I'm here because my mathematics skills are lacking somewhat for a problem i'm trying to solve..
I'm trying to work out a formula for prize distribution for the top 3rd of entrants in a competition.
All entrants pay a fixed amount to enter (£4.50) (The number of entrants is not a fixed number, there's no maximum number of entrants)
totalPrizePool = entryFee * number of players
Prizes are to be paid out to the top 3rd of entrants. totalWinners = totalEntrants / 3
totalPrizePool needs to be split between the totalWinners
Where the top placed winner is a higher percentage than the lowest placed winner. and dropping for the next place, until all winners have received a percentage.
Whats the easiest way to work out the percentage drops from 1st to last place?
I hope my question makes sense and is possible to calculate.
Any pointers in the right direction would be very much appreciated.
I intend write some php code to do the calculations, just need to know what steps to take to do the calculations
Let's say there are $n$ participants and each one pays $ x$ pounds as entree fee, so we have $n\times x$ pounds collection and this you wish to distribute among $\frac{n}{3}$ top participants.
One way to distribute this amount is to use Geometrical Progression. The sum of all terms in a GP is given by $S=\frac{a(1-r^n)}{1-r}$, here $a$ is the first term, $r$ is the common ration and $n$ is the number of terms. In your case $S=n\times x$ and the number of terms $=\lfloor\frac{n}{3}\rfloor=m $ and let's say you select $r=0.75$, then $3mx=\frac{a(1-0.75^m)}{1-0.75}\Rightarrow a=\frac{0.75mx}{1-0.75^m}$.
Now, using the numbers provided by you, if $n=5000$ then $m=1666$ and $ x=4.5$, so the amount you expect to collect is $5000 \times 4.5 = 22500$ pounds, this amount you wish to distribute among $1666$ top rankers.
The first prize will be $a=\frac{0.75\times 1666 \times 4.5}{1-0.75^{1666}} \approx 5622.75$
The second prize will be $ar^{n-1}=5622.75 \times 0.75^{(2-1)} \approx4217.06$
The third prize will be $5622.75\times 0.75^{3-1} \approx 3162.79$
The fourth prize will be $5622.75\times 0.75^{4-1} \approx 2372.09$
The fifth prize will be $5622.75\times 0.75^{5-1} \approx 1779.07$
can you continue?