A note on existence of mixed solutions to equilibrium problems with equilibrium constraints

A generalization of classical Stackelberg game concept leads to a new class of equilibrium games with hierarchical structure. The existence of equilibria in this class of problems is complicated due to the lack of convexity of the problem. Therefore, under the implicit programming approach, we apply the well-known existence result for solution in mixed strategies for (one-level) Nash equilibrium games. Further, we discuss the use of these results for the case when a solution map of the lower-level problem is multi valued.