# Mutiara Sondjaja

Harvey Mudd College Mathematics 2008

Thesis Advisor: | Prof. Francis E. Su |
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Second Reader: | Prof. Jon Jacobsen |

Thesis Proposal: | Understanding Kakutani's fixed point theorem and Sperner's lemma |

Poster: | Sperner's Lemma implies Kakutani's Theorem |

Final Report | Sperner's Lemma implies Kakutani's Fixed Point Theorem |

## Understanding Kakutani's Fixed Point Theorem and Sperner's Lemma

### Abstract

Kakutani's fixed point theorem has many applications in economics and game theory. One of its most well-known applications is in John Nash's paper in 1950, where the theorem is used to prove the existence of an equilibrium in n-person games.

Sperner's lemma, on the other hand, is a combinatorial result concerning the labeling of the vertices of simplices and their triangulations. It is known that Sperner's lemma is equivalent to a result called Brouwer's fixed point theorem, of which Kakutani's theorem is a generalization. Furthermore, many existing alrgoirhtms used to find fixed points of functions also use Sperner's lemma in their construction. A natural question that arises, then, is whether we can prove Kakutani's fixed point theoreme directly using Sperner's lemma, without going through Brouwer's theorem

In this thesis project, I aim to understand Kakutani's theorem, Sperner's lemma, and how they are related. In particular, we show that Kakutani's theorem can be proven directly from Sperner's lemma.