Combinatorial Proof of Kakutani's Fixed Point Theorem

Kakutani's fixed point theorem is a generalization of Brouwer's fixed point theorem to upper semicontinuous multivalued maps and is used extensively in game theory and other areas of economics. Earlier works have shown that Sperner's lemma implies Brouwer's theorem. In this paper, a new combinatorial labeling lemma, generalizing Sperner's original lemma, is given and is used to derive a simple proof for Kakutani's fixed point theorem. The proof is constructive and can be easily applied to numerically approximate the location of fixed points. The main method of the proof is also used to obtain a generalization of Kakutani's theorem for discontinuous maps which are locally gross direction preserving.