Probabilistic Analysis of Stable Matching in Large Markets with Siblings

2025-01-22

AI Summary

This paper addresses the daycare center matching problem for families with siblings, focusing on the existence and computability of stable matchings in large random markets. Sibling-bundled applications introduce preference complementarities, complicating standard stability analysis. Method: We introduce a refined definition of stability accommodating such complementarities and propose an enhanced algorithm tailored to sibling-aware preferences. We analytically examine market conditions under which stable matchings exist with high probability. Contribution/Results: We provide the first rigorous proof that, when schools’ priority rankings are highly correlated, stable matchings exist asymptotically almost surely—i.e., their existence probability converges to one as market size grows. Empirical evaluation on both synthetic and real-world datasets demonstrates that our algorithm significantly improves the success rate of finding stable matchings under high priority correlation, offering both theoretical guarantees and practical tools for equitable public resource allocation.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Game Theory and Economic Paradigms: Auctions and Market-Based Systems
• Machine Learning: Learning Preferences or Rankings

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: The sharing economy
• Security and Privacy: Large-scale security measurements

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.

Problem

• Stable Matching
• Sibling Preference
• Daycare Allocation

Innovation

• Stable Matching
• Sibling Pairing
• Improved Algorithm