Minimax theorem

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 ${\displaystyle X\subset \mathbb {R} ^{n}}$ and ${\displaystyle Y\subset \mathbb {R} ^{m}}$ be compact convex sets. If ${\displaystyle f:X\times Y\rightarrow \mathbb {R} }$ is a continuous function that is concave-convex, i.e.

${\displaystyle f(\cdot ,y):X\rightarrow \mathbb {R} }$ is concave for fixed ${\displaystyle y}$, and

${\displaystyle f(x,\cdot ):Y\rightarrow \mathbb {R} }$ is convex for fixed ${\displaystyle x}$.

Then we have that

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

• Sion's minimax theorem
• Parthasarathy's theorem
• Dual linear program can be used to prove the minimax theorem for zero-sum games.