Five candidates — the score leader is overturned in the runoff¶
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 Boston. Only Approval differs, electing Austin: Approval counts every score of 3–5 as one equal 'approve' and ignores intensity, rewarding Austin's breadth of acceptability over Boston's stronger but more concentrated support. A threshold story about Approval, not a STAR-vs-IRV teaching case.
Winners by method¶
| Method | Winner |
|---|---|
| STAR | Boston |
| RCV-IRV | Boston |
| Ranked Robin (RCV-RR) | Boston |
| Approval | Austin |
| Range / Score | Austin |
| Condorcet | Boston |
Flags: none
Source election: 01_STAR/02_Examples/runoff_overturns_leader/cases/02_c5_b5_leader-overturned.yaml · STAR tabulated mirror: 02_c5_b5_leader-overturned_tabulated.txt
5 candidates, 5 ballots.
The ballots¶
Each row is a group of identical score ballots (0 = no support, 5 = max).
| Count | Austin | Boston | Chicago | Denver | Erie |
|---|---|---|---|---|---|
| 1 | 5 | 0 | 1 | 0 | 2 |
| 1 | 5 | 0 | 0 | 1 | 0 |
| 1 | 4 | 5 | 1 | 0 | 0 |
| 1 | 4 | 5 | 0 | 2 | 0 |
| 1 | 4 | 5 | 0 | 0 | 1 |
STAR result (official)¶
Scoring round (sum of scores): Austin 22, Boston 15, Denver 3, Erie 3, Chicago 2
Finalists (top two): Austin and Boston
Automatic runoff: Austin 2 vs Boston 3
STAR winner: Boston
Full LH STAR engine report:
--- Runoff (Preference) Matrix ---
Head-to-head / pairwise comparison
Legend: For - Equal Support - Against
* indicates Top 2 Finalist
| * Austin | * Boston |
-----------------------------------------
* Austin > | --- |2 - 0 - 3 |
* Boston > | 3 - 0 - 2 | --- |
[Divergence from STAR]
STAR = Boston
Approval = Austin (differs from STAR)
[Runoff Reversal]
- Score Round Winner(s) = (Austin)
- Runoff Round Winner = (Boston)
Candidate Austin earned the highest total score, but
Candidate Boston 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 5 ballots.
Austin,Boston,Chicago,Denver,Erie
5, 0, 1, 0, 2
5, 0, 0, 1, 0
4, 5, 1, 0, 0
4, 5, 0, 2, 0
4, 5, 0, 0, 1
[Score Distribution] (how many ballots gave each star rating)
Score
Candidate 5 4 3 2 1 0 | Total Avg
Austin 2 3 0 0 0 0 | 22 4.4
Boston 3 0 0 0 0 2 | 15 3.0
Chicago 0 0 0 0 2 3 | 2 0.4
Denver 0 0 0 1 1 3 | 3 0.6
Erie 0 0 0 1 1 3 | 3 0.6
Scoring Round
The two highest-scoring candidates advance to the next round.
Austin -- 22 -- First place
Boston -- 15 -- Second place
Denver -- 3
Erie -- 3
Chicago -- 2
Austin and Boston advance.
Automatic Runoff Round
The candidate preferred in the most head-to-head matchups wins.
Boston -- 3 -- First place
Austin -- 2
Equal Support -- 0
Boston wins.
Voters with a preference: 5 of 5 (no Equal Support).
Boston 3 (60%) vs Austin 2 (40%); majority = 3.
Winner — STAR Voting Method (single winner)
Boston
RCV-IRV — round by round¶
--- RCV / Instant-Runoff Voting (single winner) ---
Tabulating 5 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: Austin, Boston, Chicago, Denver, Erie
1 × Austin > Erie > Chicago (5, 0, 1, 0, 2)
1 × Austin > Denver (5, 0, 0, 1, 0)
1 × Boston > Austin > Chicago (4, 5, 1, 0, 0)
1 × Boston > Austin > Denver (4, 5, 0, 2, 0)
1 × Boston > Austin > Erie (4, 5, 0, 0, 1)
FINAL RESULT
Candidate Votes Status
----------- ------- --------
Boston 3 Elected
Austin 2 Rejected
Denver 0 Rejected
Erie 0 Rejected
Chicago 0 Rejected
Winner(s) — RCV / Instant-Runoff Voting (single winner)
Boston
--- 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 5): Boston
Outside (4): Austin, Chicago, Denver, Erie
One member ⇒ Boston is the Condorcet winner, beating every rival head-to-head.
RCV-IRV winner Boston 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 5 ballots (score ballots).
Ballots:
the ranking Ranked Robin reads ("=" = tied); source scores follow in () per column: Austin, Boston, Chicago, Denver, Erie
1 × Austin > Erie > Chicago > Boston=Denver (5, 0, 1, 0, 2)
1 × Austin > Denver > Boston=Chicago=Erie (5, 0, 0, 1, 0)
1 × Boston > Austin > Chicago > Denver=Erie (4, 5, 1, 0, 0)
1 × Boston > Austin > Denver > Chicago=Erie (4, 5, 0, 2, 0)
1 × Boston > Austin > Erie > Chicago=Denver (4, 5, 0, 0, 1)
Round-Robin — every pair, head-to-head (For – Against):
Boston beats Austin 3 – 2
Austin beats Chicago 5 – 0
Austin beats Denver 5 – 0
Austin beats Erie 5 – 0
Boston beats Chicago 3 – 1
Boston beats Denver 3 – 1
Boston beats Erie 3 – 1
Chicago ties Denver 2 – 2
Erie beats Chicago 2 – 1
Denver ties Erie 2 – 2
--- Pairwise (Round-Robin) Matrix ---
Head-to-head / pairwise comparison — the Ranked Robin tally
Legend: For - Equal Support - Against (row vs column)
| Austin | Boston | Chicago | Denver | Erie |
---------------------------------------------------------------------
Austin > | --- |2 - 0 - 3 |5 - 0 - 0 |5 - 0 - 0 |5 - 0 - 0 |
Boston > | 3 - 0 - 2 | --- |3 - 1 - 1 |3 - 1 - 1 |3 - 1 - 1 |
Chicago > | 0 - 0 - 5 |1 - 1 - 3 | --- |2 - 1 - 2 |1 - 2 - 2 |
Denver > | 0 - 0 - 5 |1 - 1 - 3 |2 - 1 - 2 | --- |2 - 1 - 2 |
Erie > | 0 - 0 - 5 |1 - 1 - 3 |2 - 2 - 1 |2 - 1 - 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 Boston 4–0–0 4 +7 Austin, Erie, Denver, Chicago
2 Austin 3–1–0 3 +14 Erie, Denver, Chicago
3 Erie 1–2–1 1.5 -6 Chicago
4 Denver 0–2–2 1 -7 —
5 Chicago 0–3–1 0.5 -8 —
Winner — Ranked Robin (RCV-RR): Boston
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 5): Boston
Outside (4): Austin, Chicago, Denver, Erie
One member ⇒ Boston is the Condorcet winner, beating every rival head-to-head.
Ranked Robin (RCV-RR) winner Boston 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