The Chicago Board of Education is implementing a centralized clearinghouse to assign students to schools for 2018-19 admissions. In this clearinghouse, each student can simultaneously be admitted to a selective and a nonselective school. We study this divided enrollment system and show that an alternative unified enrollment system, which assigns each student to only one school, is better for all students. We also examine systems with two stages of admissions, which has also been considered in Chicago, and establish conditions under which the unified enrollment system is better than the divided enrollment system.

In several matching markets, in order to achieve diversity, agents' priorities are allowed to vary across an institution’s available seats, and the institution is let to choose agents in a lexicographic fashion based on a predetermined ordering of the seats, called a lexicographic choice rule. Lexicographic choice rules have been particularly useful in achieving diversity at schools while allocating school seats. We provide a characterization of lexicographic choice rules, which reveals their distinguishing properties from other plausible choice rules. Moreover, we study the market design implications of using lexicographic choice rules and provide a characterization of deferred acceptance mechanisms that operate based on a lexicographic choice structure. We also discuss some implications for the Boston school choice system and show that our analysis can be helpful in applications to select among plausible choice rules.

We show that there is no consistent Pareto improvement over any stable mechanism. To overcome this impossibility, we introduce the following weak consistency requirement: Whenever a set of students, each of whom is assigned to a school that is under-demanded at the student-optimal stable matching, are removed with their assigned seats, then the assignments of the remaining students should not change. We show that EADA (Kesten, 2010) is the unique mechanism that is weakly consistent and Pareto improves over the student-optimal stable mechanism.

Each *acceptant *and *substitutable *choice rule is known to have a *maximizer-collecting* representation: there exists a list of priority orderings such that from each choice set that includes more elements than the capacity, the choice is the union of the priority orderings' maximizers (Aizerman and Malishevski, 1981). We introduce the notion of a *prime atom* and constructively prove that the number of prime atoms of a choice rule determines its smallest size maximizer-collecting representation. We show that responsive choice rules require the maximal number of priority orderings in their maximizer-collecting representations among all acceptant and substitutable choice rules. We characterize maximizer-collecting choice rules in which the number of priorities equals the capacity. We also show that if the capacity is greater than three and the number of elements exceeds the capacity by at least two, then no acceptant and substitutable choice rule has a maximizer-collecting representation of the size equal to the capacity.

Which mechanism to use to allocate school seats to students still remains a question of hot debate. Meanwhile, immediate acceptance mechanisms remain popular in many school districts. We formalize desirable properties of mechanisms when respecting the relative rank of a school among the students' preferences is crucial. We show that those properties, together with well-known desirable resource allocation properties, characterize immediate acceptance mechanisms. Moreover, we show that replacing one of the properties, consistency, with a weaker property, non-bossiness, leads to a characterization of a much larger class of mechanisms, which we call choice-based immediate acceptance mechanisms. It turns out that certain objectives that are not achievable with immediate acceptance mechanisms, such as affirmative action, can be achieved with a choice-based immediate acceptance mechanism.

For probabilistic assignment of objects, when only ordinal preference information is available, we introduce an efficiency criterion based on the following domination relation: a probabilistic assignment dominates another assignment if, whenever the latter assignment is ex-ante efficient at a utility profile consistent with the ordinal preferences, the former assignment is ex-ante efficient too; and there is a utility profile consistent with the ordinal preferences at which the latter assignment is not ex-ante efficient but the former assignment is ex-ante efficient. We provide a simple characterization of this domination relation. We revisit an extensively studied assignment mechanism, the Probabilistic Serial mechanism (Bogomolnaia and Moulin, 2001), which always chooses a "fair" assignment. We show that the Probabilistic Serial assignment may be dominated by another fair assignment. We provide an almost full characterization of the preference profiles at which the serial assignment is undominated among fair assignments.

How to control controlled school choice: comment [ssrn] [repec] [publisher]

*American Economic Review, 107, 1362-64, 2017.*

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