Forum Council

@toby-pereira you were on the right track about something: in this system, small parties can effectively sacrifice themselves to take down larger parties. This could lead to the creation of “puppet parties” that are essentially used as weapons to take down representation of another party, which is kind of a virtual form of MAD warfare.

@cfrank Sorry, I was wrong about forcing a party out. If everyone ganged up on another party so that party had zero support from everyone else it would destroy the other parties too, as it's their minimum that counts.

I think something like approval voting would make sense for this. FPTP would obviously split votes and ruin quotas. With ranks, it's unclear what would count as support.

@toby-pereira yes I agree, thanks for taking a look! I’m curious about your point about parties ganging up, do you have a generic example in mind?

Each party’s main platform representation is limited by the minimum of compliance rates with all ambassador quotas involving that party. So for example, if we have three parties A, B, C, and the compliance rates are

(A~B): 80% of ambassador seats filled
(A~C): 90% of ambassador seats filled
(B~A): 100% of ambassador seats filled
(B~C): 85% of ambassador seats filled
(C~A): 100% of ambassador seats filled

then A will only be allowed to fill 80% of its main platform seats, B will only be allowed to fill 80% of its main platform seats, and C will only be allowed to fill 85% of its main platform seats. So the lack of compromise cuts both ways.

I imagined voting being approval based but with each voter registered under a single party. I figure parties would nominate any candidates they like, and the party affiliations of voters and the party nominations of candidates would determine the filling of “quotas” in some way. For example, if a voter registers under party A, and party B nominates a candidate, I figured a vote cast by that voter to that candidate would contribute to the (A~B) ambassador quota.

I’m now thinking about whether a candidate might be able to acquire multiple seats as ambassadors across many parties. I’m not necessarily thinking of this as a formal decision procedure for how to allocate the seats, but as an apparatus or framework within which seats might be allocated by an additional, more formal mechanism. Does that make sense?

A kind of basic “support quota” for an (A~B) ambassador would be 50%, meaning at least half of the voters registered for party A approved of the B-nominated candidate. B-nominated candidates with less than the support quota from A cannot be given an (A~B) ambassador seat.

@cfrank This is interesting, but it does also seem rather complex, so I'm glad you put up the pictures! It also seems as though all the other parties could gang up on one party to force them out, for example.

Also, I'm not sure how the voting would work with this and therefore what reaching the support quota would entail.

@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

Under Independence of Universally Approved Candidates (IUAC) (see also Universally Liked Candidate Criterion, Strong PR), adding in some universally approved candidates and extra seats to accommodate them should not change the outcome for the other candidates. E.g. with 6 to elect:

2: U1-U3, A1-A3
1: U1-U3, B1-B3

The result should be U1-U3, A1-A2, B1. However, Thiele's PAV elects U1-U3, A1-A3.

Fully optimal methods like Optimal PAV and COWPEA (which allow candidates to be elected in any proportion) completely bypass the IUAC criterion by electing only the universally approved candidates. The lottery versions of these methods also pass by electing the universally approved candidates to the first positions and then electing proportionally from there.

However, one "normal" method (fixed number of candidates elected with equal weight) that does pass is the specific version of var-Phragmén where a candidate's load must be spread equally across its voters. I've always called this Ebert's Method, but I'm not sure he actually intended it to mean this. Anyway, this method is poor overall as it is non-monotonic and can give disproportional results, but I think it does give good results with certain toy examples such as the one above and can be enlightening.

Anyway, while COWPEA and Optimal PAV don't have to deal with IUAC, I did wonder what might happen if a certain group of voters did all approve a particular candidate but not the entire electorate. Would these methods pass this lesser form of IUAC? E.g.

1: U
2: A
1: B
4: UA
2: UB

The extra single U voter at the top is to stop a method from only electing A and B, and forcing it to interact with all 3 candidates. So should the A:B ratio be 2:1 here? Why? Why not? What do each of our methods do? First of all PAV's ratios:

A: 0.376697
B: 0.147247
U: 0.476056

This gives an A:B ratio of 2.558:1

COWPEA:

A: 27/75 (0.36)
B: 10/75 (0.133)
U: 38/75 (0.507)

This gives an A:B ratio of 27:10 or 2.7:1. This is even harsher than Optimal PAV.

Finally Ebert's Method:

A: 1/2
B: 1/4
U: 1/4

This gives an A:B ratio of 2:1, which is what might seem intuitively correct.

So are Optimal PAV and COWPEA "wrong" here? My thinking of late has been that these two are the ultimate in optimal methods (while differing slightly philosophically with each other) and very trustworthy, so I'd be surprised if these results are indeed "wrong" in some way. So why these results?

I wonder if it's to do with monotonicity. The result from Ebert's Method gives U a very low ratio compared to the others despite it being the most approved candidate. Obviously it would be possible to make this higher while keeping the A:B ratio the same, but there might be other reasons not to do this. Calculating the total approval rating of each slate of candidates under each result, you get:

PAV: 6.034
COWPEA: 6.107
Ebert: 5.5

So the total score from the Ebert result is just 5.5, quite a bit lower than the others with COWPEA coming out on top. I think this has been quite an interesting experiment.

This has been discussed on this forum before and I made an argument for using the KP transformation here and here, so that you can convert the scores to approvals and then just run an approval method. But I recently made a Reddit post, and I think it's worth reproducing here as well. So here it is:

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.

@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 failure of monotonicity is interesting and makes sense, since voters are effectively “forced to buy” in an algorithmically greedy fashion, whereas one could imagine them waiting strategically and hoping others put their money up first so they can freeload and increase their representation. And in practice this would be done by choosing not to approve of a desired candidate, betting on others bearing the cost.

By graph based I was thinking of some matching or coloring scheme, but in effect that is what a Phragmén method is, although it’s equivalent to an iterative greedy process. I speculate that computing a globally optimal solution to most “good” objectives is not polynomial, but sometimes there are guaranteed bounds of divergence from the global optimum. Just musing.

I still haven’t dived into this due to other time obligations. But definitely want to learn more.

@cfrank I also like the concept of the Phragmén method to a point, but because it only cares about proportionality, it's not as monotonic as other methods such as Thiele's PAV method. I mean, it passes in that an extra approval can't count against a candidate, but sometimes it makes no difference, so I call it weakly monotonic.

By graph-based do you mean like Schulze for single winner? There is also a Schulze STV method, which reduces to Schulze in the single-winner case.