Nash Equilibrium Seeking with Non-doubly Stochastic Communication Weight Matrix

We propose a distributed Nash equilibrium seeking algorithm in a networked game, where each player has incomplete information on other players’ actions. Players keep estimates and communicate over a strongly connected digraph with their neighbours according to a gossip communication protocol. Due to the asymmetric information exchange between players, a non-doubly (row)-stochastic weight matrix is defined. We prove almost-sure convergence of the algorithm to a Nash equilibrium under diminishing step-sizes. We extend the algorithm to graphical games in which players’ cost functions are dependent only on their neighbouring players in an interference digraph. Given the interference digraph, a communication digraph is designed so that players exchange only their required information. The communication digraph is a subset of the interference digraph and a superset of its transitive reduction. Finally, we verify the efficacy of the algorithm via simulation on a social media behavioural case.