Suppose I have a topic or discussion, and a number of "support" and "opposition" points on each side (You can also think of them as "upvotes" and "downvotes") and I want to calculate a score of how "controversial" a topic is. (Let $p$ be the support score, $c$ be the opposition score, and $f(p, c)$ be the function that determines the controversy score.)
It should have the following properties:
- Controversy is maximized when equal support is given to both sides. Given that some property $g(p, c)$ is held constant (such that the slope of the tangent line of the level curve of $g(p, c)$ at any point is never positive), $f(p, c)$ should be maximized when $p = c$.
- More support on both sides means that more people care and therefore there is more controversy. Given that $p/c$ is held constant, a higher value of $p$ or $c$ should result in a higher value of $f(p, c)$.
- The amount of controversy is the same for the same imbalance of support no matter which side the imbalance favours. $f(p, c)$ should equal $f(c, p)$.
- All the support being on one side means there is no controversy. Given that either $p$ or $c$ is equal to zero, $f(p, c)$ should be equal to zero.
Is there any function like this that is already in use? If not, could one be devised?