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 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, 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.

我们关注在基于代理的多集群游戏中寻找纳什均衡，其中代理被分成不同的集群。虽然每个集群内的代理协作以实现共同目标，但集群本身被认为是虚拟玩家，它们在非合作博弈中就耦合成本函数相互竞争。在这种情况下，需要同时解决集群内问题和集群之间的博弈。因此，由此产生的集群间纳什均衡也应该是集群内部社会福利问题的最小化。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，例如代理的成本函数，仍然是私密的。我们提出了一种以线性速率收敛到最优解的分布式算法。此外，我们将我们的算法应用于扩展的古诺游戏以验证我们的理论结果。