On pure equilibria for bimatrix games

In Shapley several conditions are given for the existence of pure saddlepoints for a matrix game. In this paper the authors show that only a few of these conditions, when translated to the situation of a bimatrix game guarantee the existence of pure equilibria. Further, they associate with a bimatrix game a directed graph as well as a so-called ‘binary game’. If this graph has no cycles, then the bimatrix game in question has a pure equilibrium. It is shown that the binary game for a bimatrix game without a pure equilibrium possesses a ‘fundamental’ subgame, which can be characterized by means of ‘minimal’ cycles.