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

Abstract

We consider two-stage multi-leader-follower games, called multi-leader-follower games with vertical information,where leaders in the first stage and followers\ud 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

Regularization and Approximation Methods in Stackelberg Games and Bilevel Optimization

“…For general multileader-follower games see, for example, [98,42]. The situation where each follower can observe the action of one leader has been investigated in [97] for an economic model and in [23] for deriving a selection result. An increasing literature is also devoted to the class of multi-leader-common-follower games and to its applications to electricity markets, see for example [8,9].…”

Nash equilibrium uniqueness in nice games with isotone best replies

“…6 our analysis provides new results in terms of the necessity and su¢ ciency of the conditions for both the (chain-)concavity and the isotonicity of Choice functions. To the best of our knowledge, the (chain-)concavity of optimal solutions has not been systematically studied in the literature, but results that guarantee the chain-concavity and the concavity of Choice functions can be useful also for some game-theoretic analysis of problems that are not necessarily related to Nash equilibrium uniqueness 6 . The isotonicity of Choice functions has been investigated in the literature; however, our results on this topic do not follow from well-known theorems such as [23]'s Monotonicity Theorem or other similar results of the subsequent literature (see, e.g., like those established in [29], [9], [1] and-tough in a more abstract spirit- [20]).…”

“…As f ( ; x + (1 ) z) is upper semicontinuous and also quasiconcave, Theorem 2.5.2 in [5] and (6) imply that ( x + (1 ) z) < â + (1 ) a.…”