Formula for awarding prizes on a leaderboard

[DodgerLD]

Hi,

If you have a prize pool (say 1,000 points), and a leaderboard like:

Rank / Name
1 / Dan
2 / Rachel
3 / Max
4 / Chloe
5 / Grant
etc.

Is there a formula where you can input prize pool, rank, and maybe the number of people, and it will give you the number of points (or a %) for each rank. The person ranked #1 would get like 25% of the prize pool, and each person below would get less and less until the full prize pool is paid out.

Possible? Ideas?

Thanks.

 

[Yayness]

I suppose this is what you are looking for:

Let n be the total number of people who gets a part of the prize.
Let a be the ratio between what the 1st person gets, and what the 2nd person gets.
(If a = 2, the 1st person will get twice as much as the 2nd one, and the 2nd one will get twice as much as the 3rd one etc.)

If a person has rank x, he/she will then get \frac{a^{n-x}(a-1)}{a^n-1}\cdot 100\% of the reward.

If you want, you can exchange the "100%"-part with the total number of points.

 

[DodgerLD]

Hi Yayness,

That seems similar to what someone else gave me (which I converted to Excel for testing), which seems to be working quite well:

((1 - p) / (1 - p ^ n) * p ^ (r - 1)) * l

p = a value between 0 and 1 which defines the distribution (sort of).
n = number of players
r = rank
l = prize pool

I can't get your formula working in Excel though, I'm trying:

((a ^ (n - x)) * (a - 1)) / ((a ^ n) - 1)

And it doesn't seem to be giving the correct result, maybe I'm not converting it correctly.

Your explanation of "a" is a lot clearer than my explanation of "p". :-)

Thanks for your help.

 

[Yayness]

It should be right if you multiply with the prize pool.
Let's say l is the prize pool, then a person with rank x will get:
l * (a^(n - x) * (a - 1)) / (a^n - 1)

 

[DodgerLD]

(0.7^(10 - 1) * (0.7 - 1)) / (0.7^10 - 1)

... gives me 0.012457..., but it should give me 0.31.

 

[Yayness]

The number a needs to be larger than 1 for the best one to get more than the 2nd best one etc. In this case, you can let a be 1/0.7≈1.43, which means the best one will get 1.43 times as much as the 2nd best one, and the 2nd best one will get 1.43 times as much as the 3rd best one etc.
So a should be a number above 1, which means it does not have exactly the same definition as p in the other formula. a=1/p

 

[DodgerLD]

Ah, that makes sense. Thanks Yayness!