Tournament-style payout structures using exponential growth

[rxbandits]

Hello!

Excuse me for my very basic understanding of math. I'll try and present my idea and problem clearly.

I'd like to devise a payout structure for a tournament.

20% of the entrants will be paid. The payout will be an exponentially sloping function. The payout is in percentages that equal 100%; structured (approximately) in this way:

If there were 35 entrants, 7 places would be paid.

1st would get 37%
2nd would get 26%
3rd would get 17%
4th would get 10%
5th would get 5%
6th would get 3%
7th would get 2%

How can I represent this with a formula when I have 2,000 entrants? What about 6 entrants?

To say it one other way: 20% of the field will be paid a percentage based on a formula that slopes in the way I've illustrated above with the 7 places paid example.

7 places paid with the rate of 2-3-5-10-17-26-37 is a function I came up with based loosely on 1-4-9-16-25-36 or 1²-2²-3²-4²-5²-6² (with some number fudging added to make the total equal 100% - though there has to be a more elegant way to do that...)

Thanks so much for any guidance you can offer!

 

[rxbandits]

Poker tournament payouts often follow this type of "exponential" structure. But I'm wondering if there's an acceptable formula to describe it or if the numbers poker tournament directors come up with are entirely arbitrary. http://www.wsop.com/tournaments/payouts.asp

 

[rxbandits]

Putting it an entirely different/simpler way:

On the pic of the graph attached (y=x²), there are 7 equidistant points marked. The sum of all the points = 1.

What are the values for the individual points?
What is the formula used to find the values of these points?

Thanks again, sorry if muddled.

 

[jbriggs444]

So you do not want "exponential" behavior. You want "quadratic" behavior instead?

For exponential behavior, each contestant's payout would be a fixed ratio greater than the next lower contestant's payout.

For quadratic behavior, each contestant's payout would be a linearly-increasing amount greater than the next lower contestant's payout.

1, 4, 9, 16, 25, 36 = quadratic
1, 2, 4, 8, 16, 32 = exponential (also known as a geometric series).

 

[rxbandits]

Yes, my misunderstanding. :)

 

[williamrand1]

Hi did you solve this problem??

I think i understand what you want to do. Do you want the payout to decrease exponentially from first position down to last payout position?

 

[rxbandits]

Thanks William! No, the problem is not solved. I explain it best in Post 3 of this thread, what exactly I'm struggling with. I believe I'm looking for a polynomial function.

 

[williamrand1]

Im not sure about a polynomial function. I believe I've come up with a function that works. It a density function got from integrating the following logarithmic function,

f(x) = -Ln(x/n).

x is position. 1 is first place, 2 is second place etc.

n is the total number of payouts.

The integral of f is (x/n).(1-Ln(x/n)).

When you integrate the function f between limits 0 and n you get density or area under the curve equal to 1. This corresponds to 100% total payout. so for example to get the percentage for first position you integrate f between 0 and 1. In general to get the percentage for position i you integrate f between i-1 and i.

Im going to bed now but drop me a tread and ill explain more tomorrow.

Rgds

 

[jbriggs444]

The problem reduces to finding a "quadratic" payout graph for the n people in the top 20% of the field.

That, in turn, reduces to taking the sum of the squares of the first n integers.

The sum of the squares of the first n integers is n (n+1) (2n+1) / 6. Call this "sum"

The fraction of the total payout for the lowest ranking person is 1/sum.
The fraction of the total payout for the next lowest ranking person is 4/sum.
The fraction of the total payout for the next lowest is 9/sum.
[...]
The fraction of the total payout for the highest ranking person is n²/sum

You can see that the fractions add to 1.

 

[williamrand1]

Sorry i made a typo in my function above..

f(x) = -Ln(x/n)/n