Reweighted Range Voting – a Proportional Representation voting method that feels like range voting

By Jan Kok & Warren D.Smith. (There is also an easier-to-digest[?] RRV page by Ivan Ryan.)

Reweighted Range Voting (RRV) is based on STV (it uses the STV ideas of "Droop quota" and "ballot reweighting"), and chooses multiple winners in such a way as to obtain proportional representation. The main differences between RRV and STV that would be apparent to ordinary people are:

  1. RRV uses a more-expressive Range-Voting-style ballot – voters "score" each candidate from (say) 0=bad to 9=good – while STV uses rank-order ballots in which expressing "strength of preference" is impossible;
  2. It is simpler to explain how RRV works.

How RRV works

  1. Each ballot is given an initial "weight" of 1.
  2. The weighted scores on the ballots are summed for each candidate, thus obtaining that candidate's total score.
  3. The candidate with the highest total score (who has not already won) is declared a winner. (Note that the first winner in RRV is the same as the winner of an ordinary single-winner Range Voting election using the same ballots.)
  4. When a voter "gets her way" in the sense that a candidate she rated highly wins, her ballot weight should be reduced so that she has less influence on later choices of winners. To accomplish that, each ballot is given a new weight = 1/(1+SUM/MAX), where SUM is the sum of the scores that ballot gives to the winners-so-far, while MAX is the maximum allowed score (e.g. MAX=99 if allowed scores are in the range 0 to 99).
  5. Repeat steps b-d until the desired number of winners has been chosen.
One can instead employ this formula in step d:   weight = K / (K + SUM/MAX)   where K is any positive constant. The range ½≤K≤1 seems most interesting. Our formula above had used K=1, which is analogous to Jefferson & d'Hondt notions of proportionality, which tend to favor large political parties (incentivizes smaller parties to merge); meanwhile K=½ is analogous to Webster & Sainte-Laguë notions, which tend to be "fairer" and provide much smaller merge-incentive (and perhaps not even of the same sign, it might be a slight split-incentive).

We also remark that votes incorporating "no opinion" scores on candidates could also be allowed. Ballots employing them will not affect the weighted-average scores for any candidates they rate with "no opinion" and such ballots will not be re-weighted when such a candidate wins.

Beyond RRV's obvious simplicity advantage, it has other advantages such as monotonicity. That is, with RRV, if a voter increases a rating for a candidate, that will never change that candidate from a winner to a loser. (With STV, giving a candidate a better rank can cause that candidate to lose, even in the single-winner [IRV] case.)

You may be thinking, "STV is good enough, why should we consider another PR method?" One good reason to think about RRV has to do with single-winner methods.

RRV is a PR method that doesn't require IRV as a stepping stone. Rather, it uses Range Voting (also called "score voting") as the stepping stone; i.e. score voting is the corresponding single-winner method.

Proportionality Theorem

If some voter faction (call them the "Reds"), consisting of a fraction F (where 0≤F<1) of the voters, wants to, it is capable (regardless of what the other voters do) of electing at least ⌊(1+N)F-⌋ red winners (assuming, of course, that at least this many red candidates run, and the total number of winners is to be N).

Specifically, it can accomplish that by voting MAX for all Reds and MIN for everybody else.

To say that again: if 37% of the voters are reds, they can assure at least about 37% red winners (up to rounding-to-integers effects).


Although I believed I was the inventor of RRV in 2004, it turns out that Thorvald N. Thiele invented a special case of this system – namely where the only allowed scores are 0 and 1 (approval-style voting) and with K=1 – in 1895. Thiele's system actually was used in Sweden in about 1910 (enacted 1907), but attributed to Alfred Petersson of Pådoba, a prominent MP (and at least at one time a member of the Liberal party, although I do not know what party he was in at the time he proposed this) and wealthy farmer. I would like to know more about what happened during this period of RRV use and why Sweden abandoned it in 1921 to replace it by a cruder "party list" PR system. A scholar could probably understand that by reading old Swedish government documents. I, however, am not that good a scholar and therefore resorted to reading books and papers about Sweden written in English. The closest I found to an explanation for its abandonment was the following rather mysterious sentence:

The regulations adopted in 1907 were found to be technically unsatisfactory, and so various changes have been made in the method of election and particularly in the technical aspects of the system.
  – Nils Herlitz: Proportional Representation in Sweden, American Political Science Review 19,3 (Aug 1925) 582-592. (English translation by Fred Berquist & Clarence A. Berdahl.)

Herlitz was a law/history professor at University of Stockholm, and by 1925 Sweden had switched to a party list system within multiwinner districts, instituted universal suffrage including for women, and become generally more democratic and more PR all over. What did Herlitz's sentence mean? My guess is it just meant that Thiele was a lot more difficult to count than party list PR systems. It (like STV) needs to be counted centrally; we cannot just use subtotal tables from precincts. And Thiele requires multiple counting passes, with different ballot weights, not just one. In the pre-computer era all that was a considerable burden. As far as the election results they produced are concerned, my reading did not uncover any commentator complaints about either Thiele or its party-list replacement. Both generated multiparty parliaments which tended mainly to yield "coalition governments"; and the number of parties in Sweden tended to increase over time.

Thiele's system was criticized by Lars Edvald Phragmen and Nore B. Tenow; learn more. And indeed the Thiele-like system Sweden enacted in 1907 (described in English by J.H.Humphreys in this excerpt from his 1911 book) actually incorporated a modification designed to repair the most important such criticism. Specifically, for j=1,2,3,... while this keeps working: if more than j/(j+1) fraction of the voters have approved all members of a set of j candidates, then the top-approved such set is elected. (For example if the top 2 candidates are both approved on at least a fraction 2/3 of the ballots, then they win seats.) Only after these "easy call" candidates had been elected, did Sweden begin the Thiele process, and it was used to allocate only the remaining unfilled seats – and therefore only they obeyed a proportionality theorem.


While RRV seems superior to STV both in simplicity and properties, that is not to say that it is perfect. Two flaws in RRV (which also are flaws in STV) are

  1. a multiwinner analogue of "participation failure," and
  2. the fact that it cannot be "counted in precincts" but only centrally.
    Forest Simmons in 2007 solved an open problem by showing how to design PR multiwinner voting methods that are countable in precincts – see puzzle 15 – but that is another story for another day.

To explain the former: here is a desirable-sounding property for multiwinner voting systems:

Multiwinner "participation property":
By casting an honest vote, you cannot cause X to be elected instead of Q (with all other winners staying the same) if you prefer Q over X.

The "STV" system used in Ireland and Australia definitely fails this property, since its single-winner special case (instant runoff voting, IRV) fails it.

What about our new RRV system? It obeys this property in the single-winner case (because that is just range voting). But it fails it in the following 140-voter 3-candidate 2-winner election example:

#voterstheir vote
50Z=99, X=42, Q=0
50X=99, Q=43, Z=0
40Q=99, Z=53, X=0

In the first round, the totals are Z=7070, X=7050, and Q=6110, so RRV elects Z. That deweights X so that, second round, Q wins. (The second round totals are Q=6110, and X=6000.)

Now you (an extra voter) come with a vote Q>X>Z, for example

Q=99, X=77, Z=0.
That makes X win the first round. (First round totals: X=7127, Z=7070, and Q=6209.) But that win heavily deweights Q, allowing Z to win the second round. (Second round totals: Z=5134, Q=5059.8.)

Summary of situation:
Before you vote:Z & Q win.
After your Q>X>Z vote:Z & X win.

The property fails.


RRV is now used by the OSCARs to select the 5 nominees for "Best Visual Effects" award for movies each year, according to this rule

"Five productions shall be selected using reweighted range voting to become the nominations for final voting for the Visual Effects Award."
from They use an 0-10 scale for ratings. Voting for the OSCARs is run by the accounting firm PriceWaterhouseCoopers. [Presumably, this was done as an experiment with the idea of switching other OSCAR categories to use RRV if it worked out well.] The apparently first time (2013) they employed this system, the 5 nominees were:

Published pre-election predictions included:

The difference between the actual winning 5 from RRV versus these predictions indicates (Shentrup speculates) that RRV succeeded in thus-getting more diverse nominees, thus demonstrating "proportional representation" in action.

An example election where RRV seems to greatly outperform PR-STV.

A real-world high stakes election with 9 winners and 39 voters that was carried out with RRV (you will need to view it with wide window).

Billy Tetrud writes a simulator which seems to indicate that RRV outperforms STV. Indeed STV is outperformed by various non-PR multiwinner methods! That seems fairly damning (to whatever extent Tetrud's simulator is valid).

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