A General Method for Selecting Subgame Perfect Nash Equilibria in Two-Player Stackelberg Games

Two-player Stackelberg games may have multiple Subgame Perfect Nash Equilibria (henceforth SPNEs), especially when the best reply correspondence of the follower is not a single-valued map. Aim of the paper is to investigate the issue of selection of SPNEs in two-player Stackelberg games by exploiting perturbations of the payoff functions of the game. To achieve such a goal, since the limit of “perturbed" SPNEs is not necessarily an SPNE of the initial game even for classic perturbations, first we show how to produce an SPNE starting from a sequence of SPNEs of perturbed games. This result allows to define a general selection method for SPNEs that can accommodate various behaviors of the players. More precisely, under mild assumptions on the data of the game we prove that perturbations relying on a Tikhonov regularization, on an adverse-to-move behaviour and on an altruistic behaviour fit the general method and we present the specific selection results associated to such perturbations. On the one hand, as regards to the Tikhonov regularization and the adverse-to-move behaviour, we extend or recover the results showed by Morgan and Patrone [Advances in Dynamic Games, (2006), pp. 209-221] and by Caruso, Ceparano and Morgan [Dyn. Games Appl., 9 (2019), pp. 416-432]. On the other hand, concerning the altruistic behaviour, we present a new specific selection method for SPNEs based on the slightly altruistic approach introduced by De Marco and Morgan [J. Optim. Theory Appl., 137 (2008), pp. 347-362] for simultaneous-move games. Finally, we illustrate by examples that the general method carried out under the three different “behaviours" just mentioned can select different SPNEs.