Lower Stackelberg equilibria: from bilevel optimization to Stackelberg games

Both pessimistic and optimistic bilevel optimization problems may be not stable under perturbation when the lower-level problem has not a unique solution, meaning that the limit of sequences of solutions (resp. equilibria) to perturbed bilevel problems is not necessarily a solution (resp. an equilibrium) to the original problem. In this paper, we investigate the notion of lower Stackelberg equilibrium, an equilibrium concept arising as a limit point of pessimistic equilibria and of optimistic equilibria of perturbed bilevel problems. First, connections with pessimistic equilibria and optimistic equilibria are obtained in a general setting, together with existence and closure results. Secondly, the problem of finding a lower Stackelberg equilibrium is shown to be stable under general perturbation, differently from what happens for pessimistic and optimistic bilevel problems. Then, moving to the game theory viewpoint, the set of lower Stackelberg equilibria is proved to coincide with the set of subgame perfect Nash equilibrium outcomes of the associated Stackelberg game. These results allow to achieve a comprehensive look on various equilibrium concepts in bilevel optimization and in Stackelberg games as well as to add a new interpretation in terms of game theory to the previous limit results on pessimistic equilibria and optimistic equilibria under perturbation.