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| In [[mathematical analysis]], the '''Kakutani fixed-point theorem''' is a [[fixed-point theorem]] for [[set-valued function]]s. It provides sufficient conditions for a set-valued function defined on a [[convex set|convex]], [[compact set|compact]] subset of a [[Euclidean space]] to have a [[fixed point (mathematics)|fixed point]], i.e. a point which is [[map (mathematics)|map]]ped to a set containing it. The Kakutani fixed point theorem is a generalization of [[Brouwer fixed point theorem]]. The Brouwer fixed point theorem is a fundamental result in [[topology]] which proves the existence of fixed points for [[continuous function (topology)|continuous function]]s defined on compact, convex subsets of Euclidean spaces. Kakutani's theorem extends this to set-valued functions.
| | Wilber Berryhill is the title his parents gave him and he completely digs that name. Some time ago she chose to reside in Alaska and her mothers and fathers reside close by. It's not a typical thing but what I like doing is to climb but I don't have the time lately. Invoicing is my profession.<br><br>My blog post ... love psychic readings ([http://cartoonkorea.com/ce002/1093612 cartoonkorea.com]) |
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| The theorem was developed by [[Shizuo Kakutani]] in 1941<ref name="kakutani">
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| {{cite journal
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| | last = Kakutani
| |
| | first = Shizuo
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| | authorlink = Shizuo Kakutani
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| | title = A generalization of Brouwer’s fixed point theorem
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| | journal = Duke Mathematical Journal
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| | volume = 8
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| | pages = 457–459
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| | issue = 3
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| | year = 1941
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| | doi = 10.1215/S0012-7094-41-00838-4}}</ref> and was famously used by [[John Forbes Nash, Jr.|John Nash]]<ref name="nash"/> in his description of [[Nash equilibrium|Nash equilibria]]. It has subsequently found widespread application in [[game theory]] and [[economics]].<ref>{{cite book
| |
| | last = Border
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| | first = Kim C.
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| | title = Fixed Point Theorems with Applications to Economics and Game Theory
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| | year = 1989
| |
| | publisher = Cambridge University Press
| |
| }}</ref>
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| ==Statement<ref name=Osborne>Osborne, Martin J., and [[Ariel Rubinstein]]. ''A Course in Game Theory''. Cambridge, MA: MIT, 1994. Print.</ref>==
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| Kakutani's theorem states:
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| : ''Let S be a [[empty set|non-empty]], [[compact set|compact]] and [[convex set|convex]] [[subset]] of some [[Euclidean space]] '''R'''<sup>n</sup>. Let φ: S → 2<sup>S</sup> be a [[set-valued function]] on S with a closed graph and the property that φ(x) is non-empty and convex for all x ∈ S. Then φ has a [[fixed point (mathematics)|fixed point]].
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| ==Definitions==
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| ;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 (mathematics)|function]] from ''X'' to the [[power set]] of ''Y'', written as φ: ''X''→2<sup>''Y''</sup>, such that φ(x) is non-empty for every <math>x \in X</math>. 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.
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| ;Closed graph: A set-valued function φ: ''X''→2<sup>''Y''</sup> is said to have a '''closed graph''' if the set {(''x'',''y'')| ''y'' ∈ φ(''x'')} is a [[closed set|closed]] subset of ''X''×''Y'' in the [[product topology]] i.e. for all sequences <math>\{x_{n}\}_{n\in \mathbb{N}}</math> and <math>\{y_{n}\}_{n\in \mathbb{N}}</math> such that <math>x_{n}\to x</math>, <math>y_{n}\to y</math> and <math>y_{n}\in \varphi(x_{n})</math> for all <math>n</math>, we have <math>y\in \varphi(x)</math> .
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| ;Fixed point: Let φ: ''X''→2<sup>''X''</sup> be a set-valued function. Then ''a'' ∈ ''X'' is a '''fixed point''' of φ if ''a'' ∈ φ(''a'').
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| ==Example==
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| [[File:Kakutani.svg|thumb|150px|Fixed points for f(x)=<nowiki>[1−x/2, 1−x/4]</nowiki>]]Let ''f''(''x'') be a set-valued function defined on the closed [[interval (mathematics)|interval]] [0, 1] that maps a point ''x'' to the closed interval [1 − ''x''/2, 1 − ''x''/4]. Then ''f(x)'' satisfies all the assumptions of the theorem and must have fixed points.
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| In the diagram, any point on the 45° line (dotted line in red) which intersects the graph of the function (shaded in grey) is a fixed point, so in fact there is an infinity of fixed points in this particular case. For example, ''x'' = 0.72 (dashed line in blue) is a fixed point since 0.72 ∈ [1 − 0.72/2, 1 − 0.72/4].
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| <br style="clear:both" /><!--Stops image from floating past-->
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| ==Non-example==
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| [[File:Kakutani non.svg|thumb|150px|A function without fixed points]]The requirement that φ(''x'') be convex for all ''x'' is essential for the theorem to hold.
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| Consider the following function defined on [0,1]:
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| :<math>
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| f(x)=
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| \begin{cases}
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| 3/4 & 0 \le x < 0.5 \\
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| \{ 3/4, 1/4 \} & x = 0.5 \\
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| 1/4 & 0.5 < x \le 1 \\
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| \end{cases}
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| </math>
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| The function has no fixed point. Though it satisfies all other requirements of Kakutani's theorem, its value fails to be convex at ''x'' = 0.5.
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| <br style="clear:both" /><!--Stops image from floating past-->
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| ==Alternative statement==
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| Some sources, including Kakutani's original paper, use the concept of [[Hemicontinuity#Upper_hemicontinuity|upper hemicontinuity]] while stating the theorem:
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| :''Let S be a [[empty set|non-empty]], [[compact set|compact]] and [[convex set|convex]] [[subset]] of some [[Euclidean space]] '''R'''<sup>n</sup>. Let φ: S→2<sup>S</sup> be an [[Hemicontinuity#Upper_hemicontinuity|upper hemicontinuous]] [[set-valued function]] on S with the property that φ(x) is non-empty, [[closed set|closed]] and convex for all x ∈ S. Then φ has a [[fixed point (mathematics)|fixed point]].
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| This statement of Kakutani's theorem is completely equivalent to the statement given at the beginning of this article.
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| We can show this by using the [[Closed graph theorem]] for set-valued functions,<ref name="aliprantis"/> which says that a for a compact [[Hausdorff space|Hausdorff]] range space ''Y'', a set-valued function φ: X→2<sup>Y</sup> has a closed graph if and only if it is upper hemicontinuous and φ(''x'') is a closed set for all ''x''. Since all [[Euclidean space]]s are Hausdorff (being [[metric space]]s) 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.
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| ==Applications==
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| {{See also|Mathematical economics}}
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| ===Game theory===
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| {{See also|Game theory}}
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| The Kakutani fixed point theorem can use used to prove the [[Minimax#Minimax theorem|Minimax Theorem]] in the theory of [[Zero-sum game|zero-sum games]]. This application was specifically discussed by Kakutani's original paper.<ref name="kakutani"/>
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| Mathematician [[John Forbes Nash|John Nash]] used the Kakutani fixed point theorem to prove a major result in [[game theory]].<ref name="nash">
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| {{cite journal
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| | last = Nash
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| | first = J.F., Jr.
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| | authorlink = John Forbes Nash
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| | title = Equilibrium Points in N-Person Games
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| | journal = Proc. Nat. Acad. Sci. U.S.A.
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| | volume = 36
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| | pages = 48–49
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| | year = 1950
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| | doi = 10.1073/pnas.36.1.48
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| | pmid = 16588946
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| | issue = 1
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| | pmc = 1063129 }}</ref>
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| Stated informally, the theorem implies the existence of a [[Nash equilibrium]] in every finite game with mixed strategies for any number of players. This work would later earn him a [[Nobel Prize in Economics]].
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| In this case, ''S'' is the set of [[tuple]]s of [[mixed strategy|mixed strategies]] chosen by each player in a game. The function φ(''x'') gives a new tuple where each player's strategy is her best response to other players' strategies in ''x''. Since there may be a number of responses which are equally good, φ is set-valued rather than single-valued. Then the [[Nash equilibrium]] of the game is defined as a fixed point of φ, i.e. a tuple of strategies where each player's strategy is a best response to the strategies of the other players. Kakutani's theorem ensures that this fixed point exists.
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| ===General equilibrium===
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| {{See also|General equilibrium}}
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| 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.<ref>
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| {{cite book
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| | last = Starr
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| | first = Ross M.
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| | authorlink = Ross Starr
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| | title = General Equilibrium Theory
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| | year = 1997
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| | publisher = Cambridge University Press
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| | url = http://books.google.com/?id=Lv3VtS9CcAoC&pg
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| | isbn = 978-0-521-56473-1
| |
| }}</ref> The existence of such prices had been an open question in economics going back to at least [[Léon Walras|Walras]]. The first proof of this result was constructed by [[Lionel McKenzie]].
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| In this case, ''S'' is the set of [[tuple]]s of commodity prices. φ(''x'') is chosen as a function whose result is different from its arguments as long as the price-tuple ''x'' does not equate supply and demand everywhere. The challenge here is to construct φ so that it has this property while at the same time satisfying the conditions in Kakutani's theorem. If this can be done then φ has a fixed point according to the theorem. Given the way it was constructed, this fixed point must correspond to a price-tuple which equates supply with demand everywhere.
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| ==Proof outline==
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| ===''S'' = <nowiki>[0,1]</nowiki>===
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| The proof of Kakutani's theorem is simplest for set-valued functions defined over [[interval (mathematics)|closed intervals]] of the real line. However, the proof of this case is instructive since its general strategy can be carried over to the higher dimensional case as well.
| |
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| Let φ: <nowiki>[0,1]</nowiki>→2<sup><nowiki>[0,1]</nowiki></sup> be a [[set-valued function]] on the closed interval <nowiki>[0,1]</nowiki> which satisfies the conditions of Kakutani's fixed-point theorem.
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| * '''Create a sequence of subdivisions of <nowiki>[0,1]</nowiki> with adjacent points moving in opposite directions.'''
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| Let (''a''<sub>''i''</sub>, ''b''<sub>''i''</sub>, ''p''<sub>''i''</sub>, ''q''<sub>''i''</sub>) for ''i'' = 0, 1, … be a [[sequence]] with the following properties:
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| :{|class="wikitable"
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| |-
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| |'''1.'''||width="50%"|1 ≥ ''b''<sub>i</sub> > ''a''<sub>''i''</sub> ≥ 0
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| |'''2.'''||width="40%"|(''b''<sub>''i''</sub> − ''a''<sub>''i''</sub>) ≤ 2<sup>−''i''</sup>
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| |-
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| |'''3.'''||''p''<sub>''i''</sub> ∈ φ(''a''<sub>''i''</sub>)
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| |'''4.'''||''q''<sub>''i''</sub> ∈ φ(''b''<sub>''i''</sub>)
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| |-
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| |'''5.'''||''p''<sub>''i''</sub> ≥ ''a''<sub>''i''</sub>
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| |'''6.'''||''q''<sub>''i''</sub> ≤ ''b''<sub>''i''</sub>
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| |}
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| Thus, the closed intervals <nowiki>[</nowiki>''a''<sub>''i''</sub>, ''b''<sub>''i''</sub><nowiki>]</nowiki> form a sequence of subintervals of <nowiki>[0,1]</nowiki>. Condition (2) tells us that these subintervals continue to become smaller while condition (3)–(6) tell us that the function φ shifts the left end of each subinterval to its right and shifts the right end of each subinterval to its left.
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| Such a sequence can be constructed as follows. Let ''a''<sub>0</sub> = 0 and ''b''<sub>0</sub> = 1. Let ''p''<sub>0</sub> be any point in φ(0) and ''q''<sub>0</sub> be any point in φ(1). Then, conditions (1)–(4) are immediately fulfilled. Moreover, since ''p''<sub>0</sub> ∈ φ(0) ⊂ <nowiki>[0,1]</nowiki>, it must be the case that ''p''<sub>0</sub> ≥ 0 and hence condition (5) is fulfilled. Similarly condition (6) is fulfilled by ''q''<sub>0</sub>.
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| Now suppose we have chosen ''a''<sub>''k''</sub>, ''b''<sub>''k''</sub>, ''p''<sub>''k''</sub> and ''q''<sub>''k''</sub> satisfying (1)–(6). Let,
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| :''m'' = (''a''<sub>''k''</sub>+''b''<sub>''k''</sub>)/2.
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| Then ''m'' ∈ <nowiki>[0,1]</nowiki> because <nowiki>[0,1]</nowiki> is [[convex set|convex]].
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| If there is a ''r'' ∈ φ(''m'') such that ''r'' ≥ ''m'', then we take,
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| :''a''<sub>''k''+1</sub> = ''m''
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| :''b''<sub>''k''+1</sub> = ''b''<sub>''k''</sub>
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| :''p''<sub>''k''+1</sub> = ''r''
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| :''q''<sub>''k''+1</sub> = ''q''<sub>''k''</sub>
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| Otherwise, since φ(''m'') is non-empty, there must be a ''s'' ∈ φ(''m'') such that ''s'' ≤ ''m''. In this case let,
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| :''a''<sub>''k''+1</sub> = ''a''<sub>''k''</sub>
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| :''b''<sub>''k''+1</sub> = ''m''
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| :''p''<sub>''k''+1</sub> = ''p''<sub>''k''</sub>
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| :''q''<sub>''k''+1</sub> = ''s''.
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| It can be verified that ''a''<sub>''k''+1</sub>, ''b''<sub>''k''+1</sub>, ''p''<sub>''k''+1</sub> and ''q''<sub>''k''+1</sub> satisfy conditions (1)–(6). | |
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| * '''Find a limiting point of the subdivisions.'''
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| The [[cartesian product]] <nowiki>[0,1]</nowiki>×<nowiki>[0,1]</nowiki>×<nowiki>[0,1]</nowiki>×<nowiki>[0,1]</nowiki> is a [[compact set]] by [[Tychonoff's theorem]]. Since the sequence (''a''<sub>''n''</sub>, ''p''<sub>''n''</sub>, ''b''<sub>''n''</sub>, ''q''<sub>''n''</sub>) lies in this compact set, it must have a [[limit of a sequence|convergent]] [[subsequence]] by the [[Bolzano-Weierstrass theorem]]. Let's fix attention on such a subsequence and let its limit be (''a''*, ''p''*,''b''*,''q''*). Since the graph of φ is closed it must be the case that ''p''* ∈ φ(''a''*) and ''q''* ∈ φ(''b''*). Moreover, by condition (5), ''p''* ≥ ''a''* and by condition (6), ''q''* ≤ ''b''*.
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| But since (''b''<sub>''i''</sub> − ''a''<sub>''i''</sub>) ≤ 2<sup>−''i''</sup> by condition (2),
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| :''b''* − ''a''* = (lim ''b''<sub>''n''</sub>) − (lim ''a''<sub>''n''</sub>) = lim (''b''<sub>''n''</sub> − ''a''<sub>''n''</sub>) = 0.
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| So, ''b''* equals ''a''*. Let ''x'' = ''b''* = ''a''*.
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| Then we have the situation that
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| | |
| :''q''* ∈ φ(''x'') ≤ ''x'' ≤ ''p''* ∈ φ(''x'').
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| * '''Show that the limiting point is a fixed point.'''
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| If ''p''* = ''q''* then ''p''* = ''x'' = ''q''*. Since ''p''* ∈ φ(''x''), ''x'' is a fixed point of φ.
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| Otherwise, we can write the following. Recall that we can parameterize a line between two points a and b by (1-t)a + tb. Using our finding above that q<x<p, we can create such a line between p and q as a function of x (notice the fractions below are on the unit interval). By a convenient writing of x, and since φ(''x'') is [[convex set|convex]] and
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| :<math>x=\left(\frac{x-q^*}{p^*-q^*}\right)p^*+\left(1-\frac{x-q^*}{p^*-q^*}\right)q^*</math>
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| it once again follows that ''x'' must belong to φ(''x'') since ''p''* and ''q''* do and hence ''x'' is a fixed point of φ.
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| ===''S'' is a n-simplex===
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| In dimensions greater one, [[simplex|n-simplices]] are the simplest objects on which Kakutani's theorem can be proved. Informally, a n-simplex is the higher dimensional version of a triangle. Proving Kakutani's theorem for set-valued function defined on a simplex is not essentially different from proving it for intervals. The additional complexity in the higher-dimensional case exists in the first step of chopping up the domain into finer subpieces:
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| * Where we split intervals into two at the middle in the one-dimensional case, [[barycentric subdivision]] is used to break up a simplex into smaller sub-simplices.
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| * While in the one-dimensional case we could use elementary arguments to pick one of the half-intervals in a way that its end-points were moved in opposite directions, in the case of simplices the [[combinatorics|combinatorial]] result known as [[Sperner's lemma]] is used to guarantee the existence of an appropriate subsimplex.
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| 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.
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| ===Arbitrary ''S''===
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| 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. But instead of triangles with straight edges as in the case of n-simplices, we now use triangles with curved edges. In formal terms, we find a simplex which covers ''S'' and then move the problem from ''S'' to the simplex by using a [[deformation retract]]. Then we can apply the already established result for n-simplices.
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| ==Infinite-dimensional generalizations==
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| Kakutani's fixed-point theorem was extended to infinite-dimensional [[locally convex topological vector space]]s by [[Irving Glicksberg]]<ref>
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| {{cite journal
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| | last = Glicksberg
| |
| | first = I.L.
| |
| | title = A Further Generalization of the Kakutani Fixed Point Theorem, with Application to Nash Equilibrium
| |
| | journal = Proceedings of the American Mathematical Society
| |
| | volume = 3
| |
| | issue = 1
| |
| | pages = 170–174
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| | year = 1952
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| | doi = 10.2307/2032478
| |
| | jstor = 2032478}}</ref>
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| and [[Ky Fan]].<ref>
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| {{cite journal
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| | last = Fan
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| | first = Ky
| |
| | title = Fixed-point and Minimax Theorems in Locally Convex Topological Linear Spaces
| |
| | journal = Proc Natl Acad Sci U S A.
| |
| | volume = 38
| |
| | issue = 2
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| | pages = 121–126
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| | year = 1952
| |
| | doi = 10.1073/pnas.38.2.121
| |
| | pmid = 16589065
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| | pmc = 1063516}}</ref>
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| To state the theorem in this case, we need a few more definitions:
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| ;Upper semicontinuity: A set-valued function φ: ''X''→2<sup>''Y''</sup> is '''[[upper semicontinuous]]''' if for every [[open set]] ''W'' ⊂ ''Y'', the set {''x''| φ(''x'') ⊂ ''W''} is open in ''X''.<ref name="dugundji">
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| {{cite book
| |
| | last = Dugundji
| |
| | first = James
| |
| | authorlink = James Dugundji
| |
| | coauthors = Andrzej Granas
| |
| | title = Fixed Point Theory
| |
| | year = 2003
| |
| | publisher = Springer
| |
| | chapter = Chapter II, Section 8
| |
| | url = http://books.google.com/?id=4_iJAoLSq3cC
| |
| | format = limited preview
| |
| | isbn = 978-0-387-00173-9
| |
| }}</ref>
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| ;Kakutani map: Let ''X'' and ''Y'' be [[topological vector space]]s and φ: ''X''→2<sup>''Y''</sup> be a set-valued function. If ''Y'' is convex, then φ is termed a '''Kakutani map''' if it is upper semicontinuous and φ(''x'') is non-empty, compact and convex for all ''x'' ∈ ''X''.<ref name="dugundji"/>
| |
| | |
| Then the Kakutani-Glicksberg-Fan theorem can be stated as:<ref name="dugundji"/>
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| :''Let S be a [[empty set|non-empty]], [[compact set|compact]] and [[convex set|convex]] [[subset]] of a [[locally convex topological vector space]]. Let φ: S→2<sup>S</sup> be a Kakutani map. Then φ has a fixed point.''
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| | |
| The corresponding result for single-valued functions is the [[Tychonoff fixed-point theorem]].
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| | |
| If the space on which the function is defined is [[Hausdorff space|Hausdorff]] in addition to being locally convex, then the statement of the theorem becomes the same as that in the [[Euclidean space|Euclidean]] case:<ref name="aliprantis">
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| {{cite book
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| | last = Aliprantis
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| | first = Charlambos
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| | coauthors = Kim C. Border
| |
| | title = Infinite Dimensional Analysis: A Hitchhiker's Guide
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| | year = 1999
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| | publisher = Springer
| |
| | edition = 3rd
| |
| | chapter = Chapter 17
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| | url =
| |
| }}</ref>
| |
| | |
| :''Let S be a [[empty set|non-empty]], [[compact set|compact]] and [[convex set|convex]] [[subset]] of a [[Locally convex topological vector space|locally convex]] [[Hausdorff space]]. Let φ: S→2<sup>S</sup> be a [[set-valued function]] on S which has a closed graph and the property that φ(x) is non-empty and convex for all x ∈ S. Then the set of [[fixed point (mathematics)|fixed points]] of φ is non-empty and compact.''
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| ==Anecdote==
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| In his game theory textbook,<ref>
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| {{cite book
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| | last = Binmore
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| | first = Ken
| |
| | coauthors =
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| | title = Playing for Real: A Text on Game Theory
| |
| | year = 2007
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| | publisher = Oxford
| |
| | edition = 1st
| |
| | chapter = Chapter 8
| |
| | url =
| |
| }}</ref>
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| Ken Binmore recalls that Kakutani once asked him at a conference why so many economists had attended his talk. When Binmore told him that it was probably because of the Kakutani fixed point theorem, Kakutani was puzzled and replied, "What is the Kakutani fixed point theorem?"
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| ==Further reading==
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| * {{cite book
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| | last = Border
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| | first = Kim C.
| |
| | title = Fixed Point Theorems with Applications to Economics and Game Theory
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| | year = 1989
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| | publisher = Cambridge University Press
| |
| }} <small>(Standard reference on fixed-point theory for economists. Includes a proof of Kakutani's theorem.)</small>
| |
| | |
| * {{cite book
| |
| | last = Dugundji
| |
| | first = James
| |
| | authorlink = James Dugundji
| |
| | coauthors = Andrzej Granas
| |
| | title = Fixed Point Theory
| |
| | year = 2003
| |
| | publisher = Springer
| |
| }} <small>(Comprehensive high-level mathematical treatment of fixed point theory, including the infinite dimensional analogues of Kakutani's theorem.)</small>
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| | |
| * {{cite book
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| | last = Arrow
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| | first = Kenneth J.
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| | authorlink = Kenneth Arrow
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| | coauthors = F. H. Hahn
| |
| | title = General Competitive Analysis
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| | year = 1971
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| | publisher = Holden-Day
| |
| }} <small>(Standard reference on [[general equilibrium]] theory. Chapter 5 uses Kakutani's theorem to prove the existence of equilibrium prices. Appendix C includes a proof of Kakutani's theorem and discusses its relationship with other mathematical results used in economics.)</small>
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| ==References==
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| {{Reflist|2}}
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| ==External links==
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| * [http://cepa.newschool.edu/het/essays/math/fixedpoint.htm Fixed Point Theorems]. <small>(Page on fixed point theorems from [[the New School]]'s [http://cepa.newschool.edu/het/home.htm History of Economic Thought] site.)</small>
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| * {{springer|title=Kakutani theorem|id=p/k055090}}
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| {{Functional Analysis}}
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| [[Category:Fixed-point theorems]]
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| [[Category:Functional analysis]]
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| [[Category:Theorems in convex geometry]]
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| [[Category:Theorems in topology]]
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| [[Category:Mathematical and quantitative methods (economics)]]
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| [[Category:Mathematical economics]]
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| [[Category:General equilibrium and disequilibrium]]
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