# A Passivity-Based Approach to Nash Equilibrium Seeking over Networks

## TL;DR

This paper develops a passivity-based distributed algorithm for Nash equilibrium seeking over networks, allowing players with limited local information to converge to an equilibrium through local communication and augmented dynamics.

## Contribution

It introduces a novel augmented gradient-play dynamics that enables NE seeking with partial information using passivity and Laplacian feedback over networks.

## Key findings

- Convergence to NE under strict monotonicity of the pseudo-gradient.
- Effective distributed dynamics based on local communication.
- Tradeoffs between game properties and communication graph characteristics.

## Abstract

In this paper we consider the problem of distributed Nash equilibrium (NE) seeking 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 consensus on the NE of the game. We further discuss two cases that highlight the tradeoff between properties of the game and the communication graph.