Probabilistic Analysis of Stable Matching in Large Markets with Siblings
Zhaohong Sun, Tomohiko Yokoyama, Makoto Yokoo
TL;DR
This work investigates why stable matchings persist in large daycare markets with siblings, despite complementarities that typically break stability in two-sided matching. It introduces Extended Sorted Deferred Acceptance (ESDA), a stability-aware heuristic that extends SDA to allow seat transfers among siblings, and proves that ESDA yields a stable matching whenever it terminates. The authors develop a probabilistic large-market model using Mallows priors for daycares and independent, bounded child preferences, proving a main theorem: with $\phi = O(\log n/n)$, a stable matching exists with probability tending to 1 as $n\to\infty$. Experiments on real and synthetic data show ESDA reliably finds stable matchings under high priority similarity, often performing near the benchmark CP methods while outperforming SDA in stability guarantees. Overall, the paper provides both a theoretical explanation for observed stability in practice and a practical algorithmic tool for large-scale daycare matching with siblings.
Abstract
We study a practical centralized matching problem which assigns children to daycare centers. The collective preferences of siblings from the same family introduce complementarities, which can lead to the absence of stable matchings, as observed in the hospital-doctor matching problems involving couples. Intriguingly, stable matchings are consistently observed in real-world daycare markets, despite the prevalence of sibling applicants. We conduct a probabilistic analysis of large random markets to examine the existence of stable matchings in such markets. Specifically, we examine scenarios where daycare centers have similar priorities over children, a common characteristic in real-world markets. Our analysis reveals that as the market size approaches infinity, the likelihood of stable matchings existing converges to 1. To facilitate our exploration, we refine an existing heuristic algorithm to address a more rigorous stability concept, as the original one may fail to meet this criterion. Through extensive experiments on both real-world and synthetic datasets, we demonstrate the effectiveness of our revised algorithm in identifying stable matchings, particularly when daycare priorities exhibit high similarity.