Fully Distributed Nash Equilibrium Seeking for High-order Players with Bounded Controls and Directed Graphs

TL;DR

This paper develops a fully distributed control strategy for high-order players with bounded controls on directed graphs, ensuring convergence to Nash equilibrium without global information.

Contribution

It introduces a novel distributed Nash equilibrium seeking method for high-order integrator players with bounded controls on directed graphs, using adaptive consensus protocols.

Findings

Controls remain bounded due to saturation functions

Players' actions converge to Nash equilibrium

Method works on directed communication graphs

Abstract

This paper explores distributed Nash equilibrium seeking problems for games in which the players have limited knowledge on other players' actions. In particular, the involved players are considered to be high-order integrators with their control inputs constrained within a pre-specified region. A linear transformation for the players' dynamics is firstly utilized to facilitate the design of bounded control inputs incorporating multiple saturation functions. By introducing consensus protocols with adaptive and time-varying gains, the unknown actions for players are distributively estimated. Then, a fully distributed Nash equilibrium seeking strategy is exploited, showcasing its remarkable properties: i) ensuring the boundedness of control inputs; ii) avoiding any global information/parameters; and iii) allowing the graph to be directed. Based on Lyapunov stability analysis, it is theoretically proved that the proposed distributed control strategy can lead all the players' actions to the Nash equilibrium. Finally, an illustrative example is given to validate effectiveness of the proposed method.