Say we have a system with a fixed set of participants. Each participant has a reward fraction $f_i$. Ideally, if total reward distributed over some period is $R$, each participant should get its fair share by the end which is $R_i = R \cdot f_i$.
I'd like to come up with a metric that roughly captures how successful the system is in terms of distributing its reward according to fair shares.
What I came up with is this: let $r_i$ be the actual reward of participant $i$. We first calculate its distance from fair share in terms of percentage, i.e. $d_i = \frac{|r_i - R_i|}{R_i} \cdot 100.$ Then, I simply calculate the mean of $d_i$'s., i.e., if there are $n$ participants my fairness metric $m_f$ is $m_f = \frac{\sum_{i=1}^{n} d_i}{n}$.
Is this a good metric? I assume it roughly captures how far the system is from its fair distribution but not sure if there's a better way of doing what I want.
In these cases, one usually uses the squared distance, that is,
$$d_i=(r_i-R_i)^2$$
which of course you may normalize if you wish. The idea behind is that the metric does not punish small deviations much, but punishes large deviations more severely.
For instance, if $|r_i-R_i|$ goes from $0$ to $1$, an increase of $1$, the distance $d_i$ increases by $1$ $($from $0$ to $1)$. However, if it goes from $30$ to $31$, another increase of $1$, $d_i$ increases $61$. It is in this sense that 'large deviations' are more punished by the squared distance.