Kakutani fixed-point theorem

In mathematics, the Kakutani fixed point theorem is a fixed-point theorem. It is a generalization of the Brouwer fixed point theorem, and was developed by Shizuo Kakutani in 1941. It was famously used by John Nash in his description of Nash equilibrium.

Statement

Let $j:S\rightarrow 2^{S}$ be an upper semi-continuous correspondence (that is, a function which maps points in $S$ to subsets of $S$) where $S$ is a non-empty, compact, convex set in $\mathbb {R} ^{n}$. Assume that for all $x\in S$ the set $j(x)$ is convex and non-empty. Then, $j$ has a fixed point, i.e., there is an $x^{*}\in S$ such that $x^{*}\in j(x^{*}).$

Application on Game Theory

Mathematician John Nash used the Kakutani fixed point theorem to prove a major result in Game Theory. Stated informally, the theorem implies the existence of a Nash Equilibrium in every finite game with any number of players. This work would later earn him a Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel, more commonly known as the Nobel Prize for Economics.

References

• Page at the New School web site