Felsenthal Ex.2 — STAR (ranks→scores): elects the Condorcet winner¶

▶ Live on BetterVoting: vote · results ↗ (election 6fj2kg · test BV2145) — an independent tabulator to check the winners below against.

Bucket — IRV_OUTLIER_RR_WITH_STAR: RCV-IRV is the outlier (center squeeze)

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 elects Ada — and so do Ranked Robin and Condorcet, because Ada is the head-to-head (pairwise) winner, beating every rival one-on-one. RCV-IRV instead elects Ben: Ada collects fewer top (first-choice) votes, so IRV eliminates Ada early, and those ballots transfer to Ben. This is the classic center squeeze — the broadly acceptable candidate is knocked out by round-by-round elimination even though a majority prefers them in a direct matchup. Two methods (STAR + RR) against one (RCV-IRV): IRV is the outlier.

Winners by method¶

Method	Winner
STAR	Ada
RCV-IRV	Ben
Ranked Robin (RCV-RR)	Ada
Approval	Ada
Range / Score	Ada
Condorcet	Ada

Flags: none

Source election: method_comparisons/felsenthal_paradoxes/cases/bv2145_6fj2kg_star.yaml · STAR tabulated mirror: bv2145_6fj2kg_star_tabulated.txt

3 candidates, 17 ballots.

The ballots¶

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

Count	Ada	Ben	Cleo
3	5	3	1
2	5	1	3
4	3	5	1
2	1	5	3
4	3	1	5
2	1	3	5

STAR result (official)¶

Scoring round (sum of scores): Ada 53, Ben 51, Cleo 49

Finalists (top two): Ada and Ben

Automatic runoff: Ada 9 vs Ben 8

STAR winner: Ada

Full LH STAR engine report:

--- Runoff (Preference) Matrix ---
Head-to-head / pairwise comparison
Legend: For - Equal Support - Against
        * indicates Top 2 Finalist
               |   * Ada    |  * Ben    |
-----------------------------------------
       * Ada > |    ---     |9 - 0 - 8  |
       * Ben > | 8 - 0 - 9  |   ---     |

[Divergence from STAR]
  STAR                   = Ada
  Choose-One (Plurality) = Ben   (differs from STAR)
  RCV-IRV                = Ben   (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 17 ballots.
Count × Ada,Ben,Cleo
    4 ×   3,  5,   1
    4 ×   3,  1,   5
    3 ×   5,  3,   1
    2 ×   5,  1,   3
    2 ×   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
Ada        5  0  8  0  4  0  |    53   3.1
Ben        6  0  5  0  6  0  |    51   3.0
Cleo       6  0  4  0  7  0  |    49   2.9

Scoring Round
 The two highest-scoring candidates advance to the next round.
   Ada           -- 53 -- First place
   Ben           -- 51 -- Second place
   Cleo          -- 49
 Ada and Ben advance.

Automatic Runoff Round
 The candidate preferred in the most head-to-head matchups wins.
   Ada           -- 9 -- First place
   Ben           -- 8
   Equal Support -- 0
 Ada wins.
   Voters with a preference: 17 of 17 (no Equal Support).
   Ada 9 (53%) vs Ben 8 (47%); majority = 9.

Winner — STAR Voting Method (single winner)
 Ada

RCV-IRV — round by round¶

--- RCV / Instant-Runoff Voting (single winner) ---
 Tabulating 17 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: Ada, Ben, Cleo
     3 ×   Ada > Ben > Cleo      (5, 3, 1)
     2 ×   Ada > Cleo > Ben      (5, 1, 3)
     4 ×   Ben > Ada > Cleo      (3, 5, 1)
     2 ×   Ben > Cleo > Ada      (1, 5, 3)
     4 ×   Cleo > Ada > Ben      (3, 1, 5)
     2 ×   Cleo > Ben > Ada      (1, 3, 5)

ROUND 1
Candidate      Votes  Status
-----------  -------  --------
Ben                6  Hopeful
Cleo               6  Hopeful
Ada                5  Rejected

FINAL RESULT
Candidate      Votes  Status
-----------  -------  --------
Ben                9  Elected
Cleo               8  Rejected
Ada                0  Rejected


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

--- 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 (1 of 3): Ada
   Outside (2):        Ben, Cleo
   One member ⇒ Ada is the Condorcet winner, beating every rival head-to-head.
   RCV-IRV winner Ben is OUTSIDE the Smith set. ✗
      Every member of the set (Ada) beats Ben head-to-head, yet
      RCV-IRV elected Ben anyway. RCV-IRV is not Smith-efficient (nor
      Condorcet-efficient) — this is the shape a center squeeze leaves behind.
   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 17 ballots (score ballots).

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

Round-Robin — every pair, head-to-head (For – Against):
   Ada   beats Ben    9 – 8
   Ada   beats Cleo   9 – 8
   Ben   beats Cleo   9 – 8

--- Pairwise (Round-Robin) Matrix ---
Head-to-head / pairwise comparison — the Ranked Robin tally
Legend: For - Equal Support - Against   (row vs column)
         |    Ada    |   Ben    |  Cleo    |
--------------------------------------------
   Ada > |    ---    |9 - 0 - 8 |9 - 0 - 8 |
   Ben > | 8 - 0 - 9 |   ---    |9 - 0 - 8 |
  Cleo > | 8 - 0 - 9 |8 - 0 - 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  Ada        2–0–0         2      +2  Ben, Cleo
    2  Ben        1–1–0         1      +0  Cleo
    3  Cleo       0–2–0         0      -2  —

Winner — Ranked Robin (RCV-RR): Ada
   beats every opponent head-to-head — the Condorcet winner.

--- 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 (1 of 3): Ada
   Outside (2):        Ben, Cleo
   One member ⇒ Ada is the Condorcet winner, beating every rival head-to-head.
   Ranked Robin (RCV-RR) winner Ada 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