  We consider a gossip approach for finding a Nash equilibrium in a distributed
multi-player network game. We extend previous results on Nash equilibrium
seeking to the case when the players' cost functions may be affected by the
actions of any subset of players. An interference graph is employed to
illustrate the partially-coupled cost functions and the asymmetric information
requirements. For a given interference graph, we design a generalized
communication graph so that players with possibly partially-coupled cost
functions exchange only their required information and make decisions based on
them. Using a set of standard assumptions on the cost functions, interference
and communication graphs, we prove almost sure convergence to a Nash
equilibrium for diminishing step sizes. We then quantify the effect of the
second largest eigenvalue of the expected communication matrix on the
convergence rate, and illustrate the trade-off between the parameters
associated with the communication and the interference graphs. Finally, the
efficacy of the proposed algorithm on a large-scale networked game is
demonstrated via simulation.
