This project hasn't purely been altruistic - it's been helpful to me by laying everything out for reworking my COWPEA paper!

And the new version can be seen here (as I mentioned in the separate COWPEA thread anyway).

Obviously the Thiele and Phragmén methods have been around for over a century but academic research into this type of method only really picked up again relatively recently, and dry academic papers aren't the best place for a beginner to learn about them anyway.

Edit - Maybe I could start a thread with some basic info. Others can add to it and can also take issue with anything they might disagree with!

This is a markdown file that defines the formula.

And here are some examples

I think with careful consideration, it might be possible to adjust the multiplier so that “puppeteering” actually harms a party in proportion to the harm it could cause to its largest rival. For example, maybe raising the multiplier to some power or passing it through some other function would disincentivize puppeteering more.

https://www.votingtheory.org/forum/topic/299/pr-with-ambassador-quotas-and-cake-cutting-incentives?_=1739756465750

There are various proportional methods that use approval voting and they can be turned into more general cardinal methods by allowing scores or stars instead of a simple yes/no. But as well as all the different approval methods, there are different ways to convert these methods into score voting methods, so you can end up with a proliferation of possible methods with these two essentially independent choices you have to make (which approval method, how to deal with scores).

First of all, I should say that I'm talking about methods that use the actual values of the scores, not where scores are used as a proxy for ranks.

For example, you have methods like Allocated Score, Sequential Monroe and Sequentially Spent Score. As far as I understand, if everyone voted approval-style (so only max or min scores), these methods would all be essentially the same. The highest scoring candidate is elected, and a quota of votes is removed, as so on.

All of these methods are actually quite messy, not to mention arbitrary, and you can end up with a lot of discontinuities and edge cases when you make small changes in the vote. Scores are an inconvenience in this sense (which is why all these similar but different methods were invented) and it would be much better if you could just make them behave more predictably and continuously from the start, so you can then just apply your favourite approval method knowing things will run smoothly.

And the way to do this? Well, as far as I'm concerned, it's the KP transformation. It turns the score ballots into approval ballots in a consistent manner, so you then only have to worry about what approval method you want to use. For e.g. scores out of 5, this essentially splits each ballot into 5 parts with their own approval threshold for each candidate. The "top" part will only approve those given 5, the next part will approve those given 4 and 5, and so on. The highest scoring candidate overall automatically becomes the most approved candidate, and so on. The total scores are proportional to the total approvals they've been converted to.

This makes methods far more continuous than the above ad hoc score conversions, so the weird discontinuities they cause will go away.

The KP transformation has nice properties. For example, for an approval method that passes Independence of Irrelevant Ballots, the KP transformed method will pass multiplicative and additive scale invariance. That means that if you multiply the scores on all ballots by a constant, or add a constant, or both, the result will still be the same. So you could multiply the scores by 7 and add 3. It would not affect the result.

Taking Thiele's Proportional Approval Voting as an example, Reweighted Range Voting and Single Distributed Vote are both conversions that cause a failure in one or both forms of scale invariance. However, Harmonic Voting, or its sequential variant, which both use the KP transformation, pass.

Also, this means that electing two candidates that a voter has given a 2 and a 3 respectively is not the same as a single 5 (and 0 for any others). But I see this as a feature, not a bug. It means that someone's ballot will never be "used up" by candidates they don't give their full support to. With scores out of 5, electing candidates a voter gives 3 or less to means that 2/5 of their vote will be completely protected until a 4 or 5 is elected.

With the monotonicity, Phragmén is not actually non-monotonic as I say, but only weakly monotonic. E.g. with 2 to elect:

100 voters: ABC
100 voters: ABD

Phragmén methods tend to be indifferent between AB and CD (unless modified in some way), and that leaves them open to:

99 voters: ABC
99 voters: ABD
1 voter: C
1 voter: D

Where CD would be preferred. Electing in a greedy sequential manner rather than optimally would actually lead to more monotonic results in these cases, but other examples can be found.

The same idea is covered at [https://rangevoting.org/BRmulti.html](link url)

Methods that guarantee core stability are of interest to me (see this thread, which I linked to earlier) even if it's not my priority. From what I've read, I think it's still unproven that it's guaranteed that the core is non-empty. But if you use a stability measure (as suggested in the thread) rather than an all-or-nothing, it could be workable regardless.

BTW, we should probably distinguish different-sized cores—the possibility of an empty Hare core is unknown, but Droop cores can definitely be empty (as the Condorcet paradox proves). What I'm interested in is satisfying the Hare core with high probability and satisfying the anti-Droop core guarantee with certainty; i.e. the share of voters who would prefer some other committee is less than 1 / (seats - 1).

I assume nothing about the distribution. I speak of the average s/v, over all possible numbers of quotas in an interval (as defined above).

The bias that I speak of is differing s/v averages over low & high intervals.

Nothing whatsoever to do with a distribution.

In this post on Election Methods, you wrote:

Here is what I mean by "bias". I claim that my meaning for bias is consistent with the usual understood meaning for bias::

For any two consecutive integers N and N+1, the interval between those two integers is "Interval N"

If it is equally likely to find a party with its final quotient anywhere in interval N, then determine the expected s/v for parties in interval N.

Compare that expected s/v for some small value of N, with the expected value of s/v for some large value of N.

If the latter expected s/v is greater than the former, when using a certain seat allocation method, then that allocation method is large-biased.

If the opposite is true, then the method is small-biased.

I have bolded and italicised part of your quote. It is an assumption about the distribution. You might think it's a fair assumption. But it is an assumption, something you've been denying. So I'm glad that's clarified.

[quote]

And while that might seem unrealistic, we can see the case of very small parties that never get enough votes to win a seat. A particular party might be due about 0.1 seats at every election but never win one under a particular method. Is that bias?
[/quote]

Sure, but it’s not the kind that has always been meant when speaking of bias. Fractional-quota small parties were traditionally never really wanted in PR countries.

It seems we're changing the subject here. This is about a method being objectively unbiased. We are not talking about practicalities at all and what is wanted. So do you admit to bias in the "Bias-Free" method then?

[quote]
(Michael's method involves a 0^0 in the 0 to 1 seat range, so appears to break, so I'm not sure how it is supposed to handle this case.)
[\quote]

No, it still works, though the usual formula doesn’t work. There’s a way to do the integral from that 0^0 point. It’s an exception that has to be separately integrated as a separate problem.

The answer to that problem is a rounding point equal to 1/e.

To clarify then, those parties consistently getting fewer votes than 1/e of a quota of votes are the subject of systematic bias, under the "Bias-Free" method.

[quote]
(after all Sainte-Laguë simply returns the most proportional result)…
[/quote]

…according to the difference measure of s/v-variation. …which doesn’t make as much sense as ratio. But of course your preference is entirely your business. …but if you’re going to say that Professor Huntington was wrong, you’ll need more than an assertion. You’ll need to say where you think he was wrong. Help that mathematics professor out by explaining where he made his error.

…& as I‘be explained to you many times, if by “most proportional” you mean “ having least maximum variation in s/v, that’s an entirely different matter from bias, whose meaning I’ve already told you several times.

Well, I've discussed Huntington's paper in the previous post, so that's sorted now. And you agree that Sainte-Laguë magically gives less bias than Huntington-Hill despite being worse. I also explained in a previous post why minimising the variance of s/v measured arithmetically is the best measure. s/v adds to a set number (s in fact). It is, in essence, an arithmetic sample, not a geometric one. Using geometric variance breaks if a party has zero seats. If you had a sample that multiplied to a set number, then use the geometric variance. It would make most sense. (Edit - is you were looking at v/s instead of s/v it would make sense to use the harmonic mean and variance.)

[quote]

The only way to get rid of bias under any assumptions…
[\quote]

BF has no bias.

As pointed out above, it only has no bias under certain assumptions. I could devise a voting distribution of voting behaviour (that could exist in a possible world) where it has either small or large party bias. The only way to eliminate any possibility of this would be to use a non-deterministic method. However, "Bias-Free" does have a small-party bias relative to Sainte-Laguë, which by the most sensible measure gives the most proportional result. This is itself a form of bias. But anyway, I'm repeating myself. I think we're probably done because you're not going to reply. But it's a shame. I think "Bias-Free" probably has some interesting theoretical properties, and it would be interesting to see them explained. But you asserted too much about it, and were unable to discuss it in a reasonable manner.

Well, I wanted to check this forum out, & there was talk about doing a lot of polling, which I consider very useful to demonstrate how the methods work.

But the amount of participation in the recent poll wasn’t very promising.

Because as I pointed out in one of the threads about it, it wasn't run very well.

I prefer that. But it would be regarded as a complication. Anyway tradition wanted to discourage small parties. The option to vote for two parties would be rejected as a complication.

For a first PR proposal, just propose ordinary list-PR, like 2/3 of the world’s countries use.

Later we can propose the 2-party option.

By the way, Satisfaction Approval Voting can only be described as semi-proportional. You're wasting part of your vote on candidates that aren't elected. It's like SNTV except that you can split your vote up. They both have similar problems to FPTP.
They might be easy to explain, but they're not worth explaining!

You're right, of course, but that's why I like to bring up SAV as an "obvious" system with an obvious flaw (spoilers). Then I explain how PAV/SPAV fix that flaw with a minor change--split a vote only after a candidate is elected, not before.

A group I'm tangentially connected to has decided to switch from sequential proportional approval voting to cumulative voting (voters are given a number of points equal to the number of seats, and each voter can give any amount of their available points to one or more candidates). I'm concerned because I have heard that cumulative voting is susceptible to vote-splitting and bullet voting. Where can I learn more about these effects in cumulative voting?

The main problem with cumulative voting is what happens if they don't do bullet voting. Cumulative voting produces kind-of-proportional representation if voters are strategic and perfectly informed, because minority groups can coordinate to bullet-vote. However, too much honest voting by these groups can easily result in a bare majority sweeping all the seats; in other words, requires some very complicated coordination (especially in small elections).

I don't think there are any proportional methods simpler than SPAV except party-list representation, or possibly sequential Ebert (although I'd consider sequential Ebert about as simple as SPAV).

Have you asked what makes this group think of SPAV as "too complex?"

I think the KP transformation is generally good because it gives maintains good criterion compliance when added to an approval method without adding nasty surprises. And it's simple. Essentially all of the debate between Allocated Score, Method of Equal Shares, TEA etc., are about what to do with these messy scores that are hard to deal with when you use a score method rather than an approval method. KP sorts that out simply with generally better criterion compliance. So the consideration of TEA versus those other methods then becomes irrelevant.

And as I mentioned in this thread, all the subtractive quota methods are really just weak approximations of Phragmén-based methods. So I'd use Phragmén + KP as the basic starting point of a method you have to do better than. If determinism isn't essential, you can do better still.

Essentially, 300 "candidates" are elected, which gives the potential for a high degree of PR. According to the paper, Sequential Phragmén, Sequential PAV, Method of Equal Shares and Phragmms all behave fairly similarly. I don't think I was previously aware of Phragmms. I'll have to look into it and potentially update the wiki.

I would also be interested in seeing how COWPEA Lottery performs. It is non-deterministic, but with 300 to elect, this is likely not to affect the result significantly. It also gains the advantage of being much cheaper to calculate, as well as passing the "Holy Grail" criteria of PR, strong monotonicity, Independence of Irrelevant Ballots (IIB) and the Universally liked candidate criterion (ULC).

A candidate-based system, with presumably smaller regions, would be preferable in my opinion, and there would be no need for a discussion about whether a minimum threshold was good or bad.

I personally prefer candidate based PR to party list in general, but I haven't really thought that through in a Israeli/Palestinian context. In terms of Israel they currently are at one extreme of the spectrum where voters have as little say over parties candidates as possible. I definitely think any change in the opposite direction would help, even just going from Closed list to Open List. That would allow every party to ideally veto any of their candidates and leaders parroting genocidal war crime type talking points without turning their back on the party platform entirely. This seems like an easy change to start with since it would probably be less controversial than other ideas I support.

So really, under a system of PR that works and a parliamentary system that works, a stable government would have to be around the centre of public opinion. Whether that is in fact the case in Israel, I'm not really in a position to say.

My very limited understanding of Israeli politics leads me to believe that Netanyahu has been wildly unpopular most of my life, but he maintains power because nobody has been able to successfully challenge him and hold the position. I see Israel as not stable, but stagnant. I'd love to see actual Israeli citizens chime in here with some lived experience on the ground.

Optimised PAV Lottery is another non-deterministic method. In this you work out the optimum amount of weight each candidate should have (by e.g. infinitely cloning all candidates and running an election with a very large number of seats), and then elect candidates probabilistically according to these weights. (Though you would have to work out the distribution again every time a candidate is elected). Unlike deterministic PAV, it is thought (though not known) to be proportional by passing the Perfect Representation In the Limit criterion.

I also think that sequential methods generally fail participation (for a suitable multi-winner definition), whereas optimal elect-all-at-once methods are computationally infeasible. However, I think non-deterministic sequential methods can get around this failure. Optimised PAV Lottery is computationally infeasible anyway, but COWPEA Lottery is easily runnable.

Here's a simple example to show proportionality. This example works the same no matter what the measure of support. Basically, all these measures are clones of STV but without vote splitting. The measures I'm talking about are score, smallest pairwise win, and strongest beatpath. Also, I should say that I don't really know if the measures would work for sorting.

Setup:
2 parties: A, B.
2 seats.
100 voters.
quota: 50 voters.

Round 1:
The votes are A 55, B 45.
The best quotas of 50 voters give A 50 and B 45.
A gets a seat. The quota of 50 A-voters are removed.

Round 2:
Now we count the remaining voters, A 5, B 45.
B gets a seat.

But just to be clear anyway - when I was talking about not wanting to elect the favourite-of-nobody candidates, I wasn't referring to stopping the election of such candidates in principle. But we've discussed certain specific scenarios where we all seem to agree that electing the candidates that are the favourite of nobody isn't the best thing to do. And I was discussing ways to elect the candidates we would want to in this situation.

When there's more than one winner, what happens depends on how you interpret the scores. You could measure a voter's satisfaction by adding up the scores the voters have given to the elected candidates, but I think that might be unsatisfactory in a few ways. There's always debate about how to interpret scores and what they mean, and whether absolute numerical values should really be used in their raw form.

Instead, the scores could be used as layers of approval. This basically means that a voter's satisfaction with a candidate set is determined by the single highest score they've given to a candidate in the set, next best used as a tie-break, and so on. So for scores out of 5, a single 5 is better than multiple 4s etc.

This should keep it relatively simple. Also if candidates are elected sequentially, it should be simple enough to calculate the results.

I think this should be a decent enough method and I think I'd prefer it to things like Allocated Score and Sequentially Spent Score.

Obviously COWPEA Lottery using scores as layers of approval is God-tier in terms of criterion compliance, and very simple to implement, but it is non-deterministic, which might be too much for some people, so this method could be a good compromise.

Edit - You'd have to work out exactly how to measure the stability of a candidate set though. Let's say the first 2 candidates elected are AB. Then you need to test e.g. ABC, ABD, ABE etc. to find the 3rd candidate. But I think you might be able to test them against each individual candidate not in the set. So test ABC against, D, E, F etc. separately.

Edit 2 - You'd probably have to test each potential set against all the other subsets. So ABC would go against ABD, ABE etc., plus AD, AE, BD, BE, as well as D, E etc. Still not that many in the general scheme of things.