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Rawlsian many-to-one matching with non-linear utility

Hortence Nana, Andreas Athanasopoulos, Christos Dimitrakakis

TL;DR

The paper investigates many-to-one matching with non-linear, set-based college utilities and shows that stability can fail in such settings. It adopts a Rawlsian max-min fairness objective and develops two iterative algorithms—deterministic and stochastic—to improve the worst-off college while maintaining feasibility. Through synthetic experiments, it demonstrates that both methods enhance the minimum college utility and, in many cases, improve overall efficiency, with the stochastic variant offering better scalability and stronger fairness gains at moderate computational cost. The work provides practical tools for fair allocation when stability cannot be guaranteed, and it discusses limitations and directions for real-world data and hybrid stability-fairness mechanisms.

Abstract

We study a many-to-one matching problem, such as the college admission problem, where each college can admit multiple students. Unlike classical models, colleges evaluate sets of students through non-linear utility functions that capture diversity between them. In this setting, we show that classical stable matchings may fail to exist. To address this, we propose alternative solution concepts based on Rawlsian fairness, aiming to maximize the minimum utility across colleges. We design both deterministic and stochastic algorithms that iteratively improve the outcome of the worst-off college, offering a practical approach to fair allocation when stability cannot be guaranteed.

Rawlsian many-to-one matching with non-linear utility

TL;DR

The paper investigates many-to-one matching with non-linear, set-based college utilities and shows that stability can fail in such settings. It adopts a Rawlsian max-min fairness objective and develops two iterative algorithms—deterministic and stochastic—to improve the worst-off college while maintaining feasibility. Through synthetic experiments, it demonstrates that both methods enhance the minimum college utility and, in many cases, improve overall efficiency, with the stochastic variant offering better scalability and stronger fairness gains at moderate computational cost. The work provides practical tools for fair allocation when stability cannot be guaranteed, and it discusses limitations and directions for real-world data and hybrid stability-fairness mechanisms.

Abstract

We study a many-to-one matching problem, such as the college admission problem, where each college can admit multiple students. Unlike classical models, colleges evaluate sets of students through non-linear utility functions that capture diversity between them. In this setting, we show that classical stable matchings may fail to exist. To address this, we propose alternative solution concepts based on Rawlsian fairness, aiming to maximize the minimum utility across colleges. We design both deterministic and stochastic algorithms that iteratively improve the outcome of the worst-off college, offering a practical approach to fair allocation when stability cannot be guaranteed.

Paper Structure

This paper contains 23 sections, 3 theorems, 7 equations, 6 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Definition and setting
  4. Stability may fail under set preferences
  5. Max-Min Matching Problem
  6. Deterministic greedy max-min
  7. Stochastic greedy subset max-min
  8. Background on stochastic greedy and Sakaue's modification.
  9. Our adaptation.
  10. Properties and limitations.
  11. Experiments
  12. Preference representation
  13. Experimenal setup
  14. Randomized preferences.
  15. Evaluation protocol.
  16. ...and 8 more sections

Key Result

Theorem 1

Algorithm alg:g does not always result in a stable matching under group preferences.

Figures (6)

  • Figure 1: Improvement in minimum utility for all methods. The experiment is done under 100 trials, 500 students, 10 colleges and $\lambda = 1.0$
  • Figure 2: Average utility across colleges under different matching methods
  • Figure 3: Scalability and runtime with respect to the number of students and colleges for all methods.
  • Figure 4: Fairness–Efficiency Trade-off vs. Runtime. N=2500, C=20, max_q =100
  • Figure 5: Improvement in minimum utility for $\lambda=0.3$ (ranking dominates), $\lambda=1.0$ (diversity has weight comparable to rank) and $\lambda=2.0$( diversity dominates). The experiment is done under 100 trials, 500 students and 10 colleges.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Definition 1: Matching
  • Example 1
  • Definition 2: Stable Matching
  • Definition 3: Update function
  • Theorem 1
  • Theorem 2
  • proof
  • Remark 1: Complexity bound
  • Lemma 1
  • proof