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.