School choice with controlled choice constraints: Hard bounds versus soft bounds
To encourage diversity, schools often “reserve” some slots for students of specific types. Students only care about their school assignments and contractual terms like tuition, and hence are indifferent among slots within a school. Because these indifferences can be resolved in multiple ways, they present an opportunity for novel market design. We introduce a two-sided, many-to-one matching with contracts model in which agents with unit demand match to branches, which may have multiple slots available to accept contracts. Each slot has its own linear priority order over contracts; a branch chooses contracts by filling its slots sequentially. We demonstrate that in these matching markets with slot-specific priorities, branches’ choice functions may not satisfy the substitutability conditions typically crucial for matching with contracts. Despite this complication, we are able to show that stable outcomes exist in this framework and can be found by a cumulative offer mechanism that is strategy-proof and respects unambiguous improvements in priority. Our results provide insight into the design of transparent affirmative action matching mechanisms, and show the value of a seemingly ad hoc administrative decision in the United States Military Academy’s branch-of-choice program. JEL classification: C78, D63, D78.