Equilibrium problems with equilibrium constraints are challenging both theoretically and computationally. However, they are suitable/adequate modeling formulations in a number of important areas, such as energy markets, transportation planning, and logistics. Typically, these problems are characterized as bilevel Nash-Cournot games. For instance, determin- ing the equilibrium price in an energy market involves top-level decisions of the generators that feed the bottom-level problem of an independent system operator. Taking the Karush- Kuhn-Tucker conditions of the lower-level optimization problems and inserting them into each upper-level player’s problem is one popular approach, but it has both numerical and theo- retical difficulties. To tackle the resulting highly nonlinear model we propose a primal-dual regularization with the remarkable property of yielding equilibrium prices of minimal norm. This theoretical feature can be seen as a stabilizing mechanism for price signals. It proves also useful in guiding the solution process when solving such problems computationally, via the mixed complementarity formulations. For a general energy market model we prove existence theorems for a specific equilibrium, and convergence of the proposed regularization scheme. Our numerical results using the PATH solver illustrate the proposal. In particular, we ex- hibit that, in the given context, PATH with the regularization approach computes a genuine equilibrium almost always, while without regularization the outcome is quite the opposite.