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

[Kristofer Munsterhjelm]

Raph Frank wrote:
> This method would provide guaranteed Droop proportionality while still
> have range/score like effects.  It is basically PR-STV for electing
> the candidates but RRV for deciding who to eliminate.
> 
> So,
> 
> 1) Voter casts range/score vote
> 2) Ranked ballot can be inferred
> 3) Run PR-STV as normal until an elimination step is required
> 4) Reweight all the ballots based on current winning set and eliminate
> the lowest rated candidate
> 5) goto 3
> 
> This is slightly different to RRV as it eliminates the lowest
> candidate rather than elects the most popular candidate.

Without reweighting, the method might be more monotone than STV is. 
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. Also, if one can generalize 
from apportionment methods, such a method would still be nonmonotonic, 
because in meeting quota, it also has to fail population pair monotonicity.

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
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.

Plurality could be replaced by any method that returns, for each 
candidate, information so that it's possible to determine, using that 
information alone, whether the candidate is part of a minimal Droop set 
or not. A minimal Droop set is one that at least a Droop quota ranks 
ahead of all others, where no such smaller set exists.

(As an aside: the above could also be used in constructing PR methods 
with or without reweighting. For instance, a method like this:
  - 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.

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.
)