Nash Equilibria in Games with Playerwise Concave Coupling Constraints: Existence and Computation

2025-09-17

AI Summary

This paper investigates the existence and computation of Nash equilibria in continuous static games with shared coupling constraints, focusing on the setting where each player’s utility and individual constraint functions are concave—relaxing the conventional joint convexity assumption. Theoretically, it establishes, for the first time, Nash equilibrium existence under individual concavity of constraints, bypassing joint convexity by integrating topological fixed-point theory with contractibility analysis of feasible sets, thereby constructing a novel theoretical framework for non-convex coupled-constraint games. Algorithmically, it proposes an adaptive gradient ascent method regularized by logarithmic barrier functions, which efficiently approximates constrained Nash equilibria under exact gradient feedback. The method achieves an iteration complexity of (O(varepsilon^{-3})), substantially improving convergence efficiency and practicality in non-convex feasible domains.

Technology Category

• Search and Optimization: Non-convex Optimization
• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design

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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the existence and computation of Nash equilibria in continuous static games where the players' admissible strategies are subject to shared coupling constraints, i.e., constraints that depend on their emph{joint} strategies. Specifically, we focus on a class of games characterized by playerwise concave utilities and playerwise concave constraints. Prior results on the existence of Nash equilibria are not applicable to this class, as they rely on strong assumptions such as joint convexity of the feasible set. By leveraging topological fixed point theory and novel structural insights into the contractibility of feasible sets under playerwise concave constraints, we give an existence proof for Nash equilibria under weaker conditions. Having established existence, we then focus on the computation of Nash equilibria via independent gradient methods under the additional assumption that the utilities admit a potential function. To account for the possibly nonconvex feasible region, we employ a log barrier regularized gradient ascent with adaptive stepsizes. Starting from an initial feasible strategy profile and under exact gradient feedback, the proposed method converges to an $ε$-approximate constrained Nash equilibrium within $mathcal{O}(ε^{-3})$ iterations.

Problem

• Existence of Nash equilibria under playerwise concave constraints
• Computation methods for Nash equilibria with nonconvex constraints
• Convergence analysis of gradient methods for equilibrium solutions

Innovation

• Leveraging fixed point theory for existence
• Using independent gradient methods for computation
• Employing log barrier regularization with adaptive stepsizes