# Geometric Convergence of Distributed Gradient Play in Games with Unconstrained Action Sets

## TL;DR

This paper introduces a simple distributed gradient algorithm that guarantees geometric convergence to Nash equilibria in non-cooperative games with unconstrained actions, outperforming previous methods in convergence speed.

## Contribution

It presents a standard distributed gradient play algorithm with a single step size, providing the first geometric convergence proof for such settings and comparing favorably to prior algorithms.

## Key findings

- Proves geometric convergence of the proposed algorithm.
- Demonstrates faster convergence than the GRANE algorithm.
- Requires only one parameter to ensure convergence.

## Abstract

We provide a distributed algorithm to learn a Nash equilibrium in a class of non-cooperative games with strongly monotone mappings and unconstrained action sets. Each player has access to her own smooth local cost function and can communicate to her neighbors in some undirected graph. We consider a distributed communication-based gradient algorithm. For this procedure, we prove geometric convergence to a Nash equilibrium. In contrast to our previous works [15], [16], where the proposed algorithms required two parameters to be set up and the analysis was based on a so called augmented game mapping, the procedure in this work corresponds to a standard distributed gradient play and, thus, only one constant step size parameter needs to be chosen appropriately to guarantee fast convergence to a game solution. Moreover, we provide a rigorous comparison between the convergence rate of the proposed distributed gradient play and the rate of the GRANE algorithm presented in [15]. It allows us to demonstrate that the distributed gradient play outperforms the GRANE in terms of convergence speed.