# Sion's minimax theorem

In mathematics, and in particular game theory, Sion's minimax theorem is a generalization of John von Neumann's minimax theorem, named after Maurice Sion.

It states:

Let ${\displaystyle X}$ be a compact convex subset of a linear topological space and ${\displaystyle Y}$ a convex subset of a linear topological space. If ${\displaystyle f}$ is a real-valued function on ${\displaystyle X\times Y}$ with

${\displaystyle f(x,\cdot )}$ upper semicontinuous and quasiconcave on ${\displaystyle Y}$, ${\displaystyle \forall x\in X}$, and
${\displaystyle f(\cdot ,y)}$ lower semicontinuous and quasi-convex on ${\displaystyle X}$, ${\displaystyle \forall y\in Y}$

then,

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