Coombs Ex.17 — STAR: Bree tops the scores, the Condorcet winner Arlo wins the runoff¶
▶ Live on BetterVoting: vote · results ↗ (election xbqq8t · test BV2164) — an independent tabulator to check the winners below against.
Bucket — APPROVAL_OR_MINOR: Only Approval differs
Generated by STARVote_LH_tabulation_engine/tools_adam/scripts/build_divergence_index.py — rebuilt from the election file; do not hand-edit.
What happens¶
STAR, RCV-IRV and Ranked Robin all agree on Arlo. Only Approval differs, electing Bree: Approval counts every score of 3–5 as one equal 'approve' and ignores intensity, rewarding Bree's breadth of acceptability over Arlo's stronger but more concentrated support. A threshold story about Approval, not a STAR-vs-IRV teaching case.
Winners by method¶
| Method | Winner |
|---|---|
| STAR | Arlo |
| RCV-IRV | Arlo |
| Ranked Robin (RCV-RR) | Arlo |
| Approval | Bree |
| Range / Score | Bree |
| Condorcet | Arlo |
Flags: none
Source election: method_comparisons/felsenthal_paradoxes/cases/bv2164_xbqq8t_star.yaml · STAR tabulated mirror: bv2164_xbqq8t_star_tabulated.txt
4 candidates, 33 ballots.
The ballots¶
Each row is a group of identical score ballots (0 = no support, 5 = max).
| Count | Arlo | Bree | Cole | Dana |
|---|---|---|---|---|
| 11 | 5 | 4 | 2 | 1 |
| 12 | 1 | 5 | 4 | 2 |
| 2 | 4 | 5 | 1 | 2 |
| 4 | 4 | 1 | 5 | 2 |
| 4 | 4 | 2 | 1 | 5 |
STAR result (official)¶
Scoring round (sum of scores): Bree 126, Arlo 107, Cole 96, Dana 67
Finalists (top two): Bree and Arlo
Automatic runoff: Bree 14 vs Arlo 19
STAR winner: Arlo
Full LH STAR engine report:
--- Runoff (Preference) Matrix ---
Head-to-head / pairwise comparison
Legend: For - Equal Support - Against
* indicates Top 2 Finalist
| * Arlo | * Bree |
-----------------------------------------------
* Arlo > | --- |19 - 0 - 14 |
* Bree > | 14 - 0 - 19 | --- |
[Divergence from STAR]
STAR = Arlo
Choose-One (Plurality) = Bree (differs from STAR)
Approval = Bree (differs from STAR)
[Runoff Reversal]
- Score Round Winner(s) = (Bree)
- Runoff Round Winner = (Arlo)
Candidate Bree earned the highest total score, but
Candidate Arlo won the automatic runoff — not a malfunction,
STAR working as designed: the runoff elects the finalist preferred
by the majority (of voters with a preference).
--- STAR Voting Method (single winner) ---
Tabulating 33 ballots.
Count × Arlo,Bree,Cole,Dana
12 × 1, 5, 4, 2
11 × 5, 4, 2, 1
4 × 4, 1, 5, 2
4 × 4, 2, 1, 5
2 × 4, 5, 1, 2
[Score Distribution] (how many ballots gave each star rating)
Score
Candidate 5 4 3 2 1 0 | Total Avg
Arlo 11 10 0 0 12 0 | 107 3.2
Bree 14 11 0 4 4 0 | 126 3.8
Cole 4 12 0 11 6 0 | 96 2.9
Dana 4 0 0 18 11 0 | 67 2.0
Scoring Round
The two highest-scoring candidates advance to the next round.
Bree -- 126 -- First place
Arlo -- 107 -- Second place
Cole -- 96
Dana -- 67
Bree and Arlo advance.
Automatic Runoff Round
The candidate preferred in the most head-to-head matchups wins.
Arlo -- 19 -- First place
Bree -- 14
Equal Support -- 0
Arlo wins.
Voters with a preference: 33 of 33 (no Equal Support).
Arlo 19 (58%) vs Bree 14 (42%); majority = 17.
Winner — STAR Voting Method (single winner)
Arlo
RCV-IRV — round by round¶
--- RCV / Instant-Runoff Voting (single winner) ---
Tabulating 33 ballots (converted from score ballots; 0 = unranked, equal scores broken by candidate priority).
Ballots:
the ranking RCV-IRV reads (0 = unranked, equal scores broken by priority);
the source score ballot follows in () per column: Arlo, Bree, Cole, Dana
11 × Arlo > Bree > Cole > Dana (5, 4, 2, 1)
12 × Bree > Cole > Dana > Arlo (1, 5, 4, 2)
2 × Bree > Arlo > Dana > Cole (4, 5, 1, 2)
4 × Cole > Arlo > Dana > Bree (4, 1, 5, 2)
4 × Dana > Arlo > Bree > Cole (4, 2, 1, 5)
ROUND 1
Candidate Votes Status
----------- ------- --------
Bree 14 Hopeful
Arlo 11 Hopeful
Cole 4 Rejected
Dana 4 Rejected
FINAL RESULT
Candidate Votes Status
----------- ------- --------
Arlo 19 Elected
Bree 14 Rejected
Cole 0 Rejected
Dana 0 Rejected
Winner(s) — RCV / Instant-Runoff Voting (single winner)
Arlo
--- Smith Set (the generalized Condorcet winner) ---
The smallest group whose every member beats every candidate outside it —
the honest answer to "who is even in contention?".
Smith set (1 of 4): Arlo
Outside (3): Bree, Cole, Dana
One member ⇒ Arlo is the Condorcet winner, beating every rival head-to-head.
RCV-IRV winner Arlo is INSIDE the Smith set. ✓
Not guaranteed — RCV-IRV is not Smith-efficient — but it holds here.
More: 07_Concepts/topics/smith_set.md
NOTE: a generated cross-method view of the STAR ballots, for comparison only — not the official STAR result.
Ranked Robin (RCV-RR) — every pair, head-to-head¶
--- Ranked Robin (RCV-RR / Copeland) Method (single winner) ---
Tabulating 33 ballots (score ballots).
Ballots:
the ranking Ranked Robin reads ("=" = tied); source scores follow in () per column: Arlo, Bree, Cole, Dana
11 × Arlo > Bree > Cole > Dana (5, 4, 2, 1)
12 × Bree > Cole > Dana > Arlo (1, 5, 4, 2)
2 × Bree > Arlo > Dana > Cole (4, 5, 1, 2)
4 × Cole > Arlo > Dana > Bree (4, 1, 5, 2)
4 × Dana > Arlo > Bree > Cole (4, 2, 1, 5)
Round-Robin — every pair, head-to-head (For – Against):
Arlo beats Bree 19 – 14
Arlo beats Cole 17 – 16
Arlo beats Dana 17 – 16
Bree beats Cole 29 – 4
Bree beats Dana 25 – 8
Cole beats Dana 27 – 6
--- Pairwise (Round-Robin) Matrix ---
Head-to-head / pairwise comparison — the Ranked Robin tally
Legend: For - Equal Support - Against (row vs column)
| Arlo | Bree | Cole | Dana |
-------------------------------------------------------------------
Arlo > | --- |19 - 0 - 14 |17 - 0 - 16 |17 - 0 - 16 |
Bree > | 14 - 0 - 19 | --- |29 - 0 - 4 |25 - 0 - 8 |
Cole > | 16 - 0 - 17 | 4 - 0 - 29 | --- |27 - 0 - 6 |
Dana > | 16 - 0 - 17 | 8 - 0 - 25 | 6 - 0 - 27 | --- |
Win–loss record — Copeland score = wins + ½·ties (highest score wins; ties broken by total margin, then lot order):
# Candidate W–L–T Copeland Margin Beats
1 Arlo 3–0–0 3 +7 Bree, Cole, Dana
2 Bree 2–1–0 2 +37 Cole, Dana
3 Cole 1–2–0 1 -5 Dana
4 Dana 0–3–0 0 -39 —
Winner — Ranked Robin (RCV-RR): Arlo
beats every opponent head-to-head — the Condorcet winner.
--- Smith Set (the generalized Condorcet winner) ---
The smallest group whose every member beats every candidate outside it —
the honest answer to "who is even in contention?".
Smith set (1 of 4): Arlo
Outside (3): Bree, Cole, Dana
One member ⇒ Arlo is the Condorcet winner, beating every rival head-to-head.
Ranked Robin (RCV-RR) winner Arlo is INSIDE the Smith set. ✓
Guaranteed: Ranked Robin (Copeland) is Smith-efficient — every member of
the set outscores every outsider, so the top of the win–loss table is
always inside the set, however the tie among them is then broken.
More: 07_Concepts/topics/smith_set.md