Stable Matching with Ties: Approximation Ratios and Learning

Shiyun Lin, Simon Mauras, Nadav Merlis, Vianney Perchet

Keywords: Stable Matching, Bandit Learning, Approximation Ratio, Optimal Stable Regret

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.

Primary Area: Theory (e.g., control theory, learning theory, algorithmic game theory)

Submission Number: 27282