Harmonic function equivalent for Webster/Sainte-Laguë

In the D’Hondt Thiele variant, the satisfaction score for a voter who has approved n elected candidates is 1 + 1/2 + 1/3 + … + 1/n. This is just the harmonic function so you can stick in it an online calculator like WolframAlpha. I’ll use H for the harmonic function.

With Sainte-Laguë, the satisfaction score is 1 + 1/3 + 1/5 + … + 1/(2n-1)
= H (2n-1) - (1/2 + 1/4 + … + 1/(2n-2))
= H (2n-1) - 1/2 x (1 + 1/2 + … + 1/(n-1))
= H (2n-1) - 1/2 H (n-1)

I think this is quite helpful in calculating Sainte-Laguë results.

I had an idea for something similar: a Webster-based harmonic function. However, I find Thiele proportionality to be inferior to Vote Unitarity.

While Webster’s is much better than Jefferson’s, I still find LR-Hare the best method, and the Score PR system that extends it most naturally is Sequentially Shrinking Quota, which is a modification of Vote Unitarity that limits Hylland free-riding.

At first glance, allocation systems are direct transformations of LR systems, the two main forms have significant issues: Allocated Score de-weights and even can exhaust ballots of people who give slight support to consensus candidates, and therefore has a significant issue with Hylland free-riding and Sequential Monroe can fail to elect beloved consensus winners entirely.


Toby will point out that this is the same debate as the Hamilton vs Jefferson/Webster in the party list case. This gets to the Balinski–Young theorem which means you choose between rare participation failures and rare quota failures.

I give an example on the Sequentially Spent Score page of a participation failure.

I pretty much agree with you on your points, especially about Sequential Monroe voting and Allocated Score. Although I am still waiting on a clear formulation of Sequentially Shrinking Quota. I would think there should be a clear analytic form. Until that is found I am going to stick to Sequentially Spent Score. I think that the simplicity over Sequentially Shrinking Quota could make it more viable in a referendum. At least until there is a clearer formulation.

Anyway, back to the original topic. I am not super sold on Harmonic or Psi voting because there is a lack of theoretical motivation. It could also be the case that I just do not understand it.

I recently made this post

It is not super easy to follow since I was originally slightly wrong and Parker pointed this out. But the point is that I found a better version of RRV is the the more natural score version of SPAV. I have talked to Warren (RRV’s inventor) and he agrees. I have attempted to come up with a optimal (non-sequential) version but have not been able to do so. With SPAV it is easy to come up with PAV because all the scores are the same. This means that the sorting order does not matter so you do not need to sort. With this system I could do the same as Harmonic voting and sort by the score. However, this does not seem clearly motivated and was one of the issues I had with Harmonic voting in the first place.

If you have any good ideas on how to make this into an optimal system I would love to hear it. I was thinking to do something like an inverted Sequential Phragmen. Phragmen systems distribute a load of representation while this distributes a load of ballot. I would think Phragmen could be used but with switching candidates and voters in all computations.

I do intend to make an electowiki page for this new RRV system to make the system clear. I am still trying to come up with a name. I am thinking “Single Distributed Vote” to play off of STV. Open to suggestions.

Well yes, and I certainly think that highest averages methods are best for party-list methods. Thiele has problems relating to voters not all voting along party lines, so the problem is the generalisation to score and approval rather than highest averages itself. Phragmen-based methods also tend to reduce to Webster or Jefferson (so highest averages) in the party-list case, and Thiele’s problems are quite distinct from any problems Phragmen-based methods have, which is another reason not to blame highest averages per se.

I feel LR is superior to highest averages because it follows the quota rule. Phragmen has its own set of problems, I’m pretty sure, so it doesn’t completely redeem highest averages.

LR-Hare > Webster > LR-Droop > Jefferson > non-proportional allocation systems (Imperiali, etc.)

Is there a way we can make this more explicit? Vote Unitarity was at least one way to illustrate that failing. I suspect there are others.

I think the desirability of the quota rule is governed more by intuition than mathematics. There was a long discussion about it in another thread from about this post but basically it forces failure of IIB, and where methods like Webster and Jefferson fail the quota rule, it makes mathematical sense to do so.

Thiele fails the universally liked candidate criterion, and while that might not be the end of the world in itself, it’s closely related to a case where Thiele suffers an outright proportionality failure, described here and also in my paper.

From my paper:

20 to elect

2n voters: U1-U10, A1-A3
2n voters: U1-U10, B1-B3
n voters: C1-C6

The ideal result here would seem to be U1-U10, A1-A3, B1-B3 and C1-C4. The “UA” faction and the “UB” faction have 4/5 of the voters between them, and the “C” faction has 1/5. In terms of candidate numbers this result perfectly reflects that. However, Proportional Approval Voting would elect U1-U10, A1-A2, B1-B2 and C1-C6.

This is because after all the “U” candidates are elected, Proportional Approval Voting effectively considers UA and UB to be separate factions, meaning that they should each have twice the number of candidates as the C faction (12 to 6 in this case), rather than four times the total of the C faction when considered together. The fact that they are not completely separate factions is not taken into account.

If the UA faction and the UB faction were either in full agreement with each other or in complete disagreement, then they would get 16 candidates between them and the C faction would get 4 candidates. But this partial agreement counts against the UA and UB factions, and works in favour of the C faction, leaving the UA and UB factions with 14 candidates between them and the C faction with 6. While this isn’t a failure of PR in the simple way it has been defined in this paper, it would be a failure under a more nuanced definition.

It would fail proportionality by the definition in this thread.

(If there are equal numbers of voters and candidates, then as long as voters have approved enough candidates to make it possible, then you must be able to assign to each voter a unique candidate that they approved.)

I am not really a huge fan of Proportional Representation in general. If I could chose the system for a parliament I think the best outcomes would be obtained with single member score. I of course see that Proportional Representation is a good thing but it often has cost that need to be balanced. The other clear issue is that there is no good definition of it. Thiele methods are seem proportional enough for many to not be concerned with the proportionality failures. Getting a system that always fulfils the “Hare quota” definition of PR is much less important if failures are rare and contrived.

This would imply that that Thiele system would be preferable to Monroe/Unitary systems since participation failures are worse than PR failures. However, as you said above Thiele has issued when parties do not all vote along party lines. One example of this that voters can have elected two fully supported candidates and still influence the electiion even when they are not in a 2 quota sized faction. Another is the universally liked candidate/faction problem. Both these issues seem much more “wrong” to me than the simple failures of quota criteria. My fixed RRV does not solve either of these issues so I suspect it is a deep problem in Thiele theory.

In summary, I think that the proportionality failures can have huge effects on outcomes while the participation failures can’t. The proportionality failures are not really what one would see from simple/standard definitions of proportionality. This is because the participation failures are individual effects and the proportionality failures are group effects. This is not a silver bullet to kill Thiele systems but it is enough for me to put them behind other systems like SSS and SSQ. I would still take SPAV over STV.

@Toby_Pereira I made a page containing the above content.

The more I think about it the more this seems like a criteria worth discussing. Can you have a look at the page?

Really? I’m surprised you’ve dedicated so much time and effort to proportional systems then!

One major problem with using a single-winner system to make up a national parliament is that how the votes are spread geographically can make a major difference to how many seats each party, or ideological group, get. Clumped support in certain areas does much better than broad support across the whole country. In the 1983 UK general election, the Labour Party got 27.6% of the votes and won 209 seats, whereas the SDP-Liberal Alliance got 25.4% of the votes and just 23 seats. I think this is outrageous.

I’ve had a look at the ULC page. It seems to cover it quite well. I might have another look later to see if improvements can be made, but it basically covers it.

Well there are a few reasons for that:

  1. “He who knows only his own side of the case knows little of that." - John Stuart Mill

  2. Nobody had invented Sequentially Spent Score so as its inventor I had to show why it was good. It is a solution to a long standing problem and it would not make sense to ignore it. Especially considering the next point.

  3. Most of the reform efforts insist on PR and typically choose MMP or STV. If we are going to get stuck with PR it may as well be something good

  4. It is not that I do not value PR it is just that I value local representation and simplicity more

  5. I would also want other reforms in addition to a switch to score. For example government formation discussed here

  6. My priorities for reform are a) Fix vote splitting b) Lower partisanship c) Increase accountability d) Lower Wasted Votes. MMP and STV solve d) at the cost of b) and c) so its a bad deal.

  7. There is a bit of a middle ground. When I first invented Sequentially Spent Score it was intended to be done with Local district clusters instead of Multi-member districts. I still think this might be better.

This is a problem with the party system not the electoral system. Changing government formation would fix this. Gerrymandering should also be accounted for

I’m not sure that it is down to parties, or that sorting out gerrymandering would really fix it.

If a country’s election is actually hundreds of individual constituency elections, then the make-up of parliament will be very influenced by the geographical distribution of the voters. To take a simplistic scenario, 55% of voters might be “left” and 45% might be “right”. If voters are fairly uniformly distributed across the country, then left candidates could win potentially all the seats, and certainly a vast majority, regardless of whether parties are involved. On the other hand, if there are left regions and right regions, you could end up with something like a 55/45 split.

In another scenario, you might have 30% each for left and right, and 40% for centre. In a uniformly distributed case, centre candidates could win all the seats. But in the more polarised case, they could win none, being beaten 60/40 in every seat. Obviously it wouldn’t be quite as extreme as that, but they could still win either a lot or very few.

Given that the objective of a general election is to create a national parliament rather than hundreds of unconnected regional positions, it makes sense to have a system that represents the national make-up.

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Which is a very desirable thing.

I reject fundamentally that this is a reasonable way to look at the problem.

No an election is to create a parliament of hundreds of unconnected regional positions. Read Considerations on Representative Government is a by John Stuart Mill. Most political issues are local. Local representation is more important than broad ideological stuff. What might be considered left in one region might be considered right in another. This is all tied up with the idea of Ideal Representation

I disagree because then national issues are decided by where voters happen to live.

What in particular about it? Obviously it was overly simplistic, but more nuanced examples would give the same basic result.

I don’t agree with this at all. Parliament is a national body that votes on national issues. Sure, local issues are important, but the stuff that gets passed through national parliaments is generally national stuff.

You have local councils that deal with local stuff, and maybe you could argue that they should have more power to do so (how much power they have is likely to depend on your country anyway).

Also your local representative in parliament is there to represent you, but all they are really doing is voting yes or no on the national bills that get put to parliament. In any case, if your local representative is someone who you fundamentally disagree with on most things, what are you to do? Writing to them about your concerns is likely to fall on deaf ears. Whereas if you have, say, five local representatives elected through proportional representation, you have a choice of who to write to, and you’re far more likely to have at least one who is sympathetic to your views. I don’t see any disadvantages to this.

No, the national issues are decided by the consensus of communities. It is like hierarchical clustering. Each community finds a consensus representative and then the final consensus is found by the group of representatives.

It relies on the assumption that the only political difference is ideological and there is no regional component. Consider the allocation of federal funding for building a bridge. Or building an oil pipeline. With multi-member districts suburbs tend to lose out because they are part of the city. You get 5 winners from the city. The party alignment is matched but not the regional.

Not if you look at budgets or specific issues. All problems are local because all problems are with specific people. High level general solutions tend to fail.

They have no federal influence.

No they do much more than that. First they can table bills. Second they can bring up considerations for the review of bills. They get to hear out lobbies and citizens for specific local issues. They interact with the other governments (provincial/city) for their district. They actually work pretty hard.

Yup that sucks but they will normally agree with you on the bulk of things unless you are an extremist. Most things are simple representation. The big political decisions when the country is split is a clear issue where you would want to go with the national will of the people. This is what referendums are for.

This is actually the best argument so far. However, why is preventing you from writing to one of the MPs from the 5 adjacent districts? Also, in a system without vote splitting there is insentive for a MP to at least try to placate you.

I am not saying that PR does not have its benefits I am just saying that it has consequences. If I was given the option to have Sequentially Spent Score or Single Member Score with consensus government formation I would take the latter. It is a really hard choice. The thing is though, both are infinity better than MMP so I will advocate for whichever is able to get momentum. It seems that SSS would have more momentum because it is more aligned with the MMP people. This is my point 3 from above.

Why not have both with an upper and lower chamber of the legislature?

Well even getting an elected senate in Canada would be a good step.

One thing I have been thinking about which really only applies in places like Canada is that with multimember systems a 5 member district would be huge. Our territories are pretty big as single member districts. Combining just those three into a district would be of a size larger than Europe. So regions like this should be single member score. Dense cities would be 5 member PR systems. And there would be regions with sizes in between. There are some places like North Vancouver where due to geography it would make sense to have a 3 member district. So in the end we do not need to choose between single an multi member. It can be both if done properly.

Is this the crux of it then? This seems to be the main reason you’ve given for single-winner being actually better, as opposed to just not worse. It’s not a concern I’d ignore, but on the other hand I think the advantages of PR outweigh this. And why would you necessarily get five winners from the city? If the people in the suburbs were concerned about their local issues, they would vote for local candidates.

Obviously there are things that almost everyone would agree on but the issues that get debated in parliament are the issues where people are divided (or there wouldn’t be a debate), and it’s not unrealistic for someone to generally disagree with their own MP on most of these.

There are a lot of national issues that parliaments vote on. Some are complex as well, and a simple yes/no referendum would not help matters. And with the EU referendum we had in the UK in 2016 (which led us to leave the EU in 2020 although we’re still in a “transition” phase) it shows that these things can lead to more complications. Ordinary voters cannot be expected to be experts on so many issues, and so we have elected representatives who do it for a job.

You can do, but they might take you less seriously or have less time for you especially if they have thousands of their own constituents writing in. And with a single-winner non-PR system, your five adjacent districts might have MPs with similar ideologies - this would certainly be the case for me where I live.

There is a situation where this happened that @Sara_Wolf was talking about once before. There was a different ethnic group in a specific suburb. The parties were all dominated by the dominant ethnic group so they did not put up a candidate from that area who knew their issues. I could see this happening more so with infrastructure funds being allocated disproportionately to the city center but that is harder to notice.

I disagreed with all the candidates in my riding so much that I did not even vote last election. I would think that a single member system would allow for a local grass roots independent to have a better chance.

And this is why Mill emphasizes so much in “Considerations on Rep Gov” that it should be a single person who is held accountable that each person delegates their democratic power to. You are right that it is hard to hold them to account or even to get them to answer emails. I am not sure PR makes that much better. In fact if there are four other representatives it would seem easier for them to brush you off by saying that you should talk to one of the others.

In the end there are a ton of trade-offs. It is really hard to balance them out. There are also the issues of complexity and strategic voting. PR loses out in both of these.

Its a close call but in the end I think the next step would be to a single member score system. If it did not have the desired and expected consequences then Cardinal PR would be a step further. There is no parliament in the world currently elected by single member districts without vote splitting. There are plenty with STV and MMP. For all their failings they are PR under generally well excepted definitions. Politics in those countries is not clearly better.

I can understand why it is desirable to have a geographically representative legislature. However, geography influences the composition of a legislature elected in single member districts in ways besides ensuring that local constituencies get represented, since small district majorities get overrepresented and minority interests in a district go unrepresented. Gerrymandering is an obvious example of this problem being deliberately exploited, but it can be naturally occurring. Regional nationalist movements are often overrepresented, such as Bloq Quebecois until 2011, and the Scottish National Party in the House of Commons under Sturgeon.

Some communities may be unable to come to a consensus in the span of an election cycle.

The need to adequately represent geographically large, very sparsely populated regions and dense major cities would seem to support greater decentralization of power.