Ex-post Rank Efficiency for the Random Assignment Problem

We introduce a new notion of ex-post efficiency for random assignment problems, namely ex-post rank efficiency that gives a maximal number of agents their favored objects. An ex-post rank efficient random assignment is a lottery over rank efficient deterministic assignments, in the sense of Featherstone (2020). We show that (ex-ante) rank efficiency implies ex-post rank efficiency but not vice versa. Moreover, ex-post rank efficiency and ordinal efficiency do not imply each other. Furthermore, there is no strategy-proof mechanism that always provides an ex-post rank efficient random assignment. Finally, we showed a modified version of the immediate acceptance mechanism, called rank-respected immediate acceptance, that provides an ex-post rank efficient random assignment