2018/11/20 by Yitzchak Shmalo, Shmalo, Yitzchak
Computer Science · Economics, Econometrics and Finance · Mathematics · #Advanced Graph Theory Research #Advanced Optimization Algorithms Research #Dynamical Systems (math.DS) #FOS: Mathematics #Game Theory and Voting Systems
paper · pdf · doi:10.48550/arxiv.1811.08454
openalex publication_date 2018/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.