I am looking for a function that would determine the contribution of each individual relatively to the non-dispersion (equity) of a distribution.
I have a distribution of incomes X (which, letting aside the details, are determined individually through a series of choices, but this should not be relevant here), for example $X = \left[ 1, 0.2, 1 \right]$. $X$ is dispersed, or not uniformly distributed, because 2 individuals have an income of 1, and the other one has an income of 0.2.
I would like to measure the contribution of each individual in this distribution, relatively to the equity of the distribution. In other terms, a measure close to 1 means that the individual helped obtain a more uniform distribution (because its own income is close to the others), whereas a measure close to 0 means that the individual degraded the distribution (because its own income is different from the others). In the previous example, the contribution of the 2nd individual is less than the 1st and 3rd's contributions (because, for example, they are closer to the mean of the distribution than the 2nd individual).
I have tried using well-known equity measures, such as the Gini index or the Hoover index, to compute the equity of $X$ and then comparing to some sort of counterfactual distribution $X \backslash X_i$, excluding the $i$-th individual: $$f_X(i) = (1 - Hoover(X)) - (1 - Hoover(X \backslash Xi))$$
However, this method does not work in the general case: it seems that removing a single individual does not change the distribution sufficiently, and therefore the Hoover indexes are too much similar. Perhaps these functions are lacking some sort of property and I am using them incorrectly?
I have also tried a simpler, naïve, method, based on the difference between each individual and the mean of the distribution: $$f_X(i) = 1 - |X_i - \overline{X}|$$
However, I feel like this is perhaps too naïve and there must be some pitfalls that I have not foreseen. Is there a well-known function to achieve a similar result?
