  We address the Nash equilibrium problem in a partial-decision information
scenario, where each agent can only observe the actions of some neighbors,
while its cost possibly depends on the strategies of other agents. Our main
contribution is the design of a fully-distributed, single-layer, fixed-step
algorithm, based on a proximal best-response augmented with consensus terms. To
derive our algorithm, we follow an operator-theoretic approach. First, we
recast the Nash equilibrium problem as that of finding a zero of a monotone
operator. Then, we demonstrate that the resulting inclusion can be solved in a
fully-distributed way via a proximal-point method, thanks to the use of a novel
preconditioning matrix. Under strong monotonicity and Lipschitz continuity of
the game mapping, We prove linear convergence of our algorithm to a Nash
equilibrium. Furthermore, we show that our method outperforms the fastest known
gradient-based schemes, both in terms of guaranteed convergence rate, via
theoretical analysis, and in practice, via numerical simulations.
