Nash Equilibrium Seeking for Incomplete Cluster Game in the Cooperation–Competition Network

In this article, we investigate the problem of seeking Nash equilibrium (NE) in multiagent systems within cooperation–competition networks. Each agent aims to optimize a total cost function that accounts for its own interests as well as those of its cooperators, considering both cooperative and noncooperative interactions with other agents. It is worth noting that, unlike in existing N-coalition games, the agents in this study only have knowledge of whether they are cooperative or noncooperative with their neighboring agents; they do not have information about non-neighboring agents who might be cooperators. As a result, due to potential disconnections in the communication topology within the cluster, it is not possible to consider the entire cluster as a virtual player to optimize its objective functions. To address this issue, we developed an algorithm using the singular perturbation technique, which divides the system into two distinct timescales. We propose a novel estimation algorithm to estimate the total cost function of disconnected subnetworks within the fast system. In the slow system, the search for NE is based on a gradient algorithm, while the Lyapunov stability theory is utilized to analyze the convergence of the algorithms. Furthermore, we extend the problem to accommodate scenarios where multiple subnetworks exist within the network. Numerical simulations are conducted to demonstrate their capability for resolving the noncooperative game problem in cooperation–competition networks.