We propose a type of non-cooperative game, termed multi-cluster aggregative game, which is composed of clusters as players, where each cluster consists of collaborative agents with cost functions depending on their own decisions and the aggregate quantity of each participant cluster to modeling large-scale and hierarchical multi-agent systems. This novel game model is motivated by decision-making problems in competitive-cooperative network systems with large-scale nodes, such as the Energy Internet. To address challenges arising in seeking Nash equilibrium for such network systems, we develop an algorithm with a hierarchical communication topology which is a hybrid with distributed and semi-decentralized protocols. The upper level consists of cluster coordinators estimating the aggregate quantities with local communications, while the lower level is cluster subnets composed of its coordinator and agents aiming to track the gradient of the corresponding cluster. In particular, the clusters exchange the aggregate quantities instead of their decisions to relieve the burden of communication. Under strongly monotone and mildly Lipschitz continuous assumptions, we rigorously prove that the algorithm linearly converges to a Nash equilibrium with a fixed step size.We present the applications in the context of the Energy Internet. Furthermore, the numerical results verify the effectiveness of the algorithm.

翻译：本文提出了一类非合作博弈，称为多簇聚合博弈，其中簇作为博弈参与方，每个簇由协作智能体组成，其代价函数依赖于自身决策以及各参与簇的聚合量，旨在刻画大规模、分层的多智能体系统。这一新型博弈模型源于具有大规模节点的竞合网络系统中的决策问题，例如能源互联网。为了解决此类网络系统中纳什均衡求解面临的挑战，我们设计了一种具有分层通信拓扑的算法，该算法融合了分布式与半分散式协议。上层由簇协调器组成，通过局部通信估计聚合量；下层则由各簇的协调器及其智能体组成的子网构成，旨在追踪对应簇的梯度。特别地，簇之间交换聚合量而非决策，以减轻通信负担。在强单调和温和利普希茨连续假设下，我们严格证明了该算法在固定步长下线性收敛至纳什均衡。本文展示了该算法在能源互联网场景中的应用，并通过数值结果验证了其有效性。