Pure nash equilibria in bimatrix games

We study the existence of pure Nash equilibria in bimatrix games. Shapley (1964) showed that a matrix game has a pure saddle point if every 2 x 2 subgame has one. For bimatrix games, however, a similar condition on 2 x 2 subgames is not sufficient for the existence of pure Nash equilibria. With the addition of some interesting conditions on the structure of the bimatrix game, we show that pure Nash equilibria are guaranteed to exist. We provide examples to illustrate our results as well as to show the necessity, sufficiency or otherwise of the conditions.