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 counteract 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 PhragmenType 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 Thieletype 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, Phragmenthinking 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 applepreferrers 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, Thielethinking 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 Thieletype 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 wholyeats 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:

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)

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/Monroetype methods fail 1. and Thieletype 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?
Footnote:
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).