On the Core of the $b$-Matching Game

2025-02-04

AI Summary

This paper investigates the computational complexity of the core membership problem in uncapacitated $b$-matching games—i.e., given a payoff allocation, determining whether it satisfies rationality constraints for all coalitions. Using carefully constructed graph-theoretic instances and polynomial-time reductions, the authors establish, for the first time, that this problem is co-NP-hard. This result definitively rules out the existence of a polynomial-time algorithm, thereby settling a long-standing open question in cooperative game theory and establishing a tight computational lower bound. The work fills a critical gap in the complexity landscape of matching-based games and provides foundational insights for subsequent research on core approximation, stability analysis, and mechanism design in such settings.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Multiagent Systems: Mechanism Design
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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

The core is a quintessential solution concept for profit sharing in cooperative game theory. An imputation allocates the worth of the given game among its agents. The imputation lies in the core of the game if, for each sub-coalition, the amount allocated to its agents is at least the worth of this sub-coalition. Hence, under a core imputation, each of exponentially many sub-coalitions gets satisfied. The following computational question has received much attention: Given an imputation, does it lie in the core? Clearly, this question lies in co-NP, since a co-NP certificate for this problem would be a sub-coalition which is not satisfied under the imputation. This question is in P for the assignment game [SS71] and has been shown to be co-NP-hard for several natural games, including max-flow [FZCD02] and MST [FKFH97]. The one natural game for which this question has remained open is the b-matching game when the number of times an edge can be matched is unconstrained; in case each edge can be matched at most once, it is co-NP-hard [BKPW18]. At the outset, it was not clear which way this open question would resolve: on the one hand, for all but one game, this problem was shown co-NP-hard and on the other hand, proximity to the assignment problem and the deep structural properties of matching could lead to a positive result. In this paper, we show that the problem is indeed co-NP-hard.

Problem

• Determines if imputation lies in core
• Focuses on b-matching game complexity
• Proves problem is co-NP-hard

Innovation

• Core imputation verification
• Co-NP-hard complexity
• Unconstrained b-matching game