Some things that came out of the paper:

If A, B, C and D are parties fielding multiple candidates, then with the following ballots, what proportion of the seats should each party win?

250: AC
250: AD
250: BC
250: BD
2: C
2: D

COWPEA would say that A and B would each win just under 1/4, and C and D would each win just over 1/4.

However, Optimised PAV would elect C and D with half the weight each, and A and B would not get any weight. The multi-winner Pareto efficiency criterion assumes that a voter's satisfaction with a result can be measured purely by the number of elected candidates they have approved. And going by that, any result containing any A or B candidates is Pareto dominated by CD.

This can be seen as a two-dimensional voting space with an AB axis and a CD axis. PAV would ignore the AB axis. Arguably though COWPEA makes better use of the voting space.

That was an example where any proportions are allowed, but here is another where 2 candidates are to be elected.

150: AC
100: AD
140: BC
110: BD
1: C
1: D

Basically any deterministic would elect either AB or CD. CD Pareto dominates AB in this multi-winner sense as all 502 voters have an approved candidate under CD, compared with 500 under AB (no-one has two approved candidates). However, AB is more proportional. 250 have approved A and 250 have approved B. 291 have approved C and 211 D. So under the CD result, the 211 D voters would wield a disproportionate amount of power. So what do you think? Does that matter or is it all about number of approved candidates?

If you go for AB, then you are rejecting the multi-winner Pareto efficiency criterion, but also consistency as a by-product. This is because you can have a set of ballots where the C and D approvals are just swapped round. So it's 211 for C and 291 for D but otherwise the same. Then combining the two sets of ballots together you get:

250: AC
250: AD
250: BC
250: BD
2: C
2: D

If there are 2 to elect here, it must be CD. So if you went for AB in the other examples, then you are rejecting the consistency criterion in multi-winner elections.

@wolftune OK, that's fine. I don't disagree with your point. But I was clarifying in case your post was a response to my bit about the candidates that are the favourites of nobody, but obviously it wasn't.

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.

@dominikpeters Thanks for the information on this. Do you think that MES + completion is a good method for electing a committee of a fixed size or do you think a single method that works without an add-on would be better?

Just to clarify on the Phragmén thing:

If you just have a fixed quota then the voters that get their candidates early can get a bad deal because they pay a whole quota, whereas later on, the might not be a candidate with a whole quota of votes and yet you have to elect one anyway, so the voters of this candidate get their candidate more "cheaply".

So you might then look for a quota that distributes the cost more evenly, and that's all Phragmén really does. It distributes the load or cost across the voters as evenly as it can.

I see subtractive fixed-quota methods as a cheap hack really.

@cfrank

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.

One option would be 3s/2+4s+5s

There's also a discussion of different PR philosophies on the wiki. It doesn't really cover all the ground though, as COWPEA or COWPEA Lottery wouldn't be included, for example. But perhaps non-deterministic methods deserve their own entry on there.

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.

@toby-pereira said in Israel, Proportional Representation, Polarization, and Accountability:

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.

Interestingly, the KP Transformation, which is a way of converting approval methods to score methods, does not work on the basis that you can simply add up a voter's score for each elected candidate to see how satisfied they are. If scores are out of e.g. 5, then each voter essentially has 5 utility slots and the top one will only be satisfied by a score of 5. So two candidates with scores of 2 and 3 being elected will mean that 40% of the voter's utility slots will be untouched. A 0 and a 5 mean that all slots will be satisfied to some extent but none more than once.

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.

@anniek said in Good simple semi-PR methods?:

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?"

@toby-pereira said in Easy-to-explain Proportional (Multiwinner) elections:

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.

Yes, in party-list PR it would be good to allow us to vote for 2 parties…our favorite (but maybe not seat-winnable), & a seat-winnable party we like.

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.

@democrates I would ask them what they mean by "mob rule". How could it be detected, and why is it a problem.

@A Former User said in What level of PR do different systems get?:

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.

@toby-pereira said in Variable house sizes:

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).

@toby-pereira said in Optimal cardinal proportional representation:

By the way, since PAV with infinite clones passes core (which it doesn't with a limited number of candidates), I presume the optimal version probably is properly proportional (passes perfect representation). I might update my paper with this in at some point.

I have updated the paper to mention the proportionality of Optimal PAV (with variable candidate weight allowed), which allows for a proper comparison with COWPEA - these two methods being the main candidates for a truly optimal cardinal PR method (practicalities aside).

@toby-pereira very interesting. Many ancient cultures employed sortition for allocating responsibilities or making decisions, though perhaps this will be a more difficult sell without the appeal to "the will of the gods" becoming rhetorically effective again.