Stable Matching with Ties: Approximation Ratios and Learning

2024-11-05

arXiv.org

AI Summary

This paper studies two-sided matching markets with ties in agents’ preference lists—where workers may derive identical utility from multiple jobs—rendering a globally optimal stable matching nonexistent. To address this, we introduce the *Optimal Stable Share* (OSS) ratio: a fairness metric quantifying the minimum fraction of each worker’s individually optimal utility guaranteed across all stable matchings. We prove that a randomized unstable matching mechanism achieves a tight Θ(log n) OSS ratio; this guarantee remains robust even when utilities are only approximately known. Extending to bandit learning settings with unknown utilities, our framework smoothly bridges the tie-free and statistically tied regimes. Our core contributions are: (i) a novel, theoretically grounded fairness measure for utility allocation under ties; and (ii) matching and learning mechanisms that simultaneously achieve tight theoretical bounds and practical adaptability across preference structures.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design
• Machine Learning: Online Learning & Bandits

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We study the problem of matching markets with ties, where one side of the market does not necessarily have strict preferences over members at its other side. For example, workers do not always have strict preferences over jobs, students can give the same ranking for different schools and more. In particular, assume w.l.o.g. that workers' preferences are determined by their utility from being matched to each job, which might admit ties. Notably, in contrast to classical two-sided markets with strict preferences, there is no longer a single stable matching that simultaneously maximizes the utility for all workers. We aim to guarantee each worker the largest possible share from the utility in her best possible stable matching. We call the ratio between the worker's best possible stable utility and its assigned utility the emph{Optimal Stable Share} (OSS)-ratio. We first prove that distributions over stable matchings cannot guarantee an OSS-ratio that is sublinear in the number of workers. Instead, randomizing over possibly non-stable matchings, we show how to achieve a tight logarithmic OSS-ratio. Then, we analyze the case where the real utility is not necessarily known and can only be approximated. In particular, we provide an algorithm that guarantees a similar fraction of the utility compared to the best possible utility. Finally, we move to a bandit setting, where we select a matching at each round and only observe the utilities for matches we perform. We show how to utilize our results for approximate utilities to gracefully interpolate between problems without ties and problems with statistical ties (small suboptimality gaps).

Problem

• Measures utility loss in stable matchings with tied preferences
• Designs algorithms achieving logarithmic approximation ratios
• Extends offline results to bandit learning with partial feedback

Innovation

• Introduces Optimal Stable Share ratio for tied preferences
• Designs algorithm achieving logarithmic OSS-ratio approximation
• Extends to bandit learning with worker-optimal stable regret