Asymptotic Existence of Class Envy-free Matchings
Tomohiko Yokoyama, Ayumi Igarashi
TL;DR
We study class-envy-free fairness in a one-sided matching problem where agents are partitioned into disjoint classes and item bundles are evaluated via assignment valuations. The paper introduces a distributional model with utilities drawn from a baseline distribution and analyzes two mechanisms: (i) a welfare-maximizing maximum-weight matching that is asymptotically class-envy-free when $m = \Omega(n^2)$, and (ii) a round-robin algorithm that achieves a $1/2$-CEF1, non-wasteful matching and converges to class-envy-freeness as the system scales under mild assumptions on class sizes and the number of classes. The analysis blends random-assignment theory, a special-vertex perturbation technique with reversed-exponential edges, and concentration results to bound the expected marginal gains $X_p$ and cross-class gains $X_{pq}$, ultimately bounding envy probabilities by $o(1)$ as $n \to \infty$. The results provide asymptotic fairness and efficiency guarantees for large-scale class-based allocations, with practical relevance to public housing and distribution of scarce resources, while highlighting limitations in worst-case inputs and non-additive valuations. Overall, the work advances understanding of when fairness notions like class-envy-freeness can be achieved in large, structured matching markets.
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.