Exercise 2 — The tenth ballot: the nine counted ballots¶
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=Alex, RCV-IRV=Bella, Ranked Robin=Eli (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 | Alex |
| RCV-IRV | Bella |
| Ranked Robin (RCV-RR) | Eli |
| Approval | Alex |
| Range / Score | Alex |
| Condorcet | none (cycle) |
Flags: none
Source election: 01_STAR/05_Practice/cases/ex02_nine_ballots.yaml · STAR tabulated mirror: ex02_nine_ballots_tabulated.txt
5 candidates, 9 ballots.
The ballots¶
Each row is a group of identical score ballots (0 = no support, 5 = max).
| Count | Alex | Bella | Chris | Dana | Eli |
|---|---|---|---|---|---|
| 4 | 3 | 2 | 4 | 5 | 0 |
| 3 | 3 | 5 | 0 | 0 | 4 |
| 2 | 3 | 1 | 4 | 0 | 5 |
STAR result (official)¶
Scoring round (sum of scores): Alex 27, Bella 25, Chris 24, Eli 22, Dana 20
Finalists (top two): Alex and Bella
Automatic runoff: Alex 6 vs Bella 3
STAR winner: Alex
Full LH STAR engine report:
--- Runoff (Preference) Matrix ---
Head-to-head / pairwise comparison
Legend: For - Equal Support - Against
* indicates Top 2 Finalist
| * Alex | * Bella |
-----------------------------------------
* Alex > | --- |6 - 0 - 3 |
* Bella > | 3 - 0 - 6 | --- |
[Divergence from STAR]
STAR = Alex
Choose-One (Plurality) = Dana (differs from STAR)
RCV-IRV = Bella (differs from STAR)
RCV-RR = Eli (differs from STAR)
Note: no ballots had tied scores, so RCV-IRV vs STAR here is a genuine
method difference, not a tie-breaking artifact.
--- STAR Voting Method (single winner) ---
Tabulating 9 ballots.
Count × Alex,Bella,Chris,Dana,Eli
4 × 3, 2, 4, 5, 0
3 × 3, 5, 0, 0, 4
2 × 3, 1, 4, 0, 5
[Score Distribution] (how many ballots gave each star rating)
Score
Candidate 5 4 3 2 1 0 | Total Avg
Alex 0 0 9 0 0 0 | 27 3.0
Bella 3 0 0 4 2 0 | 25 2.8
Chris 0 6 0 0 0 3 | 24 2.7
Dana 4 0 0 0 0 5 | 20 2.2
Eli 2 3 0 0 0 4 | 22 2.4
Scoring Round
The two highest-scoring candidates advance to the next round.
Alex -- 27 -- First place
Bella -- 25 -- Second place
Chris -- 24
Eli -- 22
Dana -- 20
Alex and Bella advance.
Automatic Runoff Round
The candidate preferred in the most head-to-head matchups wins.
Alex -- 6 -- First place
Bella -- 3
Equal Support -- 0
Alex wins.
Voters with a preference: 9 of 9 (no Equal Support).
Alex 6 (67%) vs Bella 3 (33%); majority = 5.
Winner — STAR Voting Method (single winner)
Alex
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: Alex, Bella, Chris, Dana, Eli
4 × Dana > Chris > Alex > Bella (3, 2, 4, 5, 0)
3 × Bella > Eli > Alex (3, 5, 0, 0, 4)
2 × Eli > Chris > Alex > Bella (3, 1, 4, 0, 5)
ROUND 1
Candidate Votes Status
----------- ------- --------
Dana 4 Hopeful
Bella 3 Hopeful
Eli 2 Rejected
Chris 0 Rejected
Alex 0 Rejected
FINAL RESULT
Candidate Votes Status
----------- ------- --------
Bella 5 Elected
Dana 4 Rejected
Eli 0 Rejected
Chris 0 Rejected
Alex 0 Rejected
Winner(s) — RCV / Instant-Runoff Voting (single winner)
Bella
--- 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 (5 of 5): Eli, Alex, Bella, Chris, Dana
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.
Note: the Copeland leaders (Eli) are only part of the set — the
win–loss table's top block understates how wide the contention is.
RCV-IRV winner Bella 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: Alex, Bella, Chris, Dana, Eli
4 × Dana > Chris > Alex > Bella > Eli (3, 2, 4, 5, 0)
3 × Bella > Eli > Alex > Chris=Dana (3, 5, 0, 0, 4)
2 × Eli > Chris > Alex > Bella > Dana (3, 1, 4, 0, 5)
Round-Robin — every pair, head-to-head (For – Against):
Alex beats Bella 6 – 3
Chris beats Alex 6 – 3
Alex beats Dana 5 – 4
Eli beats Alex 5 – 4
Chris beats Bella 6 – 3
Bella beats Dana 5 – 4
Bella beats Eli 7 – 2
Dana beats Chris 4 – 2
Eli beats Chris 5 – 4
Eli beats Dana 5 – 4
--- Pairwise (Round-Robin) Matrix ---
Head-to-head / pairwise comparison — the Ranked Robin tally
Legend: For - Equal Support - Against (row vs column)
| Alex | Bella | Chris | Dana | Eli |
-------------------------------------------------------------------
Alex > | --- |6 - 0 - 3 |3 - 0 - 6 |5 - 0 - 4 |4 - 0 - 5 |
Bella > | 3 - 0 - 6 | --- |3 - 0 - 6 |5 - 0 - 4 |7 - 0 - 2 |
Chris > | 6 - 0 - 3 |6 - 0 - 3 | --- |2 - 3 - 4 |4 - 0 - 5 |
Dana > | 4 - 0 - 5 |4 - 0 - 5 |4 - 3 - 2 | --- |4 - 0 - 5 |
Eli > | 5 - 0 - 4 |2 - 0 - 7 |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 Eli 3–1–0 3 -2 Chris, Alex, Dana
2 Chris 2–2–0 2 +3 Alex, Bella
3 Alex 2–2–0 2 +0 Bella, Dana
4 Bella 2–2–0 2 +0 Eli, Dana
5 Dana 1–3–0 1 -1 Chris
Winner — Ranked Robin (RCV-RR): Eli
the most head-to-head wins (3).
--- 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 (5 of 5): Eli, Alex, Bella, Chris, Dana
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.
Note: the Copeland leaders (Eli) are only part of the set — the
win–loss table's top block understates how wide the contention is.
Ranked Robin (RCV-RR) winner Eli 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