[EM] PR, Approval ballots, "power"

[Kevin Venzke]

(I'm rather ignorant about PR and PAV, so here's hoping no one finds this message
laughable.)

I started writing a simulation the other day to see if I could come up with a
good way to do PR with approval ballots.  (Basically the voters form a neat spectrum,
and the candidates occupy a contiguous segment of the spectrum, meaning that they
are approved by those respective voters.)

Initially I had it do random trials to pick a set of winners, attempting to
equalize voter "power."  At first, I assumed this meant that every voter should be
represented by the same number of candidates; of course I got strange results.
I modified the definition so that, for example, if I am a voter, my "power" for
a given set of winners is the sum of 

([TotalOfAllPower] / [NumberOfSeats]) / [NumberOfVotersForThisCandidate]

for all the candidates I voted for and who are winners.  Note that the amount of
"power" a voter ideally enjoys is independent of the number of seats to be filled.

This definition seems to give pretty good results, but I wonder if any EM reader
sees a flaw in it.

Then I came up with an actual method (not just a measure).  It involves reducing
the strength of a ballot, and I wonder if it's identical to something already
invented:

For every seat (to be filled):
1. Find the candidate for whom the most ballot strength is offered/available.
(Every ballot initially has strength: (TotalOfAllPower)/(NumberOfAllVoters).  This
means that the first candidate elected will be the Approval winner.)
The strength offered for a candidate is simply the total remaining ballot strength
of all voters who voted for him.
2. The strength of every ballot who voted for this candidate is reduced by:
((TotalOfAllPower)/(NumberOfSeatsTotal)) / (VotersForThisCandidate).  Strength
can go negative, and as I've presently coded it, can count AGAINST your supported
candidates for the purposes of step 1.  (Not sure if that's good or bad.  It
probably helps proportionality, but voters might hate that they can keep other
picks from winning by having voted for them.)

Note that for a given amount of "TotalPower" and number of seats to be filled,
the "total cost" (to all supporters) is constant, no matter which candidate or
which seat it is.  This means that the "offered/available ballot strength" in
step 1 doesn't actually reflect what will be paid.

The results seem pretty good, in terms of equalizing voter "power" (as I've
defined it).  It seems to do better with at least 3 or 4 seats.  It seems 
a desirable quality that the Approval winner is always elected.

As a "centering" rule I tried forcing it to elect the MinMax (Pairwise Opposition)
winner, but it's not very helpful when seats>3, and can worsen the equality.

Any thoughts?

Kevin Venzke
stepjak at yahoo.fr


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