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The Von Neumann's Minimax theorem gives the conditions that make the max-min inequality an equality.

I understand the max-min inequality, basically $\min(\max(f))\ge \max(\min(f))$.

The Von Neumann's theorem states that, for the inequality to become an equality $f(.,y)$ should always be convex for given $y$ and $f(x,.)$ should always be concave for given $x$, which also makes sense.

This video says that for a zero-sum perfect information game, the Von Neumann's theorem always holds, so that minimax always equal to maximin, which I did not quite follow.

Questions
Why zero-sum perfect information games satisfy the conditions of Von Neumann's theorem?
If we relax the rules to be non-zero-sum or non-perfect information, how would the conditions change?


This question was originally posted on the AI site and closed as off-topic.

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