[Paper Review] School Choice as a One-Sided Matching Problem: Cardinal Utilities and Optimization

Sinan G. Aksoy, Alexander Adam Azzam+5 more|arXiv (Cornell University)|Apr 28, 2013

Game Theory and Voting Systems31 references3 citations

TL;DR

This paper proposes a one-sided matching mechanism for school choice that maximizes student cardinal utilities using a modified Hungarian algorithm, prioritizing student preferences over institutional priorities. While the mechanism improves Pareto efficiency and utility optimization, it fails to ensure strategyproofness and may produce unstable matchings due to ignored school priorities.

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. Previously considered criteria for evaluating proposed mechanisms such as stability, strategyproofness and Pareto efficiency do not always translate into desirable student assignments. In this note, we explore a class of one-sided, cardinal utility maximizing matching mechanisms focused exclusively on student preferences. We adapt a well-known combinatorial optimization technique (the Hungarian algorithm) as the kernel of this class of matching mechanisms. We find that, while such mechanisms can be adapted to meet desirable criteria not met by any previously employed mechanism in the school choice literature, they are not strategyproof. We discuss the practical implications and limitations of our approach at the end of the article.

Motivation & Objective

• To address inefficiencies in existing school choice mechanisms that prioritize stability and strategyproofness over student preference satisfaction.
• To explore one-sided matching mechanisms that focus exclusively on student preferences using cardinal utility transformations.
• To adapt the Hungarian algorithm for school assignment problems under capacity constraints and incomplete preference profiles.
• To evaluate the trade-offs between utility maximization, stability, and strategyproofness in school choice mechanisms.
• To provide a practical framework for policy makers seeking student-optimal assignments in districts with weak or no formal priority systems.

Proposed method

• Transform ordinal student preferences into cardinal utilities using utility functions f1 and f2, where f1 emphasizes rank minimization and f2 prioritizes minimizing total cost.
• Apply a modified Hungarian algorithm to solve the assignment problem as a cost-minimization task, ensuring school capacity constraints are respected.
• Complete incomplete preference profiles in a fair, non-discriminatory way to avoid disadvantaging students with incomplete forms.
• Use tie-breaking rules based on rank-minimality and Pareto efficiency to select among multiple optimal matchings.
• Incorporate indifferences by repeating rank values in the utility matrix, allowing students to express equal preference for multiple schools.
• Evaluate matchings using multiple criteria: total cost (C_f), rank-minimality, and Pareto dominance.

Experimental results

Research questions

• RQ1 Can a one-sided matching mechanism that maximizes student cardinal utility produce better outcomes than traditional school choice mechanisms?
• RQ2 How does the Hungarian algorithm adapt to school choice problems with capacity constraints and incomplete preference profiles?
• RQ3 To what extent do utility-maximizing matchings improve Pareto efficiency compared to stable or strategyproof mechanisms?
• RQ4 What are the stability and strategyproofness properties of a mechanism that ignores school priority structures?
• RQ5 How can indifferences in student preferences be fairly incorporated into a cardinal utility-based matching system?

Key findings

• The Utility-Based Hungarian Mechanism produces matchings that are Pareto efficient and minimize rank-based cost, outperforming alternatives in utility and efficiency metrics.
• Matching 1, selected via the f2-cost minimization, Pareto dominates Matching 2 despite both having the same minimal rank cost, demonstrating superiority in multi-criteria evaluation.
• All minimum f2-cost matchings are rank-minimal and Pareto efficient, though not all rank-minimal matchings are Pareto efficient.
• The mechanism is not strategyproof, as students may benefit from misrepresenting preferences, especially when indifferences are involved.
• The mechanism can be adapted to incorporate school priorities at a later stage, offering flexibility for hybrid systems.
• The profile completion process avoids disadvantaging students with incomplete forms, preserving fairness in utility-based assignment.