Geometric convergence of distributed gradient play in games with unconstrained action sets

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 Tatarenko et al. (2018); Tatarenko et al. (2019), 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 Tatarenko et al. (2019). It allows us to demonstrate that the distributed gradient play outperforms the GRANE in terms of convergence speed.