[EM] A Condorcet multiwinner method based on QBS

[Kristofer Munsterhjelm]

I've recently been cleaning up the Quota Borda System article on 
Electowiki: https://electowiki.org/wiki/Quota_Borda_system

This led me to think about a less complex way to get a Condorcet 
multiwinner method. Tideman's criticism of QBS is applicable to this 
method, too, but I thought it could be interesting, and it also gives me 
an opportunity to show my "amplification" idea for general Condorcet PR.



The election proceeds in a number of rounds. We start at round one, and 
proceed until either all the winners we need have been elected, or until 
the end of round n, where n is the number of candidates.

In round k:
	For each solid coalition starting of size (cardinality) k, let q be the 
maximum number of Droop quotas its support exceeds, and let x be the 
number of candidates from this coalition that have already been elected.
	If q > x, create a derived single-winner election where the weight of 
the voters supporting this solid coalition has been amplified (see 
below). Use a Condorcet voting method that passes mutual majority, like 
ranked pairs or Schulze, to determine a social ordering (order of 
finish). Of the candidates who are members of the solid coalition but 
haven't been elected in an earlier round, elect the one that the voting 
method ranks closest to top.
	Go to the next round once every solid coalition with q>x has been 
processed.


The amplification step changes the weight of the solid coalition's 
supporters. If we do the amplification so that a Droop quota (before 
reweighting) corresponds 50%  (after), and the single-winner method 
we're using passes ISDA, then the winner must come from the solid 
coalition, but since we're amplifying to majority instead of to 
unanimity, the unamplified voters still have a say in *which* candidate 
wins if the solid coalition supporters aren't unanimous among themselves.

To do so, we can multiply the solid coalition's supporters' weight by s 
(the number of seats), which will turn a Droop quota into a majority. 
But we can also adjust the "utility" vs "proportionality" tradeoff by 
reducing the factor to something less than s. QBS uses a factor of one - 
no reweighting at all - which probably goes too far in the other direction.

In any case, no matter what this factor is, the method passes Droop 
proportionality since the winners must come from the solid coalition 
being considered.


I did a 1D Gaussian analysis and the optimum for two candidates seems to 
place the candidates at 37.5% and 62.5%, same as PAV with delta=1. The 
approach is a bit different: I set two candidates at -x and +x, and then 
two more at -x - epsilon, +x + epsilon. If the method elects the two 
latter candidates, then the point is unstable as one can move further 
away and win. The stable point is the one closest to the median where 
there's no benefit to moving further away.
To get the 33% and 67% percentiles, the proper two-candidate 
amplification factor is 3. The 25% percentile can only be made stable in 
the limit of the factor going to infinity, i.e. making the solid 
coalition unanimous; and having a factor of one, like QBS in effect 
does, places the optimum at the median.
There is definitely an argument to be had that the natural point for 
Droop is 37.5% and thus if we want to make it 33%, we should modify the 
quota instead of the amplification factor. But I'm mentioning it in case 
the factor proves useful in some other way.

(Maybe it could be used to make a method that reduces to something like 
modified Sainte-Laguë with full bloc voting, yet be closer to actual 
Sainte-Laguë in a spatial model, thus reducing the latter's incentive 
towards fragmentation, say.)

-km