Notions of Rank Efficiency for the Random Assignment Problem

An assignment is rank efficient if there is no other assignment where the expected number of agents who received one of their top choices is weakly higher. We introduce new notions of rank efficiency for the random assignment problem and illustrate a hierarchy between them. In a rank-minimizing assignment, agents receive objects with a minimum rank on average. An ex-post rank efficient random assignment has at least one lottery over only rank efficient deterministic assignments. Thus, it could still have another lottery with some rank-dominated deterministic assignments in its support. If each deterministic assignment in any decomposition of a random assignment is rank efficient, we call it a robust ex-post rank efficient assignment. We demonstrate that rank-minimizing implies rank efficiency, which indicates (robust) ex-post rank efficiency. Moreover, we introduce a mechanism that provides an ex-post rank efficient random assignment. We also prove that ex-post rank efficiency is incompatible with strategyproofness or fairness in the sense of weak envy-freeness and equal division lower bound.