Equilibrium selection in multi-leader-follower games with vertical information

We consider two-stage multi-leader-follower games, called multi-leader-follower games with vertical information, where leaders in the first stage and followers in the second stage choose simultaneously an action, but those chosen by any leader are observed by only one “exclusive” follower. This partial unobservability leads to extensive form games that have no proper subgames but may have an infinity of Nash equilibria. So it is not possible to refine using the concept of subgame perfect Nash equilibrium and, moreover, the concept of weak perfect Bayesian equilibrium could be not useful since it does not prescribe limitations on the beliefs out of the equilibrium path. This has motivated the introduction of a selection concept for Nash equilibria based on a specific class of beliefs, called passive beliefs, that each follower has about the actions chosen by the leaders rivals of his own leader. In this paper, we illustrate the effectiveness of this concept and we investigate the existence of such a selection for significant classes of problems satisfying generalized concavity properties and conditions of minimal character on possibly discontinuous data.