Editing Minimax theorem (section)

{{confuse|Min-max theorem}}
{{short description|Gives conditions that guarantee the max–min inequality holds with equality}}
In the mathematical area of [[game theory]] and of [[convex optimization]], a '''minimax theorem''' is a theorem that claims that
: <math>\max_{x\in X} \min_{y\in Y} f(x,y) = \min_{y\in Y} \max_{x\in X}f(x,y)</math>
under certain conditions on the sets <math>X</math> and <math>Y</math> and on the function <math>f</math>.<ref>{{Citation |last=Simons |first=Stephen |title=Minimax Theorems and Their Proofs |date=1995 |work=Minimax and Applications |series=Nonconvex Optimization and Its Applications |volume=4 |pages=1–23 |editor-last=Du |editor-first=Ding-Zhu |url=https://link.springer.com/chapter/10.1007/978-1-4613-3557-3_1 |access-date=2024-10-27 |place=Boston, MA |publisher=Springer US |language=en |doi=10.1007/978-1-4613-3557-3_1 |isbn=978-1-4613-3557-3 |editor2-last=Pardalos |editor2-first=Panos M.}}</ref> It is always true that the left-hand side is at most the right-hand side ([[max–min inequality]]) but equality only holds under certain conditions identified by minimax theorems. The first theorem in this sense is [[John von Neumann|von Neumann]]'s minimax theorem about two-player [[Zero-sum game|zero-sum games]] published in 1928,<ref name=":0">{{cite journal |last=Von Neumann |first=J. |year=1928 |title=Zur Theorie der Gesellschaftsspiele |journal=[[Mathematische Annalen|Math. Ann.]] |volume=100 |pages=295–320 |doi=10.1007/BF01448847|s2cid=122961988 }}</ref> which is considered the starting point of [[game theory]]. Von Neumann is quoted as saying "''As far as I can see, there could be no theory of games ... without that theorem ... I thought there was nothing worth publishing until the Minimax Theorem was proved''".<ref name="Casti">{{cite book |author=John L Casti |url=https://archive.org/details/fivegoldenrulesg00cast/page/19 |title=Five golden rules: great theories of 20th-century mathematics – and why they matter |publisher=Wiley-Interscience |year=1996 |isbn=978-0-471-00261-1 |location=New York |page=[https://archive.org/details/fivegoldenrulesg00cast/page/19 19] |url-access=registration}}</ref> Since then, several generalizations and alternative versions of von Neumann's original theorem have appeared in the literature.<ref>{{cite book |title=Minimax and Applications |date=1995 |publisher=Springer US |isbn=9781461335573 |editor1-last=Du |editor1-first=Ding-Zhu |location=Boston, MA |editor2-last=Pardalos |editor2-first=Panos M.}}</ref><ref>{{Cite journal |last1=Brandt |first1=Felix |last2=Brill |first2=Markus |last3=Suksompong |first3=Warut |year=2016 |title=An ordinal minimax theorem |journal=Games and Economic Behavior |volume=95 |pages=107–112 |arxiv=1412.4198 |doi=10.1016/j.geb.2015.12.010 |s2cid=360407}}</ref>