Multi-District School Choice: Playing on Several Fields

We extend the seminal model of Pathak and S\"onmez (2008) to a setting with multiple school districts, each running its own separate centralized match, and focus on the case of two districts. In our setting, in addition to each student being either sincere or sophisticated, she is also either constrained - able to apply only to schools within her own district of residence - or unconstrained - able to choose any single district within which to apply. We show that several key results from Pathak and S\"onmez (2008) qualitatively flip: A sophisticated student may prefer for a sincere student to become sophisticated, and a sophisticated student may prefer for her own district to use Deferred Acceptance over the Boston Mechanism, irrespective of the mechanism used by the other district. We furthermore investigate the preferences of students over the constraint levels of other students. Many of these phenomena appear abundantly in large random markets.