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, named after Maurice Sion.

It states:

Let ${\displaystyle X}$ be a

and ${\displaystyle Y}$ a convex subset of a linear topological space. If ${\displaystyle f}$ is a real-valued function on ${\displaystyle X\times Y}$ with

${\displaystyle f(x,\cdot )}$

quasi-concave

on ${\displaystyle Y}$, ${\displaystyle \forall x\in X}$, and

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

then,

${\displaystyle \min _{x\in X}\sup _{y\in Y}f(x,y)=\sup _{y\in Y}\min _{x\in X}f(x,y).}$

See also

• Parthasarathy's theorem
• Saddle point

References

• Sion, Maurice (1958). "On general minimax theorems".
  Zbl 0081.11502
  .
• Komiya, Hidetoshi (1988). "Elementary proof for Sion's minimax theorem".
  Zbl 0646.49004
  .

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