[EM] using welfare functions in election methods

[Jobst Heitzig]

Dear Paul!

Thank you for your well-informed and elaborated answer. Who are these
academics you are talking about?

By the way, what I suggested is no nonsense at all, of course. Quite to
the contrary, welfare economics and politics strongly relies on these
measures, and it is natural to see an analogy here.

Yours, Jobst



Paul Kislanko wrote:
> 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
>>
>>----
>>election-methods mailing list - see http://electorama.com/em 
>>for list info
>>
> 
> 
>