Distributed Nash Equilibrium Seeking in a Multi-Coalition Noncooperative Game Under Incomplete Decision Information

This brief studies a distributed Nash equilibrium (NE) seeking strategy for a multi-coalition noncooperative game with local decision sets. In the game model, each coalition consists of some agents and its cost funciton is determined by the sum of the local cost functions subject to the agents in the coalition, where each coalition serves as a virtual player and the actual decision-makers are the agents in the coalition. The cost of each coalition is to minimize its own cost function under the case that the agents in each coalition only know their local information and can not directly access the opponent’s decision information in other coalitions. To this end, a distributed seeking strategy that can be viewed a two-time scale system is developed to search the NE of the formulated multi-coalition game. The fast time-scale system is used to estimate each coalition’s pseudo-gradient by a connected interference graph that illustrates the interactions among the agents in each coalition. The slow time-scale system based on a projected psedudo-gradient dynamics is implemented to seek the NE, where the coalitions estimate other coalitions’ decisions by interacting only with their neighbors via a weight-balanced digraph. For this distributed NE seeking strategy, asymptotic convergence result is given by Lyapunov stability analysis. Finally, the theoretical results is demonstrated via a numerical example.