Sion's minimax theorem

Short description: Generalization of John von Neumann'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 [math]\displaystyle{ X }[/math] be a compact convex subset of a linear topological space and [math]\displaystyle{ Y }[/math] a convex subset of a linear topological space. If [math]\displaystyle{ f }[/math] is a real-valued function on [math]\displaystyle{ X\times Y }[/math] with

[math]\displaystyle{ f(x,\cdot) }[/math] upper semicontinuous and quasi-concave on [math]\displaystyle{ Y }[/math], [math]\displaystyle{ \forall x\in X }[/math], and

[math]\displaystyle{ f(\cdot,y) }[/math] lower semicontinuous and quasi-convex on [math]\displaystyle{ X }[/math], [math]\displaystyle{ \forall y\in Y }[/math]

then,

[math]\displaystyle{ \min_{x\in X}\sup_{y\in Y} f(x,y)=\sup_{y\in Y}\min_{x\in X}f(x,y). }[/math]

See also

• Parthasarathy's theorem
• Saddle point

References

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