[EM] using welfare functions in election methods

[Paul Kislanko]

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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