We use data from a platform that centralizes a day care matching process. We estimate parents' preferences and nursery priorities by analyzing parents' rank-ordered lists and nurseries' acceptance decisions. We account for strategic behavior by using a novel estimation approach inspired by the dynamic discrete choice framework. We use the estimates to evaluate centralized matching policies tailored to the day care setting. We compare mechanisms and assess the effects of subsidies, increased capacity, and affirmative action. We find that affirmative action policies are crucial for boosting the participation of disadvantaged children, though they increase segregation due to location-based preferences.