Capacity Planning in Stable Matching

Motivated by the shortage of seats that the Chilean school choice system is facing, we introduce the problem of jointly increasing school capacities and finding a student-optimal assignment in the expanded market. Because of the theoretical and practical complexity of the problem, we provide a comprehensive set of tools to solve the problem, including different mathematical programming formulations, a cutting-plane algorithm, and two heuristics that allow obtaining near-optimal solutions quickly. On the theoretical side, we show the correctness of our formulations, different properties of the objective and feasible region that facilitate computation, and also several properties of the underlying mechanism to find a student-optimal matching under capacity expansions. On the computational side, we use data from the Chilean school choice system to demonstrate the impact of our framework and derive insights that could help alleviate the problem. Our results show that each additional seat can benefit multiple students and that we can effectively target the assignment of previously unassigned students or improve the assignment of several students through improvement chains. Nevertheless, our results show that the marginal effect of each additional seat is decreasing and that simply adding seats is insufficient to ensure that every student gets assigned to some school. Finally, we discuss several extensions of our framework, showcasing its flexibility to accommodate different needs.

Funding: This work was supported by IVADO Fond de Recherche du Quebec (FRQ)-Institut de Valorisation des Données (IVADO) Research Chair in Data Science for Combinatorial Game Theory); Fonds de recherche du Québec (FRQ-IVADO Research Chair) and the Natural Sciences and Engineering Research Council of Canada [Grant 2019-04557].

Supplemental Review: The empirical results in this paper were replicated. The code, data, and files required to reproduce the results were reviewed and are available at 10.1287/opre.2023.0386.

Supplemental Material: This article includes an online appendix, computer code, and data supporting the study’s findings, and replication files. All supplemental materials, including the code, data, and files required to reproduce the results, are available at https://doi.org/10.1287/opre.2023.0386.