Optimal cardinal proportional representation

The hunt for the "Holy Grail" of cardinal PR has been long and arduous. This isn't about practical use specifically (although it could double up), but about finding a theoretical method that obeys all the right mathematical criteria so to be deemed the purest of all PR methods (one can obviously debate which criteria are the right ones and indeed whether the entire premise of this is sound). There are, as far as I can see, four pages on Warren Smith's Range Voting website dedicated this this question  one, two, three and four. Those four pages are actually I, II, unnumbered and IV. I think perhaps unnumbered should be III.
Dealing purely with approval voting to start with (I will discuss the score conversion at the end), perhaps the best known two methods that use an optimising function are Thiele's Proportional Approval Voting (PAV) and Phragmén's Voting Rules.
PAV has a very strong form of monotonicity, but there examples where it can fail basic PR, related to its failure of the Universally Liked Candidate criterion, (ULC) which disqualify it from being the Holy Grail. Phragmén, on the other hand, only looks at proportionality and ends up with only a weak form of monotonicity, making it not Holy Grail material either.
The problem is that there are essentially two orthogonal goals for a method  maximising proportionality and also being properly monotonic (as well and passing things like Independence of Irrelevant Ballots)  and there was never any guarantee that they could be seamlessly combined.
However, truly optimal PR (with no limitations related to being usable in reallife elections) is not limited to electing candidates/parties with a fixed weight. If we are allowed to elect the candidates or parties in any proportion we like, then things change and suddenly two methods emerge as viable candidates. They are PAV (again) and COWPEA. To work out the PAV result without fixed weights, you find the seat proportions in the limit as you increase the number of seats to infinity, allowing candidates to be elected multiple times.
With fixed candidate weights removed, PAV's ULC failure simply disappears (because universally liked candidates automatically take all the seats). And because its PR failure is related to its ULC failure, it is possible that PAV becomes properly proportional again. As far as I understand, this is unproven, but it hasn't failed in any of the cases I have thrown at it. It is also worth noting that PAV can use different divisors (e.g. D'Hondt and SainteLaguë), but with optimal weighting allowed and no rounding required, my hypothesis is that they end up with the same results (the examples I have tried do not contradict this).
COWPEA is more transparently proportional, and has just one definitive version, and also has the same strongly monotonic properties that PAV has. Both methods also pass IIB.
PAV and COWPEA do have slightly different philosophies and so can give different results. PAV is purely welfarist in that it looks only at the number of candidates each voter has elected, whereas COWPEA's proportionality puts more of an emphasis on using the whole voter/candidate space. I give an example here, which I'll reproduce in this post. There are 4 parties (A, B, C, D) and 1004 voters:
250: AC
250: AD
250: BC
250: BD
2: C
2: DAccording to PAV's welfarist philosophy, the voters are better off with C and D getting 50% of the weight each, with none for A or B.
However, this can be seen as a 2dimensional voting space with an AB axis and a CD axis. PAV does not use the AB axis at all. COWPEA, on the other hand will make use of this part of the voting space and elect A and B with slightly less than 0.25 of the weight each, with C and D getting slightly more than 0.25 of the weight each.
At this point, it arguably becomes a matter of preference. So from not being able to find the Holy Grail of PR at all, we suddenly find ourselves with two candidates for it  an embarrassment of riches! (Assuming that PAV is ultimately found to be fully proportional of course.)
I have only dealt with the approval case so far, so to finish off I will briefly mention the score conversion. There are several possible methods of converting an approval method to a score method, but the KPtransformation keeps the Pareto dominance relations between candidates and allows the methods to pass the multiplicative and additive versions of scale invariance, so my current thinking is that this is the optimal score conversion.
I also discuss a lot of this in my paper on COWPEA.

Thank you so much for this post! It's great
@tobypereira said in Optimal cardinal proportional representation:
There are several possible methods of converting an approval method to a score method, but the KPtransformation keeps the Pareto dominance relations between candidates and allows the methods to pass the multiplicative and additive versions of scale invariance, so my current thinking is that this is the optimal score conversion.
I'm not 100% sure about this myself—won't any transformation of the ballots discard some information? I'm not sure if applying the KP transform to range retains the coreapproximation properties that make PAV so appealing (i.e. 2approximation of the core, and satisfying core with enough similar candidates).

@lime said
I'm not 100% sure about this myself—won't any transformation of the ballots discard some information? I'm not sure if applying the KP transform to range retains the coreapproximation properties that make PAV so appealing (i.e. 2approximation of the core, and satisfying core with enough similar candidates).
Can you remind me exactly what these mean?
And I'm glad you liked the post!

@tobypereira said in Optimal cardinal proportional representation:
@lime said
I'm not 100% sure about this myself—won't any transformation of the ballots discard some information? I'm not sure if applying the KP transform to range retains the coreapproximation properties that make PAV so appealing (i.e. 2approximation of the core, and satisfying core with enough similar candidates).
Can you remind me exactly what these mean?
And I'm glad you liked the post!
Kapproximation means K Hare quotas would prefer another committee (instead of just 1).
And if every candidate has infinitely many clones, you can guarantee the method will choose from the core (see here).
I'm wondering if some method can keep these properties in the score voting case.

@lime OK, I'm not sure how the KPtransformation would affect these things. Do you specifically think it's likely to be any worse than any other transformation, or is it general concerns about any transformation that hasn't been demonstrated to pass these things?
In any case, I definitely think PAV + KP is better than RRV or SDV because of its scale invariance, and I don't see any particular advantages of these methods over it.
There is also the probabilistic transformation, which I see as inferior to KP as well. Someone might give scores of 9 and 1 (out of 10) to 2 candidates A and B respectively. KP would split the voter as follows:
0.1: 
0.8: A
0.1: ABThe probabilistic transformation would give:
0.09: 
0.81: A
0.09: AB
0.01: BThis wrecks both Pareto dominance and scale invariance.
By the way, COWPEA fails the multiwinner Pareto criterion in the example I gave above, so might have core failings as well. Certainly in the IIB version of core (where you ignore voters who are indifferent between competing sets and just look at the proportion who favour each one of those who have a preference), it would fail. But I don't see this as a failing of COWPEA, just a different PR philosophy.
By the way, since PAV with infinite clones passes core (which it doesn't with a limited number of candidates), I presume the optimal version probably is properly proportional (passes perfect representation). I might update my paper with this in at some point.
Also, in the optimal scenario, both Phragmén and Monroe would be unsuitable as contenders. Both would be indifferent between an infinite number of different candidate proportions. They are concerned only with perfect representation, and this is very easy to achieve in the optimal case with any proportions allowed, and they have nothing to say to distinguish between them. Monroe is also essentially the Hamilton version of Phragmén (which can be made D'Hondt or SainteLaguë), so essentially the same but with more IIB failures.

@tobypereira said in Optimal cardinal proportional representation:
OK, I'm not sure how the KPtransformation would affect these things. Do you specifically think it's likely to be any worse than any other transformation, or is it general concerns about any transformation that hasn't been demonstrated to pass these things?
General concern about transformations. My worry is transforming score ballots to approval ballots discards information about which voter gave which ratings, so I'm not sure it will preserve the stable winner set properties of PAV.
@tobypereira said in Optimal cardinal proportional representation:
In any case, I definitely think PAV + KP is better than RRV or SDV because of its scale invariance, and I don't see any particular advantages of these methods over it.
Oh, definitely, harmonic voting is great! (Although it needs a less intimidating name.)
@tobypereira said in Optimal cardinal proportional representation:
By the way, COWPEA fails the multiwinner Pareto criterion in the example I gave above, so might have core failings as well. Certainly in the IIB version of core (where you ignore voters who are indifferent between competing sets and just look at the proportion who favour each one of those who have a preference), it would fail. But I don't see this as a failing of COWPEA, just a different PR philosophy.
This is a much bigger hangup for me personally. If everyone agrees a different committee would be better, then levelingdown (making some people worseoff, just to make the outcome more equal/proportional) strikes me as wrong.

@lime said in Optimal cardinal proportional representation:
@tobypereira said in Optimal cardinal proportional representation:
By the way, COWPEA fails the multiwinner Pareto criterion in the example I gave above, so might have core failings as well. Certainly in the IIB version of core (where you ignore voters who are indifferent between competing sets and just look at the proportion who favour each one of those who have a preference), it would fail. But I don't see this as a failing of COWPEA, just a different PR philosophy.
This is a much bigger hangup for me personally. If everyone agrees a different committee would be better, then levelingdown (making some people worseoff, just to make the outcome more equal/proportional) strikes me as wrong.
Right, but it's debatable whether a voter's utility is purely determined by approved candidates elected. A voter is better represented in parliament if they share their representative with fewer other voters. So in the example up the thread:
250: AC
250: AD
250: BC
250: BD
2: C
2: DA and B are more attractive options to most voters.

@lime Who would you elect in the following election with 2 to elect:
150: AC
100: AD
140: BC
110: BD
1: C
1: DEdit  I can give a more extreme example:
99: AC
51: AD
99: BC
51: BD
1: C
1: D 
@tobypereira said in Optimal cardinal proportional representation:
99: AC
51: AD
99: BC
51: BD
1: C
1: DI'm not really seeing what the problem with electing C & D here is supposed to be It seems like a gain for only 2 voters, so I might be missing something, but I'm not seeing what would make that bad.
@tobypereira said in Optimal cardinal proportional representation:
Right, but it's debatable whether a voter's utility is purely determined by approved candidates elected.
I'll briefly set aside the "approved" part and focus on score voting more broadly (since voters rarely have dichotomous preferences).
I'm not sure why the distribution of like this would particularly matter. The way I'd model is that each candidate is assigned a utility equal to the (importanceweighted) probability that they'll break a tied vote in my favor. I'm not sure why it would be better for me to have a legislator who casts votes that represent my interests less often, or why it would be better for me to have a legislator supported by fewer voters.

@lime In the example I gave, electing AB would mean that 300 of the 302 voters would have approved exactly one elected candidate, whereas electing CD would mean that all 302 would have approved exactly one elected candidate. So by that measure, CD would be better.
But  under AB, 150 people have approved A and 150 have approved B. Under CD, 199 have approved C and 103 have approved D. So CD is a disproportional result in that the 103 D voters wield a disproportionate amount of power in parliament. Or perhaps more relevantly, the D party has only about 1/3 of the support but half the power. AB would be more balanced in that respect. Methods that use a measure of proportionality rather than satisfaction (e.g. Phragmen) would tend to elect AB.
This doesn't matter in the purely optimal case, because PAV would elect CD but in the respective proportions. COWPEA would elect all four in varying proportions. If these were reallife votes, it would be likely that AB and CD focus on different issues. A and B are opposed on the issues that they focus on. C and D are opposed on the issues they focus on. By electing all four, COWPEA would be making sure that the issue space is better covered.
COWPEA isn't really a voting method as such though (it's more of a theoretical thing), but COWPEA Lottery could be used as a method. Optimal PAV Lottery would be computationally too hard to be a method I think, although theoretically interesting.

@tobypereira said in Optimal cardinal proportional representation:
But  under AB, 150 people have approved A and 150 have approved B. Under CD, 199 have approved C and 103 have approved D. So CD is a disproportional result in that the 103 D voters wield a disproportionate amount of power in parliament. Or perhaps more relevantly, the D party has only about 1/3 of the support but half the power. AB would be more balanced in that respect. Methods that use a measure of proportionality rather than satisfaction (e.g. Phragmen) would tend to elect AB.
Right. I suppose that's what I meant by disliking the idea of making an underrepresented group worseoff just to make the overrepresented one even worseoff.
@tobypereira said in Optimal cardinal proportional representation:
COWPEA isn't really a voting method as such though (it's more of a theoretical thing), but COWPEA Lottery could be used as a method. Optimal PAV Lottery would be computationally too hard to be a method I think, although theoretically interesting.
That's surprising. I know there are local councils and similar that use weighted votes, but I can't imagine any legislature or council (especially a small one) using a random method.

This post is deleted! 
@lime said in Optimal cardinal proportional representation:
@tobypereira said in Optimal cardinal proportional representation:
COWPEA isn't really a voting method as such though (it's more of a theoretical thing), but COWPEA Lottery could be used as a method. Optimal PAV Lottery would be computationally too hard to be a method I think, although theoretically interesting.
That's surprising. I know there are local councils and similar that use weighted votes, but I can't imagine any legislature or council (especially a small one) using a random method.
Which bit is surprising? I'm only saying that COWPEA Lottery could be used (i.e. there would be no computational problems)  not that it's likely to be. In any case, from my point of view, I don't have a problem with nondeterministic methods in some situations.

I think that a simple model of perfectly proportional representation is to make a network flow problem and just simply set each voter's flow to be equal.
In this network flow model, a cardinal ballot would be represented as capacity for flow between a voter and each candidate. I'll put a link here to wikipedia just for general reference: https://en.wikipedia.org/wiki/Network_flow_problem
How is it possible to have each voter's flow equal? I am thinking there would have to be a change to the rules for the elected body. We might have to allow winners to have different weights. We might have to allow any amount of winners. The benefit is we get perfectly proportional representation.

@paretoman said in Optimal cardinal proportional representation:
I think that a simple model of perfectly proportional representation is to make a network flow problem and just simply set each voter's flow to be equal.
Possibly.
Personally, I'm wondering whether we could figure out optimal representation from a model that pins down what, precisely, the utility of a given committee for each voter is, under a particular definition of what the scores are supposed to mean.
One possible model is one where the score of each candidate is the probability they'll agree with the voter on some vote, and the objective would be to choose the committee that maximizes the probability of a majority vote on any given issue agreeing with a majority vote by the whole population.

@lime said in Optimal cardinal proportional representation:
One possible model is one where the score of each candidate is the probability they'll agree with the voter on some vote, and the objective would be to choose the committee that maximizes the probability of a majority vote on any given issue agreeing with a majority vote by the whole population.
(Oh, quick note @TobyPereira : I think this is a good example of why perfect location or scale invariance may not actually be ideal. If you rescale every voter's ballot to fall between 50% and 100% instead of 0100%, that might indicate a meaningfully different situation.)

@lime said in Optimal cardinal proportional representation:
@lime said in Optimal cardinal proportional representation:
One possible model is one where the score of each candidate is the probability they'll agree with the voter on some vote, and the objective would be to choose the committee that maximizes the probability of a majority vote on any given issue agreeing with a majority vote by the whole population.
(Oh, quick note @TobyPereira : I think this is a good example of why perfect location or scale invariance may not actually be ideal. If you rescale every voter's ballot to fall between 50% and 100% instead of 0100%, that might indicate a meaningfully different situation.)
Yes, that is a way that someone could vote. However, I wouldn't see it as some sort of objective standard, so would still see scale invariance as desirable. I think you mentioned this point previously in a discussion about normalising ballots, and I didn't get round to responding at the time.
The method being discussed renormalises using ratios when candidates are eliminated, so if someone gives scores of 1 and 0 (out of e.g. 5) to two candidates, this could later become 5 and 0 after some elimination, whereas scores of 5 and 1 would be locked in as that. However, even if it is assumed that people vote in the manner you described, I still don't think this is good voting method behaviour. If I give scores of 5, 1 and 0, I would still prefer the 5 to the 1 by more than I prefer the 1 to the 0, and I think voters generally would not be happy that they lose some of the normalisation power in the 51 situation but not in the 10 situation.

@Lime As mentioned upthread, there is the probabilistic transformation as well, which you might prefer from the point of view being discussed. I'm not a big fan of it though.
@tobypereira said in Optimal cardinal proportional representation:
There is also the probabilistic transformation, which I see as inferior to KP as well. Someone might give scores of 9 and 1 (out of 10) to 2 candidates A and B respectively. KP would split the voter as follows:
0.1: 
0.8: A
0.1: ABThe probabilistic transformation would give:
0.09: 
0.81: A
0.09: AB
0.01: B 
@tobypereira said in Optimal cardinal proportional representation:
By the way, since PAV with infinite clones passes core (which it doesn't with a limited number of candidates), I presume the optimal version probably is properly proportional (passes perfect representation). I might update my paper with this in at some point.
I have updated the paper to mention the proportionality of Optimal PAV (with variable candidate weight allowed), which allows for a proper comparison with COWPEA  these two methods being the main candidates for a truly optimal cardinal PR method (practicalities aside).