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Reweighted Range Voting (RRV) — the divisor method in the family

Don't spend ballots — divide them. After each seat, every ballot's weight is divided by a growing factor based on how much score it has already given to winners. It is the oldest cardinal-PR proposal and the simplest to tabulate, and it is the one member of this family that does not pass the Hare Quota Criterion.

→ Family: the three STAR-PR methods · the theory underneath: the math behind proportional STAR · the quota methods it differs from: Allocated Score · SSS

Level: 301 · deep dive

How it counts

Seat the highest scorer. Then, rather than marking some voters represented and setting them aside, re-weight every ballot at once: a ballot's weight becomes 1 / (1 + sum_of_score_given_to_winners / max_score) — a D'Hondt/Jefferson-style divisor. A ballot that gave a seated winner 5 stars is halved; one that gave 0 is untouched; everything else lands in between. Re-tally with the new weights and seat the next candidate.

Nobody is ever "used up," and no quota is ever computed. That single design choice is the whole story below.

voting_method: rrv plus num_winners: k.

Watch it count

The same 63 ballots as the other two methods:

[Divergence from STAR]
  STAR                   = Ben
  Choose-One (Plurality) = Alice   (differs from STAR)
  Approval               = Alice   (differs from STAR)

--- Reweighted Range Voting Method (3 winners) ---

[Reweighted Range Voting]
 Tabulating 63 ballots to fill 3 seats.
Count × Alice,Ben,Cara,Dan,Eve
   18 ×     5,  4,   3,  0,  0
   15 ×     0,  0,   0,  5,  4
   12 ×     4,  5,   3,  0,  0
    9 ×     3,  4,   5,  0,  0
    9 ×     0,  0,   0,  4,  5

[Reweighted Range Voting: Round 1: Score round]
 The highest-scoring candidate wins a seat.
   Ben           -- 168 -- First place
   Alice         -- 165
   Cara          -- 135
   Dan           -- 111
   Eve           -- 105
 Ben wins a seat.

[Reweighted Range Voting: Round 1: Reweighing Ballots]
 Reweighted 39 ballots:
   27 ballots reweighted from 1 to 5/9.
   12 ballots reweighted from 1 to 1/2.

[Reweighted Range Voting: Round 2: Score round]
 The highest-scoring candidate wins a seat.
   Dan           -- 111 -- First place
   Eve           -- 105
   Alice         --  89
   Cara          --  73
 Dan wins a seat.

[Reweighted Range Voting: Round 2: Reweighing Ballots]
 Reweighted 24 ballots:
   15 ballots reweighted from 1 to 1/2.
   9 ballots reweighted from 1 to 5/9.

[Reweighted Range Voting: Round 3: Score round]
 The highest-scoring candidate wins a seat.
   Alice         -- 89 -- First place
   Cara          -- 73
   Eve           -- 55
 Alice wins a seat.

[Reweighted Range Voting: Winners — Reweighted Range Voting Method (3 winners)]
 Alice
 Ben
 Dan

Alice, Ben, Dan — the same slate the quota methods reach. A clean two-coalition electorate is exactly the case where the family difference is invisible. For the case where it is not, see the divergence below.

Where the methods actually part company

On the Lackner & Skowron shadow election — the running example used across the multi-winner literature, 7 candidates and 12 ballots, four seats — the identical ballots give:

Three seats agree; the fourth does not. That last seat is the quota-versus-divisor question made concrete, and it is the case to reach for when someone asks whether the choice between these tabulations is merely academic.

Where it stands, honestly

RRV is a divisor method, and the trade cuts both ways. Classical apportionment theory (Pukelsheim, ch. 9) proves a Coherence Theorem: a method is coherent — every subset of the winners, re-solved on its own, gives the same answer — if and only if it is a divisor method. So RRV is coherent, house-size monotone and vote-ratio monotone, and structurally immune to the family of paradoxes that quota methods are prone to — including the Alabama paradox this library demonstrates for Allocated Score. That is a real advantage, and it should not be buried under the criterion failure below.

The same coherence buys a second, less-advertised advantage: it resists the cheap free ride. A quota method spends supporters one score group at a time, so a ballot's exposure is a step function of the score it gave and there is an edge to stand just past. RRV has no groups — weight is 1 / (1 + score ÷ max_score), continuous in the score. On the library's worked case, dropping a landslide winner from 5 to 4 moves a free rider from 1/2 to 5/9 rather than from 1/2 to 1, and the seat does not flip; under Allocated Score and SSS it does. Worth stating precisely, because the usual claim in this area — that reweighting makes free riding riskier than in ranked methods — is true of RRV but not of the quota methods that also reweight.

What it pays for that: it does not guarantee quota. A faction holding a quota's worth of voters cannot always force a seat by voting as a bloc, so RRV fails the Hare Quota Criterion — which is why some classify it as semi-proportional rather than proportional. Structurally it belongs to the Thiele school, which equalizes satisfaction rather than allocating quotas; the criterion failure is a different answer to what "proportional" should mean, not a bug.

It is not on Equal Vote's shortlist, and that is consistent rather than incidental. Their committee's three finalists are Allocated Score, SSS and Sequential Monroe. RRV is the one that doesn't pass the quota criterion. Their summary of the trade, with the lean stated: RRV is the mathematically simplest tabulation and the oldest cardinal-PR proposal, but tends toward more utilitarian and less diversified winners, and is less transparent to non-mathematicians.

One more sensitivity worth knowing: because the divisor is computed from score relative to the maximum, RRV is more exposed than its siblings to the ballot's granularity — changing the score scale can flip the winner.

Scenarios in this library

Six case files use rrv, and between them they make the whole point — where RRV agrees with the quota methods, where it doesn't, and where it is the only one of the three that holds:

Scenario What it shows Read · run
The 63-ballot baseline The count above. Same slate as Allocated and SSS — the family difference stays invisible on a clean two-coalition race. page · yaml
Shadow — Lackner & Skowron The divergence: A, B, C, F where Allocated Score seats D, on identical ballots. page · yaml
Free riding, resisted The one-star free ride that flips the second seat under both quota methods — and fails here. RRV's best showing in this library. lesson · count