Stable Matching with Ties: Approximation Ratios and Learning

TL;DR

This paper introduces algorithms for matching markets with ties, achieving near-optimal utility approximations and extending to learning settings, addressing the challenge of non-existence of universally stable matchings.

Contribution

We propose the OSS-ratio to measure utility loss, design algorithms with logarithmic approximation guarantees, and extend results to a bandit learning framework for markets with unknown utilities.

Findings

Algorithms achieve $O(\log N)$ OSS-ratio for stable and non-stable matchings.

In markets with unknown utilities, the algorithms guarantee logarithmic utility approximation.

Lower bounds reveal the fundamental trade-off between stability and utility in tied preference markets.

Abstract

We study matching markets with ties, where workers on one side of the market may have tied preferences over jobs, determined by their matching utilities. Unlike classical two-sided markets with strict preferences, no single stable matching exists that is utility-maximizing for all workers. To address this challenge, we introduce the \emph{Optimal Stable Share} (OSS)-ratio, which measures the ratio of a worker's maximum achievable utility in any stable matching to their utility in a given matching. We prove that distributions over only stable matchings can incur linear utility losses, i.e., an $\Omega (N)$ OSS-ratio, where $N$ is the number of workers. To overcome this, we design an algorithm that efficiently computes a distribution over (possibly non-stable) matchings, achieving an asymptotically tight $O (\log N)$ OSS-ratio. When exact utilities are unknown, our second algorithm guarantees workers a logarithmic approximation of their optimal utility under bounded instability. Finally, we extend our offline approximation results to a bandit learning setting where utilities are only observed for matched pairs. In this setting, we consider worker-optimal stable regret, design an adaptive algorithm that smoothly interpolates between markets with strict preferences and those with statistical ties, and establish a lower bound revealing the fundamental trade-off between strict and tied preference regimes.