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I am curious to find out reasonable ways of dividing a prize among $n$ people in the following situation:

To make the example specific, we have $6$ people in total who are going to share a prize of $100$ dollars, and let us denote the amount received by each person $i$ as $p_i$. In addition, each person $i$ is given a score $s_i$, and we can think of $s_i$ as a way of measuring how well person $i$ deserves some portion of the prize. The intuition here is that we would like a higher-scoring person to receive a larger portion of the prize than a lower-scoring person, that is, $p_i\geqslant p_j$ if and only if $s_i\geqslant s_j$. The problematic thing here is that $s_i$ can be either negative or positive. For example, $s_1=1.3, s_2=2.1, s_3=-0.8, s_4=-3.7, s_5=0.7, s_6=5.2$.

So, what would be the proper ways of dividing the prize given these scores?

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    Are there maximum and minimum possible scores, or is the range of potential scores unbounded?2012-01-10
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    @BrianM.Scott, yes, the range of scores is bounded. For generality, let's say $s_{min} \leqslant s_i \leqslant s_{max}$.2012-01-10

3 Answers 3