[Paper Review] A Cost-Minimizing Algorithm for School Choice

Sinan G. Aksoy, Adam Azzam+5 more|arXiv (Cornell University)|Oct 12, 2010

Game Theory and Voting Systems24 references3 citations

TL;DR

This paper proposes a cost-minimizing algorithm for school choice that prioritizes student preferences over school priorities by redefining the assignment problem as a minimum-cost matching using a modified Hungarian algorithm. The method minimizes a preference-based cost index, producing matchings that better reflect student rankings while allowing controlled trade-offs with stability through priority violations.

ABSTRACT

The school choice problem concerns the design and implementation of matching mechanisms that produce school assignments for students within a given public school district. In this note we define a simple student-optimal criterion that is not met by any previously employed mechanism in the school choice literature. We then use this criterion to adapt a well-known combinatorial optimization technique (Hungarian algorithm) to the school choice problem.

Motivation & Objective

• To address inefficiencies in existing school choice mechanisms that prioritize stability over student preferences.
• To introduce a new evaluation criterion—preference index—that quantifies how well student preferences are met in a matching.
• To adapt the Hungarian algorithm to the school choice problem by incorporating school capacities and incomplete preference profiles.
• To enable a trade-off analysis between efficiency (student preference satisfaction) and stability (priority violations).
• To provide a mechanism that produces matchings minimizing student dissatisfaction while remaining transparent and computationally feasible.

Proposed method

• The preference index is defined as the sum of ranks assigned to each student’s school, with lower values indicating better alignment with preferences.
• The Hungarian algorithm is applied to the assignment problem after transforming the preference matrix to represent costs, with school capacities enforced via appropriate matrix modifications.
• The algorithm is adapted to handle incomplete preference lists by 'completing' them with penalty values for unlisted schools.
• Multiple minimum-cost solutions are identified using extensions of the Hungarian algorithm to ensure robustness in case of ties.
• A selection rule is proposed to choose among multiple optimal solutions: minimize variance in individual student indices or minimize priority violations.
• The mechanism allows for controlled trade-offs between efficiency (low preference index) and stability (few priority violations).

Experimental results

Research questions

• RQ1 Can a school choice mechanism be designed that prioritizes student preferences while still maintaining a reasonable level of stability?
• RQ2 To what extent can efficiency gains be achieved by allowing a small number of priority violations?
• RQ3 How does the preference index compare to traditional criteria like stability and strategyproofness in evaluating matching quality?
• RQ4 What is the impact of preference list completion and matrix transformation on the output of the Hungarian algorithm in school choice?
• RQ5 How can multiple optimal solutions be identified and selected meaningfully in the context of cost-minimizing school assignment?

Key findings

• The proposed mechanism, based on the preference index, produces matchings that often outperform standard mechanisms in terms of student preference satisfaction.
• The Hungarian algorithm can be successfully adapted to school choice by transforming preference rankings into cost matrices and enforcing capacity constraints.
• Multiple minimum-cost solutions may exist, but the mechanism can identify all such solutions, ensuring consistency regardless of input ordering.
• When multiple optimal solutions exist, selecting the one with minimal variance in individual student indices promotes fairness across students.
• The mechanism allows for a quantifiable trade-off between efficiency (low preference index) and stability (few priority violations), enabling nuanced policy decisions.
• The method provides a transparent, computationally feasible alternative to existing mechanisms, particularly beneficial in districts where student preferences are a primary concern.