Distributed Nash Equilibrium Seeking with Communication Delays

This paper addresses the problem of distributed Nash equilibrium seeking in N-player games for single integrator dynamics subject to strongly connected networks and communication delays. First, we propose a distributed estimator for each player, enabling them to estimate the actions of all players. Notably, we take into account unknown bounded time delays that occur during communication between players and their neighbors. Next, we design a distributed Nash equilibrium seeking law using the gradient play technique. Then, we analyze the stability of the closed-loop system, which consists of an interconnected nonlinear subsystem and a linear time-delay subsystem. By means of designing the Lyapunov-Krasovskii functional, we demonstrate that Nash equilibrium seeking is achieved at an exponential rate, even in the presence of unknown and bounded communication delays. Finally, we provide a simulation example to illustrate the effectiveness of our proposed approach.