Coombs Ex.19 no-show — STAR: the two abstainers get their favorite (STAR flips too)¶

▶ Live on BetterVoting: vote · results ↗ (election b7b8dv · test BV2166) — an independent tabulator to check the winners below against.

Bucket — APPROVAL_OR_MINOR: Only Approval differs

Generated by STARVote_LH_tabulation_engine/tools_adam/scripts/build_divergence_index.py — rebuilt from the election file; do not hand-edit.

What happens¶

STAR, RCV-IRV and Ranked Robin all agree on Cass. Only Approval differs, electing Amy: Approval counts every score of 3–5 as one equal 'approve' and ignores intensity, rewarding Amy's breadth of acceptability over Cass's stronger but more concentrated support. A threshold story about Approval, not a STAR-vs-IRV teaching case.

Winners by method¶

Method	Winner
STAR	Cass
RCV-IRV	Cass
Ranked Robin (RCV-RR)	Cass
Approval	Amy
Range / Score	Cass
Condorcet	none (cycle)

Flags: none

Source election: method_comparisons/felsenthal_paradoxes/cases/bv2166_b7b8dv_star.yaml · STAR tabulated mirror: bv2166_b7b8dv_star_tabulated.txt

3 candidates, 13 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

STAR result (official)¶

Scoring round (sum of scores): Cass 41, Amy 39, Boone 37

Finalists (top two): Cass and Amy

Automatic runoff: Cass 9 vs Amy 4

STAR winner: Cass

Full LH STAR engine report:

--- Runoff (Preference) Matrix ---
Head-to-head / pairwise comparison
Legend: For - Equal Support - Against
        * indicates Top 2 Finalist
               |   * Amy    |  * Cass   |
-----------------------------------------
       * Amy > |    ---     |4 - 0 - 9  |
      * Cass > | 9 - 0 - 4  |   ---     |

[Divergence from STAR]
  STAR     = Cass
  Approval = Amy   (differs from STAR)

--- STAR Voting Method (single winner) ---
 Tabulating 13 ballots.
Count × Amy,Boone,Cass
    5 ×   3,    1,   5
    4 ×   5,    3,   1
    4 ×   1,    5,   3

[Score Distribution] (how many ballots gave each star rating)
                Score
Candidate  5  4  3  2  1  0  | Total   Avg
Amy        4  0  5  0  4  0  |    39   3.0
Boone      4  0  4  0  5  0  |    37   2.8
Cass       5  0  4  0  4  0  |    41   3.2

Scoring Round
 The two highest-scoring candidates advance to the next round.
   Cass          -- 41 -- First place
   Amy           -- 39 -- Second place
   Boone         -- 37
 Cass and Amy advance.

Automatic Runoff Round
 The candidate preferred in the most head-to-head matchups wins.
   Cass          -- 9 -- First place
   Amy           -- 4
   Equal Support -- 0
 Cass wins.
   Voters with a preference: 13 of 13 (no Equal Support).
   Cass 9 (69%) vs Amy 4 (31%); majority = 7.

Winner — STAR Voting Method (single winner)
 Cass

RCV-IRV — round by round¶

--- RCV / Instant-Runoff Voting (single winner) ---
 Tabulating 13 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)

ROUND 1
Candidate      Votes  Status
-----------  -------  --------
Cass               5  Hopeful
Amy                4  Hopeful
Boone              4  Rejected

FINAL RESULT
Candidate      Votes  Status
-----------  -------  --------
Cass               9  Elected
Amy                4  Rejected
Boone              0  Rejected


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

--- 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 Cass 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 13 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)

Round-Robin — every pair, head-to-head (For – Against):
   Amy    beats Boone   9 – 4
   Cass   beats Amy     9 – 4
   Boone  beats Cass    8 – 5

--- 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 - 4 |4 - 0 - 9 |
  Boone > | 4 - 0 - 9 |   ---    |8 - 0 - 5 |
   Cass > | 9 - 0 - 4 |5 - 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      +2  Amy
    2  Amy        1–1–0         1      +0  Boone
    3  Boone      1–1–0         1      -2  Cass

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