[EM] using welfare functions in election methods

[Jobst Heitzig]

Hello folks!

This is about an idea I was thinking about for several weeks now: How
the concept of "welfare function" which is frequently used in welfare
economics could fruitfully be used in the discussion of election
methods, too.

A "social welfare function" measures the "welfare" of a group of people
by aggregating in some way the "welfare" of the individual members of
the group, as measured by some "individual welfare function".


For example, a very simple social welfare function would be the average
of the individual income (the latter being an example of an individual
welfare function).

A peculiarity of this special example is that this version of "social
welfare" does not change when income is redistributed, e.g., when two
incomes of 100 and 0 are replaced by two incomes of 50 and 50. In other
words, using the average individual welfare is insensitive for
inequality in individual welfare.


For this reason, most social welfare functions replace taking the
average by some other way of aggregation that *is* sensitive for
inequality in individual welfare. The motivation for this is that
inequality is thought of inducing some "cost" for the group.

The most widely used such function is the "Gini welfare function". It
subtracts from the average individual welfare half the average absolute
difference in individual welfare. Mathematically, denoting the
individual welfare (e.g. income) of individual  i  by  w_i,  the two
examples can be written like this:

  f_ave = sum ( w_i, i=1..n ) / n

  f_Gini = f_ave - sum ( |w_i-w_j|, i=1..n, j=1..n ) / n^2 / 2

The Gini welfare function can also be expressed as

  f_Gini = f_ave * (1-G)

where G is the "Gini coefficient of inequality":

       sum ( |w_i-w_j|, i=1..n, j=1..n )
  G = -----------------------------------
          2 * n * sum ( w_i, i=1..n )

Another way to interpret the Gini welfare function is this: pick two
members of the group at random (with replacement) and take the smaller
one of their individual welfare values. Then f_Gini is the average
outcome of this. In other words:

  f_Gini = sum ( min(w_i,w_j), i=1..n, j=1..n ) / n^2

Here's some concrete examples:

  individual welfare values w_i | f_ave | f_Gini
  ------------------------------+-------+-------
  99,  0,  0                    | 33    | 11
  33, 33, 33                    | 33    | 33
  99, 99,  0                    | 66    | 44
  99, 66, 33                    | 66    | 51.3
  66, 66, 66                    | 66    | 66

I guess most of you will have an idea by now why I tell you all this...
Obviously, one could use a Gini (or other) social welfare function to
measure the "social welfare" which the election of some specific
candidate would bring.

For example, we could let  w_i  be the range value between 0 and 99
which individual  i  gave to the candidate. Given this, ordinary Range
Voting elects the candidate who maximizes the "social welfare" as
measured by the function f_ave, whereas "Gini Range Voting" would
instead elect the candidate who maximizes the function f_Gini!

Looking forward to your thoughts,
Jobst