Distributed Nash Equilibrium Seeking in N-Cluster Games with Non-Uniform Constant Step-Sizes

This paper studies a class of non-cooperative games, known as
$N$
-cluster game, which subsumes both cooperative and non-cooperative nature among multiple agents in the two problems. Moreover, we consider a partial-decision information game setup, i.e., the agents have no direct access to the decisions of other agents in all clusters, and hence need to communicate with each other. We propose a distributed NE seeking algorithm by a synthesis of consensus and gradient tracking. Unlike other existing discrete-time methods for N- cluster games where a common step-size is either publicly known by all agents or only known by agents from the same cluster, the proposed algorithm can work with non-uniform constant step-sizes, which allows the agents (both within and across the clusters) to choose their own preferred step-sizes. We prove that all agents' decisions converge linearly to their corresponding NE so long as the largest step-size and the heterogeneity of the step-sizes are small. We verify the derived results through a numerical example in a Cournot competition game.