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


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

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.
