# why is it important to have $\max_x \min_y f(x,y)=\min_y \max_x f(x,y)$?

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?