Existence and continuity theorems of α-core of multi-leader-follower games with set payoffs

We consider the α-core of multi-leader-follower (MLMF in short) games with set payoffs. By applying a generalized Scarf’s theorem, we investigate the existence of the α-core of MLMF games with set payoffs in the setting of continuity and properly cone quasi-concave of set payoffs functions of leaders and followers. We also investigate the continuity of α-core of MLMF games with set payoffs when the leaders and followers payoffs functions are perturbed. As special cases, we obtain the corresponding existence and continuity theorems of single-leader-multi-follower (SLMF in short) games and multi-leader-single-follower (MLSF in short) games, respectively. Moreover, some numerical examples illustrate the results.