[EM] Combing Reweighted range/score voting and PR-STV

[Kristofer Munsterhjelm]

Raph Frank wrote:
> On Tue, Aug 18, 2009 at 10:11 PM, Kristofer
> Munsterhjelm<km-elmet at broadpark.no> wrote:
>> Without reweighting, the method might be more monotone than STV is.
> 
> I am not sure.  However, that seems reasonable.
> 
>> However,
>> while the Droop proportionality criterion holds for such a method (as long
>> as the base method - what is used to find winners - can determine if a
>> single candidate is supported by a Droop quota), the proportionality beyond
>> the DPC might suffer.
> 
> My view is that the DPC pretty much defines a method as being
> proportional.  However, granted Droop + rankings is "hard" while
> rating based methods would be "soft" and allow some blending of edges
> (though that is somewhat vague).  It is like the effect where
> condorcet is hard and 50%+1 gives you victory, but range/score is soft
> and may allow some compromise.

Let's consider this "perverse" voting method, then: first run a Borda 
election to get a social ordering X. Then determine all Droop sets and 
pick a winner council so that the DPC is satisfied, and when more than 
one such winning council is possible, pick the one whose members' 
average rank on X is closest to the top.

Borda is very centrist-favoring, so it would be majoritarian beyond the 
DPC. Would this be sufficiently proportional still?

>> A possible reweighting-free method could work like this:
>>
>> 1. Construct a social ordering based on Range ballots. Call this ordering,
>> X.
>> 2. Count the input ballots, Plurality style
> 
> So each ballot goes to the candidate rated highest ?
Yes.

>> 3. If a candidate is supported by more than a Droop quota:
>> 3.1. Elect this candidate.
>> 3.2. Eliminate the candidate from all ballots and from X.
>> 3.3. Go to 2 unless we have the entire council.
>> 4. If no candidate is supported by more than a Droop quota:
>> 4.1. Eliminate the candidate which X ranks last, from all ballots and from
>> X.
>> 4.2. Go to 2.
> 
> I am not sure that is proportional.  Step 3.2 doesn't de-weight
> ballots that have participated in electing candidates, so they get to
> vote max.
> 
> For example
> 
> 20) A1: 100 A2: 99
> 19) B1: 100 A*:0
> 
> and electing 2 seats gives a social ordering of A1>A2>B
> 
> Round 1
> A1) 20
> A2) 0
> B) 19
> 
> Droop is 39/3 + 1 = 14
> 
> A1 is elected
> 
> Round 2
> 
> A2) 20 (as A1 removed)
> B) 19
> 
> Droop: 39/2 = 20
> 
> A2 is elected

Yes, see my reply to Warren. Note that my suggested modification would 
elect A1 and B in one go on the first round because they're both above 
quota.

Though now that I think about it, the modification may have another 
problem. Say you have a ballot set roughly like this:

1 DQ - 1: A > B > C > D > E
1 DQ - 1: E > A > B > C > D

and A scores badly according to the elimination order. Ideally, E would 
be eliminated and then A would be elected.

>> A
>> minimal Droop set is one that at least a Droop quota ranks ahead of all
>> others, where no such smaller set exists.
> 
> Sounds reasonable.  Is it basically defining the smallest "party" that
> has achieved a quota?

Yes, or rather a solid coalition, as voters can vote across party lines.

>>  - Use DAC/DSC explicit set enumeration to find a minimal Droop set.
>>  - Pick a candidate from this set according to some desirability measure.
>>  - Eliminate that candidate from all ballots and loop.
> 
> I think you need to take into account that the supporters of the
> candidate have elected a candidate.

That might be true, yes... If we have sets of the form:

1 DQ + 2: {A B C D}
1 DQ + 1: {E F}

then we shouldn't keep picking from the first to fill the council, 
because once they've got their candidate, they should have no more. One 
could patch this by noting how many have already been elected from that 
group, like:

1 DQ + 2: {A B C D} 0
1 DQ + 1: {E F}     0

then A is elected and

1 DQ + 2: {B C D}   1 <-- got our fill
1 DQ + 1: {E F}     0 <-- pick from here!

but that seems rather messy, and vulnerable to vote management. If the 
elimination caused, say

1 DQ + 50: {B C D}

then the reweighting would be ((1 DQ + 1)/(1 DQ + 50)). If the Droop 
quota is 101, that would be 101/150 = 0.6733^... but the point was to 
avoid all that.

>> This always works because even if voters vote in a completely disorganized
>> fashion, the minimal Droop set will be the set of all candidates, at which
>> point the desirability measure kicks in. The only ambiguities lie in the
>> desirability measure and how to deal with the situation where, say, > 2
>> Droop quotas prefer one set where > 1 prefer another. The DAC/DSC solution
>> would be to intersect minimal "n Droop sets" (supported by more than n Droop
>> quotas) down to one, then picking from what results, according to whatever
>> desirability measure.
>> )
> 
> The way I tend to look at the Droop proportionality criterion is that
> you use PR-STV but have discretion when an elimination step comes up.
> 
> Ofc, different PR-STV methods handle surplus transfers differently so
> maybe that isn't valid and/or doesn't cover all possibilities.

What I was trying to do was to find out what kind of methods might be 
usable instead of Plurality, while the combined "elect and eliminate" 
method still passes the DPC. Since it would still satisfy the DPC, the 
only difference would be how it acts in between the hard limits defined 
by the Droop proportionality criterion. STV acts like IRV there.

> Ideally, you want to pick from all possible winning sets where the
> Droop criterion has been satisfied.  Is that what your rule states?

Kind of - I tried to do that sequentially. It would of course be easy to 
find all the Droop sets and then try all possible combinations of 
candidates to determine which satisfy the DPC, but that is time 
consuming and only fulfills the bare minimum.

Maybe one could do something like that exhaustive search for 
Webster-based methods: Increase the divisor until the constraints only 
satisfy a single council, then pick that one. Each solid coalition would 
form a constraint of the form "elect at least round(num voters 
supporting coalition/divisor)".