Exercise 1 — Two districts, one mayor: the COMBINED city¶
▶ Live on BetterVoting: vote · results ↗ (election 923q3d · test BV2190) — an independent tabulator to check the winners below against.
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 Elena, the Condorcet (pairwise) winner. STAR instead elects Carmen, 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 | Carmen |
| RCV-IRV | Elena |
| Ranked Robin (RCV-RR) | Elena |
| Approval | Avery |
| Range / Score | Avery |
| Condorcet | Elena |
Flags: 18 tied-score ballot(s)
Source election: 01_STAR/05_Practice/cases/ex01_district_combined.yaml · STAR tabulated mirror: ex01_district_combined_tabulated.txt
5 candidates, 18 ballots.
The ballots¶
Each row is a group of identical score ballots (0 = no support, 5 = max).
| Count | Avery | Blake | Carmen | Diego | Elena |
|---|---|---|---|---|---|
| 5 | 3 | 3 | 4 | 0 | 5 |
| 3 | 5 | 5 | 3 | 0 | 0 |
| 1 | 5 | 3 | 3 | 0 | 0 |
| 5 | 3 | 0 | 4 | 3 | 5 |
| 3 | 5 | 0 | 3 | 5 | 0 |
| 1 | 5 | 0 | 3 | 3 | 0 |
STAR result (official)¶
Scoring round (sum of scores): Avery 70, Carmen 64, Elena 50, Blake 33, Diego 33
Finalists (top two): Avery and Carmen
Automatic runoff: Avery 8 vs Carmen 10
STAR winner: Carmen
Full LH STAR engine report:
--- Runoff (Preference) Matrix ---
Head-to-head / pairwise comparison
Legend: For - Equal Support - Against
* indicates Top 2 Finalist
| * Avery | * Carmen |
-----------------------------------------------
* Avery > | --- | 8 - 0 - 10 |
* Carmen > | 10 - 0 - 8 | --- |
[Divergence from STAR]
STAR = Carmen
Choose-One (Plurality) = Elena (differs from STAR)
RCV-IRV = Elena (differs from STAR)
Approval = Avery (differs from STAR)
RCV-RR (Condorcet) = Elena (differs from STAR)
Note: 18 of 18 ballots (100%) had equal non-zero scores, so their ranks
were decided by candidate priority order. The RCV-IRV result may be
an artifact of score-to-rank tie-breaking rather than a deep
difference.
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) = (Avery)
- Runoff Round Winner = (Carmen)
Candidate Avery earned the highest total score, but
Candidate Carmen 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 18 ballots.
Count × Avery,Blake,Carmen,Diego,Elena
5 × 3, 3, 4, 0, 5
5 × 3, 0, 4, 3, 5
3 × 5, 5, 3, 0, 0
3 × 5, 0, 3, 5, 0
1 × 5, 3, 3, 0, 0
1 × 5, 0, 3, 3, 0
[Score Distribution] (how many ballots gave each star rating)
Score
Candidate 5 4 3 2 1 0 | Total Avg
Avery 8 0 10 0 0 0 | 70 3.9
Blake 3 0 6 0 0 9 | 33 1.8
Carmen 0 10 8 0 0 0 | 64 3.6
Diego 3 0 6 0 0 9 | 33 1.8
Elena 10 0 0 0 0 8 | 50 2.8
Scoring Round
The two highest-scoring candidates advance to the next round.
Avery -- 70 -- First place
Carmen -- 64 -- Second place
Elena -- 50
Blake -- 33
Diego -- 33
Avery and Carmen advance.
Automatic Runoff Round
The candidate preferred in the most head-to-head matchups wins.
Carmen -- 10 -- First place
Avery -- 8
Equal Support -- 0
Carmen wins.
Voters with a preference: 18 of 18 (no Equal Support).
Carmen 10 (56%) vs Avery 8 (44%); majority = 10.
Winner — STAR Voting Method (single winner)
Carmen
RCV-IRV — round by round¶
--- RCV / Instant-Runoff Voting (single winner) ---
Tabulating 18 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: Avery, Blake, Carmen, Diego, Elena
5 × Elena > Carmen > Avery > Blake (3, 3, 4, 0, 5)
3 × Avery > Blake > Carmen (5, 5, 3, 0, 0)
1 × Avery > Blake > Carmen (5, 3, 3, 0, 0)
5 × Elena > Carmen > Avery > Diego (3, 0, 4, 3, 5)
3 × Avery > Diego > Carmen (5, 0, 3, 5, 0)
1 × Avery > Carmen > Diego (5, 0, 3, 3, 0)
FINAL RESULT
Candidate Votes Status
----------- ------- --------
Elena 10 Elected
Avery 8 Rejected
Carmen 0 Rejected
Blake 0 Rejected
Diego 0 Rejected
Winner(s) — RCV / Instant-Runoff Voting (single winner)
Elena
--- 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 5): Elena
Outside (4): Avery, Blake, Carmen, Diego
One member ⇒ Elena is the Condorcet winner, beating every rival head-to-head.
RCV-IRV winner Elena 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 18 ballots (score ballots).
Ballots:
the ranking Ranked Robin reads ("=" = tied); source scores follow in () per column: Avery, Blake, Carmen, Diego, Elena
5 × Elena > Carmen > Avery=Blake > Diego (3, 3, 4, 0, 5)
3 × Avery=Blake > Carmen > Diego=Elena (5, 5, 3, 0, 0)
1 × Avery > Blake=Carmen > Diego=Elena (5, 3, 3, 0, 0)
5 × Elena > Carmen > Avery=Diego > Blake (3, 0, 4, 3, 5)
3 × Avery=Diego > Carmen > Blake=Elena (5, 0, 3, 5, 0)
1 × Avery > Carmen=Diego > Blake=Elena (5, 0, 3, 3, 0)
Round-Robin — every pair, head-to-head (For – Against):
Avery beats Blake 10 – 0
Carmen beats Avery 10 – 8
Avery beats Diego 10 – 0
Elena beats Avery 10 – 8
Carmen beats Blake 14 – 3
Blake ties Diego 9 – 9
Elena beats Blake 10 – 4
Carmen beats Diego 14 – 3
Elena beats Carmen 10 – 8
Elena beats Diego 10 – 4
--- Pairwise (Round-Robin) Matrix ---
Head-to-head / pairwise comparison — the Ranked Robin tally
Legend: For - Equal Support - Against (row vs column)
| Avery | Blake | Carmen | Diego | Elena |
-----------------------------------------------------------------------------------
Avery > | --- |10 - 8 - 0 | 8 - 0 - 10 |10 - 8 - 0 | 8 - 0 - 10 |
Blake > | 0 - 8 - 10 | --- | 3 - 1 - 14 | 9 - 0 - 9 | 4 - 4 - 10 |
Carmen > | 10 - 0 - 8 |14 - 1 - 3 | --- |14 - 1 - 3 | 8 - 0 - 10 |
Diego > | 0 - 8 - 10 | 9 - 0 - 9 | 3 - 1 - 14 | --- | 4 - 4 - 10 |
Elena > | 10 - 0 - 8 |10 - 4 - 4 |10 - 0 - 8 |10 - 4 - 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 Elena 4–0–0 4 +16 Carmen, Avery, Blake, Diego
2 Carmen 3–1–0 3 +22 Avery, Blake, Diego
3 Avery 2–2–0 2 +16 Blake, Diego
4 Blake 0–3–1 0.5 -27 —
5 Diego 0–3–1 0.5 -27 —
Winner — Ranked Robin (RCV-RR): Elena
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 5): Elena
Outside (4): Avery, Blake, Carmen, Diego
One member ⇒ Elena is the Condorcet winner, beating every rival head-to-head.
Ranked Robin (RCV-RR) winner Elena 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