Capacity Planning in Stable Matching: An Application to School Choice

Assignment mechanisms for many-to-one matching markets with preferences revolve around the key concept of stability. Using school choice as our matching market application, we introduce the problem of jointly allocating a school capacity expansion and finding the best stable allocation for the students in the expanded market. We analyze theoretically the problem, focusing on the trade-off behind the multiplicity of student-optimal assignments, the incentive properties, and the problem's complexity. Due to the impossibility of efficiently solving the problem with classical methods, we generalize existent mathematical programming formulations of stability constraints to our setting, most of which result in integer quadratically-constrained programs. In addition, we propose a novel mixed-integer linear programming formulation that is exponentially-large on the problem size. We show that its stability constraints can be separated in linear time, leading to an effective cutting-plane method. We evaluate the performance of our approaches in a detailed computational study, and we find that our cutting-plane method outperforms mixed-integer programming solvers applied to the formulations obtained by extending existing approaches. We also propose two heuristics that are effective for large instances of the problem. Finally, we use the Chilean school choice system data to demonstrate the impact of capacity planning under stability conditions. Our results show that each additional school seat can benefit multiple students. Moreover, our methodology can prioritize the assignment of previously unassigned students or improve the assignment of several students through improvement chains. These insights empower the decision-maker in tuning the matching algorithm to provide a fair application-oriented solution.