Probabilistic Analysis of Stable Matching in Large Markets with Siblings

TL;DR

This paper analyzes the probability of stable matchings in large markets with sibling preferences, showing that stability becomes almost certain as market size grows, and introduces an improved algorithm for finding such matchings.

Contribution

It provides a probabilistic framework for understanding stability in large sibling-involved markets and refines an algorithm to reliably find stable matchings under high priority similarity.

Findings

Probability of stable matchings approaches 1 as market size increases.

Refined heuristic algorithm effectively finds stable matchings.

High priority similarity among daycare centers enhances stability detection.

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.