[EM] Optimal Cardinal Proportional Representation

[Toby Pereira]

I posted the below on the Voting Theory Forum, but thought it might be of interest to some people on this list as well. The link formatting won't work here in the same way, but URLs can simply be copied and pasted. It should still read OK, and I'd be more likely to make a mess of it by changing everything around.

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](https://rangevoting.org/QualityMulti.html), [two](https://www.rangevoting.org/NonlinQuality.html), [three](https://rangevoting.org/PRintLinprog.html) and [four](https://rangevoting.org/HolyGrailPR.html). 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](https://electowiki.org/wiki/Proportional_approval_voting) (PAV) and [Phragmén's Voting Rules](https://electowiki.org/wiki/Phragmen%27s_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](https://electowiki.org/wiki/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](https://electowiki.org/wiki/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 real-life 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](https://electowiki.org/wiki/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](https://electowiki.org/wiki/D%27Hondt_method) and [Sainte-Laguë](https://electowiki.org/wiki/Sainte-Lagu%C3%AB_method)), 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](https://www.votingtheory.org/forum/topic/379/cowpea-and-cowpea-lottery-paper-on-arxiv/2?_=1714856641427), which I'll reproduce in this post. There are 4 parties (A, B, C, D) and 1004 voters:
250: AC250: AD250: BC250: BD2: C2: D
According 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 2-dimensional 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 [KP-transformation](https://electowiki.org/wiki/Kotze-Pereira_transformation) keeps the Pareto dominance relations between candidates and allows the methods to pass the multiplicative and additive versions of [scale invariance](https://electowiki.org/wiki/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](https://arxiv.org/abs/2305.08857).
Toby
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20240504/943e1e97/attachment.htm>