Distributed Nash Equilibrium Seeking By Gossip in Games on Graphs

TL;DR

This paper introduces a gossip-based distributed algorithm for finding Nash equilibria in multi-player network games with partially-coupled cost functions, ensuring convergence and analyzing the impact of communication and interference graph properties.

Contribution

It extends Nash equilibrium seeking to partially-coupled cost functions using interference graphs and designs a communication scheme that guarantees convergence under standard assumptions.

Findings

Proves almost sure convergence to Nash equilibrium.

Quantifies the impact of communication graph eigenvalues on convergence rate.

Demonstrates effectiveness through large-scale network simulations.

Abstract

We consider a gossip approach for finding a Nash equilibrium in a distributed multi-player network game. We extend previous results on Nash equilibrium seeking to the case when the players' cost functions may be affected by the actions of any subset of players. An interference graph is employed to illustrate the partially-coupled cost functions and the asymmetric information requirements. For a given interference graph, we design a generalized communication graph so that players with possibly partially-coupled cost functions exchange only their required information and make decisions based on them. Using a set of standard assumptions on the cost functions, interference and communication graphs, we prove almost sure convergence to a Nash equilibrium for diminishing step sizes. We then quantify the effect of the second largest eigenvalue of the expected communication matrix on the convergence rate, and illustrate the trade-off between the parameters associated with the communication and the interference graphs. Finally, the efficacy of the proposed algorithm on a large-scale networked game is demonstrated via simulation.