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.
