Strategy-proof and envy-free random assignment

We study the random assignment of indivisible objects among a set agents with strict preferences. show that there exists no mechanism which is unanimous, strategy-proof and envy-free. Weakening first requirement to q-unanimity – i.e., when every agent ranks different object at top, then each shall receive his most-preferred probability least q we satisfying strategy-proofness, envy-freeness ex-post weak non-wastefulness can be q-unanimous only for q≤2n (where n number agents). To demonstrate this bound tight, introduce new mechanism, Random-Dictatorship-cum-Equal-Division (RDcED), it achieves maximal all are acceptable. In addition, three agents, RDcED characterized by properties efficiency. If may unacceptable, strategy-proofness jointly incompatible even non-wastefulness.