTC) rule is the unique deterministic rule satisfying these properties, similar positive results are scarce in the probabilistic domain. We extend Ma's result in the probabilistic setting, and as desirable properties, consider SD-efficiency, SD-individual rationality, and a weaker notion of SD-strategy-proofness -- SD-top-strategy-proofness -- which only requires agents to have no incentive to misreport if doing so increases the probability of receiving their top-ranked object. We show that under deterministic endowments, a probabilistic rule is SD-efficient, SD-individually rational, and SD-top-strategy-proof if and only if it coincides with the TTC rule. Our result highlights a positive possibility in the face of earlier impossibility results for fractional endowments (Athanassoglou and Sethuraman (2011)) and provides a first step toward reconciling desirable properties in probabilistic assignments with endowments.