On Coalitional Rationality in a Three-person Game

To determine the solution of any game in mathematical game theory, it is necessary to establish what behavior of the players should be considered optimal. In noncooperative games (games without coalitions), the concept of optimality is related, e.g., to the concepts of Nash and Berge equilibria. Optimality in the theory of cooperative games is characterized by the conditions of individual and collective rationality. This paper considers a three-person cooperative game in normal form. For this game, the concept of coalitional rationality is introduced by embracing the conditions of individual and collective rationality with some combination of the concepts of Nash and Berge equilibria. Sufficient conditions are established under which the game has a coalitional equilibrium of this type. In addition, the existence of such a solution in mixed strategies is proved in the case of continuous payoff functions and compact strategy sets of players.