  We consider the Nash equilibrium problem in a partial-decision information
scenario. Specifically, each agent can only receive information from some
neighbors via a communication network, while its cost function depends on the
strategies of possibly all agents. In particular, while the existing methods
assume undirected or balanced communication, in this paper we allow for
non-balanced, directed graphs. We propose a fully-distributed pseudo-gradient
scheme, which is guaranteed to converge with linear rate to a Nash equilibrium,
under strong monotonicity and Lipschitz continuity of the game mapping. Our
algorithm requires global knowledge of the communication structure, namely of
the Perron-Frobenius eigenvector of the adjacency matrix and of a certain
constant related to the graph connectivity. Therefore, we adapt the procedure
to setups where the network is not known in advance, by computing the
eigenvector online and by means of vanishing step sizes.
