A distributed approach to seek Nash equilibrium over jointly strongly connected switching networks

The Nash equilibrium seeking over networks of N players has been studied under the assumption that the network is static and strongly connected or switching and every time strongly connected. In this paper, we further consider the case where the network is jointly strongly connected. Since a jointly strongly connected network can be disconnected at every time instant, the existing approach cannot handle such a case. Like the literature, assuming the pseudogradient dynamics has a globally exponentially stable Nash equilibrium, we first establish a distributed estimator for actions of all players over jointly strongly connected networks. Then we compose the pseudogradient dynamics with the distributed estimator to obtain an extended gradient system with N + 1 subsystems. Under the assumption that the pseudogradient of the game is strongly monotone and Lipschitz continuous, we show that, starting from any initial condition, the state of every subsystem of the extended gradient system will exponentially converge to the Nash equilibrium of the game concerned.