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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Did you mean both to be $\max$? – copper.hat Mar 28 '14 at 07:00
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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.
mookid
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1Sorry oft the delayed comment. Is it possible to elaborate a little more? Which symmetry should be used to get $\max_x \max_y f(x,y) =\max_y \max_x f(x,y)$. – darkmoor Nov 01 '21 at 22:10
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I believe the is a mistake (or typo?): $\max_y f(x,y) \neq \max_y \max_x f(x,y)$ (the lhs is a function of $x$ (hence not constant in principle) while the rhs is a fixed number) – StarBucK Mar 01 '23 at 16:44