Narrow Bands — a paint-swatch election scored in slivers of the scale¶

▶ Live on BetterVoting: vote · results ↗ (election gyv2qt · test BV2239) — 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=Beige, RCV-IRV=Azure, Ranked Robin=Coral (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	Beige
RCV-IRV	Azure
Ranked Robin (RCV-RR)	Coral
Approval	Azure
Range / Score	Azure
Condorcet	none (cycle)

Flags: 16 tied-score ballot(s); IRV winner flips under reversed priority (fragile tie); RR conversion-sensitive (weak=Coral, strict=Azure)

Source election: 06_Other/ballot_style_lab/cases/06_c4_b24_narrow-bands.yaml · STAR tabulated mirror: 06_c4_b24_narrow-bands_tabulated.txt

4 candidates, 24 ballots.

The ballots¶

Each row is a group of identical score ballots (0 = no support, 5 = max).

Count	Azure	Beige	Coral	Dune
1	0	0	0	5
4	4	0	4	0
1	5	5	5	5
1	0	4	0	5
1	3	3	3	3
2	2	0	2	0
1	3	5	4	4
2	0	5	0	5
1	4	4	4	0
1	2	1	0	0
1	0	0	0	0
2	3	5	4	5
1	3	4	4	5
1	4	0	0	0
1	4	3	5	2
1	4	2	5	3
1	0	5	0	0
1	3	5	3	5

STAR result (official)¶

Scoring round (sum of scores): Azure 61, Beige 61, Coral 61, Dune 57

Finalists (top two): Azure and Beige

Automatic runoff: Azure 10 vs Beige 9 (5 equal/no-preference)

STAR winner: Azure

Full LH STAR engine report:

--- Runoff (Preference) Matrix ---
Head-to-head / pairwise comparison
Legend: For - Equal Support - Against
        * indicates Top 2 Finalist
        Note: Azure, Beige and Coral tied at 61 in the Scoring Round, and the
              head-to-head rung advanced Beige and Coral. The * marks who
              advanced, not who scored highest. Azure is filtered out of this
              grid, so the head-to-head that settled it is not shown — see the
              Scoring Round.

               |  * Beige   | * Coral   |
-----------------------------------------
     * Beige > |    ---     |9 - 7 - 8  |
     * Coral > | 8 - 7 - 9  |   ---     |

[Divergence from STAR]
  STAR                   = Beige
  Choose-One (Plurality) = Azure   (differs from STAR)
  RCV-IRV                = Azure   (differs from STAR)
  Approval               = Azure   (differs from STAR)
  RCV-RR                 = Coral   (differs from STAR)
  Note: 16 of 24 ballots (67%) 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.

--- STAR Voting Method (single winner) ---
 Tabulating 24 ballots.
Count × Azure,Beige,Coral,Dune
    4 ×     4,    0,    4,   0
    2 ×     2,    0,    2,   0
    2 ×     0,    5,    0,   5
    2 ×     3,    5,    4,   5
    1 ×     0,    0,    0,   5
    1 ×     5,    5,    5,   5
    1 ×     0,    4,    0,   5
    1 ×     3,    3,    3,   3
    1 ×     3,    5,    4,   4
    1 ×     4,    4,    4,   0
    1 ×     2,    1,    0,   0
    1 ×     0,    0,    0,   0
    1 ×     3,    4,    4,   5
    1 ×     4,    0,    0,   0
    1 ×     4,    3,    5,   2
    1 ×     4,    2,    5,   3
    1 ×     0,    5,    0,   0
    1 ×     3,    5,    3,   5

[Score Distribution] (how many ballots gave each star rating)
                   Score
Candidate   5   4   3   2   1   0  | Total   Avg
Azure       1   8   6   3   0   6  |    61   2.5
Beige       8   3   2   1   1   9  |    61   2.5
Coral       3   9   2   2   0   8  |    61   2.5
Dune        9   1   2   1   0  11  |    57   2.4

Scoring Round
 The two highest-scoring candidates advance to the next round.
   Azure         -- 61 -- Tied for first place
   Beige         -- 61 -- Tied for first place
   Coral         -- 61 -- Tied for first place
   Dune          -- 57
 There's a three-way tie for first.

Scoring Round: First tiebreaker
 The two candidates preferred in the most head-to-head matchups advance.
   Beige         -- 18 -- First place
   Coral         -- 14 -- Second place
   Azure         -- 12
   Equal Support --  5
 Beige and Coral advance.

Automatic Runoff Round
 The candidate preferred in the most head-to-head matchups wins.
   Beige         -- 9 -- First place
   Coral         -- 8
   Equal Support -- 7
 Beige wins.
   Voters with a preference: 17 of 24 (7 Equal Support).
   Beige 9 (53%) vs Coral 8 (47%); majority = 9.

Winner — STAR Voting Method (single winner)
 Beige

RCV-IRV — round by round¶

⚠️ This election has a fragile IRV tie (equal scores force an arbitrary tie-break). The round table below is only one realization; a different tie-break can change the winner. See the flag above.

--- RCV / Instant-Runoff Voting (single winner) ---
 Tabulating 24 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: Azure, Beige, Coral, Dune
     1 ×   Dune      (0, 0, 0, 5)
     4 ×   Azure > Coral      (4, 0, 4, 0)
     1 ×   Azure > Beige > Coral > Dune      (5, 5, 5, 5)
     1 ×   Dune > Beige      (0, 4, 0, 5)
     1 ×   Azure > Beige > Coral > Dune      (3, 3, 3, 3)
     2 ×   Azure > Coral      (2, 0, 2, 0)
     1 ×   Beige > Coral > Dune > Azure      (3, 5, 4, 4)
     2 ×   Beige > Dune      (0, 5, 0, 5)
     1 ×   Azure > Beige > Coral      (4, 4, 4, 0)
     1 ×   Azure > Beige      (2, 1, 0, 0)
     1 ×   (all unranked)      (0, 0, 0, 0)
     2 ×   Beige > Dune > Coral > Azure      (3, 5, 4, 5)
     1 ×   Dune > Beige > Coral > Azure      (3, 4, 4, 5)
     1 ×   Azure      (4, 0, 0, 0)
     1 ×   Coral > Azure > Beige > Dune      (4, 3, 5, 2)
     1 ×   Coral > Azure > Dune > Beige      (4, 2, 5, 3)
     1 ×   Beige      (0, 5, 0, 0)
     1 ×   Beige > Dune > Azure > Coral      (3, 5, 3, 5)

ROUND 1
Candidate      Votes  Status
-----------  -------  --------
Azure             11  Hopeful
Beige              7  Hopeful
Dune               3  Rejected
Coral              2  Rejected
Blank Votes        1  Rejected

FINAL RESULT
Candidate      Votes  Status
-----------  -------  --------
Azure             13  Elected
Beige              9  Rejected
Dune               0  Rejected
Coral              0  Rejected
Blank Votes        2  Rejected


Winner(s) — RCV / Instant-Runoff Voting (single winner)
  Azure

--- 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): Azure, Beige, Coral
   Outside (1):        Dune
   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 Azure 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 24 ballots (score ballots).

Ballots:
   the ranking Ranked Robin reads ("=" = tied); source scores follow in () per column: Azure, Beige, Coral, Dune
     1 × Dune > Azure=Beige=Coral      (0, 0, 0, 5)
     4 × Azure=Coral > Beige=Dune      (4, 0, 4, 0)
     1 × Azure=Beige=Coral=Dune      (5, 5, 5, 5)
     1 × Dune > Beige > Azure=Coral      (0, 4, 0, 5)
     1 × Azure=Beige=Coral=Dune      (3, 3, 3, 3)
     2 × Azure=Coral > Beige=Dune      (2, 0, 2, 0)
     1 × Beige > Coral=Dune > Azure      (3, 5, 4, 4)
     2 × Beige=Dune > Azure=Coral      (0, 5, 0, 5)
     1 × Azure=Beige=Coral > Dune      (4, 4, 4, 0)
     1 × Azure > Beige > Coral=Dune      (2, 1, 0, 0)
     1 × Azure=Beige=Coral=Dune      (0, 0, 0, 0)
     2 × Beige=Dune > Coral > Azure      (3, 5, 4, 5)
     1 × Dune > Beige=Coral > Azure      (3, 4, 4, 5)
     1 × Azure > Beige=Coral=Dune      (4, 0, 0, 0)
     1 × Coral > Azure > Beige > Dune      (4, 3, 5, 2)
     1 × Coral > Azure > Dune > Beige      (4, 2, 5, 3)
     1 × Beige > Azure=Coral=Dune      (0, 5, 0, 0)
     1 × Beige=Dune > Azure=Coral      (3, 5, 3, 5)

Round-Robin — every pair, head-to-head (For – Against):
   Azure  beats Beige   10 –  9
   Coral  beats Azure    6 –  2
   Azure  beats Dune    11 –  9
   Beige  beats Coral    9 –  8
   Beige  beats Dune     5 –  4
   Coral  beats Dune     9 –  8

--- Pairwise (Round-Robin) Matrix ---
Head-to-head / pairwise comparison — the Ranked Robin tally
Legend: For - Equal Support - Against   (row vs column)
          |    Azure     |   Beige     |   Coral     |    Dune     |
--------------------------------------------------------------------
  Azure > |     ---      |10 -  5 -  9 | 2 - 16 -  6 |11 -  4 -  9 |
  Beige > |  9 -  5 - 10 |    ---      | 9 -  7 -  8 | 5 - 15 -  4 |
  Coral > |  6 - 16 -  2 | 8 -  7 -  9 |    ---      | 9 -  7 -  8 |
   Dune > |  9 -  4 - 11 | 4 - 15 -  5 | 8 -  7 -  9 |    ---      |

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  Coral      2–1–0         2      +4  Azure, Dune
    2  Beige      2–1–0         2      +1  Coral, Dune
    3  Azure      2–1–0         2      -1  Beige, Dune
    4  Dune       0–3–0         0      -4  —

Winner — Ranked Robin (RCV-RR): Coral
   *** 3 candidates tie for the most wins (Azure, Beige, Coral) — 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): Azure, Beige, Coral
   Outside (1):        Dune
   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 Coral 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

Count	Azure	Beige	Coral	Dune
1	0	0	0	5
4	4	0	4	0
1	5	5	5	5
1	0	4	0	5
1	3	3	3	3
2	2	0	2	0
1	3	5	4	4
2	0	5	0	5
1	4	4	4	0
1	2	1	0	0
1	0	0	0	0
2	3	5	4	5
1	3	4	4	5
1	4	0	0	0
1	4	3	5	2
1	4	2	5	3
1	0	5	0	0
1	3	5	3	5

Count	Azure	Beige	Coral	Dune
1	0	0	0	5
4	4	0	4	0
1	5	5	5	5
1	0	4	0	5
1	3	3	3	3
2	2	0	2	0
1	3	5	4	4
2	0	5	0	5
1	4	4	4	0
1	2	1	0	0
1	0	0	0	0
2	3	5	4	5
1	3	4	4	5
1	4	0	0	0
1	4	3	5	2
1	4	2	5	3
1	0	5	0	0
1	3	5	3	5

Count	Azure	Beige	Coral	Dune
1	0	0	0	5
4	4	0	4	0
1	5	5	5	5
1	0	4	0	5
1	3	3	3	3
2	2	0	2	0
1	3	5	4	4
2	0	5	0	5
1	4	4	4	0
1	2	1	0	0
1	0	0	0	0
2	3	5	4	5
1	3	4	4	5
1	4	0	0	0
1	4	3	5	2
1	4	2	5	3
1	0	5	0	0
1	3	5	3	5