Solving leaderless multi-cluster games over directed graphs *

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 themselves 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 intra-cluster problem and the game between the clusters need to be solved simultaneously. Therefore, the resulting intercluster 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, remains 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. (c) 2021 European Control Association. Published by Elsevier Ltd. All rights reserved.