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Two officers — where RRV parts company with the quota methods

Three voters, three candidates, two seats. Dana takes seat 1 under every method. Then Allocated Score and SSS seat Finn, while RRV seats Eli. The three methods charge Dana's two big backers ¼, ⅓ and 5⁄9 of their weight respectively — a ladder from harshest to gentlest — and the seat flips on the last rung.

→ Family: when the STAR-PR methods disagree · the other election: Three neighbors · the methods: Allocated Score · RRV

Level: 301 · deep dive

The ballots

Dana Eli Finn
Voter 1 4 3 1 Dana first, Eli a solid second
Voter 2 4 2 2 Dana first, Eli and Finn equal behind
Voter 3 1 3 4 Finn first, barely backs Dana
Total 9 8 7

Two seats, so the Hare quota is 3 ÷ 2 = 1.5 voters.

Seat 1 is unanimous

Dana leads at 9, and every method elects her. Voters 1 and 2 put her there with 4 stars each; voter 3 gave her a single star.

Seat 2: how hard are Dana's backers charged?

Voters 1 and 2 put Dana in with 4 stars each and now owe something for it. All three methods agree they owe something; they disagree about how much, and the answer is a clean ladder.

Allocated Score — voters 1 and 2 drop to ¼. The quota is 1.5 voters and the 4-star tier holds two ballots, so it overfills; fractional surplus keeps them at a quarter. Voter 3 is never reached by the tiers at all and stays at full weight:

weight Eli Finn
Voter 1 ¼ 0.75 0.25
Voter 2 ¼ 0.50 0.50
Voter 3 1 3.00 4.00
Round 2 4¼ 4¾ ← Finn

SSS — voters 1 and 2 drop to ⅓. SSS charges against a score quota rather than a ballot quota: total score 9, Hare score quota 7½, so every supporter's stars are reduced in proportion to what they gave. Voter 3 pays too, landing at ⅚ — she gave Dana a star, so she is charged for a star:

weight Eli Finn
Voter 1 ⅓ 1.00 0.33
Voter 2 ⅓ 0.67 0.67
Voter 3 ⅚ 2.50 3.33
Round 2 4⅙ 4⅓ ← Finn

Gentler than Allocated Score, and Eli closes from 0.50 behind to 0.17 behind — but not enough.

RRV — voters 1 and 2 only drop to 5⁄9. No quota is computed at all; each ballot is simply divided by 1 + (score given to winners / max score). Voters 1 and 2 gave 4 of 5, so 1 / 1.8 = 5⁄9. Voter 3 gave 1 of 5, so 1 / 1.2 = ⅚ — the same as under SSS:

weight Eli Finn
Voter 1 5⁄9 1.67 0.56
Voter 2 5⁄9 1.11 1.11
Voter 3 ⅚ 2.50 3.33
Round 2 5.28 ← Eli 5.00

Voters 1 and 2 keep enough to carry their shared second choice, and Eli takes seat 2 instead.

What the ladder means

Line the three charges up against the same two voters:

Method Voters 1 & 2 charged down to Eli − Finn Seat 2
Allocated Score ¼ = 0.250 −0.50 Finn
SSS ⅓ = 0.333 −0.17 Finn
RRV 5⁄9 = 0.556 +0.28 Eli

Nothing else differs — same ballots, same seat 1, same arithmetic afterwards. As the charge gets gentler, the two partly-satisfied voters keep more say and their second choice climbs, until at RRV it overtakes. A quota method decides those voters have had their seat and spends them down hard; a divisor method decides they are merely partly satisfied and turns them down by a proportion. The first hands seat 2 to the one voter still hungry; the second lets the partly-fed keep bidding.

Note where the two quota methods differ from each other, since it is easy to miss: Allocated Score leaves voter 3 at full weight because the score tiers never reach her, while SSS charges her ⅚ for the single star she gave Dana. Same winner here, different books — and Three neighbors is the election where that same bookkeeping gap changes the seat.

Neither is a malfunction. This is the Balinski–Young trade showing up in three ballots: the quota methods guarantee that a quota-sized faction can force a seat and pay with the non-monotonicity behind the Alabama paradox; RRV is coherent and monotone and pays by failing the Hare Quota Criterion. Here that abstraction has a face: voter 3 is a one-person faction just over a 1.5 quota's worth of unspent weight, and only the quota methods guarantee she gets the seat.

The counts

Allocated Score → Dana, Finn

--- Allocated Score Voting Method (2 winners) ---

[Allocated Score Voting]
 Tabulating 3 ballots to fill 2 seats.
Dana,Eli,Finn
   4,  3,   1
   4,  2,   2
   1,  3,   4

[Allocated Score Voting: Round 1]
 The highest-scoring candidate wins a seat.
   Dana          -- 9 -- First place
   Eli           -- 8
   Finn          -- 7
 Dana wins a seat.

[Allocated Score Voting: Round 1: Ballot allocation round]
 Allocating 1+1/2 ballots.

[Allocated Score Voting: Round 1: Ballot allocation round: Round 1]
 Allocating 2 ballots at score 4.
 This allocation overfills the quota.  Returning fractional surplus.
 Allocating only 75.00% of these ballots.
 Keeping these ballots, but multiplying their weights by 1/4.
 2 ballots reweighted from 1 to 1/4.

[Allocated Score Voting: Round 2]
 The highest-scoring candidate wins a seat.
   Finn          -- 4+3/4 -- First place
   Eli           -- 4+1/4
 Finn wins a seat.

[Allocated Score Voting: Winners — Allocated Score Voting Method (2 winners)]
 Dana
 Finn

Sequentially Spent Score → Dana, Finn

--- Sequentially Spent Score Voting Method (2 winners) ---

[Sequentially Spent Score]
 Tabulating 3 ballots to fill 2 seats.
Dana,Eli,Finn
   4,  3,   1
   4,  2,   2
   1,  3,   4

[Sequentially Spent Score: Round 1]
 The highest-scoring candidate wins a seat.
   Dana          -- 9 -- First place
   Eli           -- 8
   Finn          -- 7
 Dana wins a seat.

[Sequentially Spent Score: Round 1: Ballot allocation round]
 Total score is 9, Hare score quota is 7+1/2, giving back surplus.
 Reducing each ballot's stars by their vote * 1/6.
 Reweighted 3 ballots:
    2 ballots voted 4, stars reduced from 5 to 5/3, reweighted to 1/3.
    1 ballot voted 1, stars reduced from 5 to 25/6, reweighted to 5/6.

[Sequentially Spent Score: Round 2]
 The highest-scoring candidate wins a seat.
   Finn          -- 4+1/3 -- First place
   Eli           -- 4+1/6
 Finn wins a seat.

[Sequentially Spent Score: Winners — Sequentially Spent Score Voting Method (2 winners)]
 Dana
 Finn

Reweighted Range Voting → Dana, Eli

--- Reweighted Range Voting Method (2 winners) ---

[Reweighted Range Voting]
 Tabulating 3 ballots to fill 2 seats.
Dana,Eli,Finn
   4,  3,   1
   4,  2,   2
   1,  3,   4

[Reweighted Range Voting: Round 1: Score round]
 The highest-scoring candidate wins a seat.
   Dana          -- 9 -- First place
   Eli           -- 8
   Finn          -- 7
 Dana wins a seat.

[Reweighted Range Voting: Round 1: Reweighing Ballots]
 Reweighted 3 ballots:
   2 ballots reweighted from 1 to 5/9.
   1 ballot reweighted from 1 to 5/6.

[Reweighted Range Voting: Round 2: Score round]
 The highest-scoring candidate wins a seat.
   Eli           -- 5+5/18 -- First place
   Finn          -- 5
 Eli wins a seat.

[Reweighted Range Voting: Winners — Reweighted Range Voting Method (2 winners)]
 Dana
 Eli