[EM] Satisfaction Approval Voting - A Better Proportional Representation Electoral Method

[fsimmons at pcc.edu]

Satisfaction Approval Voting - A Better Proportional Representation Electoral Method

One way to generalize Proportional Approval voting to range ballots is by
finding the most natural smooth extension of the function f that takes each
natural number n to the sum

f(n) = 1 + 1/2 + ... + 1/n.

It turns out that we can extend f(n) to all positive real values of n via the
integral

Integral from zero to one of (1-t^n)/(1-t) with respect to t

For PAV generalized to range ballots, first normalize the ratings to be between
zero and one.

Then for each proposed coalition C of k candidates (assuming there are to be k
winners) and each range ballot r, let g(r,C) be f(S) where S is the sum of the
ratings (according to r) of the alternatives in the coalition C.  Elect the
coalition C with the greatest sum of g(r,C) over the range ballots r. 

This reduces to PAV if all range ballots use are voted only at the extremes of
the range, i.e. are voted approval style.

Also, even though generalized PAV is not summable, it can be used to compare the
results of other methods.

One should take as many nominations of winning coalitions as anybody wants to
submit along with the results of SAV, sequential PAV, STV, and any other
multiwinner method that can be computed from range style (including approval)
ballots, and see which one of them has the highest generalized PAV score.

Forest

Original Message:

>[EM] Satisfaction Approval Voting - A Better Proportional Representation
Electoral Method
Kristofer Munsterhjelm km-elmet at broadpark.no
Thu May 20 07:04:11 PDT 2010

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Raph Frank wrote:
>> Proportional Approval voting uses a different satisfaction metric.
> 
>> Each voter is consider to have satisfaction of
> 
>> 1 + 1/2 + 1/3 + .... + 1/N
> 
> >where N is the number of approved candidates who are elected.

>Proportional approval voting also uses raw Approval scores instead of a 
cumulative ballot. However, it is hard to calculate the optimum outcome 
(i.e. the winner set that maximizes satisfaction), and it's not summable.

>SAV does approximate PAV in a sense: if a voter votes for two 
candidates, those candidates are given power 1/2 each. If a voter votes 
for three candidates, those candidates are given power 1/3 each, and so 
on. However, the approximation ends there, because the candidates may or 
may not be elected.

>One could also make a Sainte-Lague version by having the satisfaction as:

1 + 1/3 + 1/5 + ... + 1/N

>and I think there was an earlier message on this list (somewhere...) 
with the idea of generalizing this to ratings by using logarithms to 
construct a function that, for f(maxrating) = 1, f(2*maxrating) = 1 + 
1/2, f(3 * maxrating) = 1 + 1/2 + 1/3, etc., while being defined on 
positive reals in general.




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