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 quasi-concave 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,

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

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. MR 0097026. Zbl 0081.11502.
• Komiya, Hidetoshi (1988). "Elementary proof for Sion's minimax theorem". Kodai Mathematical Journal. 11 (1): 5–7. doi:10.2996/kmj/1138038812. MR 0930413. Zbl 0646.49004.