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Stable Marriage with One-Sided Preference

Seongbeom Park

TL;DR

The paper tackles school-choice problems with ties by focusing on a two-sided market where only one side has preferences. It reduces the stable, student-centric matching problem to a maximum-weight assignment problem using a rank-to-weight transform, and proves that the resulting solution is stable and maximizes a defined utility Q(M). The proposed approach achieves a student-optimal, high-quality allocation in polynomial time and scales to large datasets, with experiments showing improved distribution of students to higher-preference schools. This work offers a theoretically sound and computationally feasible method to enhance fairness and efficiency in school assignments under tie-heavy conditions.

Abstract

Many countries around the world, including Korea, use the school choice lottery system. However, this method has a problem in that many students are assigned to less-preferred schools based on the lottery results. In addition, the task of finding a good assignment with ties often has a time complexity of NP, making it a very difficult problem to improve the quality of the assignment. In this paper, we prove that the problem of finding a stable matching that maximizes the student-oriented preference utility in a two-sided market with one-sided preference can be solved in polynomial time, and we verify through experiments that the quality of assignment is improved. The main contributions of this paper are as follows. We found that stable student-oriented allocation in a two-sided market with one-sided preferences is the same as stable allocation in a two-sided market with symmetric preferences. In addition, we defined a method to quantify the quality of allocation from a preference utilitarian perspective. Based on the above two, it was proven that the problem of finding a stable match that maximizes the preference utility in a two-sided market with homogeneous preferences can be reduced to an allocation problem. In this paper, through an experiment, we quantitatively verified that optimal student assignment assigns more students to schools of higher preference, even in situations where many students are assigned to schools of low preference using the existing assignment method.

Stable Marriage with One-Sided Preference

TL;DR

The paper tackles school-choice problems with ties by focusing on a two-sided market where only one side has preferences. It reduces the stable, student-centric matching problem to a maximum-weight assignment problem using a rank-to-weight transform, and proves that the resulting solution is stable and maximizes a defined utility Q(M). The proposed approach achieves a student-optimal, high-quality allocation in polynomial time and scales to large datasets, with experiments showing improved distribution of students to higher-preference schools. This work offers a theoretically sound and computationally feasible method to enhance fairness and efficiency in school assignments under tie-heavy conditions.

Abstract

Many countries around the world, including Korea, use the school choice lottery system. However, this method has a problem in that many students are assigned to less-preferred schools based on the lottery results. In addition, the task of finding a good assignment with ties often has a time complexity of NP, making it a very difficult problem to improve the quality of the assignment. In this paper, we prove that the problem of finding a stable matching that maximizes the student-oriented preference utility in a two-sided market with one-sided preference can be solved in polynomial time, and we verify through experiments that the quality of assignment is improved. The main contributions of this paper are as follows. We found that stable student-oriented allocation in a two-sided market with one-sided preferences is the same as stable allocation in a two-sided market with symmetric preferences. In addition, we defined a method to quantify the quality of allocation from a preference utilitarian perspective. Based on the above two, it was proven that the problem of finding a stable match that maximizes the preference utility in a two-sided market with homogeneous preferences can be reduced to an allocation problem. In this paper, through an experiment, we quantitatively verified that optimal student assignment assigns more students to schools of higher preference, even in situations where many students are assigned to schools of low preference using the existing assignment method.
Paper Structure (17 sections, 2 theorems, 2 equations, 9 figures, 1 algorithm)

This paper contains 17 sections, 2 theorems, 2 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Assignment Criteria
  3. Stable matching
  4. Student-oriented
  5. Maximizing utility
  6. Stabale Marriage and Assigment Problem
  7. Assignment problem reduction
  8. Stability
  9. Maximum utility
  10. Many-to-one matching
  11. Complexity
  12. Experiment
  13. Experimental environment
  14. 100 students & 5 schools
  15. 1,000 students & 5 schools
  16. ...and 2 more sections

Key Result

Theorem 1

매칭 $M^*$는 안정적인 매칭이다.

Figures (9)

  • Figure 1: Student-optimal matching with student preference only
  • Figure 2: Possible matchings with student's strictly ordered preference
  • Figure 3: Possible matchings with tie-breaking
  • Figure 4: An instance with symmetric preferences
  • Figure 5: Preference rank matrix
  • ...and 4 more figures

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof