Decentralized Matching in a Probabilistic Environment

TL;DR

This paper analyzes a decentralized stable matching process in probabilistic environments, demonstrating it can approximate optimal online algorithms with specific ratios across various matching scenarios.

Contribution

It introduces a decentralized stable matching process that achieves provable approximation ratios for online stochastic matching problems in general and bipartite graphs.

Findings

Provides a 0.316-approximation for general graph matching.

Offers a 1/7-approximation for many-to-one bipartite matching.

Establishes a 1/11-approximation for capacitated matching and a 1/2k-approximation for team formation.

Abstract

We consider a model for repeated stochastic matching where compatibility is probabilistic, is realized the first time agents are matched, and persists in the future. Such a model has applications in the gig economy, kidney exchange, and mentorship matching. We ask whether a $decentralized$ matching process can approximate the optimal online algorithm. In particular, we consider a decentralized $stable$ $matching$ process where agents match with the most compatible partner who does not prefer matching with someone else, and known compatible pairs continue matching in all future rounds. We demonstrate that the above process provides a 0.316-approximation to the optimal online algorithm for matching on general graphs. We also provide a $\frac{1}{7}$-approximation for many-to-one bipartite matching, a $\frac{1}{11}$-approximation for capacitated matching on general graphs, and a $\frac{1}{2k}$-approximation for forming teams of up to $k$ agents. Our results rely on a novel coupling argument that decomposes the successful edges of the optimal online algorithm in terms of their round-by-round comparison with stable matching.