Monotonicity — STAR counterpart (AFTER — X still wins)¶
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=X, RCV-IRV=Z, Ranked Robin=X (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 | X |
| RCV-IRV | Z |
| Ranked Robin (RCV-RR) | X |
| Approval | X |
| Range / Score | X |
| Condorcet | none (cycle) |
Flags: none
Source election: method_comparisons/monotonicity/cases/monotonicity_star_after.yaml · STAR tabulated mirror: monotonicity_star_after_tabulated.txt
3 candidates, 34 ballots.
The ballots¶
Each row is a group of identical score ballots (0 = no support, 5 = max).
| Count | X | Y | Z |
|---|---|---|---|
| 16 | 5 | 3 | 0 |
| 8 | 0 | 5 | 3 |
| 10 | 3 | 0 | 5 |
STAR result (official)¶
Scoring round (sum of scores): X 110, Y 88, Z 74
Finalists (top two): X and Y
Automatic runoff: X 26 vs Y 8
STAR winner: X
Full LH STAR engine report:
--- Runoff (Preference) Matrix ---
Head-to-head / pairwise comparison
Legend: For - Equal Support - Against
* indicates Top 2 Finalist
| * X | * Y |
-----------------------------------------------
* X > | --- |26 - 0 - 8 |
* Y > | 8 - 0 - 26 | --- |
[Divergence from STAR]
STAR = X
RCV-IRV = Z (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.
--- STAR Voting Method (single winner) ---
Tabulating 34 ballots.
Count × X,Y,Z
16 × 5,3,0
10 × 3,0,5
8 × 0,5,3
[Score Distribution] (how many ballots gave each star rating)
Score
Candidate 5 4 3 2 1 0 | Total Avg
X 16 0 10 0 0 8 | 110 3.2
Y 8 0 16 0 0 10 | 88 2.6
Z 10 0 8 0 0 16 | 74 2.2
Scoring Round
The two highest-scoring candidates advance to the next round.
X -- 110 -- First place
Y -- 88 -- Second place
Z -- 74
X and Y advance.
Automatic Runoff Round
The candidate preferred in the most head-to-head matchups wins.
X -- 26 -- First place
Y -- 8
Equal Support -- 0
X wins.
Voters with a preference: 34 of 34 (no Equal Support).
X 26 (76%) vs Y 8 (24%); majority = 18.
Winner — STAR Voting Method (single winner)
X
RCV-IRV — round by round¶
--- RCV / Instant-Runoff Voting (single winner) ---
Tabulating 34 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: X, Y, Z
16 × X > Y (5, 3, 0)
8 × Y > Z (0, 5, 3)
10 × Z > X (3, 0, 5)
ROUND 1
Candidate Votes Status
----------- ------- --------
X 16 Hopeful
Z 10 Hopeful
Y 8 Rejected
FINAL RESULT
Candidate Votes Status
----------- ------- --------
Z 18 Elected
X 16 Rejected
Y 0 Rejected
Winner(s) — RCV / Instant-Runoff Voting (single winner)
Z
--- 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 3): X, Y, Z
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.
RCV-IRV winner Z 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 34 ballots (score ballots).
Ballots:
the ranking Ranked Robin reads ("=" = tied); source scores follow in () per column: X, Y, Z
16 × X > Y > Z (5, 3, 0)
8 × Y > Z > X (0, 5, 3)
10 × Z > X > Y (3, 0, 5)
Round-Robin — every pair, head-to-head (For – Against):
X beats Y 26 – 8
Z beats X 18 – 16
Y beats Z 24 – 10
--- Pairwise (Round-Robin) Matrix ---
Head-to-head / pairwise comparison — the Ranked Robin tally
Legend: For - Equal Support - Against (row vs column)
| X | Y | Z |
--------------------------------------------------
X > | --- |26 - 0 - 8 |16 - 0 - 18 |
Y > | 8 - 0 - 26 | --- |24 - 0 - 10 |
Z > | 18 - 0 - 16 |10 - 0 - 24 | --- |
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 X 1–1–0 1 +16 Y
2 Y 1–1–0 1 -4 Z
3 Z 1–1–0 1 -12 X
Winner — Ranked Robin (RCV-RR): X
*** 3 candidates tie for the most wins (X, Y, Z) — 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 3): X, Y, Z
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.
Ranked Robin (RCV-RR) winner X 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