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I discussed something similar here and there's a video here as well. There's a video as well which shows a potential ballot design.

In what I discussed the top voted candidate in each district was automatically elected in the first phase.

@matija Do you need the first step about the electing the candidate with most votes in each district but only if they have a Hare quota? You could just do the whole process using the PR system. Or just elect the candidate with most votes anyway and just use the PR system for the second ones to be elected.

I tend to think quotas are somewhat arbitrary.

@matija True. If individual ballot data can be used, then the funding that results from each could be split equally across the candidates approved on that ballot. But this still might not be very satisfactory.

@matija If public funding is proportional to votes, then I'd say it's easier with approval voting than ranks. With ranks, you could use top votes as you say, but then you might give less funding to the election winner (by IRV, Condorcet or whatever).

@cfrank I've seen a few as well which I've deleted, but they're not overwhelming the board or anything, so I wouldn't want to make anything worse for any new users we might get, which isn't that many anyway! So I'd probably say leave it for now, but keep an eye on the situation.

@cfrank Yes. I think parties have their place as it makes it easier to know what someone is standing for in some cases and can simplify the process for voters, but I don't think they should form an essential part of the process, and it should be just as possible for independent candidates to stand.

@gregw COWPEA doesn't pass consistency. Fairness is subjective. But COWPEA itself is just a proportional weighting thing rather than an election method itself. The lottery version could be used for elections but being non-deterministic would likely be a difficult sell.

One other method would be to pick a voter and split their representation equally among the candidates that they have approved. This would be strategy-proof. However, it fails candidate Pareto efficiency.

1 voter: AB
1 voter: A

B would get 1/4 of the weight, and A 3/4. But B is Pareto dominated by A.

Another method is called the Conditional Utility Rule. This puts all the voter's representation onto the candidate that is most approved overall (or splits it equally if there's a tie). This guarantees a the maximum total approval score among a proportional result. But it fails IIB.

2 voters: A
1 voter: B
7 voters: AB

This would weight A:B 9:1. Passing IIB would give 2:1. But despite guaranteeing the maximum total approval score for a proportional result, it still fails multi-winner Pareto efficiency. It can sometimes be possible to find a set that dominates the winning set, although the result won't be proportional. See the result on this page. This is another reason why the multi-winner Pareto efficiency criterion is not necessarily a good thing within the voting election landscape.

I think this is largely it. This project hasn't purely been altruistic - it's been helpful to me by laying everything out for reworking my COWPEA paper!

Strong multi-winner Pareto efficiency: A set of candidates S Pareto dominates set S′ if every voter has approved at least as many candidates in S as S′ and least one voter has approved more in S.

For a method to pass the criterion S′ must not be the elected set.

(For optimal methods it would refer to weight in the committee rather than number of candidates.)

COWPEA fails this.

250 voters: AC
250 voters: AD
250 voters: BC
250 voters: BD
1 voter: C
1 voter: D

Optimal PAV would simply elect C and D with half the weight each. COWPEA would elect each with about 1/4, but C and D slightly more.

This example can be seen as a 2-dimensional voting space with A and B at opposite ends of one axis and C and D at opposite ends of the other. No voter has approved both A and B or both C and D. Viewed like this, electing only C and D seems restrictive and arguably does not make best use of the voting space, which may include policy areas not considered by all candidates. This potentially calls into question the utility of the multi-winner Pareto efficiency criterion. It's certainly not a "slam dunk".

In certain allocation scenarios, it would make more sense as a criterion where utility is purely determined by number of approved things, but it's less clear for voting.

Also in normal election cases with fixed candidates, the case against it is clearer.

2 to elect

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

The winning set must be AB or CD. If we elect any other pair, then too many voters would be without any representation. Candidates A and B are each approved by 250 voters, distinct from each other, and adding up to 500. Candidate C is approved by 291, and candidate D by 211, also distinct from each other, and adding up to 502. The strong multi-winner Pareto criterion would insist on the election of CD, since every voter would have one candidate that they approved in the committee. Under AB, there would be two unrepresented voters. However, CD is a disproportional result, as the D voters, numbering only 211 wield a disproportionally large amount of power. Without the two voters that only vote for one candidate, it seems clear that AB would be the better result, as it is more proportional with no disadvantages.

It purely comes down to whether the Pareto dominance caused by the single C-only and single D-only voter is enough to overturn the better-balanced result of AB. Unless level of proportional representation is of only negligible or tie-break value, AB must be the better result. Deterministic PAV would elect CD but other methods such as Phragmén would elect AB.

Consistency is where when two elections that give the same result are combined, the overall result must still be the same. Sticking with the fixed candidate case, we could swap the C and D voters above:

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

If it was reasonable to elect AB before, it still is now. But we can combine them:

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

And clearly CD becomes the best result, meaning consistency isn't essential. Obviously COWPEA and Optimal PAV can elect candidates in different proportions so they are not directly affected by this.

However, take these election examples:

2 voters: AC
1 voter: A
3 voters: B

C is Pareto dominated by A so COWPEA would elect AB with half the weight each.

3 voters: A
2 voters: BC
1 voter: B

Similarly here, COWPEA would elect AB in the same manner. Then combine the ballots:

4 voters: A
4 voters: B
2 voters: AC
2 voters: BC

COWPEA would now elect C with 1/9 of the weight as it is no longer Pareto dominated. Combining the ballots sets has changed C’s position within the electoral landscape. It does not seem unreasonable to elect C with some weight in this election, and it is therefore not clear that passing the consistency criterion is necessary for a proportional approval method.

Next I will briefly consider a couple of other optimal methods, one of which will also show problems with the multi-winner Pareto criterion.

OK, so COWPEA:

The weight each candidate gets in parliament is the same as the probability that they would be elected in the following lottery:

Start with a list of all candidates. Pick a ballot at random and remove from the list all candidates not approved on this ballot. Pick another ballot at random, and continue with this process until one candidate is left. Elect this candidate. If the number of candidates ever goes from >1 to 0 in one go, ignore that ballot and continue. If any tie cannot be broken, then elect the tied candidates with equal probability.

I recently found in the literature that this idea does exist. In the paper Approval-Based Apportionment it's referred to as Random priority. This puts me in a slight dilemma about pushing ahead with making my COWPEA paper fit for publication and trying to get it published, but I think there's enough in it that hasn't been discussed. Plus my original formulation of it in 2016 pre-dates the general discussion of it in proportional approval method literature.

Anyway, this passes PRIL quite trivially as each voter is picked as the starting voter 1/v of the time for v voters.

It is also monotonic. For it to fail monotonicity, there would have to be a possible iteration of the lottery where A gets elected and where adding A to a ballot prevents this election. In such a case, this could only happen when this particular ballot is picked in the random process. For this ballot to get picked (without A on) and A to still get elected, none of the other remaining candidates could also be approved on that ballot, so it continues the process where A is eventually elected. Now imagine approving A on that ballot. In this scenario, it must result in the election of A. So a non-monotonic case is impossible.

It passes strong candidate Pareto efficiency. Any dominated candidate takes zero weight.

It passes IIB as picking a ballot that approves all or none of the candidates does nothing and another one is then picked.

So COWPEA passes the Holy Grail criteria for optimal methods!

COWPEA Lottery passes the same criteria and also passes IUAC as any universally approved candidates are elected to the first positions, as with Optimal PAV Lottery.

As far as I know COWPEA and COWPEA Lottery are the only known methods that pass the Holy Grail criteria.

Optimal PAV Lottery also requires complex calculations to be run. A COWPEA Lottery election can be run by just picking a few ballots at random. The overall weights do not need to be calculated. This could be useful for e.g. groups of friends picking an activity where proportionality over time can be achieved without anyone keeping count and also if not everyone is present each time.

Next I will make a couple of other comparisons between COWPEA and Optimal PAV, specifically relating to the consistency and multi-winner Pareto efficiency criteria. Optimal PAV passes these but COWPEA doesn't, but they are of debatable utility.