Kakutani fixed-point theorem

Theorem

Let $S$ be a (nonempty), compact and convex subset of $\mathbb{R}^n$, and let $f$ be an upper hemicontinuous function (u.h.c.) which assigns to each $x \in S$ a closed and convex subset of $S$. Then there exists some $x \in S$ such that $x \in f(x)$.

Alternatively,
Suppose $A \subset \mathbb{R}^N$ is nonempty, compact, convex set. $f: A \to A$ u.h.c. with the property that $f(x) \subset A$ is nonempty and convex for every $x \in A$. Then $f(\cdot)$ has a fixed point: $\exists$ an $x \in A$ such that $x \in f(x)$.

Theorem (mapping on simplex, original formulation by Kakutani 1941)

Let $\mathfrak{R}(S)$ be the family of all closed convex subsets of $S$.

If $x \to \phi(x)$ is an upper semicontinuous point-to-set mapping a $r$-dimensional closed simplex $S$ into $\mathfrak{R}(S)$, then there exist a $x_0 \in S$ such that $x_0 \in \phi(x_0)$.

Proof

#incomplete

Notes

There is an alternative definition with upper semicontinuous functions (slight difference from u.h.c.)

This differs from Brouwer fixed-point theorem in that it deals with set-valued maps, whereas Brouwer's fixed point theorem deals with continuous functions. (Kakutani, 1941) regards it as a generalization of Brouwer's theorem.

This is used by Nash's paper in demonstrating the Nash equilibrium for $n$-person games (Nash, 1950, doi: 10.1073/pnas.36.1.48).

Kakutani uses this theorem to demonstrate von Neumann's minimax theorem.

point-to-set mapping, i.e. a correspondence.

See also:

• Brouwer fixed-point theorem
• Weierstrass extreme value theorem

References

1. https://en.wikipedia.org/wiki/Kakutani_fixed-point_theorem
2. T. Başar and G.J. Olsder, Dynamic Noncooperative Game Theory, 2nd edition, Classics in Applied Mathematics, SIAM, Philadelphia, 1999.
   • Appendix C, theorem C.2
3. Kakutani, S., Duke Math. J., 8, 457– 459 (1941). DOI: 10.1215/S0012-7094-41-00838-4