  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.
