Using lotteries is a common tool for allocation of indivisible goods. Since obtaining preferences over lotteries is often difficult, real-life mechanisms usually rely on ordinal preferences.
Random priority is one such mechanism in the context of house allocation problems.
Motivated by this observation they define a random assignment to be ordinally efficient if and only if it is not stochastically dominated. In this paper we investigate the relation between ex-post efficiency and ordinal efficiency. We introduce a new notion of domination defined over sets of assignments and show that a lottery induces an ordinally efficient random assignment if and only if it is not supported by a dominated set of assignments. We also illustrate that these notions have interesting implications in other models as well.