Raph Frank wrote:
On Tue, Feb 3, 2009 at 10:25 PM, Kristofer Munsterhjelm
<[email protected]> wrote:
I would guess that PAV, being based on a divisor method (Sainte-Laguë in the
case above), must fail Droop proportionality to some extent, just like
Webster's method must fail quota.
There is also a version for d'Hondt, i.e
1+1/2+1/3+...
I imagine one could make PAV variants for any of the "denominator"
methods (D'Hondt, Sainte-Laguë, Imperiali, proportions, etc). A
Huntington-Hill variant would go like this: 1/0+ + 1/sqrt(2) + 1/sqrt(6)
+ 1/sqrt(12) + ....
Here 1/0+ is towards the limit of 1/0; at some small value x approaching
zero, 1/x is so large that the sum of the other divisions never count as
much as 1/x. That's 1/0+, I think.
If a group of voters were to vote max for 1 candidate, and min for all
the rest, I wonder is there a weighting function that will guarantee
that that candidate will be in the best winning circle.
For PAV, that would be bullet voting. What do you mean by the "best winning
circle" - the Smith set?
I meant the set of winners that maximises the total score.
The d'Hondt version of the PAV formula probably does it.
However, my question was if there was a way to do it where the weight
to each voter depends on the position of the voter when they are
ordered according to happiness with the result.
This would make the result independent if an offset is added to all
the voter's utilities.
With PAV, the zero point matters, and with utility, the zero point
shouldn't matter.
The zero point matters in (rated) PAV because you're weighing the
intensity of utility (strength) of some against the breadth of utility
(satisfaction of having many candidates) of others. Without the
reweighting intended to do this, when comparing sums, the zero point
doesn't matter, since it cancels out.
The zero point also matters if you have a diminishing returns situation.
Also, using the PAV formula with range ballots is not entirely straight forward.
If a voters approves candidates A and B, and they are both in the
winning circle, then the voter's happiness with the result is
1+1/2 = 1.5
However, if the voter rated them as
A: 100 (of 100)
B: 75 (of 100)
It isn't entirely clear how to convert that to a happiness rating.
The voter's happiness is 1.75 before conversion. That isn't quite the
same as 2 approved candidates elected, but is more than 1 approved
candidate elected.
One option would be to split each candidate out, i.e.
1/1 + 0.75/2 = 1.375
Another option is
ln( (total)/max + (offset) )
The offset changes slowly. It would need to be set so that for
integer totals, the formula matches desired the PAV formula.
Either one considers ratings to be votes of lower weight, or ratings to
be partial votes. In the former case, one would, as you put it, "split
each candidate out", since each rating is a vote, but it's of a smaller
weight. In the latter case, one would use the logarithm because one
would be dealing with partial votes that sum up to some number of votes.
Some more information on the latter concept: With D'Hondt type divisors,
the score function when one has got p full-intensity votes (or p
Approvals) is H(p), the harmonic function. This harmonic function has a
continuous analog, the digamma function, the relation of which is
digamma(x) = H(x-1) - y
where y is the Euler-Mascheroni constant. So, one could make a
continuous score function for D'Hondt type PAV applied to Range votes
this way:
DH_score(x) = digamma(x+1) + y
For Sainte-Laguë divisors (1 / 2k-1), we have
digamma(x+1/2) = -y - 2 ln(2) + SUM k = 1 to x: 2/(2k-1)
Get rid of the constants:
digamma(x+1/2) + y + 2 ln(2) = SUM k = 1 to x: 2 / (2k-1)
And divide:
(digamma(x+1/2) + y + 2 ln(2)) / 2 = SUM k = 1 to x: 1 / (2k-1)
which gives
SL_score(x) = (digamma(x+1/2) + y + 2 ln(2)) / 2
again, where x is the sum of the ratings on the ballot in question for
the candidates on the council to be evaluated.
By these definitions, in rated PAV (PRV?), if a voter had rated A as 0.7
and B as 0.5, his satisfaction (if {A, B} was the council to be checked)
would be DH_score(1.2) = 1.12151 for D'Hondt, and SL_score(1.2) =
1.08603 for Sainte-Laguë.
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