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 local information and cannot have full access to opponents' decisions. To solve the NE seeking problem of this formulated game, a discrete-time distributed algorithm, called distributed gradient tracking algorithm (DGT), 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 Fronbenius 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.

Distributed Nash Equilibrium Seeking Algorithm Design for Multi-Cluster Games with High-Order Players

“…, N }, the problem (1) is degraded into the noncooperative games of N players investigated in [13], [14], [15], [16], Therefore, the multicluster game (1) involves cooperative and competitive behaviors of the players simultaneously: players in the same cluster collectively optimize the cost function of the cluster, while players in different clusters selfishly minimize their own cost functions of the clusters that they belong to. Moreover, without involving the high-order dynamics, existing Nash equilibrium seeking algorithms for multi-cluster games (such as [19], [20], [21], [22], [23]) cannot control the high-order player (4) to accomplish multi-cluster game task (1) autonomously. Also, the high-order dynamics of players and the nonlinearity of cost functions make it difficult to design and analyze distributed game algorithms.…”

“…Nevertheless, it is noteworthy that in numerous engineering practices, cooperation and competition among agents always coexist, such as healthcare networks and transportation networks (see [17], [18]). Multi-cluster games can simultaneously characterize cooperation relationship within clusters and competition relationship between clusters, which extends the aforementioned distributed optimization problems and noncooperative game problems, and have aroused the interest of many scholars (see [19], [20], [21], [22], [23]).…”

“…In order to reduce the communication and computation costs, [21] exploited a Nash equilibrium seeking algorithm for multi-cluster games with interference graphs. For multi-cluster games with partial-decision information, [22] proposed a distributed Nash equilibrium seeking algorithm based on the intra-and intercommunication of clusters. For multi-cluster games with nonsmooth cost functions, [23] developed a Nash equilibrium seeking algorithm by Gaussian smoothing techniques.…”

“…Nevertheless, to the best of our knowledge, there are no results about multi-cluster games with high-order multi-agent systems. Moreover, without further integrating the control of high-order dynamics, existing Nash seeking algorithms for multi-cluster games, such as [19], [20], [21], [22], [23], are ineffective for the problem. These observations motivate us to study multi-cluster games of high-order multiagent systems.…”

“…The formulation extends non-cooperative games discussed in [13], [14], [15], [16] by containing the distributed optimization of players within clusters, the distributed optimization problems studied in [7], [8], [9], [10], [11], [12] by considering the noncooperative games between clusters, and the multi-cluster games investigated in [19], [20], [21], [22], [23] by adding the high-order dynamics of players. Because the players have highorder dynamics, existing algorithms for multi-cluster games, such as [19], [20], [21], [22], [23], cannot be applied to our problem. 2) We design a distributed algorithm based on state feedback and gradient descent to seek the Nash equilibrium of multi-cluster games.…”