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Capacity Planning in Stable Matching

Federico Bobbio, Margarida Carvalho, Andrea Lodi, Ignacio Rios, Alfredo Torrico

TL;DR

This work studies capacity planning in stable matching by jointly deciding capacity expansions and a student-optimal assignment. It develops two exact formulations—a compact formulation with $L$-constraints and a generalized comb formulation with a cutting-plane algorithm—and two heuristics (Greedy and LP-based Heuristic, LPH) to obtain near-optimal solutions efficiently. The authors prove correctness, derive structural properties, and demonstrate substantial practical gains using Chilean Pre-K data from Antofagasta, showing that extra seats benefit multiple students but with diminishing returns, and that the penalty on unassigned students shapes the balance between access and improvement. The framework is flexible for policy analysis and adaptable to other domains (refugee placement, tuition waivers, healthcare/resource allocation), offering actionable insights into capacity expansions and their distributional effects.

Abstract

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.

Capacity Planning in Stable Matching

TL;DR

This work studies capacity planning in stable matching by jointly deciding capacity expansions and a student-optimal assignment. It develops two exact formulations—a compact formulation with -constraints and a generalized comb formulation with a cutting-plane algorithm—and two heuristics (Greedy and LP-based Heuristic, LPH) to obtain near-optimal solutions efficiently. The authors prove correctness, derive structural properties, and demonstrate substantial practical gains using Chilean Pre-K data from Antofagasta, showing that extra seats benefit multiple students but with diminishing returns, and that the penalty on unassigned students shapes the balance between access and improvement. The framework is flexible for policy analysis and adaptable to other domains (refugee placement, tuition waivers, healthcare/resource allocation), offering actionable insights into capacity expansions and their distributional effects.

Abstract

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.

Paper Structure

This paper contains 67 sections, 17 theorems, 34 equations, 7 figures, 3 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Model and mechanism analysis.
  4. Exact solution methods.
  5. Heuristic solution methods.
  6. Practical insights and societal impact.
  7. Organization
  8. Related Work
  9. Mathematical programming formulations.
  10. Capacity expansion.
  11. School choice.
  12. Model
  13. Stable Matching Problem
  14. Capacity Expansion
  15. Compact Formulation
  16. ...and 52 more sections

Key Result

Lemma 1

Fix $\mathbf{t}\in{\mathbb Z}_+^{{\mathcal{C}}}$. Then, minimizing $\sum_{(s,c)\in \mu} r_{s,c}$ over the space of stable matchings $\mu$ in $\Gamma_{\mathbf{t}}$ is equivalent to finding the unique student-optimal stable matching in $\Gamma_{\mathbf{t}}$.

Figures (7)

  • Figure 1: Example of a subset of the admissions graph when school $c$ has capacity 3.
  • Figure 2: Effect of the Budget on Students' Outcome
  • Figure 3: Effect for Schools
  • Figure 4: Effect of Bounds on Expansion
  • Figure 5: Effect of Using Previous Preferences
  • ...and 2 more figures

Theorems & Definitions (25)

  • Lemma 1
  • Remark 1
  • Lemma 2
  • Definition 1: Comb
  • Example 1
  • Definition 2: Generalized Combs
  • Theorem 1
  • Remark 2
  • Remark 3
  • Corollary 1
  • ...and 15 more