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This paper studies the problem of assigning a set of indivisible objects to a set of agents when monetary transfers are not allowed and agents reveal only ordinal preferences, but random assignments are possible. We offer two characterizations of the probabilistic serial mechanism, which assigns lotteries over objects. We show that it is the only mechanism that satisfies non-wastefulness and ordinal fairness, and the only mechanism that satisfies sd-efficiency, sd-envy-freeness, and weak invariance or weak truncation robustness (where "sd" stands for first-order stochastic dominance). 6 We thank an anonymous referee for suggesting that we weaken HH's definition of truncation robustness to the current definition (Definition 3). Upon showing that this new definition is strong enough to characterize PS, we observed that the proof also extends to the general case where the null object may not exist. This motivated us to obtain our second characterization result using the current definition of weak invariance (Definition 2), which is the counterpart of Definition 3 in environments without the null object. 7 Also, our proof immediately implies that we can weaken sd-efficiency in Theorem 2 and Corollary 2 as in HH and BH. 8 A more recent paper by Heo and Yılmaz (2012) extends the results of BH to the case with weak preferences for Katta and Sethuraman's (2006) extended probabilistic serial correspondence. 9 This axiom was previously introduced by Heo (2013) as one of her auxiliary axioms. She referred to it as "limited invariance."

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 introducea strong one and a weak one-and we show that these mechanisms are optimal within the class of mechanisms that satisfy these respective criteria.