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I asked the same question on math.stackexchange and had some responses, but I would also like to hear some input specifically from statisticians and data analysts and I feel that you guys may have something to offer.

The question is about finding 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 $q_i$. 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, $q_i\geqslant q_j$ if and only if $s_i\geqslant s_j$. Further, the scores are bounded, so $s_{min} \le s_i \le s_{max}$. 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?

One interesting answer by @opt suggests to use the so-called Softmax function in the context of neural networks, and it is basically ${\displaystyle p_i=\frac{\exp(s_i)}{\sum^n_j\exp(s_j)}}$, and $\sum^n_ip_i=1$. In other words, $p_i$ would be the portion of the prize that $i$ should receive given her score. I would like to hear your thoughts/opinions on this method.

Many thanks.

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    $\begingroup$ There is no unique or best answer to this question. Similar questions have been debated at least since the 1200's and were simply insoluble until a definite probabilistic meaning was given to the scores (in a famous series of letters between Fermat and Pascal in the mid-17th century). To make progress and not be completely arbitrary, you must stipulate something additional about how those scores arise (as one of the math replies indicates). In particular, there's no basis to recommend softmax or anything else. $\endgroup$
    – whuber
    Jan 10 '12 at 17:00
  • $\begingroup$ @whuber, thanks for the comments. There is no particular probabilistic meaning associated with the scores. The scores are derived by another separate process which just measures the how well each individual performs in a task, and the higher the scores, the better the performances are. Now given this measure of performance, i want to devise a way to distribute the prize proportionally to performances, hence my question. Maybe I should remove "probability" tag from the question, which might be a bit misleading. $\endgroup$ Jan 10 '12 at 21:12
  • $\begingroup$ A and B sat down to eat having brought $5$ and $3$ loaves respectively to the meal. Hungry C came without any food and asked to share, offering to pay for the food he consumed. The three divided up the food and ate it in equal shares of $8/3$ loaves. C left $8$ euros on the table before departing as payment for the food. A said, "Let's split the money $5$-$3$" but B wanted a $4$-$4$ split. They took the dispute to a judge who gave A $7$ euros and B $1$ euro since A ate $5-8/3$ of his $5$ loaves and gave $7/3$ loaves to C while B ate $3 - 8/3$ of his loaves and gave only $1/3$ loaf to C. $\endgroup$ Jan 10 '12 at 22:03
  • $\begingroup$ More seriously, here is one way of doing it. If some of the $s_i$ are negative and at least one $s_j$ is positive, divide the prize $A$ into $A/[\sum_i (s_i + s_{\min})]$ pieces, give everybody with a positive score a number of pieces equal to their score. Merge together the remaining pieces of the prize and divide it into equal shares to give to those who had a positive score. $\endgroup$ Jan 10 '12 at 22:22
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    $\begingroup$ It's ok to ignore probability (or rather, uncertainty), Simon, but that doesn't make the issue go away. Another way to look at it is to note that these scores have no units of measure. Therefore it is impossible to tell whether the range from -3.7 to 5.2 is enormous or inconsequential. The resulting division could be anywhere from giving all the money to the top score (in the former case) to splitting it evenly (in the latter case). This is why any answer is arbitrary (and therefore unfair and entirely without justification) in the absence of more information about what those scores mean. $\endgroup$
    – whuber
    Jan 10 '12 at 22:35
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As whuber says, there's nothing magical about negative scores. Oftentimes a negative score can simply mean that the score was below some baseline (perhaps the average), rather than being worthless (or worth less than nothing).

In some contexts, it makes sense to simply rank participants. Some prefer to pay disproportionally for the top few spots. Most poker tournaments do this, for instance. However, even in the context of poker tournaments, there is no universal rule. Some pay the majority of the prize to 1st place while others have a flatter structure that is more uniformly distributed among top participants. Then of course there's also the Ricky Bobby philosphy ("If you ain't first, you're last!") better known as "winner take all."

So there are a number of different sensible payout schemes just where the only score is the ranking. Other times you might want to give the prize out proportionally according to scores (which might require re-centering to avoid issues with negative scores). Other times you might want to apply some transformation to that payout though. Lots of things can make sense depending upon the context. How much spread in talent is there? How much luck is involved in performance? What real significance is there between various scores? All these things matter, and there really is no universal "right" answer.

There is no "proper way"; it depends on what you are doing.

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