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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?

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

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  • $\begingroup$ Did you mean both to be $\max$? $\endgroup$ – copper.hat Mar 28 '14 at 7:00
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$$\max_x \max_y f(x,y) =\max_y \max_x f(x,y)?$$

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.

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