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 quasiconcave on $Y$, $\forall x\in X$, and

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

then,

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

See also

• Parthasarathy's theorem
• Saddle point

References

• Sion, Maurice (1958). "On general minimax theorems". Pac. J. Math. 8 (1): 171–176. MR 0097026. Zbl 0081.11502.
• Komiya, Hidetoshi (1988). "Elementary proof for Sion's minimax theorem". Kodai Math. Journal 11 (1): 5–7. MR 0930413. Zbl 0646.49004.