School choice: How transferable characteristics can improve seat allocation

How can a school system address equity and efficiency concerns while remaining fair?

Based on Research by Carmelo Rodríguez-Álvarez and Antonio Romero-Medina

Imagine you are a parent searching for the best school for your child, but the allocation system seems rigid and full of constraints. This is not just an individual concern; school seat allocation policies are the subject of ongoing debate.

The study on how a system can allocate children to schools as a mechanism design problem was introduced by Abdulkadiroglu and Sönmez (2003). In school choice systems, students are assigned to schools based on their preferences and the schools’ priorities. These priorities often rely on factors like proximity to the school, family socioeconomic characteristics, and tiebreaker devices.

The immediate acceptance (IA) algorithm was used in the Boston area at the time. According to IA, each school assigns its seats to the applicants with higher priority till its vacant positions are filled, and these allocations are definitive. In subsequent stages, students who aren’t assigned in the first stage repeat the process with the schools with available positions. The IA algorithm produces unfair allocations of students to seats, i.e., some students may prefer to be assigned to a school where another student with less priority for that school has been admitted. A version of this procedure is currently used in most Spanish cities. An alternative to IA is the most commonly used algorithm, the deferred acceptance (DA) algorithm. DA was adopted in the school choice reform in Boston and is the standard procedure in most American cities. According to DA, admissions in each round are tentative. A second round student may claim a position in a school if she has a higher priority than students tentatively admitted in previous stages. The DA algorithm always obtains fair allocations and families can safely declare their true preferences in their applications. This implies that the school districts have the correct welfare-relevant information to make the social choice under DA.

Since schools are just objects in school choice problems, a group of students could improve their assignments without harming others, but the current system does not allow it, neither under IA nor under DA. In general, allocation protocols such as DAgenerate fair but inefficient outcomes. This is where transferable characteristics come into play to find a balance.

Transferable Characteristics: A New Approach

Rodríguez-Álvarez and Romero-Medina (2024) explores a novel approach to the school choice problem: using transferable student characteristics to improve assignments. We propose a class of algorithms called Student Exchange with Transferable Characteristics (SETC).

In IA or DA student preferences and school priorities as the model’s primitives. However, school priorities are rankings of the applicants and are derived from students’ characteristics. These characteristics generally include neighborhood, number of siblings in the school, other socioeconomic factors, and, eventually, tiebreaking devices. Schools’ priorities reflect district distributional objectives in the allocation process. We generalize the model, taking student characteristics as primitives and introducing the idea that specific student characteristics can be traded among students, redefining schools’ priorities. After the trade, the resulting priorities with the new distribution of characteristics can justify a new fair allocation that is a Pareto improvement on the initial output of the DA algorithm, alleviating the tension between fairness and efficiency.

_{Figure 1: SETC algorithms}

^{The SETC algorithms begin with an initial matching of students to schools, as obtained through DA, and iteratively implement fair Pareto improvements until no further improvement can be made without violating school priorities.}

How the Algorithm Works

The SETC algorithm operates in a series of steps:

Initial Matching: Students are assigned to schools based on existing priorities and characteristics.
Fair Pareto Improvements: The algorithm identifies opportunities for two or more students to exchange characteristics to achieve mutually beneficial outcomes.
Iterative Optimization: These exchanges are repeated until no further improvements can be made without violating the rules of fairness.

Given the system’s constraints, this iterative process ensures that the final matching is fair and as efficient as possible.

What makes the SETC algorithm particularly innovative is its ability to incorporate the flexibility of exchanges while preserving the fairness of assignments. For instance, two students may exchange their lottery numbers at different schools, allowing both to secure seats at more preferred schools without generating justified envy. Each exchange respects the predefined priorities of schools, ensuring that no student with a legitimate claim to a seat is displaced unfairly. The result is a balance between fairness and efficiency that traditional models struggle to achieve.

Moreover, the SETC approach highlights an essential distinction: the difference between procedural criteria (such as tiebreaking lotteries to create strict rankings of students) and distributional constraints (such as ensuring siblings attend the same school) in the definitions of schools’ priority rankings. By allowing selective transfers of characteristics, the algorithm avoids the inefficiencies that arise from rigid rules while maintaining overall equity. Of course, implementing the SETC algorithms begs relevant ethical questions: Should any characteristic be transferable? While a lottery number may seem harmless, other characteristics, such as socioeconomic indicators or geographic preferences, could raise concerns about equity and fairness. Policymakers will need to carefully consider which characteristics are appropriate for exchange. Some characteristics may help to determine an initial allocation but are not directly related to the set of ethical and distributional concerns that the school district aims to preserve through the schools’ priorities. The tiebreaker exemplifies this distinction. Beyond the tiebreaking lottery, many school districts allow school-specific criteria, for example an extra priority for legacy students. However, those arbitrary tiebreakers or school-specific criteria have no welfare-improving characteristics as opposed to characteristics such as the walk zone, which creates a better neighborhood or the sibling’s priority, simplifying family organization. Therefore, those arbitrary tiebreakers are optimal candidates to be considered transferable characteristics to allow for welfare improvements.

Practical Applications and Policy Relevance

The SETC algorithm is computationally efficient, making it suitable for real-world applications. For example, in Amsterdam’s 2014 high school assignment reform, families sought to exchange school seats after the initial allocation. However, this proposal was dismissed due to its complexity. The SETC approach could offer a feasible and transparent solution by formalizing such exchanges into a structured algorithm.

Another example involves walk-zone redistricting. In cities like Madrid, removing walk-zone priorities created winners and losers. A model incorporating transferable characteristics could introduce efficiency improvements while respecting specific local priorities, such as proximity or sibling reunification. This flexibility could allow districts to fine-tune their allocation systems to meet community needs better while avoiding unnecessary disruptions.

In the context of school systems, the SETC algorithm could also be extended to account for more complex characteristics, such as extracurricular preferences or language requirements. This adaptability makes it a versatile tool for policymakers looking to address the diverse needs of students and schools.

Our model has potential application in other resource allocation problems, such as organ transplantation programs or housing lotteries. In these cases, exchanging characteristics such as waiting list positions or geographic preferences could improve individuals’ outcomes without compromising the integrity of the allocation process.

Toward a More Efficient and Fair School Policy

So far, we have focused on fair improvements over DA outcomes. Many school districts, as in Madrid, still use mechanisms based on IA that may propose unfair allocations. Almost 90% of families in Madrid obtain a school seat in their first application. However, the IA–based mechanism is intricate for parents to play with. It makes the families strategize, and family applications cannot be considered representative of their true preferences. Since families have incentives to avoid applying to their preferred schools to secure a seat in a sufficiently satisfying alternative, the allocation may not respect students’ priorities, and there may be room for Pareto improvements. Moreover, new slots appear after the allocation of seats closes because either the school districts open new vacant positions in over demanded schools or some students liberate their positions when moving to the private education system. Then school districts run an informal adjustment process to distribute those new vacant positions in a scramble phase. The SETC algorithm can be implemented as that post-allocation adjustment phase, offering a possibility for improvement to the students who wish to participate. The SETC algorithm can improve fairness by offering the willing participant fair Pareto improvements through a transparent protocol with a minimal adjustment to the current design, much in the spirit of Sönmez (2024).

Ultimately, the SETC framework provides a powerful tool for addressing long-standing inefficiencies in allocation systems. Its adaptability and fairness make it a promising solution for school choice and broader policy challenges. Innovative solutions like the SETC model are a step in the right direction, providing a road map for balancing fairness and efficiency in complex allocation problems.

About the Authors:

Carmelo Rodríguez-Álvarez is an Associate Professor at ICAE – Universidad Complutense de Madrid. His research focuses on Social Choice, Strategic Voting, Market Design, and Matching.
https://produccioncientifica.ucm.es/investigadores/141315/detalle

Antonio Romero-Medina is an Associate Professor at Universidad Carlos III de Madrid. His research interests are Matching and Market Design and their applications to education and health.
https://sites.google.com/site/antonioromeromedina/arm