====================================================================== SOURCE FILE: ex09_game_night_cycle.yaml TABULATED FILE: ex09_game_night_cycle_tabulated.txt ====================================================================== election_title: "Exercise 9 — Game night: nobody is unbeatable (a Ranked Robin ladder drill)" scenario_description: |- Ten friends rank four board games. Azul beats Boggle 7-3, Boggle beats Catan 7-3, Catan beats Azul 6-4 — a rock-paper-scissors cycle, so there is NO Condorcet winner — and everything beats Dominion 10-0. Ranked Robin walks its ladder: most pairwise wins first (Azul, Boggle, and Catan all tie at 2-1), then TOTAL WIN MARGIN: Azul +12, Boggle +10, Catan +8 — Azul wins, deterministically, no lot needed. This is an LH-only exercise on purpose: BetterVoting resolves a 3-way wins tie at RANDOM (the BV2142 caveat), so a frozen BV result cannot exist. Exercise: ex09_game_night_cycle.md. Ballots and cast are this repo's own. voting_method: RankedRobin num_winners: 1 lot_numbers: [Azul, Boggle, Catan, Dominion] ballots: |- Azul>Boggle>Catan>Dominion Azul>Boggle>Catan>Dominion Azul>Boggle>Catan>Dominion Azul>Boggle>Catan>Dominion Boggle>Catan>Azul>Dominion Boggle>Catan>Azul>Dominion Boggle>Catan>Azul>Dominion Catan>Azul>Boggle>Dominion Catan>Azul>Boggle>Dominion Catan>Azul>Boggle>Dominion expected_winners: - Azul # file: ex09_game_night_cycle.yaml ====================================================================== TABULATION RESULTS ====================================================================== --- Ranked Robin (RCV-RR / Copeland) Method (single winner) --- Tabulating 10 ballots (ranked ballots). Ballots: 4 × Azul > Boggle > Catan > Dominion 3 × Boggle > Catan > Azul > Dominion 3 × Catan > Azul > Boggle > Dominion Round-Robin — every pair, head-to-head (For – Against): Azul beats Boggle 7 – 3 Catan beats Azul 6 – 4 Azul beats Dominion 10 – 0 Boggle beats Catan 7 – 3 Boggle beats Dominion 10 – 0 Catan beats Dominion 10 – 0 --- Pairwise (Round-Robin) Matrix --- Head-to-head / pairwise comparison — the Ranked Robin tally Legend: For - Equal Support - Against (row vs column) | Azul | Boggle | Catan | Dominion | ----------------------------------------------------------------------- Azul > | --- | 7 - 0 - 3 | 4 - 0 - 6 |10 - 0 - 0 | Boggle > | 3 - 0 - 7 | --- | 7 - 0 - 3 |10 - 0 - 0 | Catan > | 6 - 0 - 4 | 3 - 0 - 7 | --- |10 - 0 - 0 | Dominion > | 0 - 0 - 10 | 0 - 0 - 10 | 0 - 0 - 10 | --- | 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 Azul 2–1–0 2 +12 Boggle, Dominion 2 Boggle 2–1–0 2 +10 Catan, Dominion 3 Catan 2–1–0 2 +8 Azul, Dominion 4 Dominion 0–3–0 0 -30 — Winner — Ranked Robin (RCV-RR): Azul *** 3 candidates tie for the most wins (Azul, Boggle, Catan) — a Condorcet cycle (no candidate beats all others). Resolved by total margin, then lot order. (This is where Minimax / Ranked Pairs / Schulze differ — see 05_Ranked_Robin/01_Learn/cycle_resolution.md.) --- 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 (3 of 4): Azul, Boggle, Catan Outside (1): Dominion 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. Ranked Robin (RCV-RR) winner Azul 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