A New Relaxation of Fairness in Two-Sided Matching Respecting Acquaintance Relationships

TL;DR

This paper introduces a new fairness relaxation called local envy-freeness in two-sided matchings, focusing on envy among acquaintances, and analyzes conditions under which Pareto-efficient matchings can satisfy this fairness.

Contribution

It proposes the concept of local envy-freeness based on acquaintance graphs and analyzes Pareto efficiency under this fairness relaxation with respect to graph structure and preferences.

Findings

Local envy-freeness can be achieved in Pareto-efficient matchings under certain graph restrictions.

The level of local envy-freeness achievable depends on the graph's proximity to trees.

Preferences with single-peakedness influence the feasibility of Pareto-efficient local envy-free matchings.

Abstract

Two-sided matching, such as matching between students and schools, has been applied to various aspects of real life and has been the subject of much research, however, it has been plagued by the fact that efficiency and fairness are incompatible. In particular, Pareto efficiency and justified-envy-freeness are known to be incompatible even in the simplest one-to-one matching, i.e., the stable marriage problem. In previous research, the primary approach to improving efficiency in matchings has been to tolerate students' envy, thereby relaxing fairness constraints. In this study, we take a different approach to relaxing fairness. Specifically, it focuses on addressing only the envy that students may experience or prioritize more highly and seeks matchings without such envy. More specifically, this study assumes that envy towards students who are not acquaintances has less impact compared to envy towards students who are acquaintances. Accordingly, we assume that the students know each other or not, represented by an undirected graph, and define a local envy as a justified envy toward an acquaintance or a neighbor in the graph. We then propose the property that there is no local envy as a new relaxed concept of fairness, called local envy-freeness. We analyze whether Pareto-efficient matching can be achieved while maintaining local envy-freeness by meaningfully restricting the graph structure and the school's preferences. To analyze in detail the fairness that can achieve Pareto-efficient matching, we introduce a local version of the relaxed fairness recently proposed by Cho et al. (AAMAS 2024), which parameterizes the level of local envy-freeness by nonnegative integers. We then clarify the level of local envy-freeness that can be achieved by Pareto-efficient mechanisms for graphs that are ``close'' to trees and single-peaked preferences on the graphs.