A theory of school-choice lotteries

We introduce a new notion of ex ante stability (or fairness) that would be desirable for a school-choice mechanism to satisfy. Our criterion stipulates that a mechanism must be stable based solely on the probabilities that each student will be assigned to different schools, i.e., the assignment must be viewed as stable even before students know which school they will end up going to. This is in contrast to much of the existing literature, which has instead focused on ex post stability, meaning that assignments are deemed stable after students are assigned to schools. Armed with this criterion for evaluating mechanisms, we show that one of the mechanisms that has attracted the most attention—deferred acceptance with random tie-breaking—is not ex ante stable and under some circumstances can lead to ex ante discrimination among some students. We then propose two new mechanisms, which satisfy two notions of ex ante stability we introduce— a strong one and a weak one—and we show that these mechanisms are optimal within the class of mechanisms that satisfy these respective criteria.