Bilevel Problems with Nash Equilibrium Constraints under perturbations

We consider bilevel problems where a leader solves a MPEC or a MinSup problem with constraints defined by Nash equilibria of a non-cooperative game. We show that, in general, the infimal values of these problems are not stable under perturbations, in the sense that the sequence of the infimal values for the perturbed problems may not converge to the infimal value of the original problem even in presence of nice data. So, we introduce different types of approximate values, close to the exact value under suitable assumptions, and we investigate their asymptotic behavior under perturbations.