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


 * $$f(x,\cdot)$$ upper semicontinuous and quasi-concave on $$Y$$, $$\forall x\in X$$, and
 * $$f(\cdot,y)$$ lower semicontinuous and quasi-convex on $$X$$, $$\forall y\in Y$$

then,


 * $$\min_{x\in X}\sup_{y\in Y} f(x,y)=\sup_{y\in Y}\min_{x\in X}f(x,y).$$