Asymptotic ordinal inefficiency of random serial dictatorship

We establish that the fraction of preference profiles for which the random serial dictatorship allocation is ordinally efficient vanishes for allocation problems with many object types. We consider also a probabilistic setting where in expectation agents have moderately similar preferences reflecting varying popularity across objects. In this setting we show that the probability that the random serial dictatorship mechanism is ordinally efficient converges to zero as the number of object types becomes large. We provide results with similarly negative content for allocation problems with many objects of each type. One corollary is that ordinal efficiency is a strict refinement of ex-post efficiency at most preference profiles.