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Stable Matching with Ties: Approximation Ratios and Learning

Shiyun Lin, Simon Mauras, Nadav Merlis, Vianney Perchet

TL;DR

This work introduces the OSS-ratio, which measures the ratio of a worker's maximum achievable utility in any stable matching to their utility in a given matching, and designs an adaptive algorithm that smoothly interpolates between markets with strict preferences and those with statistical ties.

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 $Ω(N)$ 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.

Stable Matching with Ties: Approximation Ratios and Learning

TL;DR

This work introduces the OSS-ratio, which measures the ratio of a worker's maximum achievable utility in any stable matching to their utility in a given matching, and designs an adaptive algorithm that smoothly interpolates between markets with strict preferences and those with statistical ties.

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 OSS-ratio, where 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 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.

Paper Structure

This paper contains 41 sections, 26 theorems, 65 equations, 3 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Techniques Involved and Developed
  4. Related Work
  5. Preliminaries
  6. Approximation Ratios for Stable Matching with Ties
  7. Lower Bound
  8. Upper Bound
  9. Robustness and $\epsilon$-Stability
  10. Bandit Learning in Matching Markets
  11. Algorithm: Explore-then-Choose-Oracle
  12. Theoretical Analysis
  13. Conclusion
  14. Further Related Work
  15. Fractional Matchings
  16. ...and 26 more sections

Key Result

Theorem 1

There exists an instance, such that for any distribution over stable matchings, one worker only receives a $2/N$ fraction of their optimal stable share, i.e., $R_{{\mathcal{S}}} \geq \frac{N}{2} = \Omega(N)$.

Figures (3)

  • Figure 1: Lower bound on $R_{\mathcal{S}}$. All jobs have the same ordering over workers, from top to bottom. Any stable matching can be obtained by letting the first worker pick a job, then the second, etc. Hence, each stable matching contains at most one blue edge.
  • Figure 2: Lower bound on $R_{{\mathcal{M}}}$. In each example, left nodes represent workers while right nodes represent jobs. If there is an edge connecting a left node $w$ and a right node $a$, we have ${\bm{U}}(w, a) = 1$, and ${\bm{U}}(w, a) = 0$ otherwise. All the right nodes without edges connecting to them are hidden from the graph.
  • Figure 3: The graph $G_\mu = (V_\mu, E_\mu)$ is a directed forest. The matching $\tilde{\mu}$, computed in Algorithm \ref{['alg:ism']} matches each worker to a single (copy of) job in $\tilde{\mu}$. In the stable matching $\mu$, each worker $w$ is connected to all copies of $\mu(w)$ which have lower index.

Theorems & Definitions (62)

  • Definition 1: Weak Stability
  • Definition 2: Internal Stability
  • Example 1: Stable matching with indifference
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • Remark 2
  • Theorem 4
  • Definition 3: $\epsilon$-Stability
  • ...and 52 more