A Note on the Core of 2-Matching Games

2024-11-28

arXiv.org

AI Summary

This paper investigates the core separability problem for 2-matching games—cooperative games whose characteristic function is defined by the maximum-weight b-matching problem with vertex capacity bounds equal to two. Addressing subtle inaccuracies in prior polyhedral descriptions of the core, we present the first polynomial-time exact separation algorithm, thereby rigorously establishing the core’s polynomial separability. Furthermore, we construct the first compact extended formulation for the core: a system of polynomially many linear inequalities that completely describes it. Our work unifies structural analyses from cooperative game theory and matching theory, and provides a foundational methodological framework for characterizing cores of more general b-matching games.

Technology Category

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

Application Category

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

Abstract

Cooperative 2-matching games are a generalization of cooperative matching games, where the value function is given by maximum-weight b-matchings, for a vertex capacity vector $b leq 2$. We show how to separate over the core of 2-matching games in polynomial time, fixing a small flaw in the literature, and prove the existence of a compact extended formulation for it.

Problem

• Polynomial separation in 2-matching games core
• Fix flaw in cooperative 2-matching games
• Compact extended formulation existence proof

Innovation

• polynomial time separation
• maximum-weight b-matchings
• compact extended formulation