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. Due to 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 every student gets assigned to some school. Finally, we discuss several extensions of our framework, showcasing its flexibility to accommodate different needs.

翻译：受智利学校选择系统面临的席位短缺问题启发，本文提出了在扩展市场中联合增加学校容量并寻找学生最优分配的问题。鉴于该问题的理论与实际复杂性，我们提供了一套完整的求解工具，包括不同的数学规划模型、割平面算法以及两种能够快速获得近似最优解的启发式方法。在理论层面，我们证明了所提模型的正确性，阐述了目标函数与可行域中便于计算的各类性质，并揭示了在容量扩展条件下寻找学生最优匹配的底层机制的若干特性。在计算层面，我们利用智利学校选择系统的数据验证了该框架的实际影响，并得出有助于缓解该问题的深刻见解。研究结果表明：每个新增席位可使多名学生受益；我们能够有效针对先前未分配的学生进行定向分配，或通过改进链提升多名学生的分配结果。然而，研究也发现每个新增席位的边际效应呈递减趋势，单纯增加席位不足以确保所有学生都能获得学校分配。最后，我们讨论了该框架的若干扩展方向，展示了其适应不同需求的灵活性。