Nash Equilibrium Seeking Over Directed Graphs

TL;DR

This paper develops distributed algorithms for directed graphs to find Nash equilibria in noncooperative games, ensuring exponential convergence even in unbalanced, strongly connected directed networks.

Contribution

It introduces a novel distributed algorithm with a proportional gain for weight-balanced graphs and extends it to unbalanced graphs using a distributed eigenvector estimator.

Findings

Exact Nash equilibrium is reached with exponential convergence.

Algorithms work on both weight-balanced and unbalanced directed graphs.

Theoretical results are validated with an illustrative example.

Abstract

In this paper, we aim to develop distributed continuous-time algorithms over directed graphs to seek the Nash equilibrium in a noncooperative game. Motivated by the recent consensus-based designs, we present a distributed algorithm with a proportional gain for weight-balanced directed graphs. By further embedding a distributed estimator of the left eigenvector associated with zero eigenvalue of the graph Laplacian, we extend it to the case with arbitrary strongly connected directed graphs having possible unbalanced weights. In both cases, the Nash equilibrium is proven to be exactly reached with an exponential convergence rate. An example is given to illustrate the validity of the theoretical results.