Cardinal proportional representation

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Cardinal proportional representation (or "Cardinal PR") is a class containing cardinal voting systems used for proportional representation in multi-seat elections.

Because of the nature of rated ballots, it is possible to make assumptions that allow us to examine many different variations of what it means to "represent" voters in the multi-winner context, and to observe to what degree they are all represented.


Also see the following section for some categories.

When investigating cardinal PR, it is often categorized into optimal PR methods, which generally work by assigning every possible winner set a score based on how good it is, and picking the best winner set out of all possible winner sets, and sequential PR methods, which elect one candidate at a time. Optimal PR has the issue of being non-hand-countable and very computationally expensive and complex (in fact, with large committees, they may be both completely impossible to compute and very, very vulnerable to strategic voting[1]), so in practice, many sequential cardinal PR methods are designed to approximate certain optimal PR methods. When simulating the quality of various cardinal PR methods, sometimes it's common to use optimal PR methods more as "benchmarks" of how good the winner set chosen by the sequential method is, rather than as an actual way of running an election.

The KP transform can be very useful in allowing Approval PR methods (Approval voting-based cardinal PR methods) to work with rated ballots with more than two allowed scores.

Optimal methods

There is a certain parametrization of optimal PR methods that are in effect based on something like a highest averages method. Psi voting and harmonic voting are two voting methods that fall in this category. Psi voting becomes harmonic voting using the KP transform. Logarithms and the harmonic function appear prominently when discussing these methods.

Monroe's method is based on the theory that each voter should only have one representative.

Sequential methods

Sequential Monroe voting, Sequentially Spent Score, and Reweighted Range Voting are the most common such methods. They are each based on different philosophies of what PR should be and differ in the details of how best to optimize for their particular philosophies.

Often-discussed properties

Universally liked candidate criterion: Candidates who are given maximal support by all voters shouldn't affect the proportionality of the remaining candidates in the winner set when elected. Scale invariance: Multiplying all score by a constant leaves the result unaffected.


  • Under the Phragmén interpretation, voting is a distribution problem where the representation weight of candidates must be fairly spread across the different voters to produce the most equitable representation possible. The winner set composed of candidates which best distribute the candidates representation is the most proportional.
  • Under the Monroe or Eneström-Phragmén interpretation, voting is an attribution problem where every candidate has a quota of voters to be filled with specific voters. The winner set composed of candidates which maximizes the sum of score for the voters in that candidate’s quota is the most proportional. The voting method is impartial to how anybody outside of that candidate’s quota rates them.
  • Under the Thiele interpretation, voters have vote weight which should be distributed across candidates. The proportion of ballot weight assigned to each winner is the amount which that candidate supports their election. Under this interpretation, the more an outcome maximizes the sum of all score when reweighted by ballot weight, the more proportional it is.
  • Under the Unitary interpretation interpretation of each voter has an fixed amount of utility to be spent on candidates. When a candidate is elected their power to elect subsequent candidates is lower directly proportionally to the amount of utility previously spend on prior candidates. This interpretation can be thought of as an additional constraint on the Monroe interpretation but since the philosophy is about voters spending points on candidates rather than voters themselves being assigned to candidates it is a distinct interpretation of proportional representation. The Unitary interpretation is in some way the inverse interpretation of the Phragmén interpretation. In the former each voter has a conserved amount of vote weight to spend on candidates and in the latter the each candidate has a conserved amount of representation weight to distribute over the voters.
  • Under the COWPEA interpretation, the weight received by a candidate approved on a particular ballot would not be equal to the other candidates also approved on that ballot, but in a proportional manner according to the rest of the electorate.


To compare, Proportionality for Solid Coalitions can be thought of to some extent as a separate philosophy to Monroe because rather than trying to look at utility, it requires coherent groups to have a certain number of seats. PSC and Monroe can be made to conflict with examples where a solid coalition has some differences within itself, while another, smaller group is more unified; see PSC#Examples.

Proportionality for Solid Coalitions is praised for ensuring that groups of voters get what would intuitively be considered an at least somewhat proportional outcome, even if they disagree about what candidate best represents their faction. Most methods passing this criterion behave like Monroe and don't use preferences outside the faction to decide who will be a faction's best representative, but exceptions like STV-CLE exist.

Example systems

System Philosophy Comment
Sequential Monroe voting Monroe interpretation -
Method of Equal Shares Monroe interpretation -
Sequentially Spent Score Unitary interpretation -
Sequentially Shrinking Quota Unitary interpretation May not be strictly Unitary but follows from the theory
Sequential proportional approval voting Thiele Interpretation Approval ballots only
Reweighted Range Voting Thiele Interpretation May not be strictly Thiele but follows from the theory
Single distributed vote Thiele Interpretation A more Thiele implementation of Reweighted Range Voting
Sequential proportional score voting Thiele Interpretation
Harmonic Voting Thiele Interpretation
Sequential Phragmen Phragmén interpretation
Sequential Ebert Phragmén interpretation
PAMSAC Phragmén interpretation
COWPEA Lottery COWPEA interpretation

The backstory

Thiele, a Danish statistician, and Phragmen, a mathematician 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 afterward. Phragmen believed there were 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, 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 was lost to history until about a century later when they were independently rediscovered.

The Monroe interpretation named after the first first person to formalize the concept, Burt Monroe.[2] Single transferable vote is a Monroe type system which predates this formalization so it is clear that the core idea had existed for some time.

Keith Edmonds saw a unification of Proportional Representation and the concept of one person one vote which was maintained throughout winner the winner selection method. He coined the term "vote unitarity" for the second concept[3] and designed a score reweighting system which satisfied both Hare Quota Criterion and Vote Unitarity. As such it would preserve the amount of score used through sequential rounds while attributing representation in a partitioned way similar to Monroe. It would assign Hare Quotas of score to winners which allowed for a voters influence to be spread over multiple winners as opposed to Monroe which assigns a whole ballot with no spreading. Since score is a conserved quantity which is spent like money there is a natural analogy to Market based voting. This concept was heavily influence by economic theory not the Monroe interpretation even though the resultant mathematical formulation is quite similar.

Phragmen and Monroe share many desirable and undesirable properties. Most importantly a lack of convexity, the ability for votes that give every candidate the same score to affect the outcome. There are also election scenarios where both philosophies pick what is clearly the wrong winner. Further details can be found in 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)

However neither[clarification needed] fail the Universally liked candidate criterion which is a criterion that Thiele type methods fail.

Benefits of the Phragmen/Monroe/Unitary 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. Warren's multi-winner participation criteria.

Criticisms of the Phragmen metric:

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:

The Universally liked candidate criterion can be exemplified with the following example. 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!

The trouble with this is, politicians are not like tap water and oranges. That reasoning would make sense if politicians were “wholly owned” by the Blues, just as Peter wholly-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.

Incompatibility of the philosophies

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 affect 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 than 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 affect 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.

Peters and Skowron determined other properties that Phragmén passes but no Thiele-type method can pass, further indicating an incompatibility between the Phragmén and Theile philosophies.[4]


Because rated voting methods allow a voter to give no candidate the highest score, it is possible to give some voters less power to their ballots if they choose it. See normalization for discussion on this.

Just about all cardinal PR methods are immune to Woodall free riding, though their vulnerability to Hylland free riding varies. Some, like Sequentially Shrinking Quota, are maximally resistant.

It is possible to use a cardinal PR method to fill all but one of the seats in an election, and then use either STAR voting or a Condorcet-cardinal hybrid method to fill the final seat. For example, SPAV could be used to fill the first four of five seats, and then with the ballots in their reweighted forms, the Smith//Approval winner could be elected to the final seat. This can be done with ranked or rated ballots using approval thresholds to find out both approvals and head-to-head matchups. One advantage this holds over using STV#Deciding the election of the final seat or any ranked PR method with a Condorcet method for the final seat is that there appear to be no simple, hand-countable ranked PR methods that reduce to D'Hondt in their party list case, whereas SPAV and other cardinal PR methods do.

It should be noted that these methods follow different types and philosophies of proportionality than most other proportional representation methods. They all fail the "Proportionality for Solid Coalitions" criterion, though Sequential Monroe voting comes closest.

See also

Monotonicity appears very often in discussions of cardinal PR; it is a point of pride that Score voting passes every imaginable generalized form of monotonicity (which practically no other voting methods can), and cardinal PR advocates actively search for PR methods that imitate and extend that feat as much as possible in the multi-winner context.


Though most sequential cardinal PR methods aren't summable, they usually terminate in as many rounds as there are winners. When there is one winner, this means a precinct can submit its vote totals without needing to receive data back from a central authority, and when there are two winners, it need only be noted who won the first round for the precincts to determine who won the second. In general, one less two-way communication must occur than the number of rounds. This makes cardinal PR methods more feasible to be counted using two-way communication than something like STV, which can theoretically require [number of candidates - [number of winners]] rounds.


  1. Smith, Warren D. (October 2015). ""Optimal proportional representation" multiwinner voting systems I: methods, algorithms, advantages, and inherent flaws".
  2. Monroe, Burt L. (1995). "Fully Proportional Representation". American Political Science Review. Cambridge University Press (CUP). 89 (4): 925–940. doi:10.2307/2082518. ISSN 0003-0554. Retrieved 2020-02-09.
  4. Peters, Dominik; Skowron, Piotr (2019-11-26). "Proportionality and the Limits of Welfarism". arXiv:1911.11747 [cs.GT].