Cycle resolution, counted — where the Condorcet family stops agreeing

The tabulatable evidence behind Cycle Resolution — why Minimax, Ranked Pairs, and Schulze exist. When a Condorcet winner exists, every Condorcet method elects them and there is nothing to argue about. When majority preference cycles, the family splits — and these elections are the split, made runnable.

All are LH-only: the LH engine's Ranked Robin is Copeland, and in a cycle Copeland usually ties, which LH breaks by margin then lot and BetterVoting breaks at random. A tie-deciding result can't be frozen on BV, so these cases stay in the library.

The last two rows are one election counted twice — 999 ballots converted from an outside engine's published cast vote record, which is why they break this folder's usual keep-it-small rule. See RCV Lab for where the data came from and what was verified against it.

Case (source)	Ballots	What it shows
page · cycle_copeland_ties_c4_b21.yaml	21	Copeland ties Alder, Birch, Cedar at 2–1 each — the simple count can't pick. All four refined rules then agree on Alder, whose only defeat (margin 1) is the mildest in the cycle.
page · cycle_schulze_vs_ranked_pairs_c4_b40.yaml	40	The two "serious" cycle-resolvers disagree outright: Schulze elects Ana, Ranked Pairs elects Bruno, on identical ballots. Split Cycle returns both, on the grounds that the ballots don't separate them.
page · cycle_family_splits_c5_b77.yaml	77	Five candidates, Smith set = everyone, no Condorcet winner. Ranked Pairs alone picks Ben; Ranked Robin (margin tiebreak), Minimax, Schulze and Stable Voting all pick Ava; Split Cycle returns both. Replaces an earlier unsourced "Heitzig" profile — this one is search-built and pref_voting-verified.
page · cycle_vote_on_the_rule_rr_c5_b999.yaml	999	The candidates are the cycle-breaking rules, and the ballots cycle. Minimax, Ranked Pairs, Schulze, Split Cycle, Stable Voting and RCV-IRV are unanimous for Ranked Pairs — Schulze and Minimax each voting for a rival over itself. Only Copeland's Rule can't decide (all three go 3–1), so Ranked Robin's margin tiebreak lands on Schulze Method instead. Converted from RCV Lab's published CVR.
page · cycle_vote_on_the_rule_irv_c5_b999.yaml	999	The same ballots as an RCV-IRV count, kept as the fidelity check: every tally reproduces RCV Lab's published report. Winner Ranked Pairs on 492 — a majority of the 886 ballots still live, and 49.2% of the 999 cast.

Running them

The LH engine tabulates the Copeland/Ranked Robin column and writes the _tabulated mirror:

python STARVote_LH_tabulation_engine/starvote_larry_hastings.py method_comparisons/cycle_resolution/cases/cycle_schulze_vs_ranked_pairs_c4_b40.yaml

The other five rules have no LH implementation. This repo tool prints them all at once, via pref_voting:

uv run STARVote_LH_tabulation_engine/tools_adam/pref_voting_tabulation_engine/cycle_resolution_report.py method_comparisons/cycle_resolution/cases/cycle_schulze_vs_ranked_pairs_c4_b40.yaml

Pairwise margins (winner's margin over loser):
   Bruno beats Ana by 4
   Ana beats Chloe by 18
   Ana beats Diego by 12
   Bruno beats Chloe by 18
   Diego beats Bruno by 10
   Chloe beats Diego by 12

Condorcet winner: NONE — majority preference cycles.
Smith set: Ana, Bruno, Chloe, Diego

Winners by method:
   Copeland (= Ranked Robin)  [C1]  Ana, Bruno
   Minimax                    [C2]  Ana
   Ranked Pairs               [C2]  Bruno
   Schulze (beat path)        [C2]  Ana
   Split Cycle                [C2]  Ana, Bruno
   Stable Voting              [C2]  Ana

The [C1] / [C2] tags are Fishburn's classification: C1 rules read only who-beat-whom, C2 rules read the margins too.

Concept pages: cycle resolution · Split Cycle, claim-checked · Ranked Robin · Smith set · up: method_comparisons