[EM] Criterion compliance of loser elimination and weighted positional methods

[Kristofer Munsterhjelm]

Raph Frank wrote:
> On Sat, Jan 31, 2009 at 4:17 PM, Kristofer Munsterhjelm
> <km-elmet at broadpark.no> wrote:
>> Therefore, it's useful to know what election methods one can combine with
>> loser elimination so that the result passes mutual majority. Now, it might
>> be that my intuition is wrong here and you can get a good multiwinner method
>> out of something that doesn't pass mutual majority, but I don't quite see
>> how; it probably won't be much like STV.
> 
> Assuming you give the voter's vote to their first choice and
> automatically elect any candidate once he meets the Droop quota, then
> the elimination method doesn't matter.
> 
> No matter what order you eliminate the candidates in the faction,
> eventually there will be only one of them left.
> 
> That candidate will thus receive the faction's entire vote and since
> the faction is larger than a Droop quota (by definition), the
> candidate will automatically be elected.
> 
> Reading the rest of your post, I think you have come to the same
> conclusion, but for single seaters.

It might be. I was going to say that perhaps vote splitting could 
invalidate that, but I'm not sure. The difference between a 
single-winner method and the multi-winner method is that the 
single-winner method can afford to eliminate all but the single mutual 
majority winner that becomes a (plain) Majority winner. A multiwinner 
method, on the other hand, has to elect all the winners dictated by the 
DPC, so it can only afford to eliminate extraneous candidates.

The vote splitting would go like this: Say a Droop quota votes { A B C } 
in each permutation with equal probability, then a bunch of other 
candidates. The other voters vote the other candidates randomly before 
any of A B C. Then a method using Plurality with random elimination 
might end up eliminating one of A, B, or C, before it finds out that a 
Droop quota supports the set {A B C}. I don't know if this is actually 
possible, though.

Strictly speaking, it's not that difficult to meet the DPC. Just make a 
DSC/DAC variant where you count all possible subsets. Elect the 
candidates that the DPC say must win, then do whatever you want after 
that. Such a method would be nearly useless in practice, since it would 
have a great discontinuity - it wouldn't elect near-DPC-eligible 
candidates any more often than it would elect candidates far from the 
DPC (unless the base method somehow had this property).

For STV, I think the reweighting matters in letting the method discover 
the Droop quotas. Again, I'm not sure. If elimination order doesn't 
matter, then one could make a DPC version of Random Ballot: pick the 
candidate/s above quota if there is/are any, then eliminate the 
candidate that ranks last on a random uninspected ballot. Afterwards, 
mark that ballot inspected. If the vote-splitting argument above works, 
elimination order *does* matter, though.