Distributed Interview Selection for Stable Matching in Large Random Markets

In real-world settings of the Deferred Acceptance stable matching algorithm, such as the American medical residency match (NRMP), school choice programs, and various national university entrance systems, candidates need to decide which programs to list. In many of these settings there is an initial phase of interviews or information gathering which affect the preferences on one or both sides. We ask: which interviews should candidates seek? We study this question in a model, introduced by Lee (2016) and modified by Allman and Ashlagi (2023), with preferences based on correlated cardinal utilities. We describe a distributed, low-communication strategy for the doctors and students, which lead to non-match rates of $e^{(-\widetilde{O}(\sqrt{k}))}$ in the residency setting and $e^{(-\widetilde{O}(k))}$ in the school-choice setting, where $k$ is the number of interviews per doctor in the first setting, and the number of proposals per student in the second setting; these bounds do not apply to the agents with the lowest public ratings, the bottommost agents, who may not fare as well. We also obtain bounds on the expected utilities each non-bottommost agent obtains. These results are parameterized by the capacity of the hospital programs and schools. Larger capacities improve the outcome for the hospitals and schools, but don't significantly affect the outcomes of the doctors or students. Finally, in the school choice setting we obtain an $\epsilon$-Nash type equilibrium for the students apart from the bottommost ones; importantly, the equilibrium holds regardless of the actions of the bottommost students. We also discuss to what extent this result extends to the residency setting. We complement our theoretical results with an experimental study that shows the asymptotic results hold for real-world values of $n$.