Exercise 2 — The tenth ballot: Bella withdraws¶
Bucket — STAR_OUTLIER_RR_WITH_IRV: STAR is the outlier
Generated by STARVote_LH_tabulation_engine/tools_adam/scripts/build_divergence_index.py — rebuilt from the election file; do not hand-edit.
What happens¶
RCV-IRV and Ranked Robin both elect Eli, the Condorcet (pairwise) winner. STAR instead elects Chris, because its score-then-automatic-runoff can reward a candidate with broad, intense scored support even when a different candidate wins every head-to-head. Here STAR is the outlier — an honest reminder that STAR does not always elect the Condorcet candidate.
Winners by method¶
| Method | Winner |
|---|---|
| STAR | Chris |
| RCV-IRV | Eli |
| Ranked Robin (RCV-RR) | Eli |
| Approval | Alex |
| Range / Score | Alex |
| Condorcet | Eli |
Flags: none
Source election: 01_STAR/05_Practice/cases/ex02_bella_exits.yaml · STAR tabulated mirror: ex02_bella_exits_tabulated.txt
4 candidates, 9 ballots.
The ballots¶
Each row is a group of identical score ballots (0 = no support, 5 = max).
| Count | Alex | Chris | Dana | Eli |
|---|---|---|---|---|
| 4 | 3 | 4 | 5 | 0 |
| 3 | 3 | 0 | 0 | 4 |
| 2 | 3 | 4 | 0 | 5 |
STAR result (official)¶
Scoring round (sum of scores): Alex 27, Chris 24, Eli 22, Dana 20
Finalists (top two): Alex and Chris
Automatic runoff: Alex 3 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 > | --- |3 - 0 - 6 |
* Chris > | 6 - 0 - 3 | --- |
[Divergence from STAR]
STAR = Chris
Choose-One (Plurality) = Eli (differs from STAR)
RCV-IRV = Eli (differs from STAR)
Approval = Alex (differs from STAR)
RCV-RR (Condorcet) = 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.
Note: Ranked Robin (RCV-RR) sides with RCV-IRV, so STAR is the outlier
here — STAR need not elect the Condorcet candidate.
[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 9 ballots.
Count × Alex,Chris,Dana,Eli
4 × 3, 4, 5, 0
3 × 3, 0, 0, 4
2 × 3, 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
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
Chris -- 24 -- Second place
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 -- 3
Equal Support -- 0
Chris wins.
Voters with a preference: 9 of 9 (no Equal Support).
Chris 6 (67%) vs Alex 3 (33%); majority = 5.
Winner — STAR Voting Method (single winner)
Chris
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, Chris, Dana, Eli
4 × Dana > Chris > Alex (3, 4, 5, 0)
3 × Eli > Alex (3, 0, 0, 4)
2 × Eli > Chris > Alex (3, 4, 0, 5)
FINAL RESULT
Candidate Votes Status
----------- ------- --------
Eli 5 Elected
Dana 4 Rejected
Chris 0 Rejected
Alex 0 Rejected
Winner(s) — RCV / Instant-Runoff Voting (single winner)
Eli
--- 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 4): Eli
Outside (3): Alex, Chris, Dana
One member ⇒ Eli is the Condorcet winner, beating every rival head-to-head.
RCV-IRV winner Eli 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, Chris, Dana, Eli
4 × Dana > Chris > Alex > Eli (3, 4, 5, 0)
3 × Eli > Alex > Chris=Dana (3, 0, 0, 4)
2 × Eli > Chris > Alex > Dana (3, 4, 0, 5)
Round-Robin — every pair, head-to-head (For – Against):
Chris beats Alex 6 – 3
Alex beats Dana 5 – 4
Eli beats Alex 5 – 4
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 | Chris | Dana | Eli |
--------------------------------------------------------
Alex > | --- |3 - 0 - 6 |5 - 0 - 4 |4 - 0 - 5 |
Chris > | 6 - 0 - 3 | --- |2 - 3 - 4 |4 - 0 - 5 |
Dana > | 4 - 0 - 5 |4 - 3 - 2 | --- |4 - 0 - 5 |
Eli > | 5 - 0 - 4 |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 3–0–0 3 +3 Chris, Dana, Alex
2 Chris 1–2–0 1 +0 Alex
3 Dana 1–2–0 1 +0 Chris
4 Alex 1–2–0 1 -3 Dana
Winner — Ranked Robin (RCV-RR): Eli
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 4): Eli
Outside (3): Alex, Chris, Dana
One member ⇒ Eli is the Condorcet winner, beating every rival head-to-head.
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