Kakutani fixed-point theorem
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 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.
[2] It has subsequently found widespread application in game theory and economics.
The requirement that φ(x) be convex for all x is essential for the theorem to hold.
Though it satisfies all other requirements of Kakutani's theorem, its value fails to be convex at x = 0.5.
Though it satisfies all other requirements of Kakutani's theorem, its graph is not closed; for example, consider the sequences xn = 0.5 - 1/n, yn = 3/4.
Some sources, including Kakutani's original paper, use the concept of upper hemicontinuity while stating the theorem: This statement of Kakutani's theorem is completely equivalent to the statement given at the beginning of this article.
Since all Euclidean spaces are Hausdorff (being metric spaces) and φ is required to be closed-valued in the alternative statement of the Kakutani theorem, the Closed Graph Theorem implies that the two statements are equivalent.
This application was specifically discussed by Kakutani's original paper.
[1] Mathematician John Nash used the Kakutani fixed point theorem to prove a major result in game theory.
[2] Stated informally, the theorem implies the existence of a Nash equilibrium in every finite game with mixed strategies for any finite number of players.
In this case: In general equilibrium theory in economics, Kakutani's theorem has been used to prove the existence of set of prices which simultaneously equate supply with demand in all markets of an economy.
[6] The existence of such prices had been an open question in economics going back to at least Walras.
[7] In this case: Kakutani's fixed-point theorem is used in proving the existence of cake allocations that are both envy-free and Pareto efficient.
The proof of Kakutani's theorem is simplest for set-valued functions defined over closed intervals of the real line.
Let φ: [0,1]→2[0,1] be a set-valued function on the closed interval [0,1] which satisfies the conditions of Kakutani's fixed-point theorem.
Let (ai, bi, pi, qi) for i = 0, 1, … be a sequence with the following properties: Thus, the closed intervals [ai, bi] form a sequence of subintervals of [0,1].
Now suppose we have chosen ak, bk, pk and qk satisfying (1)–(6).
We have a pair of sequences of intervals, and we would like to show them to converge to a limiting point with the Bolzano-Weierstrass theorem.
This lies in the cartesian product [0,1]×[0,1]×[0,1]×[0,1], which is a compact set by Tychonoff's theorem.
Since our sequence (an, pn, bn, qn) lies in a compact set, it must have a convergent subsequence by Bolzano-Weierstrass.
Using our finding above that q In dimensions greater one, n-simplices are the simplest objects on which Kakutani's theorem can be proved. The additional complexity in the higher-dimensional case exists in the first step of chopping up the domain into finer subpieces: Once these changes have been made to the first step, the second and third steps of finding a limiting point and proving that it is a fixed point are almost unchanged from the one-dimensional case. Kakutani's theorem for n-simplices can be used to prove the theorem for an arbitrary compact, convex S. Once again we employ the same technique of creating increasingly finer subdivisions. Kakutani's fixed-point theorem was extended to infinite-dimensional locally convex topological vector spaces by Irving Glicksberg[9] and Ky Fan. There is another version that the statement of the theorem becomes the same as that in the Euclidean case:[5] In his game theory textbook,[12] Ken Binmore recalls that Kakutani once asked him at a conference why so many economists had attended his talk.
