Random Serial Dictatorship: The One and Only

Fix a Pareto-optimal, strategy-proof, and nonbossy deterministic matching mechanism and define a random matching mechanism by assigning agents to the roles in the mechanism via a uniform lottery. Given a profile of preferences, the lottery over outcomes that arises under the random matching mechanism is identical to the lottery that arises under random serial dictatorship, where the order of dictators is uniformly distributed. This result extends the celebrated equivalence between the core from random endowments and random serial dictatorship to the grand set of all Pareto-optimal, strategy-proof, and nonbossy matching mechanisms.