  A distributed Nash equilibrium seeking algorithm is presented for networked
games. We assume an incomplete information available to each player about the
other players' actions. The players communicate over a strongly connected
digraph to send/receive the estimates of the other players' actions to/from the
other local players according to a gossip communication protocol. Due to
asymmetric information exchange between the players, a non-doubly (row)
stochastic weight matrix is defined. We show that, due to the non-doubly
stochastic property, the total average of all players' estimates is not
preserved for the next iteration which results in having no exact convergence.
We present an almost sure convergence proof of the algorithm to a Nash
equilibrium of the game. Then, we extend the algorithm for graphical games in
which all players' cost functions are only dependent on the local neighboring
players over an interference digraph. We design an assumption on the
communication digraph such that the players are able to update all the
estimates of the players who interfere with their cost functions. It is shown
that the communication digraph needs to be a superset of a transitive reduction
of the interference digraph. Finally, we verify the efficacy of the algorithm
via a simulation on a social media behavioral case.
