In my course (https://www.cs.bgu.ac.il/~ppl182/Guidelines) we have a grading policy in which you can gain a bonus to your final grade if you improve your grade between the midterm exam and the final exam.
The formula for the bonus $b$ is $$b = \frac{0.1(f - m)^2}{100 - m},$$
where $f$ is the final exam and $m$ is the midterm.
I've noticed very similar bonus calculations in other courses. My question is: what is the intuition behind this formula? It seems to have some clear internal idea behind it, but I can't figure out what it is.
PS The general grade formula for the course is = 0.60*(final exam grade) + 0.20*(midterm grade) + 0.20*(assignments grade) + bonus
Consider the two extremes: when $f = m$, the bonus is $0$; when $f=100$, the bonus is $$\frac{0.1(100 - m)^2}{100 - m} = 0.1(100-m) = 10 - 0.1m.$$
At this latter extreme, your final grade will be \begin{align} 0.60(100) + 0.20m + 0.20a + b &= 60 + 0.20m + 0.20a + 10 - 0.10m\\ &=70 + 0.10m +0.20a. \end{align}
Notice that this is the average of (i) your normal weighted grade with (ii) the result of shifting all the weight of the midterm onto the final: $$\frac12\left[(0.60(100)+0.20m+0.20a) + (0.80(100)+0.20a)\right] = 70+0.10m+0.20a$$
The curve connecting these two extremes is quadratic and has the same slope as the normal grade (without the bonus) at $f=m$. As a result, it only really makes a difference to those students who show a substantial improvement.
In the end, it's hard to see the point of such a scheme. It's unnecessarily complicated and doesn't really help students all that much. It would be far simpler (and slightly more generous) to give two marking schemes (in this case $60 / 20 / 20$ and $70 / 10 / 20$) and take the higher of the two.
Here's a graph.