Exercise 5 — The squeezed bridge-builder (IRV vs STAR)¶

▶ Live on BetterVoting: vote · results ↗ (election 6bry7c · test BV2192) — 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 Brook — and so do Ranked Robin and Condorcet, because Brook is the head-to-head (pairwise) winner, beating every rival one-on-one. RCV-IRV instead elects Avi: Brook collects fewer top (first-choice) votes, so IRV eliminates Brook early, and those ballots transfer to Avi. 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	Brook
RCV-IRV	Avi
Ranked Robin (RCV-RR)	Brook
Approval	Brook
Range / Score	Brook
Condorcet	Brook

Flags: none

Source election: 01_STAR/05_Practice/cases/ex05_center_squeeze.yaml · STAR tabulated mirror: ex05_center_squeeze_tabulated.txt

3 candidates, 9 ballots.

The ballots¶

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

Count	Avi	Brook	Cole
4	5	3	0
3	0	3	5
1	3	5	0
1	0	5	3

STAR result (official)¶

Scoring round (sum of scores): Brook 31, Avi 23, Cole 18

Finalists (top two): Brook and Avi

Automatic runoff: Brook 5 vs Avi 4

STAR winner: Brook

Full LH STAR engine report:

--- Runoff (Preference) Matrix ---
Head-to-head / pairwise comparison
Legend: For - Equal Support - Against
        * indicates Top 2 Finalist
               |   * Avi    | * Brook   |
-----------------------------------------
       * Avi > |    ---     |4 - 0 - 5  |
     * Brook > | 5 - 0 - 4  |   ---     |

[Divergence from STAR]
  STAR                   = Brook
  Choose-One (Plurality) = Avi   (differs from STAR)
  RCV-IRV                = Avi   (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 9 ballots.
Count × Avi,Brook,Cole
    4 ×   5,    3,   0
    3 ×   0,    3,   5
    1 ×   3,    5,   0
    1 ×   0,    5,   3

[Score Distribution] (how many ballots gave each star rating)
                Score
Candidate  5  4  3  2  1  0  | Total   Avg
Avi        4  0  1  0  0  4  |    23   2.6
Brook      2  0  7  0  0  0  |    31   3.4
Cole       3  0  1  0  0  5  |    18   2.0

Scoring Round
 The two highest-scoring candidates advance to the next round.
   Brook         -- 31 -- First place
   Avi           -- 23 -- Second place
   Cole          -- 18
 Brook and Avi advance.

Automatic Runoff Round
 The candidate preferred in the most head-to-head matchups wins.
   Brook         -- 5 -- First place
   Avi           -- 4
   Equal Support -- 0
 Brook wins.
   Voters with a preference: 9 of 9 (no Equal Support).
   Brook 5 (56%) vs Avi 4 (44%); majority = 5.

Winner — STAR Voting Method (single winner)
 Brook

RCV-IRV — round by round¶

--- RCV / Instant-Runoff Voting (single winner) ---
 Tabulating 9 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: Avi, Brook, Cole
     4 ×   Avi > Brook      (5, 3, 0)
     3 ×   Cole > Brook      (0, 3, 5)
     1 ×   Brook > Avi      (3, 5, 0)
     1 ×   Brook > Cole      (0, 5, 3)

ROUND 1
Candidate      Votes  Status
-----------  -------  --------
Avi                4  Hopeful
Cole               3  Hopeful
Brook              2  Rejected

FINAL RESULT
Candidate      Votes  Status
-----------  -------  --------
Avi                5  Elected
Cole               4  Rejected
Brook              0  Rejected


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

--- 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): Brook
   Outside (2):        Avi, Cole
   One member ⇒ Brook is the Condorcet winner, beating every rival head-to-head.
   RCV-IRV winner Avi is OUTSIDE the Smith set. ✗
      Every member of the set (Brook) beats Avi head-to-head, yet
      RCV-IRV elected Avi 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 9 ballots (score ballots).

Ballots:
   the ranking Ranked Robin reads ("=" = tied); source scores follow in () per column: Avi, Brook, Cole
     4 × Avi > Brook > Cole      (5, 3, 0)
     3 × Cole > Brook > Avi      (0, 3, 5)
     1 × Brook > Avi > Cole      (3, 5, 0)
     1 × Brook > Cole > Avi      (0, 5, 3)

Round-Robin — every pair, head-to-head (For – Against):
   Brook  beats Avi     5 – 4
   Avi    beats Cole    5 – 4
   Brook  beats Cole    6 – 3

--- Pairwise (Round-Robin) Matrix ---
Head-to-head / pairwise comparison — the Ranked Robin tally
Legend: For - Equal Support - Against   (row vs column)
          |    Avi    |  Brook   |  Cole    |
---------------------------------------------
    Avi > |    ---    |4 - 0 - 5 |5 - 0 - 4 |
  Brook > | 5 - 0 - 4 |   ---    |6 - 0 - 3 |
   Cole > | 4 - 0 - 5 |3 - 0 - 6 |   ---    |

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  Brook      2–0–0         2      +4  Avi, Cole
    2  Avi        1–1–0         1      +0  Cole
    3  Cole       0–2–0         0      -4  —

Winner — Ranked Robin (RCV-RR): Brook
   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): Brook
   Outside (2):        Avi, Cole
   One member ⇒ Brook is the Condorcet winner, beating every rival head-to-head.
   Ranked Robin (RCV-RR) winner Brook 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