# Minimax theorem

Proven by von Neumann in 1928, this is a cornerstone of duality (and of game theory). It states that there exists $LaTeX: x \in S_m$ and $LaTeX: y \in S_n$ such that $LaTeX: x^T A y$ is a saddlepoint of the bilinear form:

$LaTeX: \min \left\{ \max \{ u^T A v: v \in S_n\}: u \in S_m \right} = \max \left\{ \min \{ u^T A v: u \in S_m\}: v \in S_n \right\} = x^TAy.$

This extends to the following.

Let $LaTeX: F:X\timesY\rightarrow \mathbb{R}$ be such that $LaTeX: X$ and $LaTeX: Y$ are non-empty, convex, compact sets, $LaTeX: F(.,y)$ is convex on $LaTeX: X$ for each $LaTeX: y \in Y$, and $LaTeX: F(x,.)$ is concave on $LaTeX: Y$ for each $LaTeX: x \in X$. Then, there exists a saddlepoint, $LaTeX: (x^*,y^*) \in X\times Y$ such that

$LaTeX: \min \left\{ \max \{F(x,y): y \in Y\}: x \in X\right\} = \max \left\{ \min \{F(x,y): x \in X\}: y \in Y\right\} = F(x^*,y^*).$