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.
Best Answer
$$\max_{x \in X} f(x)$$ is the notation we use for "the maximum value of $f(x)$ when $x$ is allowed to vary throughout the set $X$".
For example, $$\max_{x \in \{1,2,3\}} x = 3$$ $$\max_{x \in \{1,2,3\}} \frac{5}{x} = 5$$ $$\max_{\theta \in \mathbb{R}} \sin(\theta) = 1$$