Continuous-time Distributed Dynamics for Nash Equilibrium over Networks via a Passivity-Based Control Approach

In this paper we consider the problem of distributed Nash equilibrium (NE) seeking for a class of games over networks, a setting in which players have limited local information. We start from a continuous-time gradient-play dynamics that converges to an NE under strict monotonicity of the pseudo-gradient and assumes perfect information, i.e., instantaneous all-to-all player communication. We consider how to modify this gradient-play dynamics in the case of partial, or networked information between players. We propose an augmented gradient-play dynamics with correction in which players communicate locally only with their neighbours to compute an estimate of the other players' actions. We derive the new dynamics based on the reformulation as a multi-agent coordination problem over an undirected graph. We exploit incremental passivity properties and show that a synchronizing, distributed Laplacian feedback can be designed using relative estimates of the neighbours. Under a strict monotonicity property of the pseudo-gradient, we show that the augmented gradient-play dynamics converges to the consensus subspace, and moreover the action components converge to an NE of the game. We further discuss two cases that highlight the tradeoff between properties of the game and the communication graph.