Distributed Nash equilibrium seeking with a dynamic set of players

This paper formulates a new distributed Nash equilibrium seeking problem with a dynamic set of players in which players are allowed to join and leave the network in a free manner during the decision-making process. To accommodate the dynamic joining and leaving behaviors of the players, a status estimation mechanism, which is capable of estimating in a finite time whether the players are active or inactive, is introduced. Based on the status estimation mechanism, a gradient play based algorithm is developed for distributed Nash equilibrium seeking in the dynamic environment. It is shown that under strongly connected communication graphs, players’ actions are convergent to a small neighborhood of the new Nash equilibrium linearly every time the player set changes. Moreover, the convergence accuracy and convergence rate can be adjusted by suitably tuning the step-size. To cover more general communication scenarios, strongly connected graphs are further relaxed to be B-jointly connected graphs, under which the convergence properties of the proposed algorithm are analytically studied. Furthermore, the upper bound of the average tracking error is quantified to evaluate the dynamic performance of the proposed algorithm. In the last, a simulation study on energy consumption games is given to verify the effectiveness of the proposed algorithm.