Thiele vs. Phragmen/Monroe: Two very different interpretations of proportionality


There are two/three main competing philosophies between what is and is not proportional.

Under the most Phragmen interpretation, voting is a balancing problem where the weights of candidates must be balanced between the different voters and the outcomes composed of candidates that best balance these weights are the most proportional.

Under the most Monroe interpretation, every candidate has a quota, and the more an outcome maximizes the scores voters in that candidate’s quota gives them, the more proportional the voting method is regardless of how anybody outside of that candidate’s quota rates them.

Under the most Thiele interpretation, every voter has an honest utility of each candidate, and even if you completely resent a candidate, it is statistically impossible for your honest utility of any individual candidate to equal 0 exactly. Under this interpretation, the more an outcome maximizes the sum among all voters: ln( the sum of utilities that voter gave to each winner ), the more proportional it is. Now obviously since while candidates can’t chose their honest utilities, they can chose the scores they give to candidates which means that it is much more likely that a candidate will give a set of candidates all zero scores which will blow up the natural log function (see footnote), so to counter-act this, the most Thiele voting methods instead use the partial sums of the harmonic function, which are closely related to the natural log (The natural log is the integral of 1/t from t=1 to t=x and the partial sums of the harmonic series are the summation of 1/n from n=1 to n=x).

The backstory: Thiele, a danish statistician, and Phragmen, a mathematician (and yes, oddly enough Thiele was the statistician and Phragmen was the mathematician, not the other way around), have been debating these two philosophies in Sweden. Thiele originally proposed sequential proportional approval voting in 1900 and it was adopted in Sweden in 1909 before Sweden switched to party list voting afterwards in order to make the number of seats parties won match their support even more closely. Phragmen believed there was flaws in Thiele’s method, and came up with his own sequential method to correct these flaws, and that started a debate about what was the ideal metric of proportionality. Thiele also came up with the approval ballot version of harmonic voting, and it would take about a century for both his sequential proportional approval voting method, however during that time, the harmonic method was too computationally exhaustive to be used in a governmental election. Both his sequential proportional approval voting and his approval ballot version of the harmonic method were lost to history until about a century later when they were independently rediscovered.

I lumped Phragmen and Monroe together in the title, because these two philosophies share many desirable and undesirable properties: mainly a lack of convexity (i.e. the weak monotonicity described on the document), the ability for votes that give every candidate the same score to effect the outcome, this (Read the “Pereira’s Complaints about Monroe” section of Monroe’s method or the “Major defect pointed out by Toby Pereira” section of this Phragmen-Type method) haunting election scenario where both philosophies pick what is clearly the wrong winner, and not failing the universally liked candidate criterion (which is a criterion that Thiele type methods fail).

Benefits of the Phragmen/Monroe measure of proportionality:

Passes the ULC criteria. For Thiele-type methods, because they fail ULC, every time a candidate that every voter gave a max rating to wins, the distribution of the remaining winners becomes more majoritarian/utilitarian.

Benefits of the Thiele measure of proportionality:

Adding ballots that give every candidate the same score can’t change which outcome is considered the best.
Convexity (Defined as Warren’s weak monotonicity criteria on the document).
Weak participation (also defined on the document).

Criticisms of the Phragmen metric (from Warren’s Thiele vs. Phragmen debate section):

“Taken to its limits, Phragmen-thinking would say, once the 50% Reds elected a red MP, and the 50% Blues elected a blue MP, there was no benefit whatever to replacing the red MP by somebody approved by the entire populace!”

Criticisms of the Thiele metric (from Warren’s Thiele vs. Phragmen debate section):

“Three people share a house and two prefer apples and one prefers oranges. One of the apple-preferrers does the shopping and buys three pieces of fruit. But instead of buying two apples and an orange, he buys three apples. Why? Because they all have tap water available to them already and he took this into account in the proportional calculations. And his reasoning was that the larger faction (of two) should have twice as much as the smaller faction (of one) when everything is taken into account, not just the variables. Taken to its logical conclusion, Thiele-thinking would always award the largest faction everything because there is so much that we all share – air, water, public areas, etc!”

Warren also gave a defense of this criticism of Thiele-type methods:

“The trouble with this is, politicians are not like tap water and oranges. That reasoning would make sense if politicians were “wholy owned” by the Blues, just as Peter wholy-eats an apple. But even the most partisan politicians in Canada do a lot of work to help Joe Average constituent whose political leanings they do not even know. At least, so I am told.”

Pick your poison: it seems that all proportional voting methods must fail one of two closely related properties:

  1. If a group of voters gives all the candidates the same score, that cannot effect the election results (ex: if you gave every candidate a max score, your vote shouldn’t change who is and isn’t a winner any more so then you would change the results by just not voting)

  2. If some of the winners are given the same score by all voters, that cannot effect the proportionality of the election results among the remaining winners (ex: if you removed a candidate that is given a max score by all voters, and ran the election again such that you were electing 1 less winner, the only difference between that election result and the original election result should be that it does not contain the universally liked candidate).

Phragmen/Monroe-type methods fail 1. and Thiele-type methods fail 2. and as of this point, it doesn’t seem possible to have them both without giving up PR.

I am on the Thiele side of this Thiele vs. Phragmen debate, and I believe Warren is also on this side of the debate, and that Jameson lies on the Phragmen side of this debate.

What version of proportionality do you prefer and what reasons do you have for preferring it?


In addition, maximizing the natural log favors small parties a little too much to pass proportional criteria and when a voter’s satisfaction is zero is just the most extreme example of that. The partial sums of the harmonic series equation does however pass the proportional criteria that a maximization of the natural log can’t. I personally think that the partial sums of the harmonic series are better for determining the winners of an election, but the natural log of summed utilities is a better tool for measuring proportionality in computer simulations even if those simulations are skewed to representing small parties too much (which may or may not be a bad thing).


Thanks, this is a good summary of this issue.

I think Phragmen has the best mathematical properties (both aesthetically and practically), but I favor Monroe for both strategic and philosophical reasons.

Strategic: In Monroe, your ballot will almost certainly be fully “used up” by somebody you rate highly. This means there’s less of a strategic incentive on ratings further down on the ballot. I think this would lead to less free-riding in practice, and to better expressivity. I also think that less strategic voting would lead to better voter satisfaction, not just in the VSE sense of outcomes (less risk of strategic backfire and dead-weight-loss), but in the sense of voter perceptions of the process as “fair” and rewarding to participate in.

Philosophical: I think that if you assume honest voting, Monroe is the best deterministic approximation of a “variance-minimizing sampling procedure”, which is the ideal for representing a distribution.

(Side comment 1: In terms of a rough analogy to sparse inference, I think that Monroe is analagous to the ideal-but-nonconvex-and-thus-intractable L0 penalty, Phragmen to the mathematically-clean-though-nonideal L1 LASSO penalty, and Thiele to the optimizing-a-different-thing-whose-philosophical-basis-I-like-less-but-which-does-have-practical-advantages L2 ridge penalty.)

(Side comment 2: The “ideal” nondeterministic thing to do, in the sense of variance-minimizing unbiased sampling, would be: assign all seats possible using Monroe/Hare, and then use random-ballot-without-replacement on the non-exhausted votes to assign the rest. If the random draw would assign two seats to one candidate, redraw, don’t just move down the second ballot’s preference order.)

Note that none of these three proportional paradigms (Phragmen, Thiele, or Monroe) puts any value on majoritarian “accountability”. I believe that purely optimizing for any proportional paradigm will probably lead to “Israel syndrome” — incentives to divide parties into tiny single-issue splinters, so that minority views hold the balance of power. (For instance, in Israel, most voters support the possibility of interfaith or nonreligious marriages, but fundamentalist parties who literally profit from holding a monopoly on marriages hold “kingmaker” status.) So I think that some small gestures towards majoritarian principles/accountablity, such as the local threshold in PLACE, are a good idea.


Between Monroe and Thiele would be a utilitarian view that cares about inequality. We should try to optimize global utility, but every voter only gets one unit of utility. Once they have one candidate they pretty well like elected then electing more like that doesn’t increase the global utility contributed from that voter. I think STV variants with fractional vote transfer enact this pretty well.


Yes, STV is the most-familiar example of a Monroe-based method.


As Jameson says, this is a good summary of the issues, so thanks for posting it.

Some of the problems that Phragmen and Monroe have (if you consider them to be of the same philosophy), I tried to address with my PAMSAC method, which pushes things in a generally more “monotonic” direction. Using Phragmen philosophy, where it’s all about balance, sometimes an additional approval for an elected candidate can upset the balance (if it means they now have more support than the other elected candidates when previously it was equal) and this can actually cause the slate of candidates to get a lower score and then cease to be the “best” slate of candidates according to what is being optimised.

In PAMSAC, any approvals that cause the score of a slate of candidates to be lowered are discounted when considering that slate. This restores at least weak monotonicity. However, this still might not seem entirely satisfactory, as you’d probably expect every approval to count positively rather than merely not count negatively.

Also, Phragmen/Monroe philosophy would consider a universally approved slate of candidates to be equal to one where half the voters each approve half of the candidates. PAMSAC also addresses this with a slightly weird transformation that halves the total amount of approvals given.

I do think that PAMSAC takes Phragmen philosophy about as far as it can be taken, but it’s still certainly not perfect. As Parker mentioned, extra voters who approve every candidate can change the result (although I think it might weakly pass this criterion) and it’s only weakly monotonic. It is also complex and unwieldy, and you can’t quickly check what would be the results in all but the simplest of election scenarios. There is a discussion of it here and a link to the main article:!msg/electionscience/UNy1f7Jsf1Y/gqvlcqenGQAJ

In terms of looking purely at the results that get churned out, I prefer PAMSAC to Thiele, but unfortunately that’s not the only thing to look at, and I think that pretty much anyone could understand how Thiele works in about five minutes. Well, they wouldn’t necessarily understand why it works, but they would understand what the method actually does. But on the other hand, with Phragmen/Monroe philosophy, I think it’s intuitively clear why it produces proportional representation. But with Thiele (and the divisor methods of PR like D’Hondt and Sainte-Laguë) I think it is intuitively less clear. Obviously reducing the power of someone’s vote once they have a candidate elected looks like it might be working in that general direction, but why does the 1/2, 1/3, 1/4 etc. (or 1/3, 1/5, 1/7 etc.) work exactly? From an intuitive point of view I cannot see it at all, although it can be shown fairly simply with the maths.

Also where Thiele fails “strong PR” (where the addition of universally approved candidates changes the proportions in which the other parties are elected), I think it was at least originally thought to be a more majoritarian approach - i.e. the total approvals of the slate of candidates elected under Thiele would be higher than that for PAMSAC (or another Phragmen method). But it was found not to be the case - there are examples where by passing strong PR, PAMSAC also elects a more majoritarian slate than Thiele. And at the time, I considered this to be the final nail in the coffin for Thiele, from a purely results (as opposed to implementation) point of view at least.

But anyway, might there not be a third way that solves the problems of both, or do you think this is impossible?


Its worth pointing out that this is for multiplicative reweighing. You can do it with subtraction reweigting by subtracting the score given to the winner from the some amount each voter has. This has lead to some pretty good results so far. And may end up being resulting in another class of “Holy Grails”. Current optimal version is defined as follows:

There is also a sequential version which looks at it in terms of Hare Quotas worth of score in the Monroe perspective. Then subtracts the score spent.


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How does Asset rate on the philosophies of proportionality?