Kakutani fixed-point theorem explained

In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing it. The Kakutani fixed point theorem is a generalization of the Brouwer fixed point theorem. The Brouwer fixed point theorem is a fundamental result in topology which proves the existence of fixed points for continuous functions defined on compact, convex subsets of Euclidean spaces. Kakutani's theorem extends this to set-valued functions.

The theorem was developed by Shizuo Kakutani in 1941,^[1] and was used by John Nash in his description of Nash equilibria. It has subsequently found widespread application in game theory and economics.^[2]

Statement

Kakutani's theorem states:^[3]

Let S be a non-empty, compact and convex subset of some Euclidean space Rⁿ.

Let φ: S → 2^S be a set-valued function on S with the following properties:

• φ has a closed graph;

• φ(x) is non-empty and convex for all x ∈ S.

Then φ has a fixed point.

Definitions

Set-valued function: A set-valued function φ from the set X to the set Y is some rule that associates one or more points in Y with each point in X. Formally it can be seen just as an ordinary function from X to the power set of Y, written as φ: X → 2^Y, such that φ(x) is non-empty for every

x \in X . Some prefer the term correspondence, which is used to refer to a function that for each input may return many outputs. Thus, each element of the domain corresponds to a subset of one or more elements of the range.

Closed graph: A set-valued function φ: X → 2^Y is said to have a closed graph if the set is a closed subset of X × Y in the product topology i.e. for all sequences

\{x _{n\}} _{n\in} and \{y _{n\}} _{n\in} such that x _{n\to} x , y _{n} \to y and y _{n} \in \varphi(x _{n)} for all n , we have y\in \varphi(x) .

Fixed point: Let φ: X → 2^X be a set-valued function. Then a ∈ X is a fixed point of φ if a ∈ φ(a).

Notes and References

1. Kakutani . Shizuo . Shizuo Kakutani . A generalization of Brouwer's fixed point theorem . Duke Mathematical Journal . 8 . 457–459 . 3 . 1941 . 10.1215/S0012-7094-41-00838-4.
2. Book: Border , Kim C. . Fixed Point Theorems with Applications to Economics and Game Theory . 1989 . Cambridge University Press . 0-521-38808-2 .
3. Book: Osborne . Martin J. . Ariel Rubinstein . Ariel . Rubinstein . A Course in Game Theory . Cambridge, MA . MIT . 1994 .