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

[Raph Frank]

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.

> 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 ?

> 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

> 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?

>  - 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.

> 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.

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