Minimax theorem

Minimax theorem Not to be confused with Min-max theorem.

In the mathematical area of game theory, a minimax theorem is a theorem providing conditions that guarantee that the max–min inequality is also an equality. The first theorem in this sense is von Neumann's minimax theorem from 1928, which was considered the starting point of game theory. Since then, several generalizations and alternative versions of von Neumann's original theorem have appeared in the literature.[1][2] Contents 1 Zero-sum games 1.1 Examples 2 See also 3 References Zero-sum games The function f(x,y)=y2-x2 is concave-convex.

The minimax theorem was first proven and published in 1928 by John von Neumann,[3] who 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".[4] Formally, von Neumann's minimax theorem states: Let {displaystyle Xsubset mathbb {R} ^{n}} and {displaystyle Ysubset mathbb {R} ^{m}} be compact convex sets. If {displaystyle f:Xtimes Yrightarrow mathbb {R} } is a continuous function that is concave-convex, i.e.

{displaystyle f(cdot ,y):Xrightarrow mathbb {R} } is concave for fixed {displaystyle y} , and {displaystyle f(x,cdot ):Yrightarrow mathbb {R} } is convex for fixed {displaystyle x} .

Then we have that {displaystyle max _{xin X}min _{yin Y}f(x,y)=min _{yin Y}max _{xin X}f(x,y).} Examples If {displaystyle f(x,y)=x^{T}Ay} for a finite matrix {displaystyle Ain mathbb {R} ^{ntimes m}} , we have: {displaystyle max _{xin X}min _{yin Y}x^{T}Ay=min _{yin Y}max _{xin X}x^{T}Ay.} See also Sion's minimax theorem Parthasarathy's theorem — a generalization of Von Neumann's minimax theorem Dual linear program can be used to prove the minimax theorem for zero-sum games. Yao's minimax principle References ^ Du, Ding-Zhu; Pardalos, Panos M., eds. (1995). Minimax and Applications. Boston, MA: Springer US. ISBN 9781461335573. ^ Brandt, Felix; Brill, Markus; Suksompong, Warut (2016). "An ordinal minimax theorem". Games and Economic Behavior. 95: 107–112. arXiv:1412.4198. doi:10.1016/j.geb.2015.12.010. ^ Von Neumann, J. (1928). "Zur Theorie der Gesellschaftsspiele". Math. Ann. 100: 295–320. doi:10.1007/BF01448847. ^ John L Casti (1996). Five golden rules: great theories of 20th-century mathematics – and why they matter. New York: Wiley-Interscience. p. 19. ISBN 978-0-471-00261-1.

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