Networked games have wide applications and have received attention from many researchers. Nonaggregative games are a kind of networked games in which the payoff function of each player depends on the individual decisions of all players other than an aggregative term of other players' decisions. The proposed asynchronous broadcast-based gossip algorithm seeks the Nash equilibrium of nonaggregative games. Our algorithm is easier to implement and has a faster convergence rate than gossip-based algorithms, which provide a more effective way to a kind of widely existing games, such as the channel optical signal-to-noise ratio optimization problem.

In this article, we focus on seeking the Nash equilibrium (NE) of nonaggregative games over a directed communication network. An asynchronous broadcast-based gossip algorithm is proposed, where each player broadcasts its own decision and estimates of other players' decisions to its out-neighbors. Then, the neighboring players update its respective decisions using its independent stepsize and estimates of other players' decisions by weighting its own and the received decision estimates from $(0,\,1)$. The almost surely convergence of the proposed algorithm with the diminishing stepsizes is presented by using a lemma and the property of the Kronecker product. Furthermore, the expected error bound between the NE and decisions for the proposed algorithm with constant stepsizes is presented with the increasing active probability of each player. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed algorithm and the convergence rate comparison.