@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?

@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 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.

@lime I think that since multiwinner elections are generally for electing legislatures, we can say that generally the goal is for the set of winners to be able to vote on laws in such a way that the laws passed maximize voter utility (i.e. come closest to their preferences on the issues). Now since this is difficult to simulate, how about simplifying it down to an election-based process: have the winners of the multiwinner election conduct a single-winner election among themselves to choose a leader (presumably using whatever single-winner voting method the original multiwinner election's voting method simplifies to), and then see how much utility the original voters in the multiwinner election receive from that leader compared to the amount of utility those original voters would get from the leader they would have elected if they had themselves voted in the single-winner election.
Essentially, test how much utility voters would get from the Prime Minister chosen by Parliament versus their utility if they directly elected a President instead.

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

@cfrank Yeah, there's always the incentive not to vote for a preferred candidate in PR for the reasons you give.

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.

@toby-pereira this seems like a tricky thing to deal with. I appreciate your posts here, PR methods seem to be pretty complex. The whole process seems to be a “cake cutting” problem, and a while back, I was trying to think about how to pose the situation in a way that enables the optimal cake cutting protocol to work. It might not be clear, fully hashed out or easy to follow, but I wonder what your thoughts about the concept are:

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

@Toby-Pereira I tried to fix this problem, and came up with some formulas to define the quota compliance multiplier using entropy. It’s kind of complicated, but it’s well-defined, and it encourages diverse compromises between parties. It probably now gives more bargaining power to larger parties, and “puppeteering” can still be strategic, but I think those things are in direct conflict unfortunately.

This is a markdown file that defines the formula.

# Proposed Compliance Multiplier Formula This document presents a proposed formula for the compliance multiplier in the proposed system, which compares the observed distribution of ambassador seats with the ideal distribution based solely on voter shares. ## 1. Partition Functions The ideal partition function for a given party \(X\) is defined as: [ Z_P(X) = P(X \sim X) + \sum_{i \neq X} P(X \sim i) + \sum_{j \neq X} P(j \sim X) ] The observed partition function is defined as: [ Z_Q(X) = P(X \sim X) + \sum_{i \neq X} Q(X \sim i) + \sum_{j \neq X} Q(j \sim X) ] *Note: For party consistency, we impose that \(Q(X \sim X) = P(X \sim X)\) for every party \(X\).* ## 2. Entropy–like Quantities The ideal (maximum) entropy is given by: [ E_P(X) = \frac{-P(X \sim X)\ln P(X \sim X) - \sum_{i \neq X} P(X \sim i)\ln P(X \sim i) - \sum_{j \neq X} P(j \sim X)\ln P(j \sim X)}{Z_P(X)} + \ln Z_P(X) ] The observed entropy is given by: [ E_Q(X) = \frac{-P(X \sim X)\ln P(X \sim X) - \sum_{i \neq X} Q(X \sim i)\ln Q(X \sim i) - \sum_{j \neq X} Q(j \sim X)\ln Q(j \sim X)}{Z_Q(X)} + \ln Z_Q(X) ] ## 3. Compliance Multiplier The compliance multiplier for party \(X\) is then defined as: [ \text{Multiplier}(X) = \frac{E_Q(X)}{E_P(X)} ] Since for every off-diagonal entry we have \(Q(X \sim Y) \leq P(X \sim Y)\) (with equality on the diagonal), the normalized observed entropy \(E_Q(X)\) is less than or equal to the ideal entropy \(E_P(X)\). Therefore, it follows that: [ \text{Multiplier}(X) \leq 1. ] This guarantees that a party's compliance multiplier never exceeds 1, reflecting that the observed (normalized) diversity of ambassador nominations cannot surpass the ideal (maximally spread) distribution.

And here are some examples

# Four-Party Examples of Compliance Multipliers This document presents computed examples for a 4-party system under different scenarios. We consider four parties—A, B, C, and D—with the following voter shares: - **Party A:** 0.4 - **Party B:** 0.3 - **Party C:** 0.2 - **Party D:** 0.1 The ideal ambassador seat allocation is given by: [ P(X \sim Y)=P(X) \times P(Y) ] Thus, the **ideal matrix** \(P\) is: | From \(\backslash\) To | A | B | C | D | |------------------------|------|------|------|------| | **A** | 0.16 | 0.12 | 0.08 | 0.04 | | **B** | 0.12 | 0.09 | 0.06 | 0.03 | | **C** | 0.08 | 0.06 | 0.04 | 0.02 | | **D** | 0.04 | 0.03 | 0.02 | 0.01 | For each party \(X\), we define the **partition functions** as follows: - **Ideal:** [ Z_P(X)=P(X\sim X)+\sum_{Y\neq X}P(X\sim Y)+\sum_{Y\neq X}P(Y\sim X) ] - **Observed:** [ Z_Q(X)=P(X\sim X)+\sum_{Y\neq X}Q(X\sim Y)+\sum_{Y\neq X}Q(Y\sim X) ] with the assumption that for all parties, \(Q(X \sim X)=P(X \sim X)\). We then define the **entropy–like quantities**: [ E_P(X)=\frac{-\sum_Y P(X\sim Y)\ln P(X\sim Y)}{Z_P(X)}+\ln Z_P(X) ] [ E_Q(X)=\frac{-\sum_Y Q(X\sim Y)\ln Q(X\sim Y)}{Z_Q(X)}+\ln Z_Q(X) ] and the **compliance multiplier** is given by: [ \text{Multiplier}(X)=\frac{E_Q(X)}{E_P(X)}. ] Under the condition that for every off-diagonal entry \(Q(X\sim Y)\leq P(X\sim Y)\), the normalized observed entropy cannot exceed the ideal one—so \(\text{Multiplier}(X)\leq 1\). --- ## Scenario 1: Full Compliance **Situation:** Every party fills its ambassador seats exactly as in the ideal, so for all \(X, Y\): [ Q(X\sim Y)=P(X\sim Y). ] **Results:** - **Party A:** Multiplier = 1.000 - **Party B:** Multiplier = 1.000 - **Party C:** Multiplier = 1.000 - **Party D:** Multiplier = 1.000 --- ## Scenario 2: Sabotage by Party B Against Party A **Modification:** Party B refuses to elect any ambassadors from A. In our observed matrix, we set: [ Q(B\sim A)=0 \quad \text{(instead of the ideal }0.12\text{)}. ] All other entries remain ideal. ### Computed Values **For Party A:** - **Ideal Partition Function:** \(Z_P(A)=0.16+ (0.12+0.08+0.04)+(0.12+0.08+0.04)=0.16+0.24+0.24=0.64.\) - **Observed Partition Function:** - Row A remains: \(0.16+0.12+0.08+0.04=0.40.\) - Column A: Instead of \(0.12+0.08+0.04=0.24,\) we have \(0+0.08+0.04=0.12.\) So, \(Z_Q(A)=0.16+0.24+0.12=0.40.\) - **Entropy–like Quantities (Approximate):** - \(E_P(A) \approx 1.840.\) - \(E_Q(A) \approx 1.472.\) - **Compliance Multiplier:** [ \text{Multiplier}(A)\approx \frac{1.472}{1.840}\approx 0.800. ] **For Party B:** - **Ideal Partition Function:** \(Z_P(B)\approx 0.51.\) - **Observed Partition Function:** \(Z_Q(B)\approx 0.27.\) - **Entropy–like Quantities (Approximate):** - \(E_P(B) \approx 1.824.\) - \(E_Q(B) \approx 1.524.\) - **Compliance Multiplier:** [ \text{Multiplier}(B)\approx \frac{1.524}{1.824}\approx 0.834. ] **For Parties C and D:** No sabotage occurs, so: - **Multiplier(C) = 1.000.** - **Multiplier(D) = 1.000.** --- ## Scenario 3: Puppet Scenario – Party D as a Puppet for Party A **Modification:** Party D acts as a puppet for Party A to harm Party B. We set: [ Q(D\sim B)=0 \quad \text{(instead of the ideal }0.03\text{)}. ] All other entries remain ideal. ### Computed Values **For Party A:** - \(Z_Q(A)\) remains nearly ideal, so **Multiplier(A) \(\approx 1.000\).** **For Party B:** - Losing support from D reduces its observed diversity, so **Multiplier(B) \(\approx 0.910\).** **For Party C:** - Fully compliant, so **Multiplier(C) = 1.000.** **For Party D (the puppet):** - Due to its refusal to elect from B, its observed diversity is reduced: **Multiplier(D) \(\approx 0.940\).** ### Effective Representation for the A Coalition If effective main-platform representation is given by the product of voter share and multiplier, then: - **Party A:** \(0.4 \times 1.000 = 0.400.\) - **Party D:** \(0.1 \times 0.940 \approx 0.094.\) Thus, the total effective representation for the A coalition is: [ R(A \text{ coalition}) = 0.400 + 0.094 \approx 0.494. ] This is slightly less than the 0.5 (50%) of the vote they would control if D were fully compliant, reflecting the cost of splitting support. --- ## Summary of Computed Multipliers - **Scenario 1 (Full Compliance):** - A: 1.000 - B: 1.000 - C: 1.000 - D: 1.000 - **Scenario 2 (Sabotage by B Against A):** - A: ≈ 0.800 - B: ≈ 0.834 - C: 1.000 - D: 1.000 - **Scenario 3 (Puppet – D as Puppet for A):** - A: ≈ 1.000 - B: ≈ 0.910 - C: 1.000 - D: ≈ 0.940 --- ## Interpretation - **Full Compliance:** All parties achieve ideal diversity, so their compliance multipliers are 1. - **Sabotage:** When Party B refuses to elect ambassadors from Party A, both A and B see their normalized diversity reduced (multipliers drop to ≈0.800 and ≈0.834, respectively). - **Puppet Scenario:** Using Party D as a puppet to harm Party B, while A remains nearly ideal, both B and D are penalized—the puppet (D) suffers a lower multiplier (≈0.940) and B’s multiplier drops to about 0.910. Moreover, the A coalition's effective representation becomes slightly diluted (≈0.494 instead of 0.500). These examples illustrate that while parties might attempt sabotage or use puppets to undermine competitors, the system’s reliance on normalized diversity ensures that such strategies come at a cost to all parties involved.

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.