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Let $x,y \in \mathbb{R}$ and consider the problem

$$f(x,y) = x^2 + xy + y^2$$

For example, when people say Is Minimax equals to Maximin?, does it mean

$$ \min_{x} \max_{y} f(x,y) = \max_y \min_x f(x,y) ?$$

or that the solution sets are equal?

My confusion stems from, if I were to compute $\min_{x} \max_{y} f(x,y)$, then this value equals to $\infty$ at $(r, \infty)$ for any $r \in \mathbb{R}$. (It does not matter what $x$ is, the value of the problem will always be $\infty$).

Similarly, if I were to compute $ \max_y \min_x f(x,y)$, then the value is $(0, \infty)$. $x$ variable is at minimum at $0$. The value of $ \max_y \min_x f(x,y) $ is yet again $\infty$.

So does this equality mean??

The value of $\min\max f$ and $\max\min f$ are all $\infty$.

But the solution set is different, one is $(r, \infty), r \in \infty$ and the other is $(0, \infty)$.

So when we say minimax and maximin are "equal", do we mean that the solution sets are equal, or just the value at those solutions?

Rushabh Mehta
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Norman
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  • First, don't use tags that are irrelevant. Second, in this case, since both sides of the equality are "infinite", we can trivially say they are equal, but this case isn't interesting. We generally minimax functions which are bounded, as the result has a non-trivial result for these functions. So yes, the values at those solutions. – Rushabh Mehta May 21 '21 at 18:48
  • In order to give an even more precise condition than Don 1000, you have to be on a bounded set, for example the set of $(x,y)$ whose norm is less than a certain $a$. As a consequence, if you deal with continuous functions, their range will be bounded. – Jean Marie May 21 '21 at 20:11

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