# Creating an exponential or power law distribution of prizes for a competition

I want to create a function that will output the prize distributions for N players given the following parameters:

1) Minimum prize distribution value (prize distribution cannot be less than this number)

2) Total money available for distribution

Here are some other conditions that must be met:

1) Sum (or, integral) of prize distributions is equal to total money available for distribution (obviously)

2) The largest prize distribution goes to the best player, smallest to the worst player (also obvious)

3) The distribution must resemble a power law or exponential distribution in that top players will get disproportionately rewarded.

The function must be able to take ANY N players, any P pot size and any M minimum prize distribution and still output the desired distribution.

I'm a bit stumped. Any advice?

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How about a simple geometric distribution? With $\alpha$ such that $$\frac{P}{M} = \frac{\alpha^N-1}{\alpha-1}$$ you can give prizes of $M, M\alpha, M\alpha^2,...,M\alpha^{N-1}$ which total $P$.

You can solve the equation above for $\alpha$ using, for instance, Newton's method.

For example, with $P=1000$, $M=100$, and $N=6$, this yields $\alpha=1.2027936546$ and prizes of 100, 120.28, 144.67, 174.01, 209.30, and 251.74.

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What you need is the Pareto distribution. It has the power law behaviour you require, you can set a minimum prize. If you still need to tweak a bit more, look at the end of the page, there are some generalizations that you can use, like a version with an upper bound.

Since your problem is discrete, it might be more useful to take (a possibly modified) Zipf's law.

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