Exercise 2 — The tenth ballot: all ten 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=Chris, RCV-IRV=Bella, Ranked Robin=Chris (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 | Chris |
| RCV-IRV | Bella |
| Ranked Robin (RCV-RR) | Chris |
| Approval | Alex |
| Range / Score | Alex |
| Condorcet | none (cycle) |
Flags: none
Source election: 01_STAR/05_Practice/cases/ex02_tenth_ballot.yaml · STAR tabulated mirror: ex02_tenth_ballot_tabulated.txt
5 candidates, 10 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 |
| 1 | 5 | 0 | 2 | 0 | 0 |
STAR result (official)¶
Scoring round (sum of scores): Alex 32, Chris 26, Bella 25, Eli 22, Dana 20
Finalists (top two): Alex and Chris
Automatic runoff: Alex 4 vs Chris 6
STAR winner: Chris
Full LH STAR engine report:
--- Runoff (Preference) Matrix ---
Head-to-head / pairwise comparison
Legend: For - Equal Support - Against
* indicates Top 2 Finalist
| * Alex | * Chris |
-----------------------------------------
* Alex > | --- |4 - 0 - 6 |
* Chris > | 6 - 0 - 4 | --- |
[Divergence from STAR]
STAR = Chris
Choose-One (Plurality) = Dana (differs from STAR)
RCV-IRV = Bella (differs from STAR)
Approval = Alex (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.
Note: Ranked Robin (RCV-RR) agrees with STAR, so RCV-IRV is the lone
outlier — the classic center-squeeze signature.
[Runoff Reversal]
- Score Round Winner(s) = (Alex)
- Runoff Round Winner = (Chris)
Candidate Alex earned the highest total score, but
Candidate Chris 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 10 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
1 × 5, 0, 2, 0, 0
[Score Distribution] (how many ballots gave each star rating)
Score
Candidate 5 4 3 2 1 0 | Total Avg
Alex 1 0 9 0 0 0 | 32 3.2
Bella 3 0 0 4 2 1 | 25 2.5
Chris 0 6 0 1 0 3 | 26 2.6
Dana 4 0 0 0 0 6 | 20 2.0
Eli 2 3 0 0 0 5 | 22 2.2
Scoring Round
The two highest-scoring candidates advance to the next round.
Alex -- 32 -- First place
Chris -- 26 -- Second place
Bella -- 25
Eli -- 22
Dana -- 20
Alex and Chris advance.
Automatic Runoff Round
The candidate preferred in the most head-to-head matchups wins.
Chris -- 6 -- First place
Alex -- 4
Equal Support -- 0
Chris wins.
Voters with a preference: 10 of 10 (no Equal Support).
Chris 6 (60%) vs Alex 4 (40%); majority = 6.
Winner — STAR Voting Method (single winner)
Chris
RCV-IRV — round by round¶
--- RCV / Instant-Runoff Voting (single winner) ---
Tabulating 10 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)
1 × Alex > Chris (5, 0, 2, 0, 0)
ROUND 1
Candidate Votes Status
----------- ------- --------
Dana 4 Hopeful
Bella 3 Hopeful
Eli 2 Rejected
Alex 1 Rejected
Chris 0 Rejected
FINAL RESULT
Candidate Votes Status
----------- ------- --------
Bella 5 Elected
Dana 4 Rejected
Eli 0 Rejected
Alex 0 Rejected
Chris 0 Rejected
Blank Votes 1 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): Alex, Chris, Bella, Eli, 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 (Alex, Chris) 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.
Fine print: this set contains a pairwise DRAW, and a draw is enough to keep a
candidate in the Smith set but not in the tighter Schwartz set — so Schwartz
may be smaller 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 10 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)
1 × Alex > Chris > Bella=Dana=Eli (5, 0, 2, 0, 0)
Round-Robin — every pair, head-to-head (For – Against):
Alex beats Bella 7 – 3
Chris beats Alex 6 – 4
Alex beats Dana 6 – 4
Alex ties Eli 5 – 5
Chris beats Bella 7 – 3
Bella beats Dana 5 – 4
Bella beats Eli 7 – 2
Dana beats Chris 4 – 3
Chris ties Eli 5 – 5
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 > | --- |7 - 0 - 3 |4 - 0 - 6 |6 - 0 - 4 |5 - 0 - 5 |
Bella > | 3 - 0 - 7 | --- |3 - 0 - 7 |5 - 1 - 4 |7 - 1 - 2 |
Chris > | 6 - 0 - 4 |7 - 0 - 3 | --- |3 - 3 - 4 |5 - 0 - 5 |
Dana > | 4 - 0 - 6 |4 - 1 - 5 |4 - 3 - 3 | --- |4 - 1 - 5 |
Eli > | 5 - 0 - 5 |2 - 1 - 7 |5 - 0 - 5 |5 - 1 - 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 Chris 2–1–1 2.5 +5 Alex, Bella
2 Alex 2–1–1 2.5 +4 Bella, Dana
3 Bella 2–2–0 2 -2 Eli, Dana
4 Eli 1–1–2 2 -4 Dana
5 Dana 1–3–0 1 -3 Chris
Winner — Ranked Robin (RCV-RR): Chris
*** 2 candidates tie on the highest Copeland score (2.5): Alex, Chris — tied on the tally, not a cycle (some of them beat others head-to-head, but no loop closes). Resolved by total margin, then lot order.
--- 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): Alex, Chris, Bella, Eli, 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 (Alex, Chris) are only part of the set — the
win–loss table's top block understates how wide the contention is.
Ranked Robin (RCV-RR) winner Chris 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.
Fine print: this set contains a pairwise DRAW, and a draw is enough to keep a
candidate in the Smith set but not in the tighter Schwartz set — so Schwartz
may be smaller here.
More: 07_Concepts/topics/smith_set.md