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.

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

Y

, $\forall x\in X$ , and

$f(\cdot,y)$ is lower semicontinuous and quasi-convex on

X

, $\forall y\in Y$

then,

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

M. Sion. On general minimax theorems, Pac. J. Math. 8 (1958) pp. 171--176
Hidetoshi Komiya 1988. Elementary proof for Sion's minimax theorem. Kodai Math. Journal, volume 11, number 1, pages 5-7.