[EM] using welfare functions in election methods
Paul Kislanko
kislanko at airmail.net
Sun May 14 16:58:27 PDT 2006
Obviously some academics have too much time on their hands, 'cause this is
nonsense.
> -----Original Message-----
> From: election-methods-bounces at electorama.com
> [mailto:election-methods-bounces at electorama.com] On Behalf Of
> Jobst Heitzig
> Sent: Sunday, May 14, 2006 5:47 PM
> To: Election Methods Mailing List
> Subject: [EM] using welfare functions in election methods
>
> 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
>
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