====================================================================== SOURCE FILE: coombs_ex20_district1.yaml TABULATED FILE: coombs_ex20_district1_tabulated.txt ====================================================================== election_title: "Coombs Ex.20 — District I: 34 voters, Coombs elects B" scenario_description: |- District I of Felsenthal's Coombs reinforcement example. Source: Dan S. Felsenthal (2010), Appendix A7, Example 20. 34 voters, three candidates: 9×(A>B>C), 9×(B>C>A), 11×(C>A>B), 5×(C>B>A). Nobody holds a majority of first places (C leads with 16 of 34), so Coombs deletes the most-hated: A is ranked last on 14 ballots, more than B's 11 or C's 9. With A gone, B holds 18 of 34 and is elected. District II (coombs_ex20_district2.yaml) also elects B. Their union (coombs_ex20_amalgamated.yaml) elects A. That is the reinforcement paradox — a method that agrees with itself in every district and contradicts itself over the whole. A note on the source: Felsenthal prints District II as "6 voters" but lists 7, and the amalgamated table sums to 41. We take the tables as authoritative and flag the typo rather than silently picking one reading. Labels are Felsenthal's own A/B/C so the case can be read side by side with the paper's table; this is an academic reproduction, not a scenario with a cast. Coombs has no tabulator in the LH engine or on BetterVoting, so this file is tabulated as RCV-IRV, the mirror-image count. This engine elects B here, and IRV elects B in the other two files as well — but read the next paragraph before treating that as IRV agreeing with itself across all three, because in THIS district its answer is not determinate at all. The reinforcement failure still belongs to Coombs alone, on Coombs' own untied eliminations, not on the IRV control. For the Coombs count run tools_adam/pref_voting_tabulation_engine/coombs_report.py. One caveat on that IRV result, which is easy to miss because no engine prints it: in THIS district the first round is a tie. C leads with 16, and A and B are level on 9, so which of them is eliminated is a coin flip — and it decides the election. Drop A and its 9 A>B>C ballots elect B 18–16; drop B and its 9 B>C>A ballots elect C 25–9. This engine drops A (silently) and rcv-lab.org drops B, which is how the tie was noticed. So "IRV elects B" holds for our tiebreak, not for IRV as such; districts II and the amalgamated file have no such tie and elect B and A outright, so the reinforcement paradox itself is unaffected. A third engine has since confirmed it. RCTab 2.1.0 — the VVSG-tested tabulator US jurisdictions run — takes the same branch as rcv-lab.org: it drops B and elects C 25–9, and unlike either of the others it PRINTS the tie: "Candidate B lost a tie-breaker in round 1 against A. Each candidate had 9 vote(s)." Sweeping its declared candidate order over all six permutations returns C three times and B three times, so both answers are legal executions of the same rules on the same ballots. This is the single disagreement in the whole 69-case IRV corpus, and it is not an engine defect on either side — it is the documented tie, surfacing. paradoxes: [multiple-districts] voting_method: RCV_IRV num_winners: 1 ballots: |- 9:A>B>C 9:B>C>A 11:C>A>B 5:C>B>A expected_winners: - B # file: coombs_ex20_district1.yaml ====================================================================== TABULATION RESULTS ====================================================================== --- RCV / Instant-Runoff Voting (single winner) --- Coombs Ex.20 — District I: 34 voters, Coombs elects B Tabulating 34 ballots (ranked ballots). ROUND 1 Candidate Votes Status ----------- ------- -------- C 16 Hopeful B 9 Hopeful A 9 Rejected FINAL RESULT Candidate Votes Status ----------- ------- -------- B 18 Elected C 16 Rejected A 0 Rejected Winner(s) — RCV / Instant-Runoff Voting (single winner) B --- 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 3): A, B, C Outside (0): — 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. RCV-IRV winner B is INSIDE the Smith set. ✓ Not guaranteed — RCV-IRV is not Smith-efficient — but it holds here. More: 07_Concepts/topics/smith_set.md