  We study distributed algorithms for finding a Nash equilibrium (NE) in a
class of non-cooperative convex games under partial information. Specifically,
each agent has access only to its own smooth local cost function and can
receive information from its neighbors in a time-varying directed communication
network. To this end, we propose a distributed gradient play algorithm to
compute a NE by utilizing local information exchange among the players. In this
algorithm, every agent performs a gradient step to minimize its own cost
function while sharing and retrieving information locally among its neighbors.
The existing methods impose strong assumptions such as balancedness of the
mixing matrices and global knowledge of the network communication structure,
including Perron-Frobenius eigenvector of the adjacency matrix and other graph
connectivity constants. In contrast, our approach relies only on a reasonable
and widely-used assumption of row-stochasticity of the mixing matrices. We
analyze the algorithm for time-varying directed graphs and prove its
convergence to the NE, when the agents' cost functions are strongly convex and
have Lipschitz continuous gradients. Numerical simulations are performed for a
Nash-Cournot game to illustrate the efficacy of the proposed algorithm.
