Ex-Ante vs. Ex-Post Efficiency in Matching

Various forms of efficiency exist in the context of two-sided matching. Following Hylland and Zeckhauser (1979), an assignment is called ex-post efficient if no other deterministic assignment is improving on it; and ex-ante efficient if no lottery over deterministic assignments is. In the context of the "housing problem," where only one side of the market has preferences, it can be shown that these two notions are equivalent for deterministic assignments, and we show that efficient assignments (either ex-ante or ex-post) maximize a weighted sum of the individual utilities.Surprisingly, this is no longer the case when the two sides of the markets have preferences, in the "marriage problem." In this case ex-ante efficiency implies ex-post efficiency, but the converse fails: there are assignments that are ex-post efficient that are not ex-ante efficient. To explain this, we show that in the marriage problem, ex-ante efficient assignments still maximize a weighted sum of the individual utilities; but ex-post efficient assignments maximize a weighted sum of the minimum of the utilities of the partners across a pair. The results obtained in this paper offer a new perspective on the bridge (and the gap) between Transferable Utility (TU) and Non-Transferable Utility (NTU) matching models