Reinforcement — combined 9 voters, counted by STAR (scoring round → Ada, runoff → Cara)¶
▶ Live on BetterVoting: vote · results ↗ (election t4by6x · test BV2254) — 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 Cara. Only Approval differs, electing Ada: Approval counts every score of 3–5 as one equal 'approve' and ignores intensity, rewarding Ada's breadth of acceptability over Cara's stronger but more concentrated support. A threshold story about Approval, not a STAR-vs-IRV teaching case.
Winners by method¶
| Method | Winner |
|---|---|
| STAR | Cara |
| RCV-IRV | Cara |
| Ranked Robin (RCV-RR) | Cara |
| Approval | Ada |
| Range / Score | Ada |
| Condorcet | Cara |
Flags: none
Source election: method_comparisons/reinforcement_paradox/cases/reinf_combined_c3_b9_star.yaml · STAR tabulated mirror: reinf_combined_c3_b9_star_tabulated.txt
3 candidates, 9 ballots.
The ballots¶
Each row is a group of identical score ballots (0 = no support, 5 = max).
| Count | Ada | Ben | Cara |
|---|---|---|---|
| 2 | 5 | 3 | 0 |
| 2 | 0 | 5 | 3 |
| 3 | 3 | 0 | 5 |
| 2 | 5 | 0 | 3 |
STAR result (official)¶
Scoring round (sum of scores): Ada 29, Cara 27, Ben 16
Finalists (top two): Ada and Cara
Automatic runoff: Ada 4 vs Cara 5
STAR winner: Cara
Full LH STAR engine report:
--- Runoff (Preference) Matrix ---
Head-to-head / pairwise comparison
Legend: For - Equal Support - Against
* indicates Top 2 Finalist
| * Ada | * Cara |
-----------------------------------------
* Ada > | --- |4 - 0 - 5 |
* Cara > | 5 - 0 - 4 | --- |
[Divergence from STAR]
STAR = Cara
Choose-One (Plurality) = Ada (differs from STAR)
Approval = Ada (differs from STAR)
[Runoff Reversal]
- Score Round Winner(s) = (Ada)
- Runoff Round Winner = (Cara)
Candidate Ada earned the highest total score, but
Candidate Cara 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 9 ballots.
Count × Ada,Ben,Cara
3 × 3, 0, 5
2 × 5, 3, 0
2 × 0, 5, 3
2 × 5, 0, 3
[Score Distribution] (how many ballots gave each star rating)
Score
Candidate 5 4 3 2 1 0 | Total Avg
Ada 4 0 3 0 0 2 | 29 3.2
Ben 2 0 2 0 0 5 | 16 1.8
Cara 3 0 4 0 0 2 | 27 3.0
Scoring Round
The two highest-scoring candidates advance to the next round.
Ada -- 29 -- First place
Cara -- 27 -- Second place
Ben -- 16
Ada and Cara advance.
Automatic Runoff Round
The candidate preferred in the most head-to-head matchups wins.
Cara -- 5 -- First place
Ada -- 4
Equal Support -- 0
Cara wins.
Voters with a preference: 9 of 9 (no Equal Support).
Cara 5 (56%) vs Ada 4 (44%); majority = 5.
Winner — STAR Voting Method (single winner)
Cara
RCV-IRV — round by round¶
--- RCV / Instant-Runoff Voting (single winner) ---
Tabulating 9 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: Ada, Ben, Cara
2 × Ada > Ben (5, 3, 0)
2 × Ben > Cara (0, 5, 3)
3 × Cara > Ada (3, 0, 5)
2 × Ada > Cara (5, 0, 3)
ROUND 1
Candidate Votes Status
----------- ------- --------
Ada 4 Hopeful
Cara 3 Hopeful
Ben 2 Rejected
FINAL RESULT
Candidate Votes Status
----------- ------- --------
Cara 5 Elected
Ada 4 Rejected
Ben 0 Rejected
Winner(s) — RCV / Instant-Runoff Voting (single winner)
Cara
--- 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): Cara
Outside (2): Ada, Ben
One member ⇒ Cara is the Condorcet winner, beating every rival head-to-head.
RCV-IRV winner Cara 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 9 ballots (score ballots).
Ballots:
the ranking Ranked Robin reads ("=" = tied); source scores follow in () per column: Ada, Ben, Cara
2 × Ada > Ben > Cara (5, 3, 0)
2 × Ben > Cara > Ada (0, 5, 3)
3 × Cara > Ada > Ben (3, 0, 5)
2 × Ada > Cara > Ben (5, 0, 3)
Round-Robin — every pair, head-to-head (For – Against):
Ada beats Ben 7 – 2
Cara beats Ada 5 – 4
Cara beats Ben 5 – 4
--- Pairwise (Round-Robin) Matrix ---
Head-to-head / pairwise comparison — the Ranked Robin tally
Legend: For - Equal Support - Against (row vs column)
| Ada | Ben | Cara |
--------------------------------------------
Ada > | --- |7 - 0 - 2 |4 - 0 - 5 |
Ben > | 2 - 0 - 7 | --- |4 - 0 - 5 |
Cara > | 5 - 0 - 4 |5 - 0 - 4 | --- |
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 Cara 2–0–0 2 +2 Ada, Ben
2 Ada 1–1–0 1 +4 Ben
3 Ben 0–2–0 0 -6 —
Winner — Ranked Robin (RCV-RR): Cara
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): Cara
Outside (2): Ada, Ben
One member ⇒ Cara is the Condorcet winner, beating every rival head-to-head.
Ranked Robin (RCV-RR) winner Cara 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