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 generalise the coalition structure core to partition function games. Our definition relies only on one crucial assumption, namely that there is some internal consistency in the game: residuals of the deviation play a game similar to the initial one, and –whenever this is possible– they...