Abstract
In 1964 Shapley observed that a matrix has a saddle point whenever every 2 ×2 submatrix of it has one. In contrast, a bimatrix game may have no Nash equilibrium (NE) even when every 2 ×2 subgame of it has one. Nevertheless, Shapley’s claim can be generalized for bimatrix games in many ways as follows. We partition all 2 ×2 bimatrix games into fifteen classes C = {c1, ..., c15} depending on the preference pre-orders of the two players. A subset t ⊆ C is called a NE-theorem if a bimatrix game has a NE whenever it contains no subgame from t. We suggest a general method for getting all minimal (that is, strongest) NE-theorems based on the procedure of joint generation of transversal hypergraphs given by a special oracle. By this method we obtain all (six) minimal NE-theorems.