[EM] Simulations with social welfare functions

[David Cary]

--- bql at bolson.org wrote:

> To answer my own question, I think the attached perl script nicely
> shows 
> the difference between std-dev and gini by this output:


The Gini Coefficient is invariant under scaling, but not under
translation.  Standard deviation is invariant under translation, but
not under scaling.  If you want a better comparison between the two,
you might try comparing Gini to stdev/mean.  Also, Gini, like
standard deviation, can be calculated for a population or a sample. 
The Perl code is inconsistently using the formulas for the Gini of a
population, but the standard deviation of a sample.

The Wikipedia article is confusing to the point of being erroneous. 
To calculate the Gini coefficient for a population, use the Brown
formula with:
   X_k = k / n
   Y_k = S_k / S_n
where:
   S_k = sum of w_i for i = 1 to k
   (w_i) i = 1 to n, is the sequence of values (e.g. income) for each
member of the population, sorted in increasing order.

With this setup, the Brown formula can be restated as:
   T = sum of w_i * (n+1-i) for i = 1 to n
   G = 1 - 2*(T/S_n - 0.5) / n

This is algebraically equivalent to the formula given earlier on this
list.


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