Summability demo — STAR district B (Oak wins — a runoff reversal)¶
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 Oak. Only Approval differs, electing Pine: Approval counts every score of 3–5 as one equal 'approve' and ignores intensity, rewarding Pine's breadth of acceptability over Oak's stronger but more concentrated support. A threshold story about Approval, not a STAR-vs-IRV teaching case.
Winners by method¶
| Method | Winner |
|---|---|
| STAR | Oak |
| RCV-IRV | Oak |
| Ranked Robin (RCV-RR) | Oak |
| Approval | Pine |
| Range / Score | Pine |
| Condorcet | Oak |
Flags: none
Source election: method_comparisons/summability_demo/cases/star_district_B.yaml · STAR tabulated mirror: star_district_B_tabulated.txt
3 candidates, 3 ballots.
The ballots¶
Each row is a group of identical score ballots (0 = no support, 5 = max).
| Count | Maple | Oak | Pine |
|---|---|---|---|
| 1 | 1 | 5 | 4 |
| 1 | 0 | 5 | 3 |
| 1 | 4 | 0 | 5 |
STAR result (official)¶
Scoring round (sum of scores): Pine 12, Oak 10, Maple 5
Finalists (top two): Pine and Oak
Automatic runoff: Pine 1 vs Oak 2
STAR winner: Oak
Full LH STAR engine report:
--- Runoff (Preference) Matrix ---
Head-to-head / pairwise comparison
Legend: For - Equal Support - Against
* indicates Top 2 Finalist
| * Oak | * Pine |
-----------------------------------------
* Oak > | --- |2 - 0 - 1 |
* Pine > | 1 - 0 - 2 | --- |
[Divergence from STAR]
STAR = Oak
Approval = Pine (differs from STAR)
[Runoff Reversal]
- Score Round Winner(s) = (Pine)
- Runoff Round Winner = (Oak)
Candidate Pine earned the highest total score, but
Candidate Oak 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.
Maple,Oak,Pine
1, 5, 4
0, 5, 3
4, 0, 5
[Score Distribution] (how many ballots gave each star rating)
Score
Candidate 5 4 3 2 1 0 | Total Avg
Maple 0 1 0 0 1 1 | 5 1.7
Oak 2 0 0 0 0 1 | 10 3.3
Pine 1 1 1 0 0 0 | 12 4.0
Scoring Round
The two highest-scoring candidates advance to the next round.
Pine -- 12 -- First place
Oak -- 10 -- Second place
Maple -- 5
Pine and Oak advance.
Automatic Runoff Round
The candidate preferred in the most head-to-head matchups wins.
Oak -- 2 -- First place
Pine -- 1
Equal Support -- 0
Oak wins.
Voters with a preference: 3 of 3 (no Equal Support).
Oak 2 (67%) vs Pine 1 (33%); majority = 2.
Winner — STAR Voting Method (single winner)
Oak
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: Maple, Oak, Pine
1 × Oak > Pine > Maple (1, 5, 4)
1 × Oak > Pine (0, 5, 3)
1 × Pine > Maple (4, 0, 5)
FINAL RESULT
Candidate Votes Status
----------- ------- --------
Oak 2 Elected
Pine 1 Rejected
Maple 0 Rejected
Winner(s) — RCV / Instant-Runoff Voting (single winner)
Oak
--- 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): Oak
Outside (2): Maple, Pine
One member ⇒ Oak is the Condorcet winner, beating every rival head-to-head.
RCV-IRV winner Oak 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: Maple, Oak, Pine
1 × Oak > Pine > Maple (1, 5, 4)
1 × Oak > Pine > Maple (0, 5, 3)
1 × Pine > Maple > Oak (4, 0, 5)
Round-Robin — every pair, head-to-head (For – Against):
Oak beats Maple 2 – 1
Pine beats Maple 3 – 0
Oak beats Pine 2 – 1
--- Pairwise (Round-Robin) Matrix ---
Head-to-head / pairwise comparison — the Ranked Robin tally
Legend: For - Equal Support - Against (row vs column)
| Maple | Oak | Pine |
---------------------------------------------
Maple > | --- |1 - 0 - 2 |0 - 0 - 3 |
Oak > | 2 - 0 - 1 | --- |2 - 0 - 1 |
Pine > | 3 - 0 - 0 |1 - 0 - 2 | --- |
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 Oak 2–0–0 2 +2 Pine, Maple
2 Pine 1–1–0 1 +2 Maple
3 Maple 0–2–0 0 -4 —
Winner — Ranked Robin (RCV-RR): Oak
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): Oak
Outside (2): Maple, Pine
One member ⇒ Oak is the Condorcet winner, beating every rival head-to-head.
Ranked Robin (RCV-RR) winner Oak 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