need help making an equation for fantasy football

P

Psilo

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so here is the deal. I am trying to make a new kind of fantasy football format for my friends to play which involves assigning fictitious contract values to NFL players based on Fantasy Projections.

After many many different variations of how to suss out a fair, but reality reflective set of values I have found that the formula for compound interest works the best.
basically, if Adrian Peterson is ranked #1 and worth approx $10,000,000.00 , and Anthony Allen is ranked #100 and worth $400,000, plotted along a curve of 3.3% ranks 99-2 come out pretty well.

A=400,000(1+.033/1)^100= $10,282,344.33 that would be the equation for figuring out Adrian peterson's fictitious salary. But! now what i need is to get the curve looking just slightly more logarithmic.

As it stands right now, the 51st ranked player is worth $2,028,037.90 and I need him to be worth approx $1,000,000.

What can i do to this formula to make these points match up and have everything else fall into place? How can i "flare" out the edges of the curve a little better?

-thanks
 
also, i would like to try and get that "interest" rate down to Pi. I think people would just feel more comfortable knowing that the magical number is involved and these values aren't just completely pulled out of the air.

and for some clarification A=P(1+r/n)^nt

A= Salary
P= $400,000 (since that is the league minimum or "floor" aka starting amount)
r= "interest rate" (curve)
n= rank of the player (but it has to be inverse so #1= 100 and #100=1)
t= is just 1, since I am not really trying to find compound interest over time.
 
Psilo said:
so here is the deal. I am trying to make a new kind of fantasy football format for my friends to play which involves assigning fictitious contract values to NFL players based on Fantasy Projections.

After many many different variations of how to suss out a fair, but reality reflective set of values I have found that the formula for compound interest works the best.
basically, if Adrian Peterson is ranked #1 and worth approx $10,000,000.00 , and Anthony Allen is ranked #100 and worth $400,000, plotted along a curve of 3.3% ranks 99-2 come out pretty well.

A=400,000(1+.033/1)^100= $10,282,344.33 that would be the equation for figuring out Adrian peterson's fictitious salary. But! now what i need is to get the curve looking just slightly more logarithmic.

As it stands right now, the 51st ranked player is worth $2,028,037.90 and I need him to be worth approx $1,000,000.

What can i do to this formula to make these points match up and have everything else fall into place? How can i "flare" out the edges of the curve a little better?

-thanks
Click to expand...
The feature of compound interest that you are making us of is that the argument appears in an exponent. Try this formula, in which #1 gets $10M and each next player gets 96.8% of the previous. That percentage is the 99th root of a factor of 1/25.

\(\displaystyle A(n)=$10,000,000×0.968^{(n−1)}\)

\(\displaystyle \log(A(n)) = 7 - (n-1)(0.01412)\)

Now the question is how to add an additional parameter so that you can make A(51) = $1,000,000. That means a smaller ratio for the low end of the scale. This comes closer to what you want:

\(\displaystyle \log(A(n)) = 7 - (\sqrt{n}-1)(0.15533)\)

Then A(1) = $10 M
.........A(55) = $1.0 M
.........A(100) = $0.4 M

 
hey wow, awesome. thank you. While waiting for a reply i did a much more crude version...

any chance you could walk me through the steps of that?

where are you getting the 7? all I have is a calculator on my phone so I can't really put in a balanced equation (probably can but it is beyond me). any chance you could give me that equation as A= .... ? and not log(A(n))=....?

I think your equation is much more elegant than what I have come up with and would reflect a much smoother and realistic curve.

also, I am confused if that equation is supposed to be applied to all projected fantasy rankings, or if you are using two different equations: one for the top 50 and one for the lower 50.

thanks.
 
Psilo said:
hey wow, awesome. thank you. While waiting for a reply i did a much more crude version...

any chance you could walk me through the steps of that?

where are you getting the 7? all I have is a calculator on my phone so I can't really put in a balanced equation (probably can but it is beyond me). any chance you could give me that equation as A= .... ? and not log(A(n))=....?

I think your equation is much more elegant than what I have come up with and would reflect a much smoother and realistic curve.

also, I am confused if that equation is supposed to be applied to all projected fantasy rankings, or if you are using two different equations: one for the top 50 and one for the lower 50.

thanks.
Click to expand...
\(\displaystyle \displaystyle \log_{10}10^7 = 7\), so it comes from your maximum salary.

\(\displaystyle \displaystyle A(n) = 10^7 \times 0.6993^{\sqrt{n} - 1} \)

I knew you wanted the scale compressed at the low end, and I thought replacing n by sqrt(n) would do that. With that substitution, the range of the exponent is 0 to 9 instead of 0 t0 99, but the range of the salaries is still a factor of 1/25. Taking the 9th root of (1/25) gives 0.6993 as the multiplier of the salary corresponding to a change of 1 in the exponent. [Instead of the 99th root when the exponent was (n-1).]

This is a single smooth formula .. without further diddling the value of n that leads to a salary of $1M is n=55
 
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