Distributed Seeking of Time-Varying Nash Equilibrium for Non-Cooperative Games

In this note, we address a Nash equilibrium seeking problem for non-cooperative games. In contrast to previous works on Nash equilibrium seeking, the Nash equilibrium under consideration can be time-varying. A non-model-based seeking scheme is proposed to achieve time-varying Nash equilibrium seeking, where each player updates its strategy by employing an extremum seeking method. The proposed Nash seeking scheme consists of a gradient estimation algorithm and a gradient search algorithm, which can be designed in a modular fashion. For symmetric quadratic games, the proposed Nash equilibrium seeking method enables the estimated strategy to globally asymptotically converge to the Nash equilibrium. For general quadratic games that are not necessarily symmetric, the estimated strategy converges to a neighborhood of the Nash equilibrium. For more general non-quadratic games that may admit multiple equilibria, local convergence to the Nash equilibrium is proven.