Dining cryptographers problem

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 $Y$, $\mathrm{\forall }x\in X$, and

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

then,

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

See also

• Parthasarathy's theorem
• Saddle point

References

• M. Sion. On general minimax theorems, Pac. J. Math. 8 (1958) pp. 171—176
• Hidetoshi Komiya 1988. Elementary proof for Sion's minimax theorem. Kodai Math. Journal, volume 11, number 1, pages 5–7.

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