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We propose a type of non-cooperative game, termed multi-cluster aggregativegame, which is composed of clusters as players, where each cluster consists ofcollaborative agents with cost functions depending on their own decisions andthe aggregate quantity of each participant cluster to modeling large-scale andhierarchical multi-agent systems. This novel game model is motivated bydecision-making problems in competitive-cooperative network systems withlarge-scale nodes, such as the Energy Internet. To address challenges arisingin seeking Nash equilibrium for such network systems, we develop an algorithmwith a hierarchical communication topology which is a hybrid with distributedand semi-decentralized protocols. The upper level consists of clustercoordinators estimating the aggregate quantities with local communications,while the lower level is cluster subnets composed of its coordinator and agentsaiming to track the gradient of the corresponding cluster. In particular, theclusters exchange the aggregate quantities instead of their decisions torelieve the burden of communication. Under strongly monotone and mildlyLipschitz continuous assumptions, we rigorously prove that the algorithmlinearly converges to a Nash equilibrium with a fixed step size.We present theapplications in the context of the Energy Internet. Furthermore, the numericalresults verify the effectiveness of the algorithm.

Multi-Cluster Aggregative Games: A Linearly Convergent Nash Equilibrium Seeking Algorithm and its Applications in Energy Management