# Percentage with negative numbers

I have several competitors in an economic market, each one with a score of a strength indicator. Scores can vary in [-inf,inf]. For a subset of competitors, I need to calculate their relative importance.

With all positive scores (like in CASE A), I've been calculating relative importance using the following formula (like in percentages):

CASE A

Score 1 = 1 -- > r = 0.1111

Score 2 = 2 --> r = 0.2222

Score 3 = 6 --> r = 0.6667

How can I replicate my intent in CASE B, where I have some negative scores? Here, the above-mentioned formula does not reach my goal anymore.

CASE B

Score 1 = 1 -- > r = ?

Score 2 = 2 --> r = ?

Score 3 = 6 --> r = ?

Score 4 = -100 --> r = ?

• You should specify what is the meaning of a negative score. Do you expect that a competitor with negative score has a negative relative importance? It would be helpful if you could provide an example of the situation that you want to model. – Dmitry Nov 19 '18 at 8:14
• Thank you Dmitry. Negative numbers come out from the fact that the original scores are standardized with respect to all the market players and not only to the competitors in my subset. Standardization has been achieved subtracting the mean and dividing by the standard deviation. So it means below the average, but not in every case I can access the original average. Not sure if I explained myself clearly. – Forinstance Nov 19 '18 at 8:18
• why wouldn't you just shift all the scores by the lowest value? Say, $\tilde{r}_i=r_i - \min_{j}(r_j)$. For your example this will give: $r_1=101,\dots,r_4=0$ – Dmitry Nov 19 '18 at 8:24
• Because this would give me a problem, I think: I would have 101,102,106 and 0, in case B. Doing the percentage afterwards, I would get: 0.3269, 0.3301, 0.3430 and 0. In orginal scores score 2 was the double of score 1 (2 vs 1), which is however not represeted in these final numbers (0.3269 vs 0.3301). On the other hand, this is represented in case A (0.11 vs 0.22). – Forinstance Nov 19 '18 at 8:35
• OK, I'll try to come up with some plausible expression later. – Dmitry Nov 19 '18 at 8:46

Let's stick to your notation and define the score of the $$i$$th competitor by $$x_i$$ and the relative importance by $$r_i$$.
The definition $$r_i=\frac{x_i}{\sum_{j=1}^n |x_j|}$$ has the following drawbacks:
1. The relative importance can be negative if $$x_i<0$$. This is not easy to interpret. Let's say you have two competitors with relative importance $$r_i=0.1$$ and $$r_j=-0.1$$. How would you compare these numbers?
One possible option would be to define the relative importance using an exponential function: $$r_i=\frac{\alpha^{x_i}}{\sum_{j=1}^n \alpha^{x_j}},$$ with the base $$\alpha>1$$. This formulation has an advantage of penalizing very small (negative) scores. However, the relative importance grows nonlinearly with $$x_i$$, i.e., $$x_i=2x_j$$ does not imply $$r_i=2r_j$$.