A New Impossibility Result for Random Assignments

We know that ex-post Pareto efficiency is possible with strategy-proofness and weak envy-freeness while ordinally efficiency is incompatible with them. We strengthen this impossibility for random assignments and show that it prevails if ordinal efficiency is weakened to robust ex-post Pareto efficiency, an intermediate notion of efficiency weaker than ordinal efficiency and stronger than ex-post Pareto efficiency. An assignment is robust ex-post Pareto efficient whenever for all of its lottery decomposition, each deterministic assignment in its support is Pareto efficient