Obviously some academics have too much time on their hands, 'cause this is nonsense.
> -----Original Message----- > From: [EMAIL PROTECTED] > [mailto:[EMAIL PROTECTED] 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 > > ---- > election-methods mailing list - see http://electorama.com/em > for list info > ---- election-methods mailing list - see http://electorama.com/em for list info