Priority Design in Centralized Matching Markets

In many centralized matching markets, priorities take the form of a monotone transformation of an underlying order. Prominent examples include the distance-based system employed by Boston Public Schools, where students who lived within a walk-zone were prioritized over all others, and the income-based system used in New York affordable housing allocation, where eligibility is determined by a sharp income cutoff. Motivated by this, we study optimal priority design subject to not reversing an exogenously-given underlying order. Our main result is that, under stable matching mechanisms, the optimal design can be attained by splitting agents into at most three object-specific indifference classes. We apply our framework to provide insights into optimal priority design and rationalizations of the pursued priorities in three applications: distance-based priorities in Boston Public Schools, admissions tests for Chicago exam schools, and income-based priorities in New York public housing allocation.