Gradient-Tracking over Directed Graphs for solving Leaderless Multi-Cluster Games

TL;DR

This paper introduces a distributed gradient-tracking algorithm for multi-cluster games over directed graphs, enabling agents to find Nash equilibria while preserving privacy and achieving linear convergence.

Contribution

It proposes a novel gradient-tracking method tailored for leaderless multi-cluster games on directed graphs, ensuring convergence to Nash equilibria with privacy considerations.

Findings

Algorithm converges linearly to the optimal solution.

Effective in solving extended Cournot game scenarios.

Preserves agent privacy during distributed optimization.

Abstract

We are concerned with finding Nash Equilibria in agent-based multi-cluster games, where agents are separated into distinct clusters. While the agents inside each cluster collaborate to achieve a common goal, the clusters are considered to be virtual players that compete against each other in a non-cooperative game with respect to a coupled cost function. In such scenarios, the inner-cluster problem and the game between the clusters need to be solved simultaneously. Therefore, the resulting inter-cluster Nash Equilibrium should also be a minimizer of the social welfare problem inside the clusters. In this work, this setup is cast as a distributed optimization problem with sparse state information. Hence, critical information, such as the agent's cost functions, remain private. We present a distributed algorithm that converges with a linear rate to the optimal solution. Furthermore, we apply our algorithm to an extended cournot game to verify our theoretical results.