We use data from an online platform that centralizes a daycare (0-3-year-olds) matching process. We estimate the preferences of parents and the priorities of nurseries using data from parents’ rank-ordered lists and the acceptance decisions of nurseries. To allow parents to be strategic, we provide a novel estimation approach using insights from the dynamic discrete choice framework. We then use the estimates to compare recently developed matching policies that are adapted to the specific nature of daycare. We find that disadvantaged families benefit from reducing the role of nursery priorities in the matching process. Affirmative action policies strengthen this further. Advantaged families can be compensated for their loss by a modest increase in the available capacity. Affirmative action also increases segregation between both groups as we find that this is mainly driven by geographic segregation and the demand for nearby day care options.