Asymptotic Existence of Class Envy-free Matchings

2025-02-20

AI Summary

This paper studies fairness in one-sided matching with multiple agent types, formalizing the requirement that each type’s aggregate utility must be at least its maximum attainable utility under any alternative type assignment—termed *type-wise envy-freeness*. As this property is infeasible in the worst case, we adopt a stochastic utility model where agents’ utilities are independent and identically distributed. Under this assumption, we conduct asymptotic analysis: we prove that, as the number of agents grows to infinity, the round-robin algorithm yields a matching that satisfies type-wise envy-freeness almost surely. This constitutes the first asymptotic existence guarantee, overcoming a fundamental impossibility barrier in the worst-case setting. Moreover, the algorithm is inherently distributed and scalable, enabling efficient implementation in large-scale settings. Our framework thus provides a theoretically grounded and practically viable paradigm for fair resource allocation—particularly for sensitive attributes such as race or age—in public policy and platform design.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

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.

Problem

• Ensures fair treatment in class-partitioned matchings
• Achieves class envy-free utility allocations
• Proves asymptotic existence via round-robin algorithm

Innovation

• Class envy-free matchings
• Round-robin algorithm
• Asymptotic existence analysis