Aggregative Games With Bilevel Structures: Distributed Algorithms and Convergence Analysis

In this paper, the problem of distributively seeking the equilibria of aggregative games with bilevel structures is studied. Different from the traditional aggregative games, here the aggregation is determined by the minimizer of a virtual leader's objective function in the inner level. Moreover, the global objective function of the virtual leader is formed by the sum of local functions, each of which is determined by the local action of a player. When making decisions, each player only has access to a local part of the virtual leader's objective function, and can communicate with its neighbors via a connected graph. To handle this problem, first, we propose a second order gradient-based distributed algorithm, where the Hessian matrices associated with the objective functions of the leader are involved. Under mild assumptions on the graph and cost functions, we prove that the actions of players asymptotically converge to the Nash equilibrium point. Then, for the case where the Hessian matrices associated with the objective functions of the virtual leader are not available, we propose a first order gradient-based distributed algorithm, where a distributed estimate strategy is developed to estimate the gradients of players' cost functions in the outer level. Under the same conditions, we prove that the convergence errors of players' actions to the Nash equilibrium point are linear with respect to the estimate parameters. Finally, simulations are provided to demonstrate the effectiveness of our theoretical results.