A note on a class of equilibrium problems with equilibrium constraints

The behaviour of firms on an oligopolistic market is usually modeled via the CournotNash equilibrium concept ([13, 18]). In fact, it is the classical Nash equilibrium, where each player (firm) maximizes its profit subject to production constraints. Assuming that one from the firms has a temporal advantage over the others, this firm can increase its profit by replacing a Cournot-Nash strategy by a Stackelberg stratgy ([6, 17, 18]). This firm (called Leader) computes its new strategy under the assumption that the remaining firms (Followers) will share the rest of the market again in the noncooperative way. One obtains a bilevel structure with a CournotNash equilibrium (parametrized by the Leader's strategy) on the lower level. To compute the Leader's strategy, one has thus to solve a so-called mathematical program with equilibrium constraints (MPEC) ([8]). It might happen, however, that the standard Cournot-Nash strategy is simultaneously deserted by two or more firms. In such a case each of them has to make some assumptions not only about the behaviour of the Followers, but also about the behaviour of the remaining Leaders. Concerning this behaviour, two "extreme" situations can be distinguished: