[EM] Three rounds

[Kristofer Munsterhjelm]

Raph Frank wrote:
>> If you mean the Droop proportionality criterion: no, it doesn't. Since no
>> reweighting is done in the first round, it elects the Condorcet winner then,
>> and that's incompatible with the DPC.
> 
> What about running the process for double the number of steps as there
> are seats.
> 
> If there are 5 seats, then the first 4 rounds would be 'setup' rounds.
> 
> Assuming the winners in each round is A, B, C ..., then the election
> would proceed as
> 
> Setup stage
> Round 1:  A
> Round 2:  AB
> Round 3:  ABC
> Round 4:  ABCD
> 
> Election stage
> Round 5:
> E elected: ABCDE
> 
> Round 6:
> A eliminated and then F elected:
> Winners: BCDEF
> 
> Round 7:
> B eliminated and then G elected
> Winners: CDEFG
> 
> Round 8:
> C eliminated and then H elected
> Winners: DEFGH
> 
> Round 9:
> D eliminated and then I elected
> Winners: EFGHI

> Note: a candidate may be referred to by more than 1 letter. A
> candidate might be eliminated in round 6 but then re-elected in round
> 8, so that candidate is both A and H.
> 
> I wonder if it can be shown that if there is at least one solid
> coalition with a) a Droop quota and b) none of  them elected, then
> they one of them is guaranteed to get elected for the final.  If that
> was true, then each of the winners in rounds 5-9 would meet the
> criteron.  Effectively, if a candidate who is part of a solid
> coalition is eliminated, he would be reelected immediately, or
> replaced by another candidate who also meets the criteron.

I don't think so. Though I haven't investigated this method, I'm 
thinking that since it uses a divisor method (Sainte-Laguë), there would 
be instances where it breaks quota, just like ordinary Sainte-Laguë 
breaks quota, since quota (no candidate or party should need more than a 
quota worth of votes to get a seat, or get a seat with less than a 
quota's worth) is incompatible with the two criteria Sainte-Laguë meets 
(population pair and house monotonicity).

On the other hand, quota violations are very rare in ordinary 
Sainte-Laguë/Webster, so it might not matter. Yet it does seem to matter 
when we port divisor methods directly to single-winner methods (e.g 
RRV), as quota methods outperform them in my simulations.


Perhaps there's a multiwinner analog of the Condorcet criterion. If so, 
we would have a base on which to construct a method instead of having to 
guess blindly. Perhaps something like "if the method, when electing k 
winners, returns the set X, and there is a way of partitioning the 
ballots into k piles so that each pile has a CW, and each CW is in X, 
then the method passes this criterion".
Or, is there something that is to the Droop proportionality criterion as 
the Smith criterion is to mutual majority?

None of this is really applicable to the runoff (since we don't want DPC 
there), but since we were discussing methods that do meet the DPC, my 
mind wanders :-)