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First, let me start off by saying I'm not a mathematician so I'm going to need this explained to me at a pretty basic, almost intuitive level. I've taken Calculus but it's been some time so I do have some math background.

I was reading a book tonight and there was a section on the minimax principle in game theory. There was some notation in the book that I don't know what it mean. Can someone explain, in words, what something like the following would mean?

$\underset{\theta\in\Theta}{\max}R_{T}(\theta)$

Does this mean the value of theta in the parameter space that maximizes the function $R_T(\theta)?$ Could you provide a simple example?

Then, in the full context, the book reads that T is the minimax if:

$\underset{T_{1}}{\min}\,\underset{\theta\in\Theta}{\max}R_{T_{1}}(\theta)$

Thanks.

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    $\begingroup$ I think you might have meant to write $max_{\theta\in\Theta} R_T(\theta)$ for the first expression? $\endgroup$ – Paul Aljabar May 17 '17 at 6:25
  • $\begingroup$ I've double checked, and that's what is shown in the material I'm reading. That's not to say sloppy nation isn't used. I'm basically just trying to figure out what max (parameter) function of that parameter really means. If I can understand that, I can figure it out, I think. $\endgroup$ – StatsStudent May 17 '17 at 6:27
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$$\max_{x \in X} f(x)$$ is the notation we use for "the maximum value of $f(x)$ when $x$ is allowed to vary throughout the set $X$".

For example, $$\max_{x \in \{1,2,3\}} x = 3$$ $$\max_{x \in \{1,2,3\}} \frac{5}{x} = 5$$ $$\max_{\theta \in \mathbb{R}} \sin(\theta) = 1$$

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The notation $\max_{x \in X}f(x)$ means the number $\max \{ f(x) | x \in X \}$. Likewise, we have $\min_{x \in X}f(x) = \min \{ f(x)| x \in X \}$.

If you want to denote a point $y$ at which $f$ reaches its extremum, use $\arg \max _{x \in X}f(x)$ or $\arg \min_{x \in X}f(x)$ accordingly.

The notation $\min_{t \in T}\max_{x \in X}f_{t\in T}(x)$ simply means the number $\min \{ \max_{x \in X}f_{t}(x) | t \in T \}$; intuitively, first find the maximum for each $t \in T$ and then find the minimum of the maxima of these $f_{t}$.

Note that supremum and infimum are a generalization of maximum and minimum, respectively. If $f$ is continuous and if $X$ is compact, it can be proved that $\sup f(X)$ and $\inf f(X)$ are the functional values of $f$ somewhere in $X$; in this case we have $\sup f(X) = \max f(X)$ and $\inf f(X) = \min f(X)$.

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    $\begingroup$ Thanks, @Eric Clapton. In reality all the answers were great and I had difficulty accepting one out of them all (they all were admissible ;-) for an acceptance). Thanks again and to everyone else who helped me finally understand this. $\endgroup$ – StatsStudent May 17 '17 at 7:05
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    $\begingroup$ "compact" ​ -> ​ "compact and non-empty" ​ ​ ​ ​ $\endgroup$ – user57159 May 17 '17 at 8:12
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    $\begingroup$ For what it's worth, I think I once saw $\min {\rm arg} \min_x f(x)$ to denote the least value $x$ among those making $f(x)$ least. This is more precisely defined than only ${\rm arg} \min_x f(x)$, but it may fail to exist, in general. $\endgroup$ – chi May 17 '17 at 21:06
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$$\max_{\theta\in\Theta}R_T(\theta)$$means the maximum value that $R_T(\theta)$ can take as $\theta$ varies over all possible values in $\Theta$. This quantity is still dependent on $T$, and its least value for all possible values of $T$ is $$\min_{T}\max_{\theta\in\Theta}R_T(\theta).$$ Here the possible values for $T$ are assumed to be understood, and not made explicit. Also, the meaning is the same if the letter $T$ is replaced by $T_1$ or some other appropriate symbol.

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For the first question, it might be the author's intention to refer to the value of the parameter that achieves the max, but this would be incorrect. Strictly, the expression refers to the value of the function at the maximum. One should use argmax to be explicit if we refer to the parameter.

For the second expression, in effect we consider all possible $T$, for each one, determine the $\theta$ that maximises R, then, out of all $T$ identify the one that minimises R.

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    $\begingroup$ No: using the expression to mean the value of the parameter that achieves the maximum is simply wrong. The meaning is the value of the function at the maximum. Only argmax is the correct notation for the former meaning. $\endgroup$ – John Bentin May 17 '17 at 6:47
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    $\begingroup$ I agree, however not all texts represent this correctly. $\endgroup$ – Paul Aljabar May 17 '17 at 6:58
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    $\begingroup$ @Paul, can you give an example of a text using max for argmax? I don't think it's out of the question or that you're mistaken, I just don't think I've seen one myself. $\endgroup$ – Mark S. May 17 '17 at 19:39
  • $\begingroup$ Hard to search for incorrect usages - but have edited in a way that hopefully addresses these points ... $\endgroup$ – Paul Aljabar May 19 '17 at 10:55

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