Felsenthal Ex.3 Combined — STAR: Bruno, consistent with both districts¶

▶ Live on BetterVoting: vote · results ↗ (election byk9v2 · test BV2149) — 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=Bruno, RCV-IRV=Alma, Ranked Robin=Cora (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	Bruno
RCV-IRV	Alma
Ranked Robin (RCV-RR)	Cora
Approval	Cora
Range / Score	Cora
Condorcet	none (cycle)

Flags: none

Source election: method_comparisons/felsenthal_paradoxes/cases/bv2149_byk9v2_star.yaml · STAR tabulated mirror: bv2149_byk9v2_star_tabulated.txt

3 candidates, 32 ballots.

The ballots¶

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

Count	Alma	Bruno	Cora
4	5	3	1
6	5	1	3
1	3	5	1
13	1	5	3
7	3	1	5
1	1	3	5

STAR result (official)¶

Scoring round (sum of scores): Cora 102, Bruno 98, Alma 88

Finalists (top two): Cora and Bruno

Automatic runoff: Cora 14 vs Bruno 18

STAR winner: Bruno

Full LH STAR engine report:

--- Runoff (Preference) Matrix ---
Head-to-head / pairwise comparison
Legend: For - Equal Support - Against
        * indicates Top 2 Finalist
                 |   * Bruno    |   * Cora    |
-----------------------------------------------
       * Bruno > |     ---      |18 -  0 - 14 |
        * Cora > | 14 -  0 - 18 |    ---      |

[Divergence from STAR]
  STAR     = Bruno
  RCV-IRV  = Alma   (differs from STAR)
  Approval = Cora   (differs from STAR)
  RCV-RR   = Cora   (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.

[Runoff Reversal]
 - Score Round Winner(s) = (Cora)
 - Runoff Round Winner   = (Bruno)
  Candidate Cora earned the highest total score, but
  Candidate Bruno 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 32 ballots.
Count × Alma,Bruno,Cora
   13 ×    1,    5,   3
    7 ×    3,    1,   5
    6 ×    5,    1,   3
    4 ×    5,    3,   1
    1 ×    3,    5,   1
    1 ×    1,    3,   5

[Score Distribution] (how many ballots gave each star rating)
                   Score
Candidate   5   4   3   2   1   0  | Total   Avg
Alma       10   0   8   0  14   0  |    88   2.8
Bruno      14   0   5   0  13   0  |    98   3.1
Cora        8   0  19   0   5   0  |   102   3.2

Scoring Round
 The two highest-scoring candidates advance to the next round.
   Cora          -- 102 -- First place
   Bruno         --  98 -- Second place
   Alma          --  88
 Cora and Bruno advance.

Automatic Runoff Round
 The candidate preferred in the most head-to-head matchups wins.
   Bruno         -- 18 -- First place
   Cora          -- 14
   Equal Support --  0
 Bruno wins.
   Voters with a preference: 32 of 32 (no Equal Support).
   Bruno 18 (56%) vs Cora 14 (44%); majority = 17.

Winner — STAR Voting Method (single winner)
 Bruno

RCV-IRV — round by round¶

--- RCV / Instant-Runoff Voting (single winner) ---
 Tabulating 32 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: Alma, Bruno, Cora
     4 ×   Alma > Bruno > Cora      (5, 3, 1)
     6 ×   Alma > Cora > Bruno      (5, 1, 3)
     1 ×   Bruno > Alma > Cora      (3, 5, 1)
    13 ×   Bruno > Cora > Alma      (1, 5, 3)
     7 ×   Cora > Alma > Bruno      (3, 1, 5)
     1 ×   Cora > Bruno > Alma      (1, 3, 5)

ROUND 1
Candidate      Votes  Status
-----------  -------  --------
Bruno             14  Hopeful
Alma              10  Hopeful
Cora               8  Rejected

FINAL RESULT
Candidate      Votes  Status
-----------  -------  --------
Alma              17  Elected
Bruno             15  Rejected
Cora               0  Rejected


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

--- 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): Alma, Bruno, Cora
   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 Alma 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 32 ballots (score ballots).

Ballots:
   the ranking Ranked Robin reads ("=" = tied); source scores follow in () per column: Alma, Bruno, Cora
     4 × Alma > Bruno > Cora      (5, 3, 1)
     6 × Alma > Cora > Bruno      (5, 1, 3)
     1 × Bruno > Alma > Cora      (3, 5, 1)
    13 × Bruno > Cora > Alma      (1, 5, 3)
     7 × Cora > Alma > Bruno      (3, 1, 5)
     1 × Cora > Bruno > Alma      (1, 3, 5)

Round-Robin — every pair, head-to-head (For – Against):
   Alma   beats Bruno   17 – 15
   Cora   beats Alma    21 – 11
   Bruno  beats Cora    18 – 14

--- Pairwise (Round-Robin) Matrix ---
Head-to-head / pairwise comparison — the Ranked Robin tally
Legend: For - Equal Support - Against   (row vs column)
          |     Alma     |   Bruno     |    Cora     |
------------------------------------------------------
   Alma > |     ---      |17 -  0 - 15 |11 -  0 - 21 |
  Bruno > | 15 -  0 - 17 |    ---      |18 -  0 - 14 |
   Cora > | 21 -  0 - 11 |14 -  0 - 18 |    ---      |

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  Cora       1–1–0         1      +6  Alma
    2  Bruno      1–1–0         1      +2  Cora
    3  Alma       1–1–0         1      -8  Bruno

Winner — Ranked Robin (RCV-RR): Cora
   *** 3 candidates tie for the most wins (Alma, Bruno, Cora) — 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): Alma, Bruno, Cora
   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 Cora 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