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 $X\; imes\; Y$ with

: $f(x,cdot)$ upper semicontinuous and quasiconcave on $Y$, $forall\; xin\; X$, and: $f(cdot,y)$ is lower semicontinuous and quasi-convex on $X$, $forall\; yin\; Y$

then,

: $min\_\{xin\; X\}max\_\{yin\; Y\}\; f(x,y)=max\_\{yin\; Y\}min\_\{xin\; X\}f(x,y).$

See also Parthasarathy's theorem.

References

* M. Sion. " [http://projecteuclid.org/euclid.pjm/1103040253 On general minimax theorems] ", Pac. J. Math. 8 (1958) pp. 171--176
* Hidetoshi Komiya 1988. " [http://projecteuclid.org/euclid.kmj/1138038812 Elementary proof for Sion's minimax theorem] ". Kodai Math. Journal, volume 11, number 1, pages 5-7.