$\max_x \max_y f(x,y) = \max_y \max_x f(x,y)?$

Question

Just come across a question regarding sequential maximization and simultaneous maximization, and I do not recall whether there are any established conditions for the equivalence. Anyone has some idea?

$$\max_x \max_y f(x,y) =\max_y \max_x f(x,y)?$$

Answer 1

$$\max_x \max_y f(x,y) =\max_y \max_x f(x,y)?$$

Let $(x,y)\in X\times Y$. $$ f(x,y) \le \max_x f(x,y) \le \max_y \max_x f(x,y);$$as this is true for every $y\in Y$, $$ \max_y f(x,y) =\max_y \max_x f(x,y); $$and as this is true for every $x\in X$: $$\max_x \max_y f(x,y) \le \max_y \max_x f(x,y).$$

Now use the symetry to get the conclusion.