On the Efficiency of Stable Matchings in Large Markets

Stability is often the goal for matching clearinghouses, such as those matching residents to hospitals, students to schools, etc. We study the wedge between stability and utilitarian efficiency in large one-to-one matching markets. We distinguish between stable matchings’ average efficiency (or, efficiency per-person), which is maximal asymptotically for a rich preference class, and their aggregate efficiency, which is not. The speed at which average efficiency of stable matchings converges to its optimum depends on the underlying preferences. Furthermore, for severely imbalanced markets governed by idiosyncratic preferences, or when preferences are sub-modular, stable outcomes may be average inefficient asymptotically. Our results can guide market designers who care about efficiency as to when standard stable mechanisms are  desirable and when new mechanisms, or the availability of transfers, might be useful.