Coombs Ex.19 full — STAR: Boone wins the runoff 8–7¶

▶ Live on BetterVoting: vote · results ↗ (election 9vxcj7 · test BV2165) — 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=Boone, RCV-IRV=Boone, Ranked Robin=Cass (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	Boone
RCV-IRV	Boone
Ranked Robin (RCV-RR)	Cass
Approval	Cass
Range / Score	Cass
Condorcet	none (cycle)

Flags: none

Source election: method_comparisons/felsenthal_paradoxes/cases/bv2165_9vxcj7_star.yaml · STAR tabulated mirror: bv2165_9vxcj7_star_tabulated.txt

3 candidates, 15 ballots.

The ballots¶

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

Count	Amy	Boone	Cass
4	5	3	1
4	1	5	3
5	3	1	5
2	1	3	5

STAR result (official)¶

Scoring round (sum of scores): Cass 51, Boone 43, Amy 41

Finalists (top two): Cass and Boone

Automatic runoff: Cass 7 vs Boone 8

STAR winner: Boone

Full LH STAR engine report:

--- Runoff (Preference) Matrix ---
Head-to-head / pairwise comparison
Legend: For - Equal Support - Against
        * indicates Top 2 Finalist
               |  * Boone   |  * Cass   |
-----------------------------------------
     * Boone > |    ---     |8 - 0 - 7  |
      * Cass > | 7 - 0 - 8  |   ---     |

[Divergence from STAR]
  STAR                   = Boone
  Choose-One (Plurality) = Cass   (differs from STAR)
  Approval               = Cass   (differs from STAR)
  RCV-RR                 = Cass   (differs from STAR)

[Runoff Reversal]
 - Score Round Winner(s) = (Cass)
 - Runoff Round Winner   = (Boone)
  Candidate Cass earned the highest total score, but
  Candidate Boone 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 15 ballots.
Count × Amy,Boone,Cass
    5 ×   3,    1,   5
    4 ×   5,    3,   1
    4 ×   1,    5,   3
    2 ×   1,    3,   5

[Score Distribution] (how many ballots gave each star rating)
                Score
Candidate  5  4  3  2  1  0  | Total   Avg
Amy        4  0  5  0  6  0  |    41   2.7
Boone      4  0  6  0  5  0  |    43   2.9
Cass       7  0  4  0  4  0  |    51   3.4

Scoring Round
 The two highest-scoring candidates advance to the next round.
   Cass          -- 51 -- First place
   Boone         -- 43 -- Second place
   Amy           -- 41
 Cass and Boone advance.

Automatic Runoff Round
 The candidate preferred in the most head-to-head matchups wins.
   Boone         -- 8 -- First place
   Cass          -- 7
   Equal Support -- 0
 Boone wins.
   Voters with a preference: 15 of 15 (no Equal Support).
   Boone 8 (53%) vs Cass 7 (47%); majority = 8.

Winner — STAR Voting Method (single winner)
 Boone

RCV-IRV — round by round¶

--- RCV / Instant-Runoff Voting (single winner) ---
 Tabulating 15 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: Amy, Boone, Cass
     4 ×   Amy > Boone > Cass      (5, 3, 1)
     4 ×   Boone > Cass > Amy      (1, 5, 3)
     5 ×   Cass > Amy > Boone      (3, 1, 5)
     2 ×   Cass > Boone > Amy      (1, 3, 5)

ROUND 1
Candidate      Votes  Status
-----------  -------  --------
Cass               7  Hopeful
Boone              4  Hopeful
Amy                4  Rejected

FINAL RESULT
Candidate      Votes  Status
-----------  -------  --------
Boone              8  Elected
Cass               7  Rejected
Amy                0  Rejected


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

--- 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): Amy, Boone, Cass
   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 Boone 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 15 ballots (score ballots).

Ballots:
   the ranking Ranked Robin reads ("=" = tied); source scores follow in () per column: Amy, Boone, Cass
     4 × Amy > Boone > Cass      (5, 3, 1)
     4 × Boone > Cass > Amy      (1, 5, 3)
     5 × Cass > Amy > Boone      (3, 1, 5)
     2 × Cass > Boone > Amy      (1, 3, 5)

Round-Robin — every pair, head-to-head (For – Against):
   Amy    beats Boone    9 –  6
   Cass   beats Amy     11 –  4
   Boone  beats Cass     8 –  7

--- Pairwise (Round-Robin) Matrix ---
Head-to-head / pairwise comparison — the Ranked Robin tally
Legend: For - Equal Support - Against   (row vs column)
          |     Amy      |   Boone     |    Cass     |
------------------------------------------------------
    Amy > |     ---      | 9 -  0 -  6 | 4 -  0 - 11 |
  Boone > |  6 -  0 -  9 |    ---      | 8 -  0 -  7 |
   Cass > | 11 -  0 -  4 | 7 -  0 -  8 |    ---      |

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  Cass       1–1–0         1      +6  Amy
    2  Boone      1–1–0         1      -2  Cass
    3  Amy        1–1–0         1      -4  Boone

Winner — Ranked Robin (RCV-RR): Cass
   *** 3 candidates tie for the most wins (Amy, Boone, Cass) — 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): Amy, Boone, Cass
   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 Cass 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