Fairness and Efficiency in Online Class Matching

TL;DR

This paper introduces a randomized algorithm for fair online bipartite matching that balances class fairness, proportionality, and social welfare, providing theoretical guarantees and analyzing trade-offs between fairness and efficiency.

Contribution

It presents the first non-wasteful randomized algorithm achieving simultaneous approximation to class envy-freeness, proportionality, and social welfare in online matching.

Findings

Achieves 1/2 approximation for class envy-freeness

No non-wasteful algorithm can surpass 0.761 approximation for CEF

Demonstrates the trade-off between fairness and social welfare

Abstract

The online bipartite matching problem, extensively studied in the literature, deals with the allocation of online arriving vertices (items) to a predetermined set of offline vertices (agents). However, little attention has been given to the concept of class fairness, where agents are categorized into different classes, and the matching algorithm must ensure equitable distribution across these classes. We here focus on randomized algorithms for the fair matching of indivisible items, subject to various definitions of fairness. Our main contribution is the first (randomized) non-wasteful algorithm that simultaneously achieves a $1/2$ approximation to class envy-freeness (CEF) while simultaneously ensuring an equivalent approximation to the class proportionality (CPROP) and utilitarian social welfare (USW) objectives. We supplement this result by demonstrating that no non-wasteful algorithm can achieve an $\alpha$-CEF guarantee for $\alpha > 0.761$. In a similar vein, we provide a novel input instance for deterministic divisible matching that demonstrates a nearly tight CEF approximation. Lastly, we define the ``price of fairness,'' which represents the trade-off between optimal and fair matching. We demonstrate that increasing the level of fairness in the approximation of the solution leads to a decrease in the objective of maximizing USW, following an inverse proportionality relationship.