====================================================================== SOURCE FILE: bv2279_qywq7d_star.yaml TABULATED FILE: bv2279_qywq7d_star_tabulated.txt ====================================================================== election_title: Three brothers, one fruit — the majoritarian winner is not the utilitarian one voting_method: STAR num_winners: 1 bv_test_id: BV2279 bv_election_id: qywq7d bv_results_url: https://bettervoting.com/qywq7d/results scenario_description: | Race 1 of 3 in the three-brothers election (BV2279, bvid qywq7d; BV-confirmed). Live results: https://bettervoting.com/qywq7d/results Companion races: bv2279_qywq7d_ranked_robin.yaml (also elects Banana) and bv2279_qywq7d_approval.yaml (elects Orange). The smallest election in which the two deepest ideals of a "good winner" name different candidates. The construction is Warren Smith's "three brothers split one fruit" (rangevoting.org), quoted for years in this repo as a prose table of happiness numbers on an arbitrary 0-11 scale. This file is that table made runnable: the utilities are rescaled x5/11 onto a real 0-5 STAR ballot, which preserves every relation the example turns on — the ordering of the totals, and every head-to-head. original utility Apple Orange Banana this ballot Apple Orange Banana boy 1 2 7 8 boy 1 1 3 4 boy 2 3 9 10 boy 2 1 4 5 boy 3 4 11 0 boy 3 2 5 0 average 3 9 6 total 4 12 9 Two answers, both defensible: UTILITARIAN — Orange maximizes total satisfaction (12 points to Banana's 9). Banana is the favorite of two brothers but is worth literally nothing to the third, and that zero is the whole story. Orange is everyone's good-enough fruit. MAJORITARIAN — Banana is the Condorcet winner. It beats Orange 2-1 head-to-head and Apple 2-1, so it wins every one-on-one matchup. A majority prefers it, and majority preference does not care that boy 3's loss is far larger than boys 1 and 2's gain. What the engine shows is which ideal each stage of STAR is chasing. The SCORING ROUND is a utilitarian count and Orange leads it. The AUTOMATIC RUNOFF is a majoritarian check and it overturns that result: Banana 2, Orange 1. STAR elects Banana. So this is a case where STAR does NOT elect the utilitarian winner — by design. The runoff exists precisely to make the score leader survive a majority vote, and here it doesn't. Score voting and Approval elect Orange; STAR, Ranked Robin, IRV and Plurality all elect Banana. Only the two methods that never take a majority vote reach the utilitarian answer. Apple is the Condorcet LOSER and finishes last everywhere. It is in the file to keep the pairwise matrix from being a two-candidate echo of the runoff — with only Orange and Banana there is no "beats everyone" ideal to distinguish, because head-to-head and majority rule coincide. Neither winner is wrong. They optimize different things, which is the point of the page this case backs. video_script: | Read the scoring round out loud first — Orange 12, Banana 9 — and say "Orange makes everyone happiest." Then the runoff line, and say "and a majority still prefers Banana." That pause is the lesson: STAR shows you both answers and tells you which one it acted on. Ranked methods can only ever compute the second, because the ballot never recorded boy 3's zero. Don't sell this as a STAR win. It's an honest limit — the utilitarian optimum is on screen and STAR walks past it. The sales point is that you can SEE it walk past. ballots: |- Apple,Orange,Banana 1,3,4 # Boy 1 — banana by a nose, orange close behind 1,4,5 # Boy 2 — banana best, orange nearly as good 2,5,0 # Boy 3 — orange is everything, banana is worthless expected_winners: [Banana] # file: bv2279_qywq7d_star.yaml ====================================================================== TABULATION RESULTS ====================================================================== === Three brothers, one fruit — the majoritarian winner is not the utilitarian one === Race 1 of 3 in the three-brothers election (BV2279, bvid qywq7d; BV-confirmed). Live results: https://bettervoting.com/qywq7d/results Companion races: bv2279_qywq7d_ranked_robin.yaml (also elects Banana) and bv2279_qywq7d_approval.yaml (elects Orange). The smallest election in which the two deepest ideals of a "good winner" name different candidates. The construction is Warren Smith's "three brothers split one fruit" (rangevoting.org), quoted for years in this repo as a prose table of happiness numbers on an arbitrary 0-11 scale. This file is that table made runnable: the utilities are rescaled x5/11 onto a real 0-5 STAR ballot, which preserves every relation the example turns on — the ordering of the totals, and every head-to-head. original utility Apple Orange Banana this ballot Apple Orange Banana boy 1 2 7 8 boy 1 1 3 4 boy 2 3 9 10 boy 2 1 4 5 boy 3 4 11 0 boy 3 2 5 0 average 3 9 6 total 4 12 9 Two answers, both defensible: UTILITARIAN — Orange maximizes total satisfaction (12 points to Banana's 9). Banana is the favorite of two brothers but is worth literally nothing to the third, and that zero is the whole story. Orange is everyone's good-enough fruit. MAJORITARIAN — Banana is the Condorcet winner. It beats Orange 2-1 head-to-head and Apple 2-1, so it wins every one-on-one matchup. A majority prefers it, and majority preference does not care that boy 3's loss is far larger than boys 1 and 2's gain. What the engine shows is which ideal each stage of STAR is chasing. The SCORING ROUND is a utilitarian count and Orange leads it. The AUTOMATIC RUNOFF is a majoritarian check and it overturns that result: Banana 2, Orange 1. STAR elects Banana. So this is a case where STAR does NOT elect the utilitarian winner — by design. The runoff exists precisely to make the score leader survive a majority vote, and here it doesn't. Score voting and Approval elect Orange; STAR, Ranked Robin, IRV and Plurality all elect Banana. Only the two methods that never take a majority vote reach the utilitarian answer. Apple is the Condorcet LOSER and finishes last everywhere. It is in the file to keep the pairwise matrix from being a two-candidate echo of the runoff — with only Orange and Banana there is no "beats everyone" ideal to distinguish, because head-to-head and majority rule coincide. Neither winner is wrong. They optimize different things, which is the point of the page this case backs. --- Runoff (Preference) Matrix --- Head-to-head / pairwise comparison Legend: For - Equal Support - Against * indicates Top 2 Finalist | Apple | * Orange | * Banana | ----------------------------------------------------- Apple > | --- |0 - 0 - 3 |1 - 0 - 2 | * Orange > | 3 - 0 - 0 | --- |1 - 0 - 2 | * Banana > | 2 - 0 - 1 |2 - 0 - 1 | --- | [Condorcet Winner] Condorcet Winner: Banana — matches the STAR winner [Condorcet Loser] Condorcet Loser: Apple — loses every head-to-head matchup [Divergence from STAR] STAR = Banana Approval = Orange (differs from STAR) [Runoff Reversal] - Score Round Winner(s) = (Orange) - Runoff Round Winner = (Banana) Candidate Orange earned the highest total score, but Candidate Banana 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) --- [STAR Voting] Tabulating 3 ballots. Apple,Orange,Banana 1, 3, 4 1, 4, 5 2, 5, 0 [Score Distribution] (how many ballots gave each star rating) Score Candidate 5 4 3 2 1 0 | Total Avg Apple 0 0 0 1 2 0 | 4 1.3 Orange 1 1 1 0 0 0 | 12 4.0 Banana 1 1 0 0 0 1 | 9 3.0 [STAR Voting: Scoring Round] The two highest-scoring candidates advance to the next round. Orange -- 12 -- First place Banana -- 9 -- Second place Apple -- 4 Orange and Banana advance. [STAR Voting: Automatic Runoff Round] The candidate preferred in the most head-to-head matchups wins. Banana -- 2 -- First place Orange -- 1 Equal Support -- 0 Banana wins. Runoff math: 3 ballots cast − 0 Equal Support (no preference between the two finalists) ─ 3 voters with a preference (majority = 2) Banana 2 (67%) · Orange 1 (33%) [STAR Voting: Winner — STAR Voting Method (single winner)] Banana