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

By Jan Kok & Warren D.Smith

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 Range-Voting-style ballot, while STV uses ranked ballots;
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 a 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 = 1 / (K + SUM/MAX)   where K is any positive constant. The range ½≤K≤1 seems most interesting. Our formula above had used K=1.

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 as the stepping stone; RV 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).

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).

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.)

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

The property fails.

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

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