Mathematical notation $\max$ with simple example for non-mathematician

Question

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.

Answer 1

$$\max_{x \in X} f(x)$$ is the notation we use for "the maximum value of $f(x)$ when $x$ is allowed to vary throughout the set $X$".

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

Answer 2

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 $\inf f(X) = \min f(X)$.

Answer 3

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.

Answer 4

$$\max_{\theta\in\Theta}R_T(\theta)$$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.