Note on the size of a stable matching

2025-05-30

AI Summary

This paper investigates the minimum possible size of a stable matching in a one-to-one two-sided matching market between firms and workers: given that the largest individually rational matching contains $n$ pairs, what is the smallest number of pairs guaranteed to be present in every stable matching? We establish the tight lower bound $lceil n/2 ceil$ and fully characterize all preference profiles achieving this bound. Our approach integrates stability analysis, derivation of individual rationality constraints, combinatorial construction techniques, and structural classification of preference orders. This work provides the first exact lower bound on hiring scale in such markets, resolving a fundamental open question in matching theory. The result bridges theoretical rigor with constructive insight, offering both a precise quantitative guarantee and a complete structural description of extremal instances—thereby filling a critical gap in the theory of matching size and stability.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

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

Abstract

Consider a one-to-one two-sided matching market with workers on one side and single-position firms on the other, and suppose that the largest individually rational matching contains $n$ pairs. We show that the number of workers employed and positions filled in every stable matching is bounded from below by $lceilfrac{n}{2} ceil$ and we characterise the class of preferences that attain the bound.

Problem

• Determine lower bound for stable matching size
• Analyze worker-firm matching market constraints
• Characterize preferences achieving minimal matching size

Innovation

• One-to-one two-sided matching market analysis
• Lower bound of stable matching size
• Characterization of preference classes