Sufficient conditions for the existence of Nash equilibria in bimatrix games in terms of forbidden \(2 \times 2\) subgames

In 1964 Shapley observed that a matrix has a saddle point in pure strategies whenever every its 2 × 2 submatrix has one. In contrast, a bimatrix game may have no pure strategy Nash equilibrium (NE) even when every 2 × 2 subgame has one. Nevertheless, Shapley’s claim can be extended to bimatrix games as follows. We partition all 2×2 bimatrix games into fifteen classes C = {c1, . . . , c15} depending on the preferences of 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) strongest NE-theorems.