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

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?

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

• Did you mean both to be $\max$? – copper.hat Mar 28 '14 at 7:00

$$\max_x \max_y f(x,y) =\max_y \max_x f(x,y)?$$
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 $x\in X$: $$\max_x \max_y f(x,y) \le \max_y \max_x f(x,y).$$