On the linear convergence of distributed Nash equilibrium seeking for multi-cluster games under partial-decision information

Abstract

This paper considers the distributed strategy design for Nash equilibrium (NE) seeking in multi-cluster games under a partial-decision information scenario. In the considered game, there are multiple clusters and each cluster consists of a group of agents. A cluster is viewed as a virtual noncooperative player that aims to minimize its local payoff function and the agents in a cluster are the actual players that cooperate within the cluster to optimize the payoff function of the cluster through communication via a connected graph. In our setting, agents have only partial-decision information, that is, they only know their own local cost functions, local feasible strategy sets and strategies of neighboring agents. To solve the NE seeking problem of this formulated game, a discrete-time distributed algorithm, called distributed projected gradient tracking algorithm (DPGT), is devised based on the inter- and intra-communication of clusters. In the designed algorithm, each agent is equipped with strategy variables including its own strategy and estimates of other clusters’ strategies. With the help of a weighted Frobenius norm and a weighted Euclidean norm, theoretical analysis is presented to rigorously show the linear convergence of the algorithm. Finally, a numerical example is given to illustrate the proposed algorithm.