  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.
