  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.
