Nash equilibrium seeking under partial-decision information over directed communication networks

TL;DR

This paper develops a distributed algorithm for finding Nash equilibria in games over directed, possibly unbalanced communication networks, with guaranteed linear convergence under certain conditions.

Contribution

It introduces a fully-distributed pseudo-gradient method that converges linearly in directed networks, extending prior work which focused on undirected or balanced graphs.

Findings

Algorithm guarantees linear convergence to Nash equilibrium.

Method works with directed, non-balanced communication graphs.

Adapted for unknown network structures using online eigenvector computation.

Abstract

We consider the Nash equilibrium problem in a partial-decision information scenario. Specifically, each agent can only receive information from some neighbors via a communication network, while its cost function depends on the strategies of possibly all agents. In particular, while the existing methods assume undirected or balanced communication, in this paper we allow for non-balanced, directed graphs. We propose a fully-distributed pseudo-gradient scheme, which is guaranteed to converge with linear rate to a Nash equilibrium, under strong monotonicity and Lipschitz continuity of the game mapping. Our algorithm requires global knowledge of the communication structure, namely of the Perron-Frobenius eigenvector of the adjacency matrix and of a certain constant related to the graph connectivity. Therefore, we adapt the procedure to setups where the network is not known in advance, by computing the eigenvector online and by means of vanishing step sizes.