Here is one by Piotr Skowron as well titled "Consistent Approval-Based Multi-Winner Rules".

And here's one called "Proportionality and the Limits of Welfarism" (comparing Phragmén and Thiele). I didn't actually catch the guy's name. It didn't sound like the names in the paper - Dominik Peters or Piotr Skowron (and it doesn't look like Piotr Skowron) - and the subtitles have it as Scott Anand, which is a name I'm not familiar with.

If this was extended to score voting, I think it should elect the Condorcet winner in the single-winner case, if there is one. Otherwise, obviously it would have to choose between the candidates in some way, like other Condorcet methods do so it's not a big problem.

When there's more than one winner, what happens depends on how you interpret the scores. You could measure a voter's satisfaction by adding up the scores the voters have given to the elected candidates, but I think that might be unsatisfactory in a few ways. There's always debate about how to interpret scores and what they mean, and whether absolute numerical values should really be used in their raw form.

Instead, the scores could be used as layers of approval. This basically means that a voter's satisfaction with a candidate set is determined by the single highest score they've given to a candidate in the set, next best used as a tie-break, and so on. So for scores out of 5, a single 5 is better than multiple 4s etc.

This should keep it relatively simple. Also if candidates are elected sequentially, it should be simple enough to calculate the results.

I think this should be a decent enough method and I think I'd prefer it to things like Allocated Score and Sequentially Spent Score.

Obviously COWPEA Lottery using scores as layers of approval is God-tier in terms of criterion compliance, and very simple to implement, but it is non-deterministic, which might be too much for some people, so this method could be a good compromise.

Edit - You'd have to work out exactly how to measure the stability of a candidate set though. Let's say the first 2 candidates elected are AB. Then you need to test e.g. ABC, ABD, ABE etc. to find the 3rd candidate. But I think you might be able to test them against each individual candidate not in the set. So test ABC against, D, E, F etc. separately.

Edit 2 - You'd probably have to test each potential set against all the other subsets. So ABC would go against ABD, ABE etc., plus AD, AE, BD, BE, as well as D, E etc. Still not that many in the general scheme of things.

Some things that came out of the paper:

If A, B, C and D are parties fielding multiple candidates, then with the following ballots, what proportion of the seats should each party win?

250: AC
250: AD
250: BC
250: BD
2: C
2: D

COWPEA would say that A and B would each win just under 1/4, and C and D would each win just over 1/4.

However, Optimised PAV would elect C and D with half the weight each, and A and B would not get any weight. The multi-winner Pareto efficiency criterion assumes that a voter's satisfaction with a result can be measured purely by the number of elected candidates they have approved. And going by that, any result containing any A or B candidates is Pareto dominated by CD.

This can be seen as a two-dimensional voting space with an AB axis and a CD axis. PAV would ignore the AB axis. Arguably though COWPEA makes better use of the voting space.

That was an example where any proportions are allowed, but here is another where 2 candidates are to be elected.

150: AC
100: AD
140: BC
110: BD
1: C
1: D

Basically any deterministic would elect either AB or CD. CD Pareto dominates AB in this multi-winner sense as all 502 voters have an approved candidate under CD, compared with 500 under AB (no-one has two approved candidates). However, AB is more proportional. 250 have approved A and 250 have approved B. 291 have approved C and 211 D. So under the CD result, the 211 D voters would wield a disproportionate amount of power. So what do you think? Does that matter or is it all about number of approved candidates?

If you go for AB, then you are rejecting the multi-winner Pareto efficiency criterion, but also consistency as a by-product. This is because you can have a set of ballots where the C and D approvals are just swapped round. So it's 211 for C and 291 for D but otherwise the same. Then combining the two sets of ballots together you get:

250: AC
250: AD
250: BC
250: BD
2: C
2: D

If there are 2 to elect here, it must be CD. So if you went for AB in the other examples, then you are rejecting the consistency criterion in multi-winner elections.

@wolftune OK, that's fine. I don't disagree with your point. But I was clarifying in case your post was a response to my bit about the candidates that are the favourites of nobody, but obviously it wasn't.

But just to be clear anyway - when I was talking about not wanting to elect the favourite-of-nobody candidates, I wasn't referring to stopping the election of such candidates in principle. But we've discussed certain specific scenarios where we all seem to agree that electing the candidates that are the favourite of nobody isn't the best thing to do. And I was discussing ways to elect the candidates we would want to in this situation.

@dominikpeters Thanks for the information on this. Do you think that MES + completion is a good method for electing a committee of a fixed size or do you think a single method that works without an add-on would be better?

Just to clarify on the Phragmén thing:

If you just have a fixed quota then the voters that get their candidates early can get a bad deal because they pay a whole quota, whereas later on, the might not be a candidate with a whole quota of votes and yet you have to elect one anyway, so the voters of this candidate get their candidate more "cheaply".

So you might then look for a quota that distributes the cost more evenly, and that's all Phragmén really does. It distributes the load or cost across the voters as evenly as it can.

I see subtractive fixed-quota methods as a cheap hack really.

@cfrank

Here's a simple example to show proportionality. This example works the same no matter what the measure of support. Basically, all these measures are clones of STV but without vote splitting. The measures I'm talking about are score, smallest pairwise win, and strongest beatpath. Also, I should say that I don't really know if the measures would work for sorting.

Setup:
2 parties: A, B.
2 seats.
100 voters.
quota: 50 voters.

Round 1:
The votes are A 55, B 45.
The best quotas of 50 voters give A 50 and B 45.
A gets a seat. The quota of 50 A-voters are removed.

Round 2:
Now we count the remaining voters, A 5, B 45.
B gets a seat.