  School choice is the two-sided matching market where students (on one side)
are to be matched with schools (on the other side) based on their mutual
preferences. The classical algorithm to solve this problem is the celebrated
deferred acceptance procedure, proposed by Gale and Shapley. After both sides
have revealed their mutual preferences, the algorithm computes an optimal
stable matching. Most often in practice, notably when the process is
implemented by a national clearinghouse and thousands of schools enter the
market, there is a quota on the number of applications that a student can
submit: students have to perform a partial revelation of their preferences,
based on partial information on the market. We model this situation by drawing
each student type from a publicly known distribution and study Nash equilibria
of the corresponding Bayesian game. We focus on symmetric equilibria, in which
all students play the same strategy. We show existence of these equilibria in
the general case, and provide two algorithms to compute such equilibria under
additional assumptions, including the case where schools have identical
preferences over students.
