Sion's minimax theorem explained

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 x Y withf(x, ⋅ ) upper semicontinuous and quasi-concave on Y , \forall x\in X , andf( ⋅ ,y) lower semicontinuous and quasi-convex on X , \forall y\in Y

then,

min _{x\in} \sup _{y\in} f(x,y) = \sup _{y\in} min _{x\in} f(x,y).

See also

• Parthasarathy's theorem
• Saddle point

References

• Maurice . Sion . On general minimax theorems . . 8 . 1 . 1958 . 171–176 . 0081.11502 . 0097026 . 10.2140/pjm.1958.8.171. free .
• Hidetoshi . Komiya . 1988 . Elementary proof for Sion's minimax theorem . . 11 . 1 . 5–7 . 0930413 . 0646.49004 . 10.2996/kmj/1138038812.