How to distribute a prize among a group of people given their scores? 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.
 A: 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.
