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)

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).

References

Sion, Maurice (1958). "On general minimax theorems". Pacific Journal of Mathematics. 8 (1): 171–176. MR 0097026. Zbl 0081.11502. doi:10.2140/pjm.1958.8.171.
Komiya, Hidetoshi (1988). "Elementary proof for Sion's minimax theorem". Kodai Mathematical Journal. 11 (1): 5–7. MR 0930413. Zbl 0646.49004. doi:10.2996/kmj/1138038812.

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