Bilevel Nash equilibrium problems: Numerical approximation via direct-search methods

We address the numerical approximation of bilevel problems where a Nash equilibrium has to be determined both in the upper level and in the lower level. Widely applied in engineering and economic frameworks, such models are an extension of the well-known Stackelberg duopoly model and of the classical bilevel optimization problem. In this paper, the lower level involves a non-parametric ratio-bounded game (as introduced by Caruso, Ceparano and Morgan in CSEF Working Papers 593, 2020) and the upper level involves a potential game (as introduced by Monderer and Shapley in Games Econ. Behav. 14, 1996). After presenting existence and uniqueness results for the solutions of such bilevel Nash equilibrium problems, we define a numerical method relying on a derivative-free unconstrained optimization technique connected to direct-search methods. The associate algorithm is shown to globally converge towards a solution; error estimations, rates of convergence and illustrative examples are also provided.