Playing on a Level Field: Sincere and Sophisticated Players in the Boston Mechanism with a Coarse Priority Structure

Who gains and who loses from a manipulable school choice mechanism? We examine this question with a focus on the outcomes for sincere and sophisticated students, and present results concerning their absolute and relative gains under the manipulable Boston Mechanism (BM) as compared with the strategy-proof Deferred Acceptance (DA). The absolute gain of a student of a certain type is the difference between her expected utility under (an equilibrium of) BM and her utility under (the dominant-strategy equilibrium of) DA. Holding everything else constant, one type of a player has relative gain with respect to another type if her absolute gain is higher. Prior theoretical works presented inconclusive results regarding the absolute gains of both types of students, and predicted (or assumed) positive relative gains for sophisticated types compared to sincere types. The empirical evidence is also mixed, with different markets exhibiting very different behaviors. We extend the previous results and explain the inconsistent empirical findings using a large random market approach. We provide robust and generic results of the "anything goes" variety for markets with a coarse priority structure. That is, in such markets there are many sincere and sophisticated students who prefer BM to DA (positive absolute gain), and vice versa (negative absolute gain). Furthermore, some populations may even get a relative gain from being sincere (and being perceived as such). We conclude by studying market forces that can influence the choice between the two mechanisms.