Exercise 13 — Where do you draw the line? The honest opinions (STAR)¶
▶ Live on BetterVoting: vote · results ↗ (election qdtqf2 · test BV2200) — an independent tabulator to check the winners below against.
Bucket — CYCLE_OR_THREE_WAY: Cycle / three-way split
Generated by STARVote_LH_tabulation_engine/tools_adam/scripts/build_divergence_index.py — rebuilt from the election file; do not hand-edit.
What happens¶
There is no Condorcet winner — the head-to-head results form a cycle (X beats Y beats Z beats X). With no pairwise anchor the methods split: STAR=Cora, RCV-IRV=Cora, Ranked Robin=Ash (RR breaks the cycle by total margin). No rule is clearly 'right'; a genuinely hard election, not an indictment of any one method.
Winners by method¶
| Method | Winner |
|---|---|
| STAR | Cora |
| RCV-IRV | Cora |
| Ranked Robin (RCV-RR) | Ash |
| Approval | Ash |
| Range / Score | Ash |
| Condorcet | none (cycle) |
Flags: none
Source election: 01_STAR/05_Practice/cases/ex13_opinions.yaml · STAR tabulated mirror: ex13_opinions_tabulated.txt
3 candidates, 9 ballots.
The ballots¶
Each row is a group of identical score ballots (0 = no support, 5 = max).
| Count | Ash | Beck | Cora |
|---|---|---|---|
| 3 | 5 | 4 | 0 |
| 2 | 0 | 5 | 3 |
| 4 | 3 | 0 | 5 |
STAR result (official)¶
Scoring round (sum of scores): Ash 27, Cora 26, Beck 22
Finalists (top two): Ash and Cora
Automatic runoff: Ash 3 vs Cora 6
STAR winner: Cora
Full LH STAR engine report:
--- Runoff (Preference) Matrix ---
Head-to-head / pairwise comparison
Legend: For - Equal Support - Against
* indicates Top 2 Finalist
| * Ash | * Cora |
-----------------------------------------
* Ash > | --- |3 - 0 - 6 |
* Cora > | 6 - 0 - 3 | --- |
[Divergence from STAR]
STAR = Cora
Approval = Ash (differs from STAR)
RCV-RR = Ash (differs from STAR)
[Runoff Reversal]
- Score Round Winner(s) = (Ash)
- Runoff Round Winner = (Cora)
Candidate Ash earned the highest total score, but
Candidate Cora 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 × Ash,Beck,Cora
4 × 3, 0, 5
3 × 5, 4, 0
2 × 0, 5, 3
[Score Distribution] (how many ballots gave each star rating)
Score
Candidate 5 4 3 2 1 0 | Total Avg
Ash 3 0 4 0 0 2 | 27 3.0
Beck 2 3 0 0 0 4 | 22 2.4
Cora 4 0 2 0 0 3 | 26 2.9
Scoring Round
The two highest-scoring candidates advance to the next round.
Ash -- 27 -- First place
Cora -- 26 -- Second place
Beck -- 22
Ash and Cora advance.
Automatic Runoff Round
The candidate preferred in the most head-to-head matchups wins.
Cora -- 6 -- First place
Ash -- 3
Equal Support -- 0
Cora wins.
Voters with a preference: 9 of 9 (no Equal Support).
Cora 6 (67%) vs Ash 3 (33%); majority = 5.
Winner — STAR Voting Method (single winner)
Cora
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: Ash, Beck, Cora
3 × Ash > Beck (5, 4, 0)
2 × Beck > Cora (0, 5, 3)
4 × Cora > Ash (3, 0, 5)
ROUND 1
Candidate Votes Status
----------- ------- --------
Cora 4 Hopeful
Ash 3 Hopeful
Beck 2 Rejected
FINAL RESULT
Candidate Votes Status
----------- ------- --------
Cora 6 Elected
Ash 3 Rejected
Beck 0 Rejected
Winner(s) — RCV / Instant-Runoff Voting (single winner)
Cora
--- 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 (3 of 3): Ash, Beck, Cora
Outside (0): —
More than one member ⇒ NO Condorcet winner: the top of the tournament is a
cycle, so the strongest "candidate" is a set, not a person. Which member of
the set should win is exactly what Minimax / Ranked Pairs / Schulze disagree
about — see 05_Ranked_Robin/01_Learn/cycle_resolution.md.
RCV-IRV winner Cora 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: Ash, Beck, Cora
3 × Ash > Beck > Cora (5, 4, 0)
2 × Beck > Cora > Ash (0, 5, 3)
4 × Cora > Ash > Beck (3, 0, 5)
Round-Robin — every pair, head-to-head (For – Against):
Ash beats Beck 7 – 2
Cora beats Ash 6 – 3
Beck beats Cora 5 – 4
--- Pairwise (Round-Robin) Matrix ---
Head-to-head / pairwise comparison — the Ranked Robin tally
Legend: For - Equal Support - Against (row vs column)
| Ash | Beck | Cora |
--------------------------------------------
Ash > | --- |7 - 0 - 2 |3 - 0 - 6 |
Beck > | 2 - 0 - 7 | --- |5 - 0 - 4 |
Cora > | 6 - 0 - 3 |4 - 0 - 5 | --- |
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 Ash 1–1–0 1 +2 Beck
2 Cora 1–1–0 1 +2 Ash
3 Beck 1–1–0 1 -4 Cora
Winner — Ranked Robin (RCV-RR): Ash
*** 3 candidates tie for the most wins (Ash, Beck, Cora) — a Condorcet cycle (no candidate beats all others). Resolved by total margin, then lot order. (This is where Minimax / Ranked Pairs / Schulze differ — see 05_Ranked_Robin/01_Learn/cycle_resolution.md.)
--- 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 (3 of 3): Ash, Beck, Cora
Outside (0): —
More than one member ⇒ NO Condorcet winner: the top of the tournament is a
cycle, so the strongest "candidate" is a set, not a person. Which member of
the set should win is exactly what Minimax / Ranked Pairs / Schulze disagree
about — see 05_Ranked_Robin/01_Learn/cycle_resolution.md.
Ranked Robin (RCV-RR) winner Ash 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