Nash Equilibrium Seeking with Non-doubly Stochastic Communication Weight Matrix

TL;DR

This paper introduces a distributed Nash equilibrium seeking algorithm for networked games with asymmetric communication, proving convergence despite non-doubly stochastic weights, and extends it to graphical games with local interference.

Contribution

It presents a novel algorithm handling non-doubly stochastic communication matrices and extends it to graphical games with local interference dependencies.

Findings

Algorithm converges almost surely to Nash equilibrium.

Non-doubly stochastic weights prevent exact average preservation.

Effective in social media behavioral simulation.

Abstract

A distributed Nash equilibrium seeking algorithm is presented for networked games. We assume an incomplete information available to each player about the other players' actions. The players communicate over a strongly connected digraph to send/receive the estimates of the other players' actions to/from the other local players according to a gossip communication protocol. Due to asymmetric information exchange between the players, a non-doubly (row) stochastic weight matrix is defined. We show that, due to the non-doubly stochastic property, the total average of all players' estimates is not preserved for the next iteration which results in having no exact convergence. We present an almost sure convergence proof of the algorithm to a Nash equilibrium of the game. Then, we extend the algorithm for graphical games in which all players' cost functions are only dependent on the local neighboring players over an interference digraph. We design an assumption on the communication digraph such that the players are able to update all the estimates of the players who interfere with their cost functions. It is shown that the communication digraph needs to be a superset of a transitive reduction of the interference digraph. Finally, we verify the efficacy of the algorithm via a simulation on a social media behavioral case.