No Condorcet Winner — STAR (ranks→scores): Brad wins the runoff¶

▶ Live on BetterVoting: vote · results ↗ (election cxrf8v · test BV2138) — 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=Brad, RCV-IRV=Dave, Ranked Robin=Abby (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	Brad
RCV-IRV	Dave
Ranked Robin (RCV-RR)	Abby
Approval	Erin
Range / Score	Abby
Condorcet	none (cycle)

Flags: none

Source election: method_comparisons/no_condorcet_bv2138/cases/bv2138_cxrf8v_star.yaml · STAR tabulated mirror: bv2138_cxrf8v_star_tabulated.txt

5 candidates, 921 ballots.

The ballots¶

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

Count	Abby	Brad	Cora	Dave	Erin
98	5	1	4	2	3
64	4	5	2	1	3
12	4	5	1	2	3
98	3	5	2	1	4
13	3	5	1	2	4
125	2	5	1	3	4
124	4	1	5	2	3
76	3	1	5	2	4
21	4	3	1	5	2
30	3	4	1	5	2
98	1	4	2	5	3
139	3	2	4	5	1
23	2	3	4	5	1

STAR result (official)¶

Scoring round (sum of scores): Abby 2836, Brad 2780, Cora 2761, Dave 2738, Erin 2700

Finalists (top two): Abby and Brad

Automatic runoff: Abby 458 vs Brad 463

STAR winner: Brad

Full LH STAR engine report:

--- Runoff (Preference) Matrix ---
Head-to-head / pairwise comparison
Legend: For - Equal Support - Against
        * indicates Top 2 Finalist
                    |     * Abby      |    * Brad      |
--------------------------------------------------------
           * Abby > |       ---       |458 -   0 - 463 |
           * Brad > | 463 -   0 - 458 |      ---       |

[Divergence from STAR]
  STAR     = Brad
  RCV-IRV  = Dave   (differs from STAR)
  Approval = Erin   (differs from STAR)
  RCV-RR   = Abby   (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) = (Abby)
 - Runoff Round Winner   = (Brad)
  Candidate Abby earned the highest total score, but
  Candidate Brad 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 921 ballots.
Count × Abby,Brad,Cora,Dave,Erin
  139 ×    3,   2,   4,   5,   1
  125 ×    2,   5,   1,   3,   4
  124 ×    4,   1,   5,   2,   3
   98 ×    5,   1,   4,   2,   3
   98 ×    3,   5,   2,   1,   4
   98 ×    1,   4,   2,   5,   3
   76 ×    3,   1,   5,   2,   4
   64 ×    4,   5,   2,   1,   3
   30 ×    3,   4,   1,   5,   2
   23 ×    2,   3,   4,   5,   1
   21 ×    4,   3,   1,   5,   2
   13 ×    3,   5,   1,   2,   4
   12 ×    4,   5,   1,   2,   3

[Score Distribution] (how many ballots gave each star rating)
                      Score
Candidate    5    4    3    2    1    0  | Total   Avg
Abby        98  221  356  148   98    0  |  2836   3.1
Brad       312  128   44  139  298    0  |  2780   3.0
Cora       200  260    0  260  201    0  |  2761   3.0
Dave       311    0  125  323  162    0  |  2738   3.0
Erin         0  312  396   51  162    0  |  2700   2.9

Scoring Round
 The two highest-scoring candidates advance to the next round.
   Abby          -- 2836 -- First place
   Brad          -- 2780 -- Second place
   Cora          -- 2761
   Dave          -- 2738
   Erin          -- 2700
 Abby and Brad advance.

Automatic Runoff Round
 The candidate preferred in the most head-to-head matchups wins.
   Brad          -- 463 -- First place
   Abby          -- 458
   Equal Support --   0
 Brad wins.
   Voters with a preference: 921 of 921 (no Equal Support).
   Brad 463 (50%) vs Abby 458 (50%); majority = 461.

Winner — STAR Voting Method (single winner)
 Brad

RCV-IRV — round by round¶

--- RCV / Instant-Runoff Voting (single winner) ---
 Tabulating 921 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: Abby, Brad, Cora, Dave, Erin
    98 ×   Abby > Cora > Erin > Dave > Brad      (5, 1, 4, 2, 3)
    64 ×   Brad > Abby > Erin > Cora > Dave      (4, 5, 2, 1, 3)
    12 ×   Brad > Abby > Erin > Dave > Cora      (4, 5, 1, 2, 3)
    98 ×   Brad > Erin > Abby > Cora > Dave      (3, 5, 2, 1, 4)
    13 ×   Brad > Erin > Abby > Dave > Cora      (3, 5, 1, 2, 4)
   125 ×   Brad > Erin > Dave > Abby > Cora      (2, 5, 1, 3, 4)
   124 ×   Cora > Abby > Erin > Dave > Brad      (4, 1, 5, 2, 3)
    76 ×   Cora > Erin > Abby > Dave > Brad      (3, 1, 5, 2, 4)
    21 ×   Dave > Abby > Brad > Erin > Cora      (4, 3, 1, 5, 2)
    30 ×   Dave > Brad > Abby > Erin > Cora      (3, 4, 1, 5, 2)
    98 ×   Dave > Brad > Erin > Cora > Abby      (1, 4, 2, 5, 3)
   139 ×   Dave > Cora > Abby > Brad > Erin      (3, 2, 4, 5, 1)
    23 ×   Dave > Cora > Brad > Abby > Erin      (2, 3, 4, 5, 1)

ROUND 1
Candidate      Votes  Status
-----------  -------  --------
Brad             312  Hopeful
Dave             311  Hopeful
Cora             200  Rejected
Abby              98  Rejected
Erin               0  Rejected

FINAL RESULT
Candidate      Votes  Status
-----------  -------  --------
Dave             609  Elected
Brad             312  Rejected
Cora               0  Rejected
Abby               0  Rejected
Erin               0  Rejected


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

--- 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 (4 of 5): Abby, Brad, Dave, Erin
   Outside (1):        Cora
   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.
   Note: the Copeland leaders (Abby, Brad) are only part of the set — the
   win–loss table's top block understates how wide the contention is.
   RCV-IRV winner Dave 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 921 ballots (score ballots).

Ballots:
   the ranking Ranked Robin reads ("=" = tied); source scores follow in () per column: Abby, Brad, Cora, Dave, Erin
    98 × Abby > Cora > Erin > Dave > Brad      (5, 1, 4, 2, 3)
    64 × Brad > Abby > Erin > Cora > Dave      (4, 5, 2, 1, 3)
    12 × Brad > Abby > Erin > Dave > Cora      (4, 5, 1, 2, 3)
    98 × Brad > Erin > Abby > Cora > Dave      (3, 5, 2, 1, 4)
    13 × Brad > Erin > Abby > Dave > Cora      (3, 5, 1, 2, 4)
   125 × Brad > Erin > Dave > Abby > Cora      (2, 5, 1, 3, 4)
   124 × Cora > Abby > Erin > Dave > Brad      (4, 1, 5, 2, 3)
    76 × Cora > Erin > Abby > Dave > Brad      (3, 1, 5, 2, 4)
    21 × Dave > Abby > Brad > Erin > Cora      (4, 3, 1, 5, 2)
    30 × Dave > Brad > Abby > Erin > Cora      (3, 4, 1, 5, 2)
    98 × Dave > Brad > Erin > Cora > Abby      (1, 4, 2, 5, 3)
   139 × Dave > Cora > Abby > Brad > Erin      (3, 2, 4, 5, 1)
    23 × Dave > Cora > Brad > Abby > Erin      (2, 3, 4, 5, 1)

Round-Robin — every pair, head-to-head (For – Against):
   Brad  beats Abby   463 – 458
   Abby  beats Cora   461 – 460
   Abby  beats Dave   485 – 436
   Abby  beats Erin   511 – 410
   Brad  beats Cora   461 – 460
   Dave  beats Brad   609 – 312
   Brad  beats Erin   623 – 298
   Dave  beats Cora   461 – 460
   Erin  beats Cora   461 – 460
   Erin  beats Dave   610 – 311

--- Pairwise (Round-Robin) Matrix ---
Head-to-head / pairwise comparison — the Ranked Robin tally
Legend: For - Equal Support - Against   (row vs column)
         |      Abby       |     Brad       |     Cora       |     Dave       |     Erin       |
------------------------------------------------------------------------------------------------
  Abby > |       ---       |458 -   0 - 463 |461 -   0 - 460 |485 -   0 - 436 |511 -   0 - 410 |
  Brad > | 463 -   0 - 458 |      ---       |461 -   0 - 460 |312 -   0 - 609 |623 -   0 - 298 |
  Cora > | 460 -   0 - 461 |460 -   0 - 461 |      ---       |460 -   0 - 461 |460 -   0 - 461 |
  Dave > | 436 -   0 - 485 |609 -   0 - 312 |461 -   0 - 460 |      ---       |311 -   0 - 610 |
  Erin > | 410 -   0 - 511 |298 -   0 - 623 |461 -   0 - 460 |610 -   0 - 311 |      ---       |

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  Abby       3–1–0         3    +146  Dave, Erin, Cora
    2  Brad       3–1–0         3     +34  Abby, Erin, Cora
    3  Dave       2–2–0         2     -50  Brad, Cora
    4  Erin       2–2–0         2    -126  Dave, Cora
    5  Cora       0–4–0         0      -4  —

Winner — Ranked Robin (RCV-RR): Abby
   *** 2 candidates tie for the most wins (Abby, Brad) — tied on the tally, not a cycle (some of them beat others head-to-head, but no loop closes). Resolved by total margin, then lot order.

--- 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 (4 of 5): Abby, Brad, Dave, Erin
   Outside (1):        Cora
   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.
   Note: the Copeland leaders (Abby, Brad) are only part of the set — the
   win–loss table's top block understates how wide the contention is.
   Ranked Robin (RCV-RR) winner Abby 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

Count	Abby	Brad	Cora	Dave	Erin
98	5	1	4	2	3
64	4	5	2	1	3
12	4	5	1	2	3
98	3	5	2	1	4
13	3	5	1	2	4
125	2	5	1	3	4
124	4	1	5	2	3
76	3	1	5	2	4
21	4	3	1	5	2
30	3	4	1	5	2
98	1	4	2	5	3
139	3	2	4	5	1
23	2	3	4	5	1

Count	Abby	Brad	Cora	Dave	Erin
98	5	1	4	2	3
64	4	5	2	1	3
12	4	5	1	2	3
98	3	5	2	1	4
13	3	5	1	2	4
125	2	5	1	3	4
124	4	1	5	2	3
76	3	1	5	2	4
21	4	3	1	5	2
30	3	4	1	5	2
98	1	4	2	5	3
139	3	2	4	5	1
23	2	3	4	5	1

Count	Abby	Brad	Cora	Dave	Erin
98	5	1	4	2	3
64	4	5	2	1	3
12	4	5	1	2	3
98	3	5	2	1	4
13	3	5	1	2	4
125	2	5	1	3	4
124	4	1	5	2	3
76	3	1	5	2	4
21	4	3	1	5	2
30	3	4	1	5	2
98	1	4	2	5	3
139	3	2	4	5	1
23	2	3	4	5	1