  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.
