Broadcast Gossip Algorithm for Nash Equilibrium Seeking of Nonaggregative Games

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.