Back in high-school, I learned about this concept in economics class, in relation to diminishing returns, but I forgot what it's called. Below is the illustration of its use:
Suppose you need to allocate a finite amount of points to characteristics of a game character. Suppose that some characteristics of this game character are mutually reinforcing. For example, critical hit rate and critical hit strength will benefit from improving the paired characteristic. Similarly, the strength of character's armor and the health cap will benefit each other (since every point invested in armor will make each point invested in health yield more value). So, it would be possible to describe this relationship as multiplication of some sort.
Now, when allocating points to each character, it appears that the best strategy is to split them evenly between benefiting characteristics. For example, suppose you have 10 points to spend between vitality and toughness, then your total yield would be:
1 * 9 = 9
2 * 8 = 16
3 * 7 = 21
4 * 6 = 24
5 * 5 = 25
What is this "law" called? (I'd also imagine this extends to more than two factors). Is there a better, more formal description of this phenomenon?
PS: Sorry for poor tagging! I think there used to be "naming" tag, but I cannot find it too... as you might have figured by now, the author's memory isn't as sharp as it used to be...
I don't know what it's called in economics, but in math this result follows from an inequality called the Arithmetic Mean-Geometric Mean (or AM-GM for short), which says that for any two real numbers $ x $ and $ y $, $$ xy \leq \frac{(x+y)^2}{4} $$ with equality happening precisely when $ x = y $. This inequality says that if $ x + y $ is fixed, then the strategy to maximize $ xy $ is to split them even, and then the maximum value will be equal to $ (x+y)^2/4 $. The proof is very simple: starting from $ (x-y)^2 \geq 0 $, add $ 4xy $ to both sides and you get the result.
This is always true even for more than $2$ variables, in which you have that for nonnegative reals $ x_1, \dots, x_n $, $$ \sqrt[n]{x_1 \cdots x_n} \leq \frac{x_1 + \cdots + x_n}{n}, $$ with equality precisely when $ x_1 = \cdots = x_n $. The nicest proof (in my opinion) of this generalized inequality uses Cauchy's backward induction, which you can read about on its Wikipedia page.