Multi-District School Choice: When Sincere Students Stay and Sophisticated Students Stray

Abstract

We extend the canonical school choice problem of market design by proposing a model in which there are multiple nearby school districts, which may use different matching mechanisms, and some students are able to choose which district to enroll in. Specifically, we lean on the Pathak and Sönmez (2008) formulation of sincere and sophisticated players to distinguish students: sincere students remain in their district of residence and rank the district’s schools according to their true preferences, while sophisticated students strategize by choosing which district to enroll in and how to rank the schools in that district. Using a two-district model, we show that a sophisticated student may be strictly better off when some sincere student becomes sophisticated, and that a sophisticated student may strictly prefer their district use Deferred Acceptance rather than the Boston Mechanism. These are inversions of results from the one-district setting, in which neither of these phenomena are possible. Additionally, in a large random market model, we present and prove probabilistic theorems describing the frequency with which such phenomena occur. These results highlight that district choice, which manifests as the real-life behavior of sending one’s children to school in a desirable district by either using a false address or paying a premium to move, plays an important role in shaping the playing field for both sophisticated and sincere students.