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Three neighbors — where Allocated Score and SSS part company

Three voters, three candidates, two seats. Cleo takes seat 1 under every method. Then Allocated Score seats Bo, who scored 3 points in total, while SSS and RRV seat Ana, who scored 8. The disagreement is entirely about what one lukewarm supporter's two stars cost her.

→ Family: when the STAR-PR methods disagree · the other election: Two officers · the methods: Allocated Score · SSS

Level: 301 · deep dive

The ballots

Ana Bo Cleo
Neighbor 1 4 0 5 loves Cleo, likes Ana
Neighbor 2 0 2 2 lukewarm on both, nothing for Ana
Neighbor 3 4 1 5 loves Cleo, likes Ana
Total 8 3 12

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

Seat 1 is unanimous

Cleo leads on score, 12 to Ana's 8, and every method elects her. Nothing interesting yet — the interesting part is the bill.

Seat 2: who paid for Cleo?

Cleo's support came from three different places: two 5-star ballots and one 2-star ballot. Each method charges for that differently.

Allocated Score allocates by score tier. It takes the 5-star group first — neighbors 1 and 3, two ballots — and that alone overfills the 1.5-voter quota. Fractional surplus returns the excess, leaving those two at quarter weight; the 4-star tier is never reached, and neither is the 2-star tier. So neighbor 2 is never allocated at all and keeps her full weight of 1.0:

weight Ana Bo
Neighbor 1 0.25 1.00 0.00
Neighbor 2 1.00 0.00 2.00
Neighbor 3 0.25 1.00 0.25
Round 2 total 2.00 2.25

Neighbor 2 outweighs the other two combined and elects her best remaining candidate. Bo takes seat 2 — on three total points, against Ana's eight.

SSS charges every supporter in proportion to the score they gave. Neighbor 2's two stars for Cleo were real support, so they cost her real weight. She can no longer outweigh what neighbors 1 and 3 have left, and their 4-star backing carries Ana in instead.

RRV divides rather than spends, and lands with SSS: Ana.

Is Bo's win a bug?

No — and this is the part worth sitting with. Neighbor 2 is a minority of one who got only partial representation from Cleo, and Allocated Score's answer is that she is therefore still owed a say. Handing seat 2 to the voter who is least represented is exactly what a proportional method is supposed to do. Score totals are not the standard here; Bloc STAR would elect Ana and Cleo and leave neighbor 2 with nothing at all.

The genuine disagreement is narrower, and it is about accounting: did neighbor 2's two stars for Cleo buy her anything? Allocated Score says no — she never reached a tier, so she is untouched and fully owed. SSS says yes — she got partial value and should pay partially for it. Both are defensible readings of "represented to a degree", which is why both are on Equal Vote's shortlist.

The counts

Allocated Score → Bo, Cleo

[Divergence from STAR]
  STAR     = Cleo
  Approval = Ana   (differs from STAR)

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

[Allocated Score Voting]
 Tabulating 3 ballots to fill 2 seats.
Ana,Bo,Cleo
  4, 0,   5
  0, 2,   2
  4, 1,   5

[Allocated Score Voting: Round 1]
 The highest-scoring candidate wins a seat.
   Cleo          -- 12 -- First place
   Ana           --  8
   Bo            --  3
 Cleo 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 5.
 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.
   Bo            -- 2+1/4 -- First place
   Ana           -- 2
 Bo wins a seat.

[Allocated Score Voting: Winners — Allocated Score Voting Method (2 winners)]
 Bo
 Cleo

Sequentially Spent Score → Ana, Cleo

[Divergence from STAR]
  STAR     = Cleo
  Approval = Ana   (differs from STAR)

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

[Sequentially Spent Score]
 Tabulating 3 ballots to fill 2 seats.
Ana,Bo,Cleo
  4, 0,   5
  0, 2,   2
  4, 1,   5

[Sequentially Spent Score: Round 1]
 The highest-scoring candidate wins a seat.
   Cleo          -- 12 -- First place
   Ana           --  8
   Bo            --  3
 Cleo wins a seat.

[Sequentially Spent Score: Round 1: Ballot allocation round]
 Total score is 12, Hare score quota is 7+1/2, giving back surplus.
 Reducing each ballot's stars by their vote * 3/8.
 Reweighted 3 ballots:
    2 ballots voted 5, stars reduced from 5 to 15/8, reweighted to 3/8.
    1 ballot voted 2, stars reduced from 5 to 15/4, reweighted to 3/4.

[Sequentially Spent Score: Round 2]
 The highest-scoring candidate wins a seat.
   Ana           -- 3     -- First place
   Bo            -- 1+7/8
 Ana wins a seat.

[Sequentially Spent Score: Winners — Sequentially Spent Score Voting Method (2 winners)]
 Ana
 Cleo

Reweighted Range Voting → Ana, Cleo

[Divergence from STAR]
  STAR     = Cleo
  Approval = Ana   (differs from STAR)

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

[Reweighted Range Voting]
 Tabulating 3 ballots to fill 2 seats.
Ana,Bo,Cleo
  4, 0,   5
  0, 2,   2
  4, 1,   5

[Reweighted Range Voting: Round 1: Score round]
 The highest-scoring candidate wins a seat.
   Cleo          -- 12 -- First place
   Ana           --  8
   Bo            --  3
 Cleo wins a seat.

[Reweighted Range Voting: Round 1: Reweighing Ballots]
 Reweighted 3 ballots:
   2 ballots reweighted from 1 to 1/2.
   1 ballot reweighted from 1 to 5/7.

[Reweighted Range Voting: Round 2: Score round]
 The highest-scoring candidate wins a seat.
   Ana           -- 4       -- First place
   Bo            -- 1+13/14
 Ana wins a seat.

[Reweighted Range Voting: Winners — Reweighted Range Voting Method (2 winners)]
 Ana
 Cleo