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Empirical and experimental evidence suggests different levels of sophistication among families in the Boston Public School student assignment plan. In this paper, we analyze the Nash equilibria of the preference revelation game induced by the Boston mechanism when there are two types of players. Sincere players are restricted to report their true preferences, while sophisticated players play a best response. We characterize the set of Nash equilibrium outcomes as the set of stable matchings of an economy with a modified priority structure, where sincere students lose their priority to sophisticated students. While there are multiple equilibrium outcomes, a sincere student receives the same assignment in all equilibria. Moreover any sophisticated student weakly prefers her assignment under the Pareto-dominant Nash equilibrium of the Boston mechanism to her assignment under the student-optimal stable mechanism, which was recently adopted by BPS for use starting with 2005-2006 school year.