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

Abstract

In 1964 Shapley observed that a matrix has a saddle point in pure strategies whenever every its $2 \times 2$ submatrix has one. In contrast, a bimatrix game may have no pure strategy Nash equilibrium (NE) even when every $2 \times 2$ subgame has one. Nevertheless, Shapley’s claim can be extended to bimatrix games as follows. We partition all $2 \times 2$ bimatrix games into fifteen classes $C = \{c_1, \ldots , c_{15}\}$ depending only on the preferences of two players. A subset $t \subseteq C$ 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”.