Exercise 8 — a small runoff reversal (sample solution, three candidates)

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

Winners by method

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

Flags: none

Source election: 01_STAR/05_Practice/cases/ex08_minimal_reversal_3c.yaml · STAR tabulated mirror: ex08_minimal_reversal_3c_tabulated.txt

3 candidates, 5 ballots.

The ballots

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

Count	A	B	C
3	5	4	0
2	0	5	1

STAR result (official)

Scoring round (sum of scores): B 22, A 15, C 2

Finalists (top two): B and A

Automatic runoff: B 2 vs A 3

STAR winner: A

Full LH STAR engine report:

--- Runoff (Preference) Matrix ---
Head-to-head / pairwise comparison
Legend: For - Equal Support - Against
        * indicates Top 2 Finalist
               |    * A     |   * B     |
-----------------------------------------
         * A > |    ---     |3 - 0 - 2  |
         * B > | 2 - 0 - 3  |   ---     |

[Divergence from STAR]
  STAR     = A
  Approval = B   (differs from STAR)

[Runoff Reversal]
 - Score Round Winner(s) = (B)
 - Runoff Round Winner   = (A)
  Candidate B earned the highest total score, but
  Candidate A 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 5 ballots.
Count × A,B,C
    3 × 5,4,0
    2 × 0,5,1

[Score Distribution] (how many ballots gave each star rating)
                Score
Candidate  5  4  3  2  1  0  | Total   Avg
A          3  0  0  0  0  2  |    15   3.0
B          2  3  0  0  0  0  |    22   4.4
C          0  0  0  0  2  3  |     2   0.4

Scoring Round
 The two highest-scoring candidates advance to the next round.
   B             -- 22 -- First place
   A             -- 15 -- Second place
   C             --  2
 B and A advance.

Automatic Runoff Round
 The candidate preferred in the most head-to-head matchups wins.
   A             -- 3 -- First place
   B             -- 2
   Equal Support -- 0
 A wins.
   Voters with a preference: 5 of 5 (no Equal Support).
   A 3 (60%) vs B 2 (40%); majority = 3.

Winner — STAR Voting Method (single winner)
 A

RCV-IRV — round by round

--- RCV / Instant-Runoff Voting (single winner) ---
 Tabulating 5 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: A, B, C
     3 ×   A > B      (5, 4, 0)
     2 ×   B > C      (0, 5, 1)

FINAL RESULT
Candidate      Votes  Status
-----------  -------  --------
A                  3  Elected
B                  2  Rejected
C                  0  Rejected


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

--- 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): A
   Outside (2):        B, C
   One member ⇒ A is the Condorcet winner, beating every rival head-to-head.
   RCV-IRV winner A 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 5 ballots (score ballots).

Ballots:
   the ranking Ranked Robin reads ("=" = tied); source scores follow in () per column: A, B, C
     3 × A > B > C      (5, 4, 0)
     2 × B > C > A      (0, 5, 1)

Round-Robin — every pair, head-to-head (For – Against):
   A  beats B   3 – 2
   A  beats C   3 – 2
   B  beats C   5 – 0

--- Pairwise (Round-Robin) Matrix ---
Head-to-head / pairwise comparison — the Ranked Robin tally
Legend: For - Equal Support - Against   (row vs column)
      |     A     |    B     |    C     |
-----------------------------------------
  A > |    ---    |3 - 0 - 2 |3 - 0 - 2 |
  B > | 2 - 0 - 3 |   ---    |5 - 0 - 0 |
  C > | 2 - 0 - 3 |0 - 0 - 5 |   ---    |

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  A          2–0–0         2      +2  B, C
    2  B          1–1–0         1      +4  C
    3  C          0–2–0         0      -6  —

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