Designing a reward decay function I can solve this problem programmatically but I expect there is a better solution using pretty basic math. Given $k$ players in a game and a total reward pool of $m$ points, I'd like a function that for any individuals rank $r$ I can get back their appropriate reward amount, knowing that the first player gets $20\%$ of the reward pool, the next gets $20\%$ of what's left and so on. For this function, giving out fractions of points is allowed. $F(r, k, m) =$ ?

Guide:

$r=1$: the first player gets $0.2$.

now, what is left is $0.8$.

$r=2$: the second player gets $0.2(0.8)$.

now, what is left is $0.8(0.8)=0.8^2$.

$r=3$: the third player gets $0.2(0.8^2)$

now, what is left is $0.8(0.8^2)$.

So the corresponding reward for $r=1,2,3$ are $0.2, 0.2(0.8), 0.2(0.8^2)$.

Can you spot the pattern from these workings?