Minimax Ex.29 — STAR: elects B, a top-cycle member, not the universal loser¶
▶ Live on BetterVoting: vote · results ↗ (election f3dxq9 · test BV2167) — 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=B, RCV-IRV=C, Ranked Robin=A (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 | B |
| RCV-IRV | C |
| Ranked Robin (RCV-RR) | A |
| Approval | B |
| Range / Score | B |
| Condorcet | none (cycle) |
Flags: none
Source election: method_comparisons/felsenthal_paradoxes/cases/bv2167_f3dxq9_star.yaml · STAR tabulated mirror: bv2167_f3dxq9_star_tabulated.txt
4 candidates, 11 ballots.
The ballots¶
Each row is a group of identical score ballots (0 = no support, 5 = max).
| Count | A | B | C | D |
|---|---|---|---|---|
| 2 | 4 | 1 | 2 | 5 |
| 3 | 2 | 4 | 1 | 5 |
| 3 | 2 | 4 | 5 | 1 |
| 1 | 4 | 5 | 2 | 1 |
| 2 | 5 | 2 | 4 | 1 |
STAR result (official)¶
Scoring round (sum of scores): B 35, A 34, C 32, D 31
Finalists (top two): B and A
Automatic runoff: B 7 vs A 4
STAR winner: B
Full LH STAR engine report:
--- Runoff (Preference) Matrix ---
Head-to-head / pairwise comparison
Legend: For - Equal Support - Against
* indicates Top 2 Finalist
| * A | * B |
-----------------------------------------
* A > | --- |4 - 0 - 7 |
* B > | 7 - 0 - 4 | --- |
[Divergence from STAR]
STAR = B
Choose-One (Plurality) = D (differs from STAR)
RCV-IRV = C (differs from STAR)
RCV-RR = A (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 11 ballots.
Count × A,B,C,D
3 × 2,4,1,5
3 × 2,4,5,1
2 × 4,1,2,5
2 × 5,2,4,1
1 × 4,5,2,1
[Score Distribution] (how many ballots gave each star rating)
Score
Candidate 5 4 3 2 1 0 | Total Avg
A 2 3 0 6 0 0 | 34 3.1
B 1 6 0 2 2 0 | 35 3.2
C 3 2 0 3 3 0 | 32 2.9
D 5 0 0 0 6 0 | 31 2.8
Scoring Round
The two highest-scoring candidates advance to the next round.
B -- 35 -- First place
A -- 34 -- Second place
C -- 32
D -- 31
B and A advance.
Automatic Runoff Round
The candidate preferred in the most head-to-head matchups wins.
B -- 7 -- First place
A -- 4
Equal Support -- 0
B wins.
Voters with a preference: 11 of 11 (no Equal Support).
B 7 (64%) vs A 4 (36%); majority = 6.
Winner — STAR Voting Method (single winner)
B
RCV-IRV — round by round¶
--- RCV / Instant-Runoff Voting (single winner) ---
Tabulating 11 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: A, B, C, D
2 × D > A > C > B (4, 1, 2, 5)
3 × D > B > A > C (2, 4, 1, 5)
3 × C > B > A > D (2, 4, 5, 1)
1 × B > A > C > D (4, 5, 2, 1)
2 × A > C > B > D (5, 2, 4, 1)
ROUND 1
Candidate Votes Status
----------- ------- --------
D 5 Hopeful
C 3 Hopeful
A 2 Rejected
B 1 Rejected
FINAL RESULT
Candidate Votes Status
----------- ------- --------
C 6 Elected
D 5 Rejected
A 0 Rejected
B 0 Rejected
Winner(s) — RCV / Instant-Runoff Voting (single winner)
C
--- 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 4): A, B, C
Outside (1): D
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 C 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 11 ballots (score ballots).
Ballots:
the ranking Ranked Robin reads ("=" = tied); source scores follow in () per column: A, B, C, D
2 × D > A > C > B (4, 1, 2, 5)
3 × D > B > A > C (2, 4, 1, 5)
3 × C > B > A > D (2, 4, 5, 1)
1 × B > A > C > D (4, 5, 2, 1)
2 × A > C > B > D (5, 2, 4, 1)
Round-Robin — every pair, head-to-head (For – Against):
B beats A 7 – 4
A beats C 8 – 3
A beats D 6 – 5
C beats B 7 – 4
B beats D 6 – 5
C beats D 6 – 5
--- Pairwise (Round-Robin) Matrix ---
Head-to-head / pairwise comparison — the Ranked Robin tally
Legend: For - Equal Support - Against (row vs column)
| A | B | C | D |
----------------------------------------------------
A > | --- |4 - 0 - 7 |8 - 0 - 3 |6 - 0 - 5 |
B > | 7 - 0 - 4 | --- |4 - 0 - 7 |6 - 0 - 5 |
C > | 3 - 0 - 8 |7 - 0 - 4 | --- |6 - 0 - 5 |
D > | 5 - 0 - 6 |5 - 0 - 6 |5 - 0 - 6 | --- |
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 A 2–1–0 2 +3 C, D
2 B 2–1–0 2 +1 A, D
3 C 2–1–0 2 -1 B, D
4 D 0–3–0 0 -3 —
Winner — Ranked Robin (RCV-RR): A
*** 3 candidates tie for the most wins (A, B, C) — 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 4): A, B, C
Outside (1): D
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 A 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