#### Graduation Year

2008

#### Document Type

Open Access Senior Thesis

#### Degree Name

Bachelor of Science

#### Department

Mathematics

#### Reader 1

Francis Edward Su

#### Reader 2

Jon T. Jacobsen

#### 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 [8], where the theorem is used to prove the existence of an equilibrium strategy in n-person games. Sperner’s lemma, on the other hand, is a combinatorial result concerning the labelling 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. A natural question that arises is whether we can prove Kakutani’s fixed point theorem directly using Sperner’s lemma without going through Brouwer’s theorem. The objective of this thesis to understand Kakutani’s theorem, Sperner’s lemma, and how they are related. In particular, I explore ways in which Sperner’s lemma can be used to prove Kakutani’s theorem and related results.

#### Recommended Citation

Sondjaja, Mutiara, "Sperner's Lemma Implies Kakutani's Fixed Point Theorem" (2008). *HMC Senior Theses*. 214.

https://scholarship.claremont.edu/hmc_theses/214