Natural Interviewing Equilibria for Stable Matching

Stable matching problems are ubiquitous, though much of the work on stable matching assumes that both sides of the market are able to fully specify their preference orderings. However, as the size of matching markets grow, this assumption becomes unrealistic, and so there has been interest in understanding how agents may use interviews to refine their preferences over subsets of alternatives. In this paper we study a market where one side (hospital residency programs) maintains a common preference master list, while the other side (residents) have idiosyncratic preferences which they can refine by conducting a limited number of interviews. The question we study is How should residents choose their interview sets, given the choices of others? We provide a payoff function for this imperfect information game, and find that this game always has a pure strategy equilibrium. Moreover, when residents are restricted to two interviews and their preferences are distributed according to a φ-Mallows model with low dispersion, there is a unique Bayesian equilibrium in which residents interview assortatively: each resident pair r2j , r2j+1 interviews with hospitals h2j , h2j+1. We observe that with high dispersion, assortative interviewing is not an equilibrium.