Distributed Nash Equilibrium Seeking in an Aggregative Game on a Directed Graph

An aggregative game with local constraint sets is studied in this article, where each player's cost function is dependent on the aggregation function that is unavailable to all players. To compute the Nash equilibrium (NE) point of the game in a distributed manner, the players are endowed with several auxiliary state variables that are used to estimate the aggregation function by exchanging their estimates with local neighbors on a directed graph. In the two cases with strongly connected weight-balanced and weight-unbalanced directed graphs, respectively, NE seeking strategies are proposed by the interconnection of projected gradient-play with average consensus dynamics. The proposed algorithms are proved to be able to reach the NE point by using tools from variational inequality theory and Lyapunov stability theory. Finally, an example is simulated to demonstrate the theoretical results.