Nash cooperative solution for two-person games in strategic form

In this paper we consider two-person games in strategic form in cases in which each player has only two moves. The related Nash competitive and cooperative solution and Nash equilibria are presented. Nash cooperative solution is the only solution such that the corresponding payoffs respect three properties (belonging to feasible set, Pareto optimality and symmetry) together with two technical axioms of independence from irrelevant alternatives and from linear transformations. In addition to the existence and uniqueness of this solution that has been proved for many cases [5, 11, 24] in this paper we also prove for the first time that if the convexity of the set X is necessary and sufficient condition also the condition that the Pareto optimal boundary of the convex hull of the individually rational feasible set coincides with the Pareto optimal boundary of X is necessary and sufficient condition i.e. X is convex. This finding implies that if the Pareto optimal boundary of the convex hull of the feasible set coincides with the Pareto optimal boundary of the last set, then the solution is unique; elsewhere the solution does not exist.