A Local Variation Method for Bilevel Nash Equilibrium Problems

We address the numerical approximation of bilevel problems consisting of one Nash equilibrium problem in the upper level and another Nash equilibrium problem in the lower level. These problems, widely employed in engineering and economic applications, are a generalization of the well-known Stackelberg (or bilevel optimization) problem. In this paper, we define a numerical method for bilevel Nash equilibrium problems where in the lower level there is a ratio-bounded game (introduced in Caruso, Ceparano, Morgan [CSEF Working Papers, 593 (2020)]) and in the upper level there is a potential game (introduced in Monderer, Shapley [Games Econ. Behav., 14 (1996)]). The method, relying on a derivative-free unconstrained optimization technique called local variation method, is shown to globally converge towards a solution of the problem and also allows to obtain error estimations.