## Illustration of Von Neumann's Minimax theorem in games?

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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))>=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?

Why won't you accept the edit? See this post on meta. And please stop rolling it back.

– Mithical – 2016-08-11T14:38:42.613

@Mithrandir see my answer on that post, at least the game-theory tag should be there – dontloo – 2016-08-11T14:39:42.407

I'll agree with the difference between game-theory and gaming. But can you please edit the other tags out? – Mithical – 2016-08-11T14:40:51.877

@Mithrandir sure – dontloo – 2016-08-11T14:43:23.333

1This is an interesting questions - you've clearly done your research - but unfortunately, it seems better for [stats.se] or [datascience.se]. For more information, visit [meta]. – Ben N – 2016-08-16T16:01:06.013

I actually like this question for AI b/c game theory is so fundamental to the field. – DukeZhou – 2017-09-13T21:34:01.880

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Here is one of many web pages devoted to the application of Minimax in Tic-Tac-Toe solving: Minimax Algorithm in Game Theory | Set 3 (Tic-Tac-Toe AI – Finding optimal move)