Capacity Modification in the Stable Matching Problem

2024-02-07

Adaptive Agents and Multi-Agent Systems

AI Summary

This paper investigates how changes in firm capacities (i.e., seat numbers) affect the set of stable matchings in many-to-one stable matching. It addresses two core questions: (i) how capacity expansions or reductions alter worker–firm assignments, and (ii) how to adjust capacities to achieve a desired stable matching—e.g., ensuring a specific pair is matched or rendering a given matching stable. To this end, the paper introduces the notion of “capacity peaks,” the first systematic, computationally tractable characterization of how capacity variations reshape matching structure. Under global capacity constraints, all considered problems admit polynomial-time algorithms; under individual capacity constraints, they are NP-hard. Furthermore, the paper identifies threshold conditions under which capacity manipulation dominates preference manipulation in terms of strategic efficacy. These results establish capacity as a novel, theoretically grounded lever for mechanism design in two-sided matching markets.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

We study the problem of capacity modification in the many-to-one stable matching of workers and firms. Our goal is to systematically study how the set of stable matchings changes when some seats are added to or removed from the firms. We make three main contributions: First, we examine whether firms and workers can improve or worsen upon changing the capacities under worker-proposing and firm-proposing deferred acceptance algorithms. Second, we study the computational problem of adding or removing seats to either match a fixed worker-firm pair in some stable matching or make a fixed matching stable with respect to the modified problem. We develop polynomial-time algorithms for these problems when only the overall change in the firms' capacities is restricted, and show NP-hardness when there are additional constraints for individual firms. Lastly, we compare capacity modification with the classical model of preference manipulation by firms and identify scenarios under which one mode of manipulation outperforms the other. We find that a threshold on a given firm's capacity, which we call its peak, crucially determines the effectiveness of different manipulation actions.

Problem

• Study how stable matchings change with firm capacity adjustments
• Develop algorithms for seat modification under capacity constraints
• Compare capacity modification effectiveness versus preference manipulation

Innovation

• Analyzes capacity changes in stable matching algorithms
• Develops polynomial-time algorithms for seat modifications
• Compares capacity and preference manipulation effectiveness