More stars, fewer voters — the runoff overturns the score leader¶
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 Brownie. Only Approval differs, electing Almond: Approval counts every score of 3–5 as one equal 'approve' and ignores intensity, rewarding Almond's breadth of acceptability over Brownie's stronger but more concentrated support. A threshold story about Approval, not a STAR-vs-IRV teaching case.
Winners by method¶
| Method | Winner |
|---|---|
| STAR | Brownie |
| RCV-IRV | Brownie |
| Ranked Robin (RCV-RR) | Brownie |
| Approval | Almond |
| Range / Score | Almond |
| Condorcet | Brownie |
Flags: none
Source election: 01_STAR/02_Examples/runoff_overturns_leader/cases/01a_c3_b3_more-stars-fewer-voters.yaml · STAR tabulated mirror: 01a_c3_b3_more-stars-fewer-voters_tabulated.txt
3 candidates, 3 ballots.
The ballots¶
Each row is a group of identical score ballots (0 = no support, 5 = max).
| Count | Almond | Brownie | Cocoa |
|---|---|---|---|
| 1 | 5 | 1 | 2 |
| 2 | 4 | 5 | 0 |
STAR result (official)¶
Scoring round (sum of scores): Almond 13, Brownie 11, Cocoa 2
Finalists (top two): Almond and Brownie
Automatic runoff: Almond 1 vs Brownie 2
STAR winner: Brownie
Full LH STAR engine report:
--- Runoff (Preference) Matrix ---
Head-to-head / pairwise comparison
Legend: For - Equal Support - Against
* indicates Top 2 Finalist
| * Almond | * Brownie |
--------------------------------------------
* Almond > | --- | 1 - 0 - 2 |
* Brownie > | 2 - 0 - 1 | --- |
[Divergence from STAR]
STAR = Brownie
Approval = Almond (differs from STAR)
[Runoff Reversal]
- Score Round Winner(s) = (Almond)
- Runoff Round Winner = (Brownie)
Candidate Almond earned the highest total score, but
Candidate Brownie 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.
Count × Almond,Brownie,Cocoa
2 × 4, 5, 0
1 × 5, 1, 2
[Score Distribution] (how many ballots gave each star rating)
Score
Candidate 5 4 3 2 1 0 | Total Avg
Almond 1 2 0 0 0 0 | 13 4.3
Brownie 2 0 0 0 1 0 | 11 3.7
Cocoa 0 0 0 1 0 2 | 2 0.7
Scoring Round
The two highest-scoring candidates advance to the next round.
Almond -- 13 -- First place
Brownie -- 11 -- Second place
Cocoa -- 2
Almond and Brownie advance.
Automatic Runoff Round
The candidate preferred in the most head-to-head matchups wins.
Brownie -- 2 -- First place
Almond -- 1
Equal Support -- 0
Brownie wins.
Voters with a preference: 3 of 3 (no Equal Support).
Brownie 2 (67%) vs Almond 1 (33%); majority = 2.
Winner — STAR Voting Method (single winner)
Brownie
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: Almond, Brownie, Cocoa
1 × Almond > Cocoa > Brownie (5, 1, 2)
2 × Brownie > Almond (4, 5, 0)
FINAL RESULT
Candidate Votes Status
----------- ------- --------
Brownie 2 Elected
Almond 1 Rejected
Cocoa 0 Rejected
Winner(s) — RCV / Instant-Runoff Voting (single winner)
Brownie
--- 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): Brownie
Outside (2): Almond, Cocoa
One member ⇒ Brownie is the Condorcet winner, beating every rival head-to-head.
RCV-IRV winner Brownie 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: Almond, Brownie, Cocoa
1 × Almond > Cocoa > Brownie (5, 1, 2)
2 × Brownie > Almond > Cocoa (4, 5, 0)
Round-Robin — every pair, head-to-head (For – Against):
Brownie beats Almond 2 – 1
Almond beats Cocoa 3 – 0
Brownie beats Cocoa 2 – 1
--- Pairwise (Round-Robin) Matrix ---
Head-to-head / pairwise comparison — the Ranked Robin tally
Legend: For - Equal Support - Against (row vs column)
| Almond | Brownie | Cocoa |
-----------------------------------------------
Almond > | --- |1 - 0 - 2 |3 - 0 - 0 |
Brownie > | 2 - 0 - 1 | --- |2 - 0 - 1 |
Cocoa > | 0 - 0 - 3 |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 Brownie 2–0–0 2 +2 Almond, Cocoa
2 Almond 1–1–0 1 +2 Cocoa
3 Cocoa 0–2–0 0 -4 —
Winner — Ranked Robin (RCV-RR): Brownie
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): Brownie
Outside (2): Almond, Cocoa
One member ⇒ Brownie is the Condorcet winner, beating every rival head-to-head.
Ranked Robin (RCV-RR) winner Brownie 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