Natural Interviewing Equilibria in Matching Settings

While matching markets are ubiquitous, much of the work on stable matching assumes that both sides of the market are able to fully specify their preferences. However, in real-world cases with many participants, this assumption becomes unreasonable. When agents are faced with a huge number of alternatives, which they only have a vague familiarity with, they focus on the alternatives they consider likely, and study them more carefully to end up with a more refined preference order between them. In this paper, we use the terminology of one of the large matching markets – hospitals and residents – which use, in many cases, Gale-Shapley’s Deferred Acceptance mechanism. More specifically, we have one side (e.g., hospitals) with a common preference master list (based on grades, for example), while the other side (e.g., residents) have individual preferences drawn from some distribution, but they have a level of uncertainty regarding them (due, for example, to things like personal chemistry with doctors in a particular hospital). The residents can choose a subset of alternatives to Email addresses: bor@cs.toronto.edu (Allan Borodin), jdrummond@cs.toronto.edu (Joanna Drummond), kate.larson@uwaterloo.ca (Kate Larson), omerlev@bgu.ac.il (Omer Lev) URL: http://www.cs.toronto.edu/~bor/ (Allan Borodin), https://cs.uwaterloo.ca/~klarson/ (Kate Larson), http://bgu.ac.il/~omerlev (Omer Lev) Preprint submitted to Games and Economic Behavior November 9, 2021 interview in, allowing them to gain information on potentially relevant hospitals and refine their preferences. We investigate the structure of the Nash equilibria of such settings when preferences are drawn from a Mallows model. One possible outcome is an assortative equilibrium, i.e., the top k residents interview with the top k hospitals, the k second tier residents interview with k second tier hospitals, and so on. Surprisingly, we prove that such an outcome is quite rare – it exists when k is a small number (2,3), but for k ≥ 4, an assortative equilibria is impossible if all agents are quite aligned (i.e., Mallows’ parameter close to 0). Moreover, we examine simulations on the possible outcomes of a Nash equilibrium, and discover that it seems that residents may be pursuing a reach/safety strategy, in which one interviews in several schools which are about their “real” level, as well as several which a rung above their own level.