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

[Kristofer Munsterhjelm]

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.