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I am currently trying to understand the minimax theorem of Von Neumann and the improved versions of this theorem.

At any case we have the property

$$\max_{x\in X} \min_{y\in Y}f(x,y)=\min_{y\in Y} \max_{x\in X} f(x,y)$$

for some compact sets $X,Y$ and a convex (in $y$), concave (in $x$) function $f(x,y)$. If this property is satisfied we say that there exists a saddle point

$$\hat{x},\hat{y}=\arg \min_{y\in Y} \max_{x\in X} f(x,y)$$

Question: What is the important of a saddle point intuitively? For example if I have a non-trivial solution to $$\min_{y\in Y} \max_{x\in X} f(x,y)$$ but if $$\max_{x\in X} \min_{y\in Y}f(x,y)<\min_{y\in Y} \max_{x\in X} f(x,y)?$$

how can I comparare this case to the sadle point case?

In which applications/situations these two cases can make a significant difference?

Thanks alot for any comments.

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