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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.

Inemesit Affia
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  • see also http://math.stackexchange.com/questions/1412988/sharing-items-to-a-particular-number-of-people-as-a-funtion-of-amount-contribute – Inemesit Affia Aug 28 '15 at 19:30
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    Are you asking how to express a large natural number as the sum of $n=11$ parts that are "as equal as possible"? – hardmath Aug 28 '15 at 19:57
  • Théophile has a reasonable solution – Inemesit Affia Aug 28 '15 at 20:22
  • if the winning contributions total $1000 I give the first person $300 for example and the second $200, When the contributions are $1 million, I might want it to be maybe 300k, 290k correspondingly – Inemesit Affia Aug 28 '15 at 20:35
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    It seems you have an opinion about how to do this, but it has not been expressed clearly enough in the body of the Question for a solution to be deduced mathematically. – hardmath Aug 28 '15 at 20:37
  • Maybe as equal as possible, but not equal – Inemesit Affia Aug 28 '15 at 20:37
  • if the winning contributions total $1000 I give the first person $300 for example and the second $200, When the contributions are $1 million, I might want it to be maybe $300k, $290k correspondingly – Inemesit Affia Aug 28 '15 at 20:43
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    Yes, there would be many ways to do this. It's not constrained mathematically enough to suggest one way is "best", but if you are distributing the prizes, it can be done by an arithmetic series as Theophile suggests, or in some other fashion where the prize differences are small in relation to the total. – hardmath Aug 28 '15 at 20:46
  • I've adjusted the question, personally I'll try different methods, then use a graph to see if it works for me – Inemesit Affia Aug 28 '15 at 20:50

1 Answers1

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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.

Théophile
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