Sion's minimax theorem

Short description: Generalization of John von Neumann'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 $Y$, $\mathrm{\forall }x\in X$, and

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

then,

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

See also

• Parthasarathy's theorem
• Saddle point

References

• Sion, Maurice (1958). "On general minimax theorems". Pacific Journal of Mathematics 8 (1): 171–176. doi:10.2140/pjm.1958.8.171. 
• Komiya, Hidetoshi (1988). "Elementary proof for Sion's minimax theorem". Kodai Mathematical Journal 11 (1): 5–7. doi:10.2996/kmj/1138038812. http://projecteuclid.org/euclid.kmj/1138038812.