Optimal Cardinal Proportional Representation
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, 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 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. 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 Sainte-Laguë), 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: 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 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.
Toby
"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,
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 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. 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
Sainte-Laguë),
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: 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
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.
Toby
Election-Methods mailing list - seehttps://electorama.com/em for list info
On 2024-05-05 09:32, Richard Lung wrote:
"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.
How would you define cumulative voting? It seems like you're using a
different definition than is common on the list.
-km
As I understand it (or misunderstand it) cumulative voting is an early
version of cardinal voting going by quantities of points or "plumping"
for a single candidate. Of course the details are different but they
share the similarity of cardinal voting obscuring an ordinal vote, a
straight-forward order of choice, in the hands of the voters, instead of
the parties. ultimately reducing to FPTP under the plumping option.
Or as HG Wells would say: enfeebling complications fruitful of corruption.
On 05/05/2024 12:25, Kristofer Munsterhjelm wrote:
On 2024-05-05 09:32, Richard Lung wrote:
"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.
How would you define cumulative voting? It seems like you're using a
different definition than is common on the list.
-km
On 2024-05-05 17:22, Richard Lung wrote:
As I understand it (or misunderstand it) cumulative voting is an early
version of cardinal voting going by quantities of points or "plumping"
for a single candidate. Of course the details are different but they
share the similarity of cardinal voting obscuring an ordinal vote, a
straight-forward order of choice, in the hands of the voters, instead of
the parties. ultimately reducing to FPTP under the plumping option.
Or as HG Wells would say: enfeebling complications fruitful of corruption.
That does seem pretty accurate, but I don't think the observation can be
generalized to all cardinal methods.
The problem with cumulative voting is that you have a given number of
points and need to spread them around. If you don't plump them all for a
single candidate, you're making your vote weaker. Hence it behaves like
FPTP.
But with approval or Range, you can set your rating of each candidate
independently, so how much of your voting power you can give to A
doesn't depend on how much you've given to B. So the incentive to plump
isn't nearly as strong.
There are definitely weaknesses to cardinal voting. But cardinal methods
aren't all the same: different rules give different weaknesses.
-km