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

In 1964 Shapley observed that a matrix has a saddle point in pure strategies whenever every its $$2 \times 2$$ 2 × 2 submatrix has one. In contrast, a bimatrix game may have no pure strategy Nash equilibrium (NE) even when every $$2 \times 2$$ 2 × 2 subgame has one. Nevertheless, Shapley’s claim can be extended to bimatrix games as follows. We partition all $$2 \times 2$$ 2 × 2 bimatrix games into fifteen classes $$C = \{c_1, \ldots , c_{15}\}$$ C = { c 1 , … , c 15 } depending only on the preferences of two players. A subset $$t \subseteq C$$ t ⊆ C is called a NE-theorem if a bimatrix game has a NE whenever it contains no subgame from t. We suggest a method to construct 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. Let us remark that the suggested approach, which may be called “math-pattern recognition”, is very general: it allows to characterize completely an arbitrary “target” in terms of arbitrary “attributes”.