# Sion's minimax theorem

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 compact convex subset of a linear topological space and {displaystyle Y} a convex subset of a linear topological space. If {displaystyle f} is a real-valued function on {displaystyle Xtimes Y} with {displaystyle f(x,cdot )} upper semicontinuous and quasi-concave on {displaystyle Y} , {displaystyle forall xin X} , and {displaystyle f(cdot ,y)} lower semicontinuous and quasi-convex on {displaystyle X} , {displaystyle forall yin Y} then, {displaystyle min _{xin X}sup _{yin Y}f(x,y)=sup _{yin Y}min _{xin X}f(x,y).} 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. 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.

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