Simultaneous social cost minimization and nash equilibrium seeking in non-cooperative games

An N-coalition non-cooperative game is formulated in this paper. In the considered game, there are N interacting coalitions and each of them includes a set of agents. Each coalition acts as a virtual player (VP) in the game that aims to minimize its own objective function, which is defined as the sum of the agents' local objective functions in the coalition. However, the actual decision-makers are not the coalitions but the agents therein. That is, the agents within each coalition collaboratively minimize the coalition's objective function while constituting an entity that serves as a self-interested player (i.e., the coalition) in the game among the interacting coalitions. A seeking strategy is designed for the agents to find the Nash equilibrium of the N-coalition non-cooperative games. The equilibrium seeking strategy is based on an adaptation of a dynamic average consensus protocol and the gradient play. The dynamic average consensus protocol is leveraged to estimate the averaged gradients of the coalitions' objective functions. The gradient play is then implemented by utilizing the estimated information to achieve the Nash equilibrium seeking. Convergence results are established by utilizing Lyapunov stability analysis. A numerical example is given in supportive of the theoretical results.