Existance Of Perfect Equilibria: A Direct Proof

The authors formulate and prove a modification of Eilenberg-Montgomery fixed-point theorem, which is a generalization of Kakutani's theorem. It enables them to provide a direct proof of the existence of perfect equilibria in finite normal form games and extensive games with perfect recall. They construct a correspondence whose fixed points are precisely the perfect equilibria of a given finite game. Existence of a fixed point is secured by the modified version of Eilenberg-Montgomery theorem.