Multi-Rank Smart Reserves

We study the school choice problem where each school has flexible multi-ranked diversity goals, and each student may belong to multiple overlapping types, and consumes only one of the positions reserved for their types. We propose a novel choice function and show that it is the unique rule that satisfies three fundamental properties: maximal diversity, non-wastefulness, and justified envy-freeness. We provide a fast polynomial-time algorithm for our choice function that is based on the Dulmage Mendelsohn Decomposition Theorem as well as new insights into the combinatorial structure of constrained rank maximal matchings. Even for the case of minimum and maximum quotas for types (that capture two ranks), ours is the first known polynomial-time approach to compute an optimally diverse choice outcome. Finally, we prove that the choice function we design for schools, satisfies substitutability and hence can be directly embedded in the generalized deferred acceptance algorithm to achieve strategyproofness and stability. Our algorithms and results have immediate policy implications and directly apply to a variety of scenarios, such as where hiring positions or scarce medical resources need to be allocated while taking into account diversity concerns or ethical principles.