Unlimited candidate-weight Thiele PAV and failures of proportionality

While I Thiele PAV doesn’t give what I would consider to be acceptable results in a normal proportional election with individual candidates elected with equal weights, I wondered whether its problems could be overcome with unlimited candidate-weight elections, or party list using these same weights.

One of the problems with Thiele Proportional Approval Voting is that it fails strong PR. For example:

6 to elect

2 voters: U1, U2, U3, A1, A2, A3
1 voter: U1, U2, U3, B1

Under strong PR, the universally approved candidates (U1-3) should not affect the ratios of the other candidates elected. So the correct result would be U1-3, A1-2, B1 (or equivalent).

Thiele PAV elects U1-3, A1-3, failing the criterion. This result gives the A faction 6 candidates and the B faction 3 candidates, and some may still argue that this is an acceptable proportional result.

However, this is not the worst case.

20 to elect

2 voters: U1-10, A1-3
2 voters: U1-10, B1-3
1 voter: C1-6

In this case the strong PR failure leads to a worse PR failing. The correct result seems to be U1-10, A1-3, B1-3, C1-4. The C faction has 1/5 of the voters so should have 1/5 of the candidates, which is 4. The other 16 should be even among the A and B factions.

This result gives the A and B factions each 13 candidates and the C faction 4, but Thiele works to give a 2:2:1 ratio even though the A and B factions aren’t truly separate. Thiele gives the result U1-10, B1-2, C1-2, giving the factions 12, 12, and 6 candidates respectively. However, this causes the C faction to be over-represented in parliament (getting 3/10 instead of 2/10 of the representation), whereas A and B considered together are under-represented getting 7/10 instead of 8/10 of the representation.

Because of this I don’t really consider Thiele PAV a suitable election method for use. And this brings us to the case where candidates can be given different weights in parliament, or where these weights are used in a party-list proportional approval election. In the above two elections, proportionality would be restored.

6 to elect

2 voters: U1, U2, U3, A1, A2, A3
1 voter: U1, U2, U3, B1

In this case the U candidates would win all the weight between them, or for party list, party U would win all the seats.

20 to elect

2 voters: U1-10, A1-3
2 voters: U1-10, B1-3
1 voter: C1-6

In this case A and B would win no seats. U would win 16 (or 4/5 of the weight) and C would win 4 (1/5 of the weight).

So far so good. Thiele’s problem is when factions overlap, but where both factions support a single party/candidate then that party/candidate would win all the weight so the problem of the overlap effectively goes away. But you can have a case like this:

44 to elect

1 voter: AB
1 voter: AC
1 voter: BC
1 voter: D

The correct proportional result is for A, B, C and D to all be elected with equal weight, so 11 seats each. However, this gives the top three voters 22 candidates each and the D voter 11. Thiele here works to give a 3:1 ratio rather than 2:1, and gives a result of 12, 12, 12, 8. There is no “universal” candidate to sort out the problem of the overlap, so this a problem both for a normal election and an unlimited-weight election.

This is also probably the most stark example I’ve seen of a plain proportionality failure with Thiele PAV. The D faction is just a normal faction with no overlap minding its own business, and it should unequivocally get 1/4 of the weight in parliament. However, Thiele PAV gives this faction 8/44 of the weight - approximately 18% instead of 25%.


Is that not inconsistent with Jefferson style? I think it may be an issue with all highest averages methods since the reweight is nonlinear. This should also apply to Harmonic Voting. When you say they get the wrong result you are applying a Hamilton Method to judge. This is the same as demanding that Vote Unitarity is satisfied. Why do we not just use a method which satisfies that? My Optimal Unitary and Sequentially Spent score would work. For am I missing something?

No, Phragmen does that. Probably the optimal method I posted here a few months ago that was intended to reconcile ULC and IIB does as well, although I may need to add some tiebreakers to make the ULC part recursive. The modification to SSS that I came up with to try to limit vote management definitely does.


Yes, as said Jefferson/D’Hondt is consistent with strong PR.

Also, what do you mean that I’m applying a Hamilton method to judge? The examples I gave don’t involve rounding of seats, so the rounding method shouldn’t matter. (One of the examples was a 44-seat election to make sure that both the “correct” result and the Thiele result gave an integer number of seats.)

I actually think that Hamilton/Largest Remainder methods aren’t as good as highest averages methods such as Jefferson/D’Hondt or Webster/Sainte-Laguë. With largest remainder methods you can end up with weird results like the Alabama Paradox. I also think that highest averages method are more “naturally proportional” and tend to come out where you are trying to make a proportional method without pre-supposing what you want it to be. Phragmen and Ebert don’t use divisors like D’Hondt and Sainte-Laguë but end up producing the same results if people vote along party lines.

So while I haven’t (so far) looked that closely into Vote Unitarity or Optimal Unitary and Sequentially Spent score, if they result in largest remainder rather than highest averages rounding, then I probably won’t be their biggest fan!

In any case, the initial point about strong PR in my opening post was just the starting point and not the main event. The unlimited candidate weight version of Thiele is supposed to bypass the problem of strong PR. Also one of the features of unlimited candidate weight methods is that there is no rounding so one’s personal favourite rounding method doesn’t have any relevance. But even so, Thiele was shown to fail basic proportionality. Not because it rounds badly or because it fails strong PR, but because beyond basic party voting, it’s not actually a properly proportional method.


Quick question: what does Thiele become in the single-winner case? Since, if Monroe becomes Score in the single-winner case, and isn’t fully proportional, and Thiele also isn’t proportional, then what could possibly qualify as being the gold standard while having a single-winner version of itself?

The concept of proportionality isn’t really meaningful in the single winner case. Anyway, Thiele isn’t exactly score in the single winner case. A single candidate’s score in Thiele would be the sum of the output of the digamma function on the score each ballot gives the candidate.

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Could you give an example of how that would be better or worse than regular Score?

I think it makes going from a 0 to 1 on someone’s ballot more valuable than from 4 to 5. It’s less transparent, though, because your average voter isn’t going to be familiar with the digamma function, so I think that’s worse.

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Is this a factor in why Thiele is more consensus-biased than Monroe, which is linear in score-to-support translation?

Monroe is not really linear. The goal of the systems I designed was to be linear.

It’s linear, but maybe we could say “localized linear” because it only cares what each voter base wants for one allotted representative. It still responds in linear fashion since, within a group of voters, one candidate getting a few more points than another candidate translates into an equivalently higher support over the other. Reweighting isn’t exactly linear, but it probably will be close when we factor in that Monroe seeks to elect candidates with maximal support, since that makes its maximal ballot spending more linear too.

SSS seems more like a more explicable version of RRV than a radical disruption to the traditional spectrum of “consensus-bias” to “fully proportional”.

Anyways, couldn’t SMV be made linear if, instead of fractional or random spending of ballots, the ballots that were to be allocated after a candidate was elected had SSS reweighting applied to them?

Not entirely, RRV is also consensus-biased, and it is much like Thiele as far as reweighting, but uses standard score selection.

It does also depend on how you translate from approval to score. I would apply the KP transformation to the ballots first rather than using the scores directly. The KP transformation gives much cleaner results generally as well as reducing to score voting in the single winner case.

If everyone scores either 10 (out of 10) or 0 to every candidate, you can end up with different results from if everyone gave a different score (say, 7) or 0 if you don’t use the KP transformation, so it’s not scale invariant.

Do you have any thoughts about the optimal method I describe here? I haven’t had a chance to investigate its properties in detail (or it’s been difficult to do so), but it has performed well in test cases of proportionality like those you use to criticize Thiele’s method in this post.

I’ll have a look into it. I haven’t been on the forum that much so there’s loads that I would have missed and people seem to be posting more about proportional score/approval systems these days.

It may need a few modifications before it works exactly as intended, but I think the idea has potential.

Having looked at it, I can’t really see why the scoring system works as it does. E.g. I have no idea what this would do and what results it would give: “The score for each switch is half the improvement of the ballot portion with the smallest gain from reassignment times the total weight of the switched ballots.”

Could you not just use the Monroe score as the score but restrict the swaps as you have done to ensure it passes the desired criteria?

  1. Just using the Monroe score of the final configuration of ballot assignments can cause lowering a candidate’s score to raise the overall set score, by allowing the ballot with the lowered score to be able to swap with another ballot when it wouldn’t be able to otherwise. The weird way swaps are scored prevents swaps made possible by lowering a candidate’s score from being more valuable than had the ballot just given the candidate the higher score.

  2. I realized recently that it does not pass ULC factions, and so the swap restrictions will likely need to be changed.

What do you suggest as an alternative to Thiele and Monroe?

With pairwise matchups, the A’s and B’s could do vote management such that the 10 U, 3 A, and 3 B candidates each have 0.25 votes, equalling the C’s best possible vote management.

I think Ebert’s measure is probably better than both, although it might require some approval removal to get it to work properly. Having said that, Monroe works by assigning each voter to just one candidate, so it’s effectively an approval-removal system, but one that leaves each voter approving a maximum of one candidate. You could do the same with Ebert. Find the set of candidates that gives the best Ebert score where each voter keeps just one approval. Not that I think this is the best method overall, but it’s better than Monroe, and Thiele isn’t even really proportional.

The problem with Monroe only allowing a quota of voters to be assigned to each candidate is that irrelevant ballots change the size of the quota. As far as I can see there are two basic ways to fail IIB, and one is worse than the other.

The worse one is where under some ballots a set of candidates is elected. Then some new ballots appear that approve some completely different candidates that have no chance of being elected. But these change the set of elected candidates anyway. Monroe can fail this because changing the number of ballots changes the quota, which has knock-on effects.

The less bad failure is where new ballots appear that approve lots of candidate. For example, we have the following ballots where voters approve along party lines:

2: Party A
1: Party B

Then along come ballots that approve both A and B. Some methods might now award A and B seats in a ratio different from 2:1, or be indifferent between several ratios. This isn’t ideal, but I see this as a weaker failure.

In any case, under Ebert the number of ballots in total doesn’t matter because there is no quota so it won’t fail the worse version of IIB. By restricting ballots to one approval that isn’t removed, some candidates will have more approvals than others in a potential winning set, so effectively candidates are “assigned” different numbers of voters. But a set that has good equality across the candidates will get a better Ebert score. I can’t see any advantages of Monroe over this version of Ebert, and I don’t see this as the best version of Ebert anyway.

But regarding the more general question of a system I’d recommend, I think that’s still open. Cardinal PR methods aren’t well established, and I think it would be a mistake to go public recommending a particular one too soon because it could get torn to shreds by critics looking for its flaws.