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… (More)