Asymptotic behavior of subgame perfect Nash equilibria in Stackelberg games

The study on how equilibria behave when perturbations occur in the data of a game is a fundamental theme, since actions and payoffs of the players may be affected by uncertainty or trembles. In this paper, we investigate the asymptotic behavior and the variational stability of the subgame perfect Nash equilibrium (SPNE) in one-leader one-follower Stackelberg games under perturbations both of the action sets and of the payoff functions. To pursue this aim, we consider a general sequence of perturbed Stackelberg games and a set of assumptions that fit the usual types of perturbations. We study if the limit of SPNEs of the perturbed games is an SPNE of the original game and if the limit of SPNE-outcomes of perturbed games is an SPNE-outcome of the original game. We fully positively answer when the follower’s best reply correspondence is single-valued. When the follower’s best reply correspondence is not single-valued, the answer is positive only for the SPNE-outcomes; whereas the answer for SPNEs may be negative, even if the perturbation does not affect the sets and affects only one payoff function. However, we show that under suitable non-restrictive assumptions it is possible to obtain an SPNE starting from the limit of SPNEs of perturbed games, possibly modifying the limit at just one point.