Characterization and computation of local Nash equilibria in continuous games

We present derivative-based necessary and sufficient conditions ensuring player strategies constitute local Nash equilibria in non-cooperative continuous games. Our results can be interpreted as generalizations of analogous second-order conditions for local optimality from nonlinear programming and optimal control theory. Drawing on this analogy, we propose an iterative steepest descent algorithm for numerical approximation of local Nash equilibria and provide a sufficient condition ensuring local convergence of the algorithm. We demonstrate our analytical and computational techniques by computing local Nash equilibria in games played on a finite-dimensional differentiable manifold or an infinite-dimensional Hilbert space.

I. Introduction

When resources are scarce, competition develops between self-interested agents. Game theory is an established technique for modeling this interaction, and it has emerged as an engineering tool for analysis and synthesis of systems comprised of dynamically-coupled decision-making agents possessing competing interests [1], [2]. We focus on games with a finite number of agents where the strategy space is continuous, either a finite-dimensional differentiable manifold or an infinite-dimensional Hilbert space.