On the Characterization of Local Nash Equilibria in Continuous Games

TL;DR

This paper introduces a comprehensive framework for identifying and analyzing local Nash equilibria in continuous, possibly infinite-dimensional, non-convex games, emphasizing conditions for stability and uniqueness.

Contribution

It provides intrinsic first- and second-order conditions for local Nash equilibria, defines differential Nash equilibria, and explores their stability and structural properties.

Findings

Necessary and sufficient first- and second-order conditions for local Nash equilibria.

Differential Nash equilibria are isolated under non-degeneracy conditions.

Structural stability of differential Nash equilibria in continuous games.

Abstract

We present a unified framework for characterizing local Nash equilibria in continuous games on either infinite-dimensional or finite-dimensional non-convex strategy spaces. We provide intrinsic necessary and sufficient first- and second-order conditions ensuring strategies constitute local Nash equilibria. We term points satisfying the sufficient conditions differential Nash equilibria. Further, we provide a sufficient condition (non-degeneracy) guaranteeing differential Nash equilibria are isolated and show that such equilibria are structurally stable. We present tutorial examples to illustrate our results and highlight degeneracies that can arise in continuous games.