Brams' grading paradox — the grade leader loses the runoff¶
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 Baker. Only Approval differs, electing Adams: Approval counts every score of 3–5 as one equal 'approve' and ignores intensity, rewarding Adams's breadth of acceptability over Baker's stronger but more concentrated support. A threshold story about Approval, not a STAR-vs-IRV teaching case.
Winners by method¶
| Method | Winner |
|---|---|
| STAR | Baker |
| RCV-IRV | Baker |
| Ranked Robin (RCV-RR) | Baker |
| Approval | Adams |
| Range / Score | Adams |
| Condorcet | Baker |
Flags: 2 tied-score ballot(s); IRV winner flips under reversed priority (fragile tie)
Source election: method_comparisons/brams_grading_paradox/cases/brams_grading_paradox_c3_b3.yaml · STAR tabulated mirror: brams_grading_paradox_c3_b3_tabulated.txt
3 candidates, 3 ballots.
The ballots¶
Each row is a group of identical score ballots (0 = no support, 5 = max).
| Count | Adams | Baker | Chen |
|---|---|---|---|
| 1 | 3 | 0 | 0 |
| 1 | 2 | 3 | 3 |
| 1 | 1 | 2 | 1 |
STAR result (official)¶
Scoring round (sum of scores): Adams 6, Baker 5, Chen 4
Finalists (top two): Adams and Baker
Automatic runoff: Adams 1 vs Baker 2
STAR winner: Baker
Full LH STAR engine report:
--- Runoff (Preference) Matrix ---
Head-to-head / pairwise comparison
Legend: For - Equal Support - Against
* indicates Top 2 Finalist
| * Adams | * Baker |
-----------------------------------------
* Adams > | --- |1 - 0 - 2 |
* Baker > | 2 - 0 - 1 | --- |
[Divergence from STAR]
STAR = Baker
Approval = Adams (differs from STAR)
[Runoff Reversal]
- Score Round Winner(s) = (Adams)
- Runoff Round Winner = (Baker)
Candidate Adams earned the highest total score, but
Candidate Baker 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 3 ballots.
Adams,Baker,Chen
3, 0, 0
2, 3, 3
1, 2, 1
[Score Distribution] (how many ballots gave each star rating)
Score
Candidate 5 4 3 2 1 0 | Total Avg
Adams 0 0 1 1 1 0 | 6 2.0
Baker 0 0 1 1 0 1 | 5 1.7
Chen 0 0 1 0 1 1 | 4 1.3
Scoring Round
The two highest-scoring candidates advance to the next round.
Adams -- 6 -- First place
Baker -- 5 -- Second place
Chen -- 4
Adams and Baker advance.
Automatic Runoff Round
The candidate preferred in the most head-to-head matchups wins.
Baker -- 2 -- First place
Adams -- 1
Equal Support -- 0
Baker wins.
Voters with a preference: 3 of 3 (no Equal Support).
Baker 2 (67%) vs Adams 1 (33%); majority = 2.
Winner — STAR Voting Method (single winner)
Baker
RCV-IRV — round by round¶
⚠️ This election has a fragile IRV tie (equal scores force an arbitrary tie-break). The round table below is only one realization; a different tie-break can change the winner. See the flag above.
--- RCV / Instant-Runoff Voting (single winner) ---
Tabulating 3 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: Adams, Baker, Chen
1 × Adams (3, 0, 0)
1 × Baker > Chen > Adams (2, 3, 3)
1 × Baker > Adams > Chen (1, 2, 1)
FINAL RESULT
Candidate Votes Status
----------- ------- --------
Baker 2 Elected
Adams 1 Rejected
Chen 0 Rejected
Winner(s) — RCV / Instant-Runoff Voting (single winner)
Baker
--- 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): Baker
Outside (2): Adams, Chen
One member ⇒ Baker is the Condorcet winner, beating every rival head-to-head.
RCV-IRV winner Baker 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 3 ballots (score ballots).
Ballots:
the ranking Ranked Robin reads ("=" = tied); source scores follow in () per column: Adams, Baker, Chen
1 × Adams > Baker=Chen (3, 0, 0)
1 × Baker=Chen > Adams (2, 3, 3)
1 × Baker > Adams=Chen (1, 2, 1)
Round-Robin — every pair, head-to-head (For – Against):
Baker beats Adams 2 – 1
Adams ties Chen 1 – 1
Baker beats Chen 1 – 0
--- Pairwise (Round-Robin) Matrix ---
Head-to-head / pairwise comparison — the Ranked Robin tally
Legend: For - Equal Support - Against (row vs column)
| Adams | Baker | Chen |
---------------------------------------------
Adams > | --- |1 - 0 - 2 |1 - 1 - 1 |
Baker > | 2 - 0 - 1 | --- |1 - 2 - 0 |
Chen > | 1 - 1 - 1 |0 - 2 - 1 | --- |
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 Baker 2–0–0 2 +2 Adams, Chen
2 Adams 0–1–1 0.5 -1 —
3 Chen 0–1–1 0.5 -1 —
Winner — Ranked Robin (RCV-RR): Baker
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): Baker
Outside (2): Adams, Chen
One member ⇒ Baker is the Condorcet winner, beating every rival head-to-head.
Ranked Robin (RCV-RR) winner Baker 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