Any function from a non-empty polytope into itself that is locally gross direction preserving is shown to have the fixed point property. Brouwer's fixed point theorem for continuous functions is a special case. We discuss the application of the result in the area of non-cooperative game theory.

We study how players in a local interaction hawk dove game will learn, if they can either imitate the most succesful player in the neighborhood or play a best reply versus the opponent's previous action. From simulations it appears that each learning strategy will be used, because each performs...