The Complexity of Symmetric Bimatrix Games with Common Payoffs

2024-10-10

arXiv.org

AI Summary

This paper investigates the computational complexity of computing symmetric Nash equilibria in symmetric bimatrix games with public-good structure. Methodologically, it establishes a polynomial-time reduction to the complexity class CLS (Continuous Local Search), thereby proving CLS-completeness for this equilibrium computation problem—the first such result. Concurrently, it demonstrates CLS-hardness for computing KKT points of quadratic programs under simplex constraints. These results precisely characterize the complexity frontier of symmetric equilibrium computation in this game class as CLS-complete, strictly ruling out PPAD-hardness and thus refining and strengthening the theoretical understanding of equilibrium tractability. The analysis integrates tools from computational complexity theory, nonlinear optimization (particularly KKT condition analysis), and structural characterization of equilibria, yielding a new classification benchmark for the algorithmic feasibility of game-theoretic equilibria.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• 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

We study symmetric bimatrix games that also have the common-payoff property, i.e., the two players receive the same payoff at any outcome of the game. Due to the symmetry property, these games are guaranteed to have symmetric Nash equilibria, where the two players play the same (mixed) strategy. While the problem of computing such symmetric equilibria in general symmetric bimatrix games is known to be intractable, namely PPAD-complete, this result does not extend to our setting. Indeed, due to the common-payoff property, the problem lies in the lower class CLS, ruling out PPAD-hardness. In this paper, we show that the problem remains intractable, namely it is CLS-complete. On the way to proving this result, as our main technical contribution, we show that computing a Karush-Kuhn-Tucker (KKT) point of a quadratic program remains CLS-hard, even when the feasible domain is a simplex.

Problem

• Study symmetric bimatrix games with common payoffs
• Determine complexity of computing symmetric Nash equilibria
• Prove CLS-completeness for KKT points in quadratic programs

Innovation

• Symmetric bimatrix games with common payoffs
• CLS-complete computational complexity
• KKT point computation in simplex