When $\min \max = \max \min$? Let $X \subset \mathbb{R}^n$ and $Y \subset \mathbb{R}^m$ be compact sets.
Consider a continuous function $f : X \times Y \rightarrow \mathbb{R}$.
Say under which condition we have
$$ \min_{x \in X}  \max_{y \in Y} f(x,y) = \max_{y \in Y} \min_{x \in X} f(x,y). $$
From this we have that $\max_{y \in Y} \min_{x \in X} f(x,y) \leq \min_{x \in X}  \max_{y \in Y} f(x,y)$. So here we are looking for conditions on $f$ such that we have the equality.
 A: If you look at this problem as a Primal Problem and its Dual Problem then you're basically asking when Strong Duality Holds.  
Basically when $ f\left( x, y \right) $ is Convex in $ x $ and Concave in $ y $.
A: A general result called Von Neumann-Fan minimax theorem states the following:

Theorem 2 (Von Neumann-Fan minimax theorem). Let $X$ and $Y$ be Banach spaces.Let $C \subset X$ be nonempty and convex, and let $D \subset Y$ be nonempty, weakly compact and convex. Let $g : X \times Y \to \Bbb{R}$ be convex with respect to $x \in C$ and concave
  and upper-semicontinuous with respect to $y \in D$, and weakly continuous in $y$ when restricted to $D$. Then
  $$d := \max_{y\in D} \inf_{x\in C} g(x, y) = \inf_{x\in C} \max_{y\in D}
g(x, y).$$

See for example the following link: https://www.carma.newcastle.edu.au/jon/minimax.pdf
A: take two examples:
$$f(x,y) = \cos(x+y)$$
and 
$$f(x,y)=2xy(x-y)$$
In the first case $\min_x \max_y f(x,y)=1 $ and $\max_y\min_x f(x,y)=-1$ for all $x,y$.
In the second one we have $y=0.5x$ and $x=0.5y$ since we have $f(x,y)^{''}>0$ for $\frac{d^2}{dx}f(x,y)$ and $f(x,y)^{''}<0$ for $\frac{d^2}{dy}f(x,y)$, they are minima and maxima respectively. As a result one gets $$\min_x \max_y f(x,y) = \max_y \min_x f(x,y)$$
for the second case.
