Probabilistic Analysis of Stable Matching in Large Markets with Siblings

Zhaohong Sun, Tomohiko Yokoyama, Makoto Yokoo

Keywords: Stable Matching, Stability, Siblings, Deferred Acceptance

Abstract:

We study a practical matching problem that involves assigning children to daycare centers. The collective preferences of siblings from the same family introduce complementarities, which can lead to the non-existence of stable matchings, as observed in the well-studied hospital-doctor matching problems involving couples. Intriguingly, stable matchings have been observed in real-world daycare markets, even with a substantial number of sibling applicants.

Our research systematically explores the presence of stable matchings in these markets. We conduct a probabilistic analysis of large random matching markets that incorporate sibling preferences. Specifically, we examine scenarios where daycares have similar priorities over children, a common characteristic in practical markets. Our analysis reveals that as the market size approaches infinity, the likelihood of stable matchings existing converges to 1.

To facilitate our investigation, we introduce significant modifications to the Sorted Deferred Acceptance algorithm proposed by ed by Ashlagi et al. [2014]. These adaptations are essential to accommodate a more stringent stability concept, as the original algorithm may yield matchings that fail to meet this criterion. By leveraging our revised algorithm, we successfully identify stable matchings in all real-life datasets examined. Additionally, we conduct comprehensive empirical investigations using synthetic datasets to validate the efficacy of our algorithm in identifying stable matchings.

Primary Area: Algorithmic game theory

Submission Number: 5978