Asymptotic Existence of Class Envy-free Matchings

TL;DR

This paper proves that in large-scale random matching problems with fairness constraints, a simple round-robin algorithm asymptotically guarantees class envy-freeness, addressing practical fairness in resource allocation.

Contribution

It introduces a distributional model for class envy-free matchings and proves asymptotic existence using a straightforward algorithm, advancing understanding of fair matchings in large systems.

Findings

Round-robin algorithm yields envy-free matchings asymptotically

As the number of agents grows, envy-freeness becomes almost certain

Model applies to real-world resource allocation scenarios

Abstract

We consider a one-sided matching problem where agents who are partitioned into disjoint classes and each class must receive fair treatment in a desired matching. This model, proposed by Benabbou et al. [2019], aims to address various real-life scenarios, such as the allocation of public housing and medical resources across different ethnic, age, and other demographic groups. Our focus is on achieving class envy-free matchings, where each class receives a total utility at least as large as the maximum value of a matching they would achieve from the items matched to another class. While class envy-freeness for worst-case utilities is unattainable without leaving some valuable items unmatched, such extreme cases may rarely occur in practice. To analyze the existence of a class envy-free matching in practice, we study a distributional model where agents' utilities for items are drawn from a probability distribution. Our main result establishes the asymptotic existence of a desired matching, showing that a round-robin algorithm produces a class envy-free matching as the number of agents approaches infinity.