Sharing items to a particular number of people

Question

If I have a game where everyone contributes money, but only $n$ ($11$ in this case) people can win. How do I share the winnings such that the prize amounts do not diverge significantly as the amount contributed increases?

If I have winnings a, b , c ... ( a + 10)

I want a/b at \$1000 to be greater than a/b at say $\1 million

Also, if they contribute differing amounts, how do I award them as a function of the contributed amount. So that it depends on their position and amount contributed.

Answer 1

You could give each winner a base prize of $p$ along with a bonus according to whether they won first, second, etc., place. For instance:

1st place: $p$ + 100
2nd place: $p$ + 90
$\cdots$
11th place: $p$ + 10
11th place: $p$ + 0

Say that $M$ dollars have been collected in total. The total prize money is $11p + 550$, so we have $$p = \frac{M-550}{11}.$$

You can probably come up with a more exciting set of bonuses; this is just to show one possible approach.