Nash-convergence of Game Dynamics and Complexity

2026-02-17

AI Summary

This work investigates the fundamental reasons why learning dynamics in games fail to globally converge to Nash equilibria, disentangling whether this limitation stems from the geometric structure of games or inherent computational complexity barriers. By integrating computational complexity theory, game theory, and equilibrium computation under affine subspace constraints, the study establishes, for the first time in non-degenerate games, deep connections between the existence of Nash-convergent dynamics and major complexity classes such as PPAD, CLS, and NP. The core contributions include formulating and substantiating an “impossibility conjecture”—that the existence of locally efficient Nash-convergent dynamics would imply P = PPAD—constructing a “proof game” that exposes the limitations of black-box reductions, and demonstrating that the computability of three specific dynamics would collapse prominent complexity classes (e.g., NP = RP or CLS = PPAD), while also precisely characterizing the complexity of computing Nash equilibria over affine subspaces.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• 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
• Social Networks and Social Media: Computational social science

Abstract

Does the failure of learning dynamics to converge globally to Nash equilibria stem from the geometry of the game or the complexity of computation? Previous impossibility results relied on game degeneracy, leaving open the case for generic, nondegenerate games. We resolve this by proving that while Nash-convergent dynamics theoretically exist for all nondegenerate games, computing them is likely intractable. We formulate the Impossibility Conjecture: if a locally efficient Nash-convergent dynamic exists for nondegenerate games, then $P=PPAD$. We validate this for three specific families of dynamics, showing their tractability would imply collapses such as $NP=RP$ or $CLS=PPAD$. En route, we settle the complexity of finding Nash equilibria of a given game that lie on a given affine subspace. Finally, we explain why the general conjecture remains open: we introduce a Proving Game to demonstrate that black-box reductions cannot distinguish between convergent and non-convergent dynamics in polynomial time. Our results suggest the barrier to Nash learning is not the non-existence of a vector field, but the intractability of computing it.

Problem

• Nash equilibrium
• game dynamics
• computational complexity
• PPAD
• nondegenerate games

Innovation

• Nash equilibrium
• game dynamics
• computational complexity
• PPAD
• nondegenerate games