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Adding local short description: "On when a function f: S→Pow(S) on a compact nonempty convex subset S⊂ℝⁿ has a fixed point", overriding Wikidata description "theorem that a function f: S→Pow(S) on a compact nonempty convex subset S⊂ℝⁿ, whose graph is closed and whose image f(x) is nonempty and convex for all x∈S, has a fixed point" (Shortdesc helper) |
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| first = Shizuo
| author-link = Shizuo Kakutani
| title = A generalization of
| journal = Duke Mathematical Journal
| volume = 8
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==Statement==
Kakutani's theorem states:<ref name=Osborne>{{cite book |
: ''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 the following properties:''
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| pmid = 16588946
| issue = 1
| pmc = 1063129 | bibcode = 1950PNAS...36...48N
| pmc = 1063129 }}</ref> 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 later earned him a [[Nobel Prize in Economics]]. In this case:▼
| doi-access = free
▲
* The base set ''S'' is the set of [[tuple]]s of [[mixed strategy|mixed strategies]] chosen by each player in a game. If each player has ''k'' possible actions, then each player's strategy is a ''k''-tuple of probabilities summing up to 1, so each player's strategy space is the [[standard simplex]] in '''''R'''''<sup>''k''</sup>''.'' Then, ''S'' is the cartesian product of all these simplices. It is indeed a nonempty, compact and convex subset of '''''R'''''<sup>''kn''</sup>''.''
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| year = 1997
| publisher = Cambridge University Press
| url = https://books.google.com/books?id=Lv3VtS9CcAoC
| 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]].<ref>{{cite journal |first=Lionel |last=McKenzie |title=On Equilibrium in Graham's Model of World Trade and Other Competitive Systems |journal=[[Econometrica]] |volume=22 |issue=2 |year=1954 |pages=147–161 |doi=10.2307/1907539 |jstor=1907539 }}</ref>
In this case:
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| doi = 10.1073/pnas.38.2.121
| pmid = 16589065
| pmc = 1063516
| doi-access = free
}}</ref>
To state the theorem in this case, we need a few more definitions:
;Upper hemicontinuity: A set-valued function φ: ''X''→2<sup>''Y''</sup> is '''[[upper hemicontinuous]]''' if for every [[open set]] ''W'' ⊂ ''Y'', the set {''x''| φ(''x'') ⊂ ''W''} is open in ''X''.<ref name="dugundji">
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| publisher = Springer
| chapter = Chapter II, Section 5.8
| url = https://books.google.com/books?id=4_iJAoLSq3cC
| format = limited preview
| isbn = 978-0-387-00173-9
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==Anecdote==
In his game theory textbook,<ref>{{cite book |last=Binmore |first=Ken |title=Playing for Real: A Text on Game Theory |year=2007 |publisher=Oxford University Press |edition=1st |chapter=When Do Nash Equilibria Exist? |page=256 |isbn=978-0-19-804114-6 |chapter-url=https://books.google.com/books?id=NycDSDcmSM4C&pg=PA256 }}</ref> [[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?"
==References==
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| year = 1971
| publisher = Holden-Day
| isbn = 9780816202751
==External links==
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