A timestamp-based projected gradient play for distributed Nash equilibrium seeking in monotone games

Abstract

In this paper, a timestamp-based projected gradient play is proposed for Nash equilibrium seeking over communication networks for strongly monotone games. Its difference from the well-known consensus-based Nash equilibrium seeking method is that each player’s local estimate of other players’ actions is updated through a timestamp-based broadcasting protocol instead of typical consensus protocols. We show that the proposed timestamp-based projected gradient play is indeed a projected gradient play with time delay, which can be further transformed into a disturbed full-information projected gradient play. Hence, we prove its convergence to the Nash equilibrium in the case of diminishing step-size and to the ϵ-approximation Nash equilibrium in the case of fixed step-size. Both convergence results are established on jointly strongly connected networks. The major advantage of the proposed approach is that it is intrinsically flexible to communication delay, disturbance, and asynchronization. Furthermore, its convergence rate is empirically comparable to the full-information projected gradient play with the same step-sizes. Illustrative numerical simulations are presented for distributed Nash equilibrium seeking in both quadratic and non-quadratic Nash–Cournot games over different types of communication networks.