Exercise 6 — Bullet voting backfires: the strategic ballots¶
▶ Live on BetterVoting: vote · results ↗ (election 7f4f7q · test BV2194) — 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 Cash. Only Approval differs, electing Ari: Approval counts every score of 3–5 as one equal 'approve' and ignores intensity, rewarding Ari's breadth of acceptability over Cash's stronger but more concentrated support. A threshold story about Approval, not a STAR-vs-IRV teaching case.
Winners by method¶
| Method | Winner |
|---|---|
| STAR | Cash |
| RCV-IRV | Cash |
| Ranked Robin (RCV-RR) | Cash |
| Approval | Ari |
| Range / Score | Cash |
| Condorcet | Cash |
Flags: none
Source election: 01_STAR/05_Practice/cases/ex06_bullet_backfire.yaml · STAR tabulated mirror: ex06_bullet_backfire_tabulated.txt
3 candidates, 9 ballots.
The ballots¶
Each row is a group of identical score ballots (0 = no support, 5 = max).
| Count | Ari | Bree | Cash |
|---|---|---|---|
| 4 | 5 | 0 | 0 |
| 4 | 0 | 2 | 5 |
| 1 | 0 | 5 | 1 |
STAR result (official)¶
Scoring round (sum of scores): Cash 21, Ari 20, Bree 13
Finalists (top two): Cash and Ari
Automatic runoff: Cash 5 vs Ari 4
STAR winner: Cash
Full LH STAR engine report:
--- Runoff (Preference) Matrix ---
Head-to-head / pairwise comparison
Legend: For - Equal Support - Against
* indicates Top 2 Finalist
| * Ari | * Cash |
-----------------------------------------
* Ari > | --- |4 - 0 - 5 |
* Cash > | 5 - 0 - 4 | --- |
[Divergence from STAR]
STAR = Cash
Choose-One (Plurality) = Ari (differs from STAR)
Approval = Ari (differs from STAR)
--- STAR Voting Method (single winner) ---
Tabulating 9 ballots.
Count × Ari,Bree,Cash
4 × 5, 0, 0
4 × 0, 2, 5
1 × 0, 5, 1
[Score Distribution] (how many ballots gave each star rating)
Score
Candidate 5 4 3 2 1 0 | Total Avg
Ari 4 0 0 0 0 5 | 20 2.2
Bree 1 0 0 4 0 4 | 13 1.4
Cash 4 0 0 0 1 4 | 21 2.3
Scoring Round
The two highest-scoring candidates advance to the next round.
Cash -- 21 -- First place
Ari -- 20 -- Second place
Bree -- 13
Cash and Ari advance.
Automatic Runoff Round
The candidate preferred in the most head-to-head matchups wins.
Cash -- 5 -- First place
Ari -- 4
Equal Support -- 0
Cash wins.
Voters with a preference: 9 of 9 (no Equal Support).
Cash 5 (56%) vs Ari 4 (44%); majority = 5.
Winner — STAR Voting Method (single winner)
Cash
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: Ari, Bree, Cash
4 × Ari (5, 0, 0)
4 × Cash > Bree (0, 2, 5)
1 × Bree > Cash (0, 5, 1)
ROUND 1
Candidate Votes Status
----------- ------- --------
Cash 4 Hopeful
Ari 4 Hopeful
Bree 1 Rejected
FINAL RESULT
Candidate Votes Status
----------- ------- --------
Cash 5 Elected
Ari 4 Rejected
Bree 0 Rejected
Winner(s) — RCV / Instant-Runoff Voting (single winner)
Cash
--- 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): Cash
Outside (2): Ari, Bree
One member ⇒ Cash is the Condorcet winner, beating every rival head-to-head.
RCV-IRV winner Cash 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: Ari, Bree, Cash
4 × Ari > Bree=Cash (5, 0, 0)
4 × Cash > Bree > Ari (0, 2, 5)
1 × Bree > Cash > Ari (0, 5, 1)
Round-Robin — every pair, head-to-head (For – Against):
Bree beats Ari 5 – 4
Cash beats Ari 5 – 4
Cash beats Bree 4 – 1
--- Pairwise (Round-Robin) Matrix ---
Head-to-head / pairwise comparison — the Ranked Robin tally
Legend: For - Equal Support - Against (row vs column)
| Ari | Bree | Cash |
--------------------------------------------
Ari > | --- |4 - 0 - 5 |4 - 0 - 5 |
Bree > | 5 - 0 - 4 | --- |1 - 4 - 4 |
Cash > | 5 - 0 - 4 |4 - 4 - 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 Cash 2–0–0 2 +4 Bree, Ari
2 Bree 1–1–0 1 -2 Ari
3 Ari 0–2–0 0 -2 —
Winner — Ranked Robin (RCV-RR): Cash
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): Cash
Outside (2): Ari, Bree
One member ⇒ Cash is the Condorcet winner, beating every rival head-to-head.
Ranked Robin (RCV-RR) winner Cash 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