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@cfrank Sorry, I was wrong about forcing a party out. If everyone ganged up on another party so that party had zero support from everyone else it would destroy the other parties too, as it's their minimum that counts.

I think something like approval voting would make sense for this. FPTP would obviously split votes and ruin quotas. With ranks, it's unclear what would count as support.

@cfrank This is interesting, but it does also seem rather complex, so I'm glad you put up the pictures! It also seems as though all the other parties could gang up on one party to force them out, for example.

Also, I'm not sure how the voting would work with this and therefore what reaching the support quota would entail.

Under Independence of Universally Approved Candidates (IUAC) (see also Universally Liked Candidate Criterion, Strong PR), adding in some universally approved candidates and extra seats to accommodate them should not change the outcome for the other candidates. E.g. with 6 to elect:

2: U1-U3, A1-A3
1: U1-U3, B1-B3

The result should be U1-U3, A1-A2, B1. However, Thiele's PAV elects U1-U3, A1-A3.

Fully optimal methods like Optimal PAV and COWPEA (which allow candidates to be elected in any proportion) completely bypass the IUAC criterion by electing only the universally approved candidates. The lottery versions of these methods also pass by electing the universally approved candidates to the first positions and then electing proportionally from there.

However, one "normal" method (fixed number of candidates elected with equal weight) that does pass is the specific version of var-Phragmén where a candidate's load must be spread equally across its voters. I've always called this Ebert's Method, but I'm not sure he actually intended it to mean this. Anyway, this method is poor overall as it is non-monotonic and can give disproportional results, but I think it does give good results with certain toy examples such as the one above and can be enlightening.

Anyway, while COWPEA and Optimal PAV don't have to deal with IUAC, I did wonder what might happen if a certain group of voters did all approve a particular candidate but not the entire electorate. Would these methods pass this lesser form of IUAC? E.g.

1: U
2: A
1: B
4: UA
2: UB

The extra single U voter at the top is to stop a method from only electing A and B, and forcing it to interact with all 3 candidates. So should the A:B ratio be 2:1 here? Why? Why not? What do each of our methods do? First of all PAV's ratios:

A: 0.376697
B: 0.147247
U: 0.476056

This gives an A:B ratio of 2.558:1

COWPEA:

A: 27/75 (0.36)
B: 10/75 (0.133)
U: 38/75 (0.507)

This gives an A:B ratio of 27:10 or 2.7:1. This is even harsher than Optimal PAV.

Finally Ebert's Method:

A: 1/2
B: 1/4
U: 1/4

This gives an A:B ratio of 2:1, which is what might seem intuitively correct.

So are Optimal PAV and COWPEA "wrong" here? My thinking of late has been that these two are the ultimate in optimal methods (while differing slightly philosophically with each other) and very trustworthy, so I'd be surprised if these results are indeed "wrong" in some way. So why these results?

I wonder if it's to do with monotonicity. The result from Ebert's Method gives U a very low ratio compared to the others despite it being the most approved candidate. Obviously it would be possible to make this higher while keeping the A:B ratio the same, but there might be other reasons not to do this. Calculating the total approval rating of each slate of candidates under each result, you get:

PAV: 6.034
COWPEA: 6.107
Ebert: 5.5

So the total score from the Ebert result is just 5.5, quite a bit lower than the others with COWPEA coming out on top. I think this has been quite an interesting experiment.

This has been discussed on this forum before and I made an argument for using the KP transformation here and here, so that you can convert the scores to approvals and then just run an approval method. But I recently made a Reddit post, and I think it's worth reproducing here as well. So here it is:

There are various proportional methods that use approval voting and they can be turned into more general cardinal methods by allowing scores or stars instead of a simple yes/no. But as well as all the different approval methods, there are different ways to convert these methods into score voting methods, so you can end up with a proliferation of possible methods with these two essentially independent choices you have to make (which approval method, how to deal with scores).

First of all, I should say that I'm talking about methods that use the actual values of the scores, not where scores are used as a proxy for ranks.

For example, you have methods like Allocated Score, Sequential Monroe and Sequentially Spent Score. As far as I understand, if everyone voted approval-style (so only max or min scores), these methods would all be essentially the same. The highest scoring candidate is elected, and a quota of votes is removed, as so on.

All of these methods are actually quite messy, not to mention arbitrary, and you can end up with a lot of discontinuities and edge cases when you make small changes in the vote. Scores are an inconvenience in this sense (which is why all these similar but different methods were invented) and it would be much better if you could just make them behave more predictably and continuously from the start, so you can then just apply your favourite approval method knowing things will run smoothly.

And the way to do this? Well, as far as I'm concerned, it's the KP transformation. It turns the score ballots into approval ballots in a consistent manner, so you then only have to worry about what approval method you want to use. For e.g. scores out of 5, this essentially splits each ballot into 5 parts with their own approval threshold for each candidate. The "top" part will only approve those given 5, the next part will approve those given 4 and 5, and so on. The highest scoring candidate overall automatically becomes the most approved candidate, and so on. The total scores are proportional to the total approvals they've been converted to.

This makes methods far more continuous than the above ad hoc score conversions, so the weird discontinuities they cause will go away.

The KP transformation has nice properties. For example, for an approval method that passes Independence of Irrelevant Ballots, the KP transformed method will pass multiplicative and additive scale invariance. That means that if you multiply the scores on all ballots by a constant, or add a constant, or both, the result will still be the same. So you could multiply the scores by 7 and add 3. It would not affect the result.

Taking Thiele's Proportional Approval Voting as an example, Reweighted Range Voting and Single Distributed Vote are both conversions that cause a failure in one or both forms of scale invariance. However, Harmonic Voting, or its sequential variant, which both use the KP transformation, pass.

Also, this means that electing two candidates that a voter has given a 2 and a 3 respectively is not the same as a single 5 (and 0 for any others). But I see this as a feature, not a bug. It means that someone's ballot will never be "used up" by candidates they don't give their full support to. With scores out of 5, electing candidates a voter gives 3 or less to means that 2/5 of their vote will be completely protected until a 4 or 5 is elected.

@cfrank Yeah, there's always the incentive not to vote for a preferred candidate in PR for the reasons you give.

With the monotonicity, Phragmén is not actually non-monotonic as I say, but only weakly monotonic. E.g. with 2 to elect:

100 voters: ABC
100 voters: ABD

Phragmén methods tend to be indifferent between AB and CD (unless modified in some way), and that leaves them open to:

99 voters: ABC
99 voters: ABD
1 voter: C
1 voter: D

Where CD would be preferred. Electing in a greedy sequential manner rather than optimally would actually lead to more monotonic results in these cases, but other examples can be found.

@cfrank I also like the concept of the Phragmén method to a point, but because it only cares about proportionality, it's not as monotonic as other methods such as Thiele's PAV method. I mean, it passes in that an extra approval can't count against a candidate, but sometimes it makes no difference, so I call it weakly monotonic.

By graph-based do you mean like Schulze for single winner? There is also a Schulze STV method, which reduces to Schulze in the single-winner case.

I think talking about Phragmén loads is over-complicating the matter. We could just say that for a method to pass (in the deterministic case), then as long as it's possible from the ballot profiles (each voter has approved enough candidates), then under any result produced by the method, it must be possible to uniquely assign each elected candidate to a voter who approved them in a way that means no voter has ≥2 candidates more than another.

For non-deterministic methods we could still talk in terms of whole candidates rather than loads and just say that the variance / total elected candidates must approach zero as the number of elected candidates increases.

@cfrank said in Proportionality criteria for approval methods:

@toby-pereira I didn’t get to check this in detail but I find it interesting. It also makes sense, although I wonder if it should be stated a bit more generally, saying maybe that there is some fixed positive constant C such that for all epsilon>0, there is some k such that for all k’>=k, C*Var(l)/k’<epsilon.

Is there a reason for choosing the normalized variance Var(l)/k’ rather than the normalized standard deviation sqrt(Var(l))/k’? Or even expressing in terms of sqrt(Var(l))/E(l)?

I could have made it specify the proportionality level more specifically, but left it more open so that it doesn't throw out non-deterministic methods such as COWPEA Lottery.

But in terms of maximum allowable load variance for a deterministic method, I think we would look at the worst case scenario of every voter approving a completely different set of candidates. We would then sequentially award a candidate to each voter until they all have one, and then start the process again. As I understand it, the variance would be highest when half the voters have an "extra" candidate and half don't.

As every candidate adds a "load" of 1, at this point every voter would have a load that is 0.5 away from the mean. So the variance would be 0.25. So instead of talking about tending towards zero, we could just say that to pass PRIL it must satisfy: var (ℓ) ≤ 0.25 (or 0.5 if we use the standard deviation instead).

I could have used other measures such as the standard deviation. I mainly used variance because of the existence of var-Phragmén. I could also have put it in terms of the leximax-Phragmén metric or Monroe.

Of course, as discussed, this doesn't define anything about the route to Perfect Representation. If there are 50 voters of party A and 50 of party B, it would allow the first 50 candidates to all go to the same party. But as also discussed, unless a method has been heavily contrived, this won't happen, and it's similar to Independence of Clones in this way.

I've been on the EndFPTP Reddit posting on this subject and I got ChatGPT to formalise the PRIL criterion. I'll copy my post from there wholesale:

On the formalisation of PRIL, I put to ChatGPT my idea of using the var-Phragmén metric and the concept of loads and told it to put the criterion in formal notation. PRIL is method agnostic though and could be used in conjunction with other methods that aim for Perfect Representation like leximax-Phragmén or Monroe. Anyway, here is what ChatGPT came up with. I make no guarantees an error hasn't slipped in:

Formal Definition of PRIL Using var-Phragmén:
Consider an election with: A set of voters N = {1, 2, . . . , n}.
A set of candidates C = {c1, c2, . . . , cm}.
Each voter i ∈ N has an approval ballot Ai ⊆ C.
A target number of winners k.
In the context of the var-Phragmén method:
Each voter i is assigned a "load" ℓi representing their share in the election of the chosen committee.

The PRIL criterion can be formalized as follows:
For any arbitrarily small positive number ϵ>0, there exists a number of winners k such that for all k′ ≥ k, the normalized variance of the voter loads satisfies:
Var (ℓ) / k′ < ϵ

This condition ensures that as the number of elected candidates k′ increases, the distribution of voter loads becomes increasingly uniform, approaching perfect representation in the limit. In essence, PRIL requires that for sufficiently large committees, the method should allocate representation so evenly among voters that the per-candidate variance of their loads becomes arbitrarily small, reflecting an ideal proportional representation.

@abel I suppose you could see the primary system as a bit of a gamble in this sense. You want to beat the other party, but you also want to do so with a candidate that is as popular as possible within the party. If your party is favourite to win anyway, you might be more inclined to just pick the most popular candidate within the party. If it's a closer election you might look at who is most likely to beat the other party. So there would be some balancing.