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.

中文翻译：

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有向图上的梯度跟踪，用于解决无领导者多集群游戏

我们担心要在基于代理的多集群游戏中找到纳什均衡，在这种游戏中，代理被分成不同的集群。尽管每个集群内的代理人协作以实现一个共同的目标，但集群却被视为在耦合成本函数方面在非合作游戏中相互竞争的虚拟玩家。在这种情况下，内部集群问题和集群之间的博弈需要同时解决。因此，由此产生的集群间Nash平衡也应该是集群内部社会福利问题的最小化者。在这项工作中，此设置被转换为具有稀疏状态信息的分布式优化问题。因此，关键信息（例如代理商的成本函数）将保持私有。我们提出了一种分布式算法，该算法以线性速率收敛到最优解。此外，我们将算法应用于扩展的Cournot博弈，以验证我们的理论结果。