Robust Ex-Post Pareto Efficiency and Fairness in Random Assignments : Two Impossibility Results

A random assignment is robust ex-post Pareto efficient whenever for any of its lottery decomposition, each deterministic assignment in its support is Pareto efficient. We show that ordinal efficiency implies robust ex-post Pareto efficiency while the reverse does not hold. We know that strategy-proof and ordinal efficient mechanisms satisfy neither equal treatment of equals nor equal division lower bound. We prove that it is not possible to avoid these two impossibilities by weakening ordinal efficiency to robust ex-post Pareto efficiency