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 be a compact convex subset of a linear topological space and a convex subset of a linear topological space. If is a real-valued function on with

upper semicontinuous and quasi-concave on , , and
lower semicontinuous and quasi-convex on ,


See also[edit]


  • Sion, Maurice (1958). "On general minimax theorems". Pacific Journal of Mathematics. 8 (1): 171–176. doi:10.2140/pjm.1958.8.171. MR 0097026. Zbl 0081.11502.
  • Komiya, Hidetoshi (1988). "Elementary proof for Sion's minimax theorem". Kodai Mathematical Journal. 11 (1): 5–7. doi:10.2996/kmj/1138038812. MR 0930413. Zbl 0646.49004.