We study random assignment of indivisible objects among a set of agents, when each agent is to receive one object and has strict preferences over the objects. Random Serial Dictatorship (RSD) satisfies equal treatment of equals, ex post efficiency, and strategy‐proofness. Answering a longstanding open question, we show that RSD is not characterized by those properties—there are other mechanisms satisfying equal treatment of equals, ex post efficiency, and strategy‐proofness which are not welfare‐equivalent to RSD. On the other hand, we show that RSD is not Pareto dominated by any mechanism that is (i) strategy‐proof and (ii) boundedly invariant. Moreover, the same holds for all mechanisms that are ex post efficient, strategy‐proof, and boundedly invariant: no such mechanism is dominated by any other mechanism that is strategy‐proof and boundedly invariant.