Fully distributed Nash equilibrium seeking over time-varying communication networks with linear convergence rate

We design a distributed algorithm for learning Nash equilibria over time-varying communication networks in a partial-decision information scenario, where each agent can access its own cost function and local feasible set, but can only observe the actions of some neighbors. Our algorithm is based on projected pseudo-gradient dynamics, augmented with consensual terms. Under strong monotonicity and Lipschitz continuity of the game mapping, we provide a simple proof of linear convergence, based on a contractivity property of the iterates. Compared to similar solutions proposed in literature, we also allow for time-varying communication and derive tighter bounds on the step sizes that ensure convergence. In fact, in our numerical simulations, our algorithm outperforms the existing gradient-based methods, when the step sizes are set to their theoretical upper bounds. Finally, to relax the assumptions on the network structure, we propose a different pseudo-gradient algorithm, which is guaranteed to converge on time-varying balanced directed graphs.