Three brothers, one fruit — the majoritarian winner is not the utilitarian one¶
▶ Live on BetterVoting: vote · results ↗ (election qywq7d · test BV2279) — 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 Banana. Only Approval differs, electing Orange: Approval counts every score of 3–5 as one equal 'approve' and ignores intensity, rewarding Orange's breadth of acceptability over Banana's stronger but more concentrated support. A threshold story about Approval, not a STAR-vs-IRV teaching case.
Winners by method¶
| Method | Winner |
|---|---|
| STAR | Banana |
| RCV-IRV | Banana |
| Ranked Robin (RCV-RR) | Banana |
| Approval | Orange |
| Range / Score | Orange |
| Condorcet | Banana |
Flags: none
Source election: method_comparisons/majoritarian_vs_utilitarian/cases/bv2279_qywq7d_star.yaml · STAR tabulated mirror: bv2279_qywq7d_star_tabulated.txt
3 candidates, 3 ballots.
The ballots¶
Each row is a group of identical score ballots (0 = no support, 5 = max).
| Count | Apple | Orange | Banana |
|---|---|---|---|
| 1 | 1 | 3 | 4 |
| 1 | 1 | 4 | 5 |
| 1 | 2 | 5 | 0 |
STAR result (official)¶
Scoring round (sum of scores): Orange 12, Banana 9, Apple 4
Finalists (top two): Orange and Banana
Automatic runoff: Orange 1 vs Banana 2
STAR winner: Banana
Full LH STAR engine report:
--- Runoff (Preference) Matrix ---
Head-to-head / pairwise comparison
Legend: For - Equal Support - Against
* indicates Top 2 Finalist
| * Orange | * Banana |
-----------------------------------------
* Orange > | --- |1 - 0 - 2 |
* Banana > | 2 - 0 - 1 | --- |
[Divergence from STAR]
STAR = Banana
Approval = Orange (differs from STAR)
[Runoff Reversal]
- Score Round Winner(s) = (Orange)
- Runoff Round Winner = (Banana)
Candidate Orange earned the highest total score, but
Candidate Banana 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 3 ballots.
Apple,Orange,Banana
1, 3, 4
1, 4, 5
2, 5, 0
[Score Distribution] (how many ballots gave each star rating)
Score
Candidate 5 4 3 2 1 0 | Total Avg
Apple 0 0 0 1 2 0 | 4 1.3
Orange 1 1 1 0 0 0 | 12 4.0
Banana 1 1 0 0 0 1 | 9 3.0
Scoring Round
The two highest-scoring candidates advance to the next round.
Orange -- 12 -- First place
Banana -- 9 -- Second place
Apple -- 4
Orange and Banana advance.
Automatic Runoff Round
The candidate preferred in the most head-to-head matchups wins.
Banana -- 2 -- First place
Orange -- 1
Equal Support -- 0
Banana wins.
Voters with a preference: 3 of 3 (no Equal Support).
Banana 2 (67%) vs Orange 1 (33%); majority = 2.
Winner — STAR Voting Method (single winner)
Banana
RCV-IRV — round by round¶
--- RCV / Instant-Runoff Voting (single winner) ---
Tabulating 3 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: Apple, Orange, Banana
1 × Banana > Orange > Apple (1, 3, 4)
1 × Banana > Orange > Apple (1, 4, 5)
1 × Orange > Apple (2, 5, 0)
FINAL RESULT
Candidate Votes Status
----------- ------- --------
Banana 2 Elected
Orange 1 Rejected
Apple 0 Rejected
Winner(s) — RCV / Instant-Runoff Voting (single winner)
Banana
--- 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 3): Banana
Outside (2): Apple, Orange
One member ⇒ Banana is the Condorcet winner, beating every rival head-to-head.
RCV-IRV winner Banana 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 3 ballots (score ballots).
Ballots:
the ranking Ranked Robin reads ("=" = tied); source scores follow in () per column: Apple, Orange, Banana
1 × Banana > Orange > Apple (1, 3, 4)
1 × Banana > Orange > Apple (1, 4, 5)
1 × Orange > Apple > Banana (2, 5, 0)
Round-Robin — every pair, head-to-head (For – Against):
Orange beats Apple 3 – 0
Banana beats Apple 2 – 1
Banana beats Orange 2 – 1
--- Pairwise (Round-Robin) Matrix ---
Head-to-head / pairwise comparison — the Ranked Robin tally
Legend: For - Equal Support - Against (row vs column)
| Apple | Orange | Banana |
----------------------------------------------
Apple > | --- |0 - 0 - 3 |1 - 0 - 2 |
Orange > | 3 - 0 - 0 | --- |1 - 0 - 2 |
Banana > | 2 - 0 - 1 |2 - 0 - 1 | --- |
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 Banana 2–0–0 2 +2 Orange, Apple
2 Orange 1–1–0 1 +2 Apple
3 Apple 0–2–0 0 -4 —
Winner — Ranked Robin (RCV-RR): Banana
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 3): Banana
Outside (2): Apple, Orange
One member ⇒ Banana is the Condorcet winner, beating every rival head-to-head.
Ranked Robin (RCV-RR) winner Banana 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