Gradient Play in $n$-Cluster Games with Zero-Order Information

TL;DR

This paper introduces a distributed gradient play algorithm for finding Nash equilibria in multi-cluster games where agents have access only to zero-order information and communicate over a network, with proven convergence guarantees.

Contribution

It proposes a novel distributed algorithm that handles zero-order information and network communication for multi-cluster game Nash equilibrium seeking.

Findings

Algorithm converges almost surely to Nash equilibrium.

Handles zero-order (derivative-free) local cost estimations.

Applicable to networks with undirected communication graphs.

Abstract

We study a distributed approach for seeking a Nash equilibrium in $n$-cluster games with strictly monotone mappings. Each player within each cluster has access to the current value of her own smooth local cost function estimated by a zero-order oracle at some query point. We assume the agents to be able to communicate with their neighbors in the same cluster over some undirected graph. The goal of the agents in the cluster is to minimize their collective cost. This cost depends, however, on actions of agents from other clusters. Thus, a game between the clusters is to be solved. We present a distributed gradient play algorithm for determining a Nash equilibrium in this game. The algorithm takes into account the communication settings and zero-order information under consideration. We prove almost sure convergence of this algorithm to a Nash equilibrium given appropriate estimations of the local cost functions' gradients.