When $\min \max = \max \min$?

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 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