Stackelberg problems: subgame perfect equilibria via Tikhonov regularization

In this chapter we consider a two-stage game with one leader and one (or more) followers and we investigate the behavior of a Tikhonov regularization when the best reply for the follower(s) is not uniquely determined. More precisely, we show, under mild assumptions in the case of one follower and sufficiently mild in the case of two followers, that a convergent sequence of solutions to regularized two-stage games generates a subgame perfect equilibrium (SPE) of the original game, providing a constructive way to approach an SPE in a continuous setting. Various elementary examples show that our results cannot be strengthened up to guaranteeing convergence to a strong or a weak Stackelberg equilibrium and that the method cannot be extended to all of the cases in which two followers play a mixed extension of a finite game.