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<p>"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.<br>
</p>
<p>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.</p>
<p>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.</p>
<p>Regards,</p>
<p>Richard Lung.</p>
<p><br>
</p>
<p><br>
</p>
<div class="moz-cite-prefix">On 04/05/2024 22:39, Toby Pereira
wrote:<br>
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<div dir="ltr" data-setdir="false">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.</div>
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<div>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](<a class="moz-txt-link-freetext" href="https://rangevoting.org/QualityMulti.html">https://rangevoting.org/QualityMulti.html</a>),
[two](<a class="moz-txt-link-freetext" href="https://www.rangevoting.org/NonlinQuality.html">https://www.rangevoting.org/NonlinQuality.html</a>),
[three](<a class="moz-txt-link-freetext" href="https://rangevoting.org/PRintLinprog.html">https://rangevoting.org/PRintLinprog.html</a>) and
[four](<a class="moz-txt-link-freetext" href="https://rangevoting.org/HolyGrailPR.html">https://rangevoting.org/HolyGrailPR.html</a>). Those
four pages are actually I, II, unnumbered and IV. I think
perhaps unnumbered should be III.</div>
<div><br>
</div>
<div>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](<a class="moz-txt-link-freetext" href="https://electowiki.org/wiki/Proportional_approval_voting">https://electowiki.org/wiki/Proportional_approval_voting</a>)
(PAV) and [Phragmén's Voting
Rules](<a class="moz-txt-link-freetext" href="https://electowiki.org/wiki/Phragmen%27s_voting_rules">https://electowiki.org/wiki/Phragmen%27s_voting_rules</a>).</div>
<div><br>
</div>
<div>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](<a class="moz-txt-link-freetext" href="https://electowiki.org/wiki/Universally_liked_candidate_criterion">https://electowiki.org/wiki/Universally_liked_candidate_criterion</a>),
(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.</div>
<div><br>
</div>
<div>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](<a class="moz-txt-link-freetext" href="https://electowiki.org/wiki/Independence_of_Irrelevant_Ballots">https://electowiki.org/wiki/Independence_of_Irrelevant_Ballots</a>))
- and there was never any guarantee that they could be
seamlessly combined.</div>
<div><br>
</div>
<div>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](<a class="moz-txt-link-freetext" href="https://electowiki.org/wiki/COWPEA">https://electowiki.org/wiki/COWPEA</a>). 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.</div>
<div><br>
</div>
<div>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](<a class="moz-txt-link-freetext" href="https://electowiki.org/wiki/D%27Hondt_method">https://electowiki.org/wiki/D%27Hondt_method</a>)
and
[Sainte-Laguë](<a class="moz-txt-link-freetext" href="https://electowiki.org/wiki/Sainte-Lagu%C3%AB_method">https://electowiki.org/wiki/Sainte-Lagu%C3%AB_method</a>)),
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).</div>
<div><br>
</div>
<div>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.</div>
<div><br>
</div>
<div>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](<a class="moz-txt-link-freetext" href="https://www.votingtheory.org/forum/topic/379/cowpea-and-cowpea-lottery-paper-on-arxiv/2?_=1714856641427">https://www.votingtheory.org/forum/topic/379/cowpea-and-cowpea-lottery-paper-on-arxiv/2?_=1714856641427</a>),
which I'll reproduce in this post. There are 4 parties (A,
B, C, D) and 1004 voters:</div>
<div><br>
</div>
<div>250: AC</div>
<div>250: AD</div>
<div>250: BC</div>
<div>250: BD</div>
<div>2: C</div>
<div>2: D</div>
<div><br>
</div>
<div>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.</div>
<div><br>
</div>
<div>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.</div>
<div><br>
</div>
<div>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.)</div>
<div><br>
</div>
<div>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](<a class="moz-txt-link-freetext" href="https://electowiki.org/wiki/Kotze-Pereira_transformation">https://electowiki.org/wiki/Kotze-Pereira_transformation</a>)
keeps the Pareto dominance relations between candidates
and allows the methods to pass the multiplicative and
additive versions of [scale
invariance](<a class="moz-txt-link-freetext" href="https://electowiki.org/wiki/Scale_invariance">https://electowiki.org/wiki/Scale_invariance</a>),
so my current thinking is that this is the optimal score
conversion.</div>
<div><br>
</div>
<div>I also discuss a lot of this in my [paper on
COWPEA](<a class="moz-txt-link-freetext" href="https://arxiv.org/abs/2305.08857">https://arxiv.org/abs/2305.08857</a>).</div>
</div>
<div><br>
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<div dir="ltr" data-setdir="false">Toby</div>
<br>
</div>
</div>
<br>
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