@toby-pereira said in General stuff about approval/cardinal PR:
This project hasn't purely been altruistic - it's been helpful to me by laying everything out for reworking my COWPEA paper!
And the new version can be seen here (as I mentioned in the separate COWPEA thread anyway).
The paper has been updated and some errors corrected.
I've just learnt from Rob Lanphier on the election methods mailing list of the death of Jameson Quinn after an accident while hiking in Guatemala. This is very sad news and a lot of you will be familiar with Jameson from his contributions to the voting method community over many years. I used to communicate with him about cardinal proportional methods and he's the only person from the community that I've ever met in person, back in 2017 I think.
Rob's original message can be seen here and the message from his mother posting the news is here.
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.
@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.
Right so moving on to finding the "Holy Grail". The criteria again:
Proportionality (so PRIL)
Monotonicity (ideally a stronger version that won't elect BC in the above example)
Independence of Irrelevant Ballots (IIB)
Independence of Universally Approved Candidates (IUAC)
On monotonicity, I would also add strong candidate Pareto efficiency as a strengthener.
Candidate A Pareto dominates candidate B if A is approved on all the ballots that B is and at least one other ballot. Passing strong candidate Pareto efficiency means that an election method must not elect a Pareto dominated candidate unless the candidate that dominates it is also elected. For an optimal method, the Pareto dominated candidate must not be elected with any weight. (Weak efficiency would allow for such a dominated set to be tied at the top.)
This criterion is being used as a proxy for a stronger form of monotonicity where adding an approval for a candidate should actively count in a candidate’s favour rather than merely not count against.
It is actually possible to pass strong candidate Pareto efficiency and fail monotonicity (as will be seen), but having both should protect against the "pathological" examples.
While normal PAV fails PRIL, Optimal PAV passes. The failure in the example a few posts above is avoided because the U candidates Pareto dominate the A and B ones, so you essentially end up with just a U and a C faction. And while this is just a simple example, it can be shown that Optimal PAV passes PRIL in general.
Optimal PAV looks like taking the clean sweep of Holy Grail criteria - except that it actually fails monotonicity, unlike normal PAV!
Optimal PAV is equivalent to the Nash Product Rule as I found out the other day. This is where you maximise the product of the voters' utilities. (By utility, I mean the number of approved candidates they have elected.) Intuitively this makes sense because product of utilities is the same as adding the logs, and the harmonic function of x converges to ln x + 0.577, and the 0.577 proportionally disappears as you get higher. A more formal proof is here. An example of monotonicity failure is here.
Anyway, Optimal PAV can be turned into Optimal PAV Lottery. This is where candidates are elected sequentially with probability equal to their weight in the PAV optimal committee. The weights are recalculated each time a candidate is elected and removed from the pool.
Unlike normal PAV, Optimal PAV Lottery passes IUAC because any universally approved candidates are elected to the first positions with probability 1, and then the rest is the same from there. But like Optimal PAV it fails monotonicity so cannot be deemed a Holy Grail method.
I'll do COWPEA in the next post.
OK, so the criteria. They're often called axioms in the literature, but I'm not sure it seems the right word.
The paper Multi-Winner Voting with Approval Preferences by Martin Lackner and Piotr Skowron is probably the best single resource on this, but there are lots of papers on the subject obviously. On page 56 it has a chart of some of the criteria and which imply which others. You'll notice that most of them imply lower quota satisfaction.
Lower quota says that a party must get at least its correct proportional number of seats rounded down. It sounds reasonable on the surface, but it disqualifies Sainte-Laguë party list (equivalently Webster apportionment), and any proportional approval method that reduces to it under party voting. The authors acknowledge the problem:
Most axiomatic notions for proportionality are only applicable to ABC rules that
extend apportionment methods satisfying lower quota (see Figure 4.1). This excludes, e.g., ABC rules that extend the Sainte-Lagu¨e method. As the Sainte-Lagu¨e
method is in certain aspects superior to the D’Hondt method (Balinski and Young
[2] discuss this in detail), it would be desirable to have notions of proportionality
that are agnostic to the underlying apportionment method.
As far as I'm concerned, the solution is not to demand specific proportionality requirements at exact points because different methods round in different ways, while still being proportional overall. Any specific requirements would likely be arbitrary and potentially disqualify reasonable methods (as seen) (edit - and also they can let in disproportional methods as PAV passes a lot of criteria). It is better instead to demand that a method becomes proportional in the limit as the number of candidates increases. And while this says nothing about the route to PR in the long haul, unless a method has been heavily contrived to do so in a "bad" way, it's likely to get there in a smooth, continuous and sensible way. (Similarly independence of clones only makes demands for candidates approved on exactly the same ballots, or ranked consecutively on all ballots, but it is generally accepted that a passing method will behave reasonably with near clones and it is a recognised criterion.)
And this takes us back to perfect representation. Closeness to it seems a good measure of PR, but as said above demanding it whenever it's possible seems too strong a requirement.
2 to elect
99 voters: AB
99 voters: AC
1 voter: B
1 voter: C
We would not want to be forced into electing BC. This is why I came up with the Perfect Representation In the Limit (PRIL) criterion. I also discuss all this in this thread. I got ChatGPT to come up with a formal definition that I put in there, but I have a better one now, though it doesn't copy and paste easily.
Also a deterministic method should pass perfect representation when the number of voters is the same as the number of elected candidates, as no rounding will be required. But PRIL is more general and works for non-deterministic methods, so it's my primary PR criterion.
So that's proportionality criteria covered. Next I want to talk about the "Holy Grail" criteria, which is something that has evolved out of internet discussions over the years. My version of the criteria are:
Proportionality (so PRIL)
Monotonicity (ideally a stronger version that won't elect BC in the above example)
Independence of Irrelevent Ballots (IIB)
Independence of Universally Approved Candidates (IUAC)
No known deterministic method passes all of these, so we have to look beyond the likes of PAV and Phragmén. So in the next post I will talk about non-deterministic methods and also about optimal methods that can elect any number of candidates with any weight. They may not be practical election methods, but they are of theoretical interest and are closely related to non-deterministic methods.
In an optimal method the "Holy Grail" criteria are the same except without IUAC. In a good optimal method, universally approved candidates should take all the weight, so there's nothing left to be independent of them. Also optimal methods are already "in the limit" so PRIL and perfect representation are the same thing.
Before I get onto PR criteria, I just want to mention quota-removal methods. As methods they are simple and intuitive to understand and can be of practical use, but I think they are less interesting from a mathematical and academic point of view. They are discussed in more detail in this thread.
In this context I will also mention the Method of Equal Shares (MES). I've seen people rave about how great this is. But as it's defined it's a method of participatory budgeting where different projects that people vote for can have different costs. There is also a set budget, rather than a set number of things to elect so in its default form, it isn't a standard proportional approval method.
Presumably to convert it, the cost for each candidate is set at the same price, which would be a quota (Droop, Hare etc.). This could still lead to not enough candidates being elected if not enough reach the quota, so something else would need to be added to the method.
In any case, it is just another quota removal method, so inherits all of their problems.
Onto Phragmén then. There are essentially two Phragmén methods - max-Phragmén and var-Phragmén. Max-Phragmén reduces to D'Hondt party list for party voting, and var-Phragmén reduces to Sainte-Laguë, so these correspond to the two different PAV versions discussed in the previous post. But despite this, PAV and Phragmén have very different PR philosophies. PAV is all about maximising approvals, although in a diminishing returns way, whereas Phragmén is all about balancing representation across voters with no particular concern for actual amount of support. Phragmén's monotonic properties are not as strong because of this (although it does technically pass the main criterion).
In Phragmén, each elected candidate has a "load" (sometimes called cost but whatever) of 1 that is spread out among the voters of that candidate. The loads does not have to be spread equally but is spread in the way that allows for the best set score.
For max-Phragmén, only the single voter with the highest load is considered. The set with the lowest single voter load is the winning set. For var-Phragmén, the variance of all the loads is calculated. The set with the lowest variance wins. I'll give a couple of examples.
2 to elect
1 voter: AB
1 voter: AC
Note that A is unanimously approved. If the load from a candidate had to be spread equally across its voters, this would cause a monotonicity failure. Take set AB. The AB voter would get a load of 0.5 from A and 1 from B, totalling 1.5. The AC voter would just get a load of 0.5 from A. So the two loads on the voters would be 1.5 and 0.5.
Now look at BC. The AB voter would get a load of 1 from B, and the AC voter would get a load of 1 from C. So the loads on both voters would be 1. This minimises the variance and the max load on a single voter. So BC would be elected. However, A is unanimously approved and would have fared better with just one approval, showing a monotonicity failure. One could argue that BC is more proportional in some sense, but I think most would agree that this would be taking it too far.
This is why loads are allowed to be spread unevenly. Under AB then, the entire A load can go to the AC voter with the B load going to the AB voter. Now both loads would be 1, minimising the max load and the variance.
But this still isn't a great victory for monotonicity. AB and AC only tie with BC. Similarly with 2 to elect:
1 voter ABC
1 voter: ABD
The unanimously approved AB is considered no better than CD.
Of course there could be some sort of tie-breaking mechanism, which would likely favour more approvals. But these are knife-edge results. Take the following example with 2 to elect:
99 voters: AB
99 voters: AC
1 voter: B
1 voter: C
Despite 99% of voters approving A, and just 50% approving B and C, BC would be elected under Phragmén. I would consider this to be an undesirable result.
When every voter has an equal load, this is known as "perfect representation". Methods that elect a candidate set that gives perfect representation whenever it is possible pass the perfect representation criterion. The Phragmén methods but not PAV pass this.
In this respect perfect representation can be seen as too strong a criterion to insist on a method passing. On the other hand, closeness to perfect representation seems a good measure of proportionality. So it's a question of getting the balance right between proportionality and support. I will be discussing different PR criteria in the next post, and which ones make most sense to use.
I'll also just point out here that two other similarish methods people talk about are Monroe (also known as Fully Proportional Representation), and Chamberlin-Courant. Under Monroe, each elected candidate is assigned an equal number of voters. The winning set is the one that allows the most voters to be assigned to a candidate that they approved. As I understand it, this behaves fairly similarly to Phragmén, except that for party voting it reduces to the Hamilton apportionment method. Essentially this is like Sainte-Laguë but with IIB failures.
Chamberlin-Courant elects the set that allows the most voters to have an approved candidate in the winning set. It makes no difference how many voters have approved each candidate. If there are 10 voters and 2 to elect, Chamberlin-Courant would not distinguish between a set where each candidate is approved by 5 distinct voters, and a set where the ratio is 9:1. So it too is similar to Phragmén and Monroe, but can give more skewed (less proportional) results.
Both Monroe and Chamberlin-Courant are discussed here. It is said that Chamberlin-Courant could give weighted voting within the committee to counteract the disproportionality, but at this point it is getting away from the usual requirements of a proportional approval method.
So this is why PAV and Phragmén are the big two out of these methods.
After a discussion with @cfrank I thought I'd start this thread to discuss some of the basic stuff and moving on to other stuff.
First of all, when people think about proportional representation, they would normally think about parties getting the right number of seats in proportion to the votes they get. But with approval or score voting, it's harder to define precisely. With ranked ballots there is STV, which makes no reference to parties, but does become party proportional if people simply vote along party lines. So approval/cardinal PR would be something similar.
It becomes even more complex when talking about scores, but there's a separate thread on that, so by outsourcing that part of the discussion to there, we can focus on the simpler approval case in this thread.
The two oldest methods people generally talk about are Thiele's Proportional Approval Voting (PAV), and Phragmén's voting rules. There is a paper by Svante Janson that describes both methods in detail in English.
There are different versions of these methods, but I will discuss the non-sequential versions, since the sequential versions are just approximations so that election results can be calculated efficiently (or at all). But mathematically, they are of less interest.
PAV is a generalisation of D'Hondt party list. Voters get a satisfaction score based on the number of elected candidates that they have approved. If they have approved j elected candidates, then their satisfaction score is 1 + 1/2 + 1/3 + ... + 1/j. The candidate set that gives the highest total satisfaction score is the winning set. Alternatively using 1 + 1/3 + 1/5 etc. would make the method a generalisation of the Sainte-Laguë party list method. If voters simply approved the candidates of one party, then PAV would reduce to D'Hondt (or Sainte-Laguë).
Because approving an extra elected candidate simply adds to a voter's satisfaction score, PAV has good monotonic properties. It also passes Independence of Irrelevant Ballots (IIB). A voter approving all or none of the candidates will make no difference to the outcome.
However, when voters do not vote purely along party lines, you can get some disproportional looking results.
20 to elect
2 voters: U1-U10; A1-A10
2 voters: U1-U10; B1-B10
1 voter: C1-C20
In this case, proportionally the UA and UB factions should have 16 elected candidates between them, and the C faction should have 4 candidates. The U candidates clearly give a higher satisfaction score than either the A or B candidates because of the Pareto dominance, so would be elected in preference to them. The winning candidate set should therefore be U1-U10, A1-A3, B1-B3, C1-C4 (assuming priority for lower numbers, without loss of generality). However, under PAV, U1-U10, A1-A2, B1-B2, C1-C6 gives a higher satisfaction score. Under this result, the 4 members of the UA and UB factions only get 14 candidates, whereas the 1 member of the C faction gets 6.
At an intuitive level, this is because the PAV-preferred result of U1-U10, A1-A2, B1-B2, C1-C6 gives the UA voters 12 elected candidates, UB voters also 12, and the C voter 6, which fits in with the faction sizes, not taking into account the overlap. The proportional result of U1-U10, A1-A3, B1-B3, C1-C4 awards 13, 13, 4, which looks disproportional to PAV. This result is not because a lack of seats has caused a rounding error. This is essentially PAV’s version of proportionality. Overlapping factions count against the voters in these factions under PAV.
This highlights the need for a specific definition of PR for approval methods. Also Phragmén's voting rules does not have this problem. And I will discuss this in the next post...
Also I've been doing some work on my COWPEA paper to see if I can get it to publication standard. So hopefully that will happen at some point.
One other post I made on Reddit is basically the topic of this thread. Again, some of the words are links in the original, so I recommend going to that for more info.
Optimal cardinal proportional representation and the "Holy Grail"
By optimal cardinal PR, I mean you remove the restriction of having to elect a fixed number of candidates with equal weight, but can elect any number with any weight. So this is a theoretical thing rather than about coming up with a practical method for use.
But by "Holy Grail", I mean a cardinal method that does elect a fixed number of candidates with equal weight (the usual requirement) and passes certain criteria. So this could be potentially used.
Although this is about cardinal PR, I will make it simpler by talking about approval methods, since I've previously argued for the KP-transformation as the best way to convert scores into approvals.
First of all optimal cardinal PR. It would need a strong form of monotonicity not present in Phragmén-based methods, which would be indifferent between the infinite number of results giving Perfect Representation. To cut a long story short, there are two candidate methods that are proportional, strongly monotonic and pass Independence of Irrelevant Ballots (IIB). They are the optimal version of Thiele's Proportional Approval Voting (Optimal PAV), and COWPEA.
To work out an Optimal PAV result (or an approximation to it), you increase the number of seats to some large number and, allowing unlimited clones, see what proportion of the seats each candidate takes. That proportion would be each candidate's weight in the elected committee. This method would be beyond calculation but exists as a theoretically nice method. If you elect using PAV sequentially it doesn't always give a good approximation, as I think it's possible to end up giving weight to candidates that would actually receive no weight under Optimal PAV, since I think it's possible for Optimal PAV to give zero weight to the most approved candidate. E.g.
150: AC
100: AD
140: BC
110: BD
1: A
1: B
If I've worked it out right, Optimal PAV would give A and B half the weight each, and C and D no weight. This is despite the fact that C has the most votes at 290 (A and B each have 251; D has 210).
COWPEA elects candidates proportionally according to the probability 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.
Because each voter would be the first ballot picked in the same proportion (1/v for v voters), each voter is guaranteed 1/v of the elected body. But where a voter approves multiple candidates, these candidates are then elected proportionally in the same manner according to the rest of the electorate. COWPEA is also beyond calculation for real elections, but can be approximated with repeated iterations of the algorithm.
Both Optimal PAV and COWPEA have the properties that makes them contenders for the optimal approval method, and ultimately it's likely a matter of preference rather than one having objectively the best properties. I compare them both in my non-peer-reviewed COWPEA paper here if you're interested. The current version is not set in stone, and I might tighten certain things up further at some point. But just to give an example of where they differ:
100: AC
100: AD
100: BC
100: BD
1: A
1: B
COWPEA would elect the candidates in roughly equal proportions (with A and B getting slightly more). Optimal PAV would only elect A and B and with half the weight each. 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. COWPEA makes more use of the voting space in this sense, whereas Optimal PAV only looks at voter satisfaction as measured by number of elected candidates, and every voter is either indifferent between AB and CD or prefers AB. This is also why the most approved candidates in the previous example gets no weight under Optimal PAV.
Without the extra two voters that approve just A and B respectively, COWPEA would elect all four equally. Optimal PAV would be indifferent between any AB to CD ratio as long as A and B are equal to each other and so are C and D.
Finally, onto the Holy Grail where a fixed number of candidates with equal weight are required. Where unlimited clones are allowed, PAV passes all the criteria, but is not fully proportional where there aren't such clones as I discussed here.
So we need the method to be proportional, strongly monotonic, pass IIB and ideally also Independence of Universally Approved Candidates (IUAC). As far as I'm aware, no known deterministic method passes all of these, but if it doesn't have to be deterministic, then two methods do. And they are versions of the methods above. Optimal PAV Lottery and COWPEA Lottery.
Under Optimal PAV Lottery, the Optimal PAV weights are used as probabilities, but these would need to be recalculated every time a candidate is elected and removed from the pool. This method is clearly not possible to calculate in practice.
COWPEA Lottery is just the lottery used in the COWPEA algorithm. This is easily runnable. And while this may be unrealistic for elections to public office, it can certainly have more informal uses. E.g. friends can use it to determine activities so that choices proportionally reflect the views of the group over time without anyone having to keep count or worrying what to do if not exactly the same people are present each time.
In conclusion, the main contenders for optimal cardinal proportional representation are Optimal PAV Lottery and COWPEA. For the Holy Grail, we have PAV where unlimited clones are allowed, but otherwise Optimal PAV Lottery or COWPEA Lottery, of which only COWPEA Lottery can be reasonably computed.