关键词: mixed game, Nash equilibirum, cooperative equilibrium, existence, generic stability

Abstract: This paper proposes a mixed game with finitely many leaders and follower populations. In such a game, two types of equilibria are defined. First, a Nash equilibrium is introduced for the scenario in which leaders and follower populations are in perfect competition, each maximizing its own payoff. Second, a cooperative equilibrium is proposed for the case where leaders and follower populations, respectively, form coalitions and cooperate. Moreover, the existence of both Nash and cooperative equilibria is proved under the condition that the payoff functions are continuous and quasi-concave. Finally, we demonstrate the generic stability of cooperative equilibria in mixed games. More concretely, in the sense of Baire category, the cooperative equilibria in most mixed games are stable under perturbations of the payoff functions. In short, this paper presents two main contributions. On the one hand, we provide a novel mixed-game framework, which differs from both classical leader-follower games and leader-follower population games. On the other hand, the Nash and cooperative equilibria in our mixed games are distinct from those in existing leader-follower population games. The results are further illustrated with examples.

Key words: mixed game, Nash equilibirum, cooperative equilibrium, existence, generic stability