Minimax theorem

In the mathematical area of game theory and of convex optimization, a minimax theorem is a theorem that claims that

${\displaystyle \max _{x\in X}\min _{y\in Y}f(x,y)=\min _{y\in Y}\max _{x\in X}f(x,y)}$

under certain conditions on the sets ${\displaystyle X}$ and ${\displaystyle Y}$ and on the function ${\displaystyle f}$.^[1] 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 von Neumann's minimax theorem about two-player zero-sum games published in 1928,^[2] 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".^[3] Since then, several generalizations and alternative versions of von Neumann's original theorem have appeared in the literature.^[4][5]