Kakutani fixed-point theorem

E3648

fixed-point theorem mathematical theorem

The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.

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All labels observed (5)

Label	Occurrences
Kakutani fixed-point theorem canonical	13
Kakutani fixed point theorem	1
Kakutani fixed-point theorem (contributions)	1
Kakutani fixed-point theorem to infinite-dimensional settings	1
Kakutani’s theorem on random ergodic theorems	1

How this entity was disambiguated

This entity first appeared as the object of triple T31596 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Kakutani fixed-point theorem
Context triple: [Non-cooperative Games, usesTool, Kakutani fixed-point theorem]

A. Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
B. Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
C. Hilbert spaces
Hilbert spaces are complete inner product spaces that provide the fundamental framework for modern functional analysis and many areas of mathematical physics.
D. Non-cooperative Games
Non-cooperative Games is John Nash’s seminal 1950 paper that founded modern non-cooperative game theory and introduced the concept now known as Nash equilibrium.
E. Oskar Morgenstern
Oskar Morgenstern was an Austrian-American economist best known as the co-founder of game theory through his seminal work "Theory of Games and Economic Behavior" with John von Neumann.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Kakutani fixed-point theorem
Target entity description: The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.

A. Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
B. Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
C. Hilbert spaces
Hilbert spaces are complete inner product spaces that provide the fundamental framework for modern functional analysis and many areas of mathematical physics.
D. Non-cooperative Games
Non-cooperative Games is John Nash’s seminal 1950 paper that founded modern non-cooperative game theory and introduced the concept now known as Nash equilibrium.
E. Hamiltonian economic program
The Hamiltonian economic program was Alexander Hamilton’s comprehensive plan to strengthen the early United States’ financial system through federal assumption of state debts, creation of a national bank, and support for manufacturing and commerce.
F. None of above. chosen

Statements (50)

Predicate	Object
instanceOf	fixed-point theorem ⓘ mathematical theorem ⓘ
appliesTo	correspondences in finite-dimensional normed spaces ⓘ set-valued maps on Euclidean spaces ⓘ
assumption	correspondence is upper hemicontinuous ⓘ domain is a nonempty compact convex subset of a Euclidean space ⓘ graph of the correspondence is closed ⓘ set-valued map has nonempty values ⓘ values of the correspondence are convex sets ⓘ
conclusion	there exists a fixed point ⓘ there exists x such that x is in F(x) ⓘ
field	functional analysis ⓘ game theory ⓘ mathematical analysis ⓘ topology ⓘ
generalizes	Brouwer fixed-point theorem ⓘ
hasCondition	domain is compact ⓘ domain is convex ⓘ graph is closed or correspondence is upper hemicontinuous ⓘ values are convex ⓘ values are nonempty ⓘ
hasProofMethod	topological arguments ⓘ use of Brouwer fixed-point theorem ⓘ
impliesExistenceOf	fixed point of a correspondence ⓘ
involvesStructure	Euclidean space ⓘ topological vector space ⓘ
isToolFor	nonconstructive existence proofs ⓘ proof of Nash's theorem on equilibria in finite games ⓘ
namedAfter	Shizuo Kakutani ⓘ
namedAfterOccupation	Japanese-American mathematician ⓘ
relatedTo	Brouwer fixed-point theorem ⓘ Glicksberg fixed-point theorem ⓘ Schauder fixed-point theorem ⓘ linked to: Brouwer fixed-point theorem
timePeriod	20th century ⓘ
usedFor	existence of Nash equilibrium ⓘ existence of competitive equilibria ⓘ existence of equilibria in finite games ⓘ existence of general equilibrium in economics ⓘ existence of solutions to differential inclusions ⓘ existence of solutions to variational inequalities ⓘ existence proofs in game theory ⓘ existence proofs in mathematical economics ⓘ
usesConcept	compactness ⓘ convexity ⓘ correspondence (set-valued map) ⓘ fixed point ⓘ multivalued function ⓘ nonempty convex compact subset ⓘ set-valued function ⓘ upper hemicontinuity ⓘ

How these facts were elicited

Referenced by (17)

Full triples — surface form annotated when it differs from this entity's canonical label.

Non-cooperative Games → usesTool → Kakutani fixed-point theorem ⓘ

Brouwer fixed-point theorem → relatedTo → Kakutani fixed-point theorem ⓘ

Glicksberg fixed-point theorem → extends → Kakutani fixed-point theorem ⓘ

Glicksberg fixed-point theorem → generalizes → Kakutani fixed-point theorem to infinite-dimensional settings ⓘ

linked to: Kakutani fixed-point theorem

Glicksberg fixed-point theorem → relatedTo → Kakutani fixed-point theorem ⓘ

Shizuo Kakutani → notableWork → Kakutani fixed-point theorem ⓘ

Shizuo Kakutani → notableWork → Kakutani’s theorem on random ergodic theorems ⓘ

linked to: Kakutani fixed-point theorem

Shizuo Kakutani → knownFor → Kakutani fixed-point theorem ⓘ

Shizuo Kakutani → hasTheoremNamedAfter → Kakutani fixed-point theorem ⓘ

Hugo Steinhaus → notableWork → Kakutani fixed-point theorem (contributions) ⓘ

linked to: Kakutani fixed-point theorem

Schauder fixed-point theorem → relatedTo → Kakutani fixed-point theorem ⓘ

Sperner's lemma → usedForProofOf → Kakutani fixed-point theorem ⓘ

Shizuo Kakutani → notableFor → Kakutani fixed-point theorem ⓘ

subject linked to: Shizuo

Knaster–Kuratowski–Mazurkiewicz lemma → underlies → Kakutani fixed-point theorem ⓘ

Gale–Nikaidō–Debreu theorem → relatedTo → Kakutani fixed point theorem ⓘ

linked to: Kakutani fixed-point theorem

Scarf’s lemma → relatedTo → Kakutani fixed-point theorem ⓘ

Nash equilibrium → mathematicallyBasedOn → Kakutani fixed-point theorem ⓘ

All labels observed (5)

How this entity was disambiguated Show

Statements (50)

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Referenced by (17)

How this entity was disambiguated

How these facts were elicited