Sion's minimax theorem

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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) is 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).

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