[EM] Optimal Cardinal Proportional Representation

[Richard Lung]

"The hunt for the "Holy Grail" of cardinal PR has been long and 
arduous." And it will no doubt remain so, for any seeking it.

I take it  you mean PR wth a cardinal number vote, as it already 
involves the usual cardinal numbers in the count: "Dealing purely with 
approval voting to start with..." a new name for cumulative voting, that 
slightly favorable system compared to FPTP, that has not gained any 
favor except as an intellectual buffer for an American Political Science 
association. To me that simply robs the voters of electing their most 
prefered candidates in their order of choice, as democracy requires.

And representative democracy (which includes self-representation as a 
special case) elects representatives and not merely "parties." (Below 
"There are 4 parties..."). Transferable voting can express unity as well 
as division, unlike the immemorial petty tribalism of party elections 
more properly called partitions rather than elections.

Regards,

Richard Lung.



On 04/05/2024 22:39, Toby Pereira wrote:
> 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: AC
> 250: AD
> 250: BC
> 250: BD
> 2: C
> 2: 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
>
>
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