Distributed nash equilibrium seeking in multi-agent games with partially coupled payoff functions

Abstract

In this paper, distributed Nash equilibrium seeking for multi-agent games, particularly for games where the players' payoff functions are partially coupled, is investigated. To model the (partial, explicit) dependence of the players' payoff functions on the players' actions, an interference graph is introduced. Besides, the players are supposed to be equipped with a communication graph to achieve Nash equilibrium seeking in a distributed fashion. Nash equilibrium seeking strategy design is firstly explored under undirected graphs (including undirected interference graphs and communication graphs). In the proposed seeking strategy, each player generates estimates on the actions of its neighbors in the undirected interference graph. However, in some games, the payoff function of player j is a function of player i's action (i ≠ j), while the payoff function of player i is explicitly independent of the action of player j. To capture the asymmetric, explicit dependence of the players' payoff functions on the players' actions, Nash equilibrium seeking under directed graphs (i.e., directed interference graphs and communication graphs) is investigated. The seeking strategy under directed graphs further reduces the number of the estimation variables. The presented convergence results are analytically studied and verified by numerical examples.