Capacity Planning in Stable Matching: An Application to School Choice
Bibliographic record
Abstract
Centralized mechanisms are becoming the standard approach to solve several assignment problems. Examples include the allocation of students to schools (school choice), high-school graduates to colleges, residents to hospitals and refugees to cities. In most of these markets, a desirable property of the assignment is stability, which guarantees that no pair of agents has incentive to circumvent the matching. Using school choice as our matching market application, we introduce the problem of jointly allocating a school capacity expansion and finding the best stable matching for the students in the expanded market. We analyze theoretically the problem, focusing on the trade-off behind the multiplicity of student-optimal assignments, and the problem complexity. Since the theoretical intractability of the problem precludes the adaptation of classical approaches to solve it efficiently, we generalize existent mathematical programming formulations of stability constraints to our setting. These generalizations result in integer quadratically-constrained programs, which are computationally hard to solve. In addition, we propose a novel mixed-integer linear programming formulation that is exponentially-large on the problem size. We show that the 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 existent formulations extended to our problem setting. 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. On the one hand, we can focus on access by prioritizing extra seats that benefit previously unassigned students; on the other hand, we can focus on merit by allocating extra seats that benefit several students via chains of improvement. These insights empower the decision-maker in tuning the matching algorithm to provide a fair application-oriented solution.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.011 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".