Payout amount dependent upon value for each person in population

Question

I am a software developer tasked with solving a problem programmatically, but I'm limited in what I remember about statistics and am not sure how to properly word my question to search for the answer. Thanks in advance for your help.

Problem I have a list of people that each have a particular salary amount. I also have a bucket of money. I want to distribute the bucket of money to everyone based on their salary, with the lowest earners getting the highest amount and the highest earners getting the least.

Example

• Person A - $4.00

• Person B - $5.00

• Person C - $6.00

• Person D - $20.00

The bucket of money is $100.00

FYI, my dataset is made up of almost 3000 records, so the above example is very simplified.

Is there a formula/algorithm that can help me figure this out ?

Answer 1

You might try making it proportional to the reciprocal of the person's earnings. In the example, we have $$S=\frac14+\frac15+\frac16+\frac1{20}=\frac23$$

So then A would get $$\frac{1/4}{2/3}=\frac38=37.5\%$$ of the pot, B would get $30\%$, C would get $25\%$ and D would get $7.5\%$.

There's no way to defend this mathematically, of course. It does guarantee that persons earning less get more of the pot, though.

Answer 2

From your reply to the first comment, I think you are not only looking for an algorithm but also for an argument for your decision on how to distribute the bucket.

I suggest you use a Lorenz curve. First, order the individuals in ascending order, from the one with the lowest salary to the one with the highest and get the cummulative proportion of your population. Use this line, which is of length one, as your x-axis. Second, for each point in x, locate the cummulative wealth that corresponds to the cummulative population in your y-axis. You will end up with a figure similar to the following

A line of 45 degrees would represent equality among your individuals. Then, from the several alternatives of distribution you have, choose the one which makes the new wealth distribution more equal.