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Cycle Resolution — why Minimax, Ranked Pairs, and Schulze exist

Status: draft / Level 301. A learning page on what happens to Condorcet methods when there's no Condorcet winner — and why a whole family of methods exists just to answer that. All examples below are verified with the pref_voting engine.

For the underlying math — tournaments as graphs, the Smith & Schwartz sets, and each method mapped to its math (Floyd–Warshall, game theory, NP-hardness) — see The Math Behind Condorcet.

For the wider field — where Minimax/Ranked Pairs/Schulze sit among all the ranked tabulations (Borda, Bucklin, Coombs, Copeland, Dodgson…), see The ranked-ballot method zoo.

One line: when a Condorcet winner exists, every Condorcet method elects them — Ranked Robin, Minimax, Ranked Pairs, Schulze all agree. They differ only when majority preference forms a cycle (A beats B, B beats C, C beats A, with no one beating all). "Cycle resolution" is the rule a method uses to pick a winner in that case — and it's the entire difference between these methods.

→ the cycle itself: BV2157 — Condorcet cycle (rock-paper-scissors) · the base method: Ranked Robin · GLOSSARY.

Both profiles on this page are now runnable — method_comparisons/cycle_resolution. Every winner below is printed by cycle_resolution_report.py, which runs all six rules through pref_voting; nothing here is asserted from memory.

The problem: majority rule can eat its own tail

Usually one candidate beats every other head-to-head — the Condorcet winner — and the choice is obvious. But majority preference isn't guaranteed to be transitive. Sometimes:

a majority prefers A > B, a majority prefers B > C, and a majority prefers C > A.

There is no "beats everyone" candidate — the result is a cycle (the Condorcet paradox, known since the 1780s). Now "elect the candidate the majority prefers" has no answer, and a method has to break the tie somehow. How it breaks it is what separates the methods.

Why Ranked Robin / Copeland is tie-prone

Ranked Robin (Copeland) scores by pairwise wins − losses. That's beautifully simple — but in a cycle, candidates tend to share the same record, so it often can't pick a unique winner. Concrete example (21 voters, 4 candidates; verified):

10 : A>B>C>D       Head-to-heads in the top cycle:
 6 : B>C>A>D            A beats B by 9      B beats C by 11      C beats A by 1
 5 : C>A>B>D       (everyone ranks D last, so A, B, C each beat D)

Win–loss record: A +1, B +1, C +1, D −3 → Copeland ties A, B, and C. The simple count throws up its hands. That tie-proneness is exactly why the refined methods below exist: they look at how strong each defeat is, not just who-beat-whom.

The cycle-resolution methods (same ballots, different rule)

On that same example, each refined method gives a unique winner — here, all four pick A (the candidate whose only loss, to C, is the smallest at margin 1):

Minimax (simplest) — elect the candidate whose worst single defeat is the least bad. A's biggest loss is just 1; B's is 9; C's is 11 → A wins. Intuition: "least strongly beaten." (Caveat: in 4+ candidate fields Minimax can occasionally pick a candidate outside the top cycle / Smith set.)

Ranked Pairs (Tideman) — sort every pairwise victory by margin, largest first, and "lock in" each one unless it would create a cycle with the ones already locked. Here: lock B>C (11), then A>B (9); C>A (1) would close a cycle, so skip it. The locked relations read A > B > C → A wins.

Schulze (beatpath) — A "beats" B if the strongest chain of defeats from A to B (its weakest link is its strength) is stronger than the strongest chain back. Follow the strong links and A wins here too. (Widely used in practice — Debian, Wikimedia, Ubuntu.)

Split Cycle (newest) — in every cycle, throw away that cycle's weakest defeat, then elect whoever is left undefeated. Here the only cycle is A>B>C>A, its weakest link is C>A (margin 1), so it goes — and A wins, undefeated. The rule comes from Holliday & Pacuit (2023), and its defining habit shows up in the next section: when the discarding leaves two candidates undefeated, Split Cycle returns both instead of picking one.

→ runnable: the 21-voter profile (cast: Alder / Birch / Cedar / Dogwood).

…but they don't always agree

In the example above all four landed on A. They needn't. A second profile — 40 voters, four candidates, runnable as Ana / Bruno / Chloe / Diego — splits them:

 7 : A>B>C>D      B beats A by 4     A beats C by 18
 8 : B>A>C>D      A beats D by 12    B beats C by 18
14 : D>B>A>C      D beats B by 10    C beats D by 12
11 : C>A>D>B
Method Winner Why
Ranked Robin / Copeland A, B (tie) both are 2–1
Minimax A A's worst defeat is 4, the field's mildest
Schulze A strongest beatpaths run A's way
Ranked Pairs B locks the 18s and 12s first, and they favor B
Split Cycle A, B discarding each cycle's weakest defeat leaves both undefeated

Same ballots, and the two "serious" cycle-resolvers disagree outright — Schulze elects A, Ranked Pairs elects B. That's the whole point of this page in one table.

Split Cycle's answer is the interesting one, and it isn't indecision: its winner set is always a superset of Schulze's and Ranked Pairs'. Where those two produce a single name, Split Cycle is claiming they did so by convention rather than by evidence — the ballots here genuinely fail to separate A from B, and it hands that back rather than resolving it silently. Whether that's honesty or buck-passing is a real disagreement, and it's the subject of its own page.

A nastier five-candidate cycle drives the point home — 77 voters, runnable as Ava / Ben / Cole / Dana / Ezra, with no Condorcet winner and a Smith set of all five:

Method Winner
Ranked Robin / Copeland Ava Copeland ties Ava & Ben (both 3–1); the margin tiebreak picks Ava (+76 vs +24)
Minimax Ava her worst defeat (to Ben, by 3) is the field's mildest
Schulze Ava strongest beatpaths run Ava's way
Ranked Pairs Ben locks the biggest margins first, and they carry Ben
Split Cycle Ava, Ben discarding each cycle's weakest defeat leaves both

Same ballots, and Ranked Pairs stands alone at Ben while every other rule leans Ava — the two "serious" cycle-resolvers, Schulze and Ranked Pairs, disagree outright again. This is the whole point: "Condorcet method" names a family, and once you're inside a cycle, the family splits. (Outside cycles — i.e. almost always — they're identical.)

An earlier draft here showed an unsourced "100-voter Heitzig" profile from memory; it's replaced by this one, built by search and verified with pref_voting. If you want a named profile from the literature, the canonical Schulze-vs-Ranked-Pairs disagreement examples live in Schulze's own paper and on electowiki.

What they share (the good news)

  • Condorcet-consistent: all elect the Condorcet winner whenever one exists — which, with many voters and realistic preferences, is the overwhelming majority of elections.
  • Smith-efficient (the good ones): Ranked Pairs, Schulze, and Copeland always elect from the Smith set (the smallest group that beats everyone outside it). Minimax can miss it.
  • Clone-independent & monotone: Ranked Pairs, Schulze and Split Cycle add these guarantees; that robustness is why they're the "serious" cycle-resolvers despite being harder to explain.
  • Where they part company on criteria: Split Cycle additionally satisfies immunity to spoilers and positive/negative involvement, which Schulze and Ranked Pairs fail — the price being those multi-winner answers. The Split Cycle page checks that trade with a tabulated election in which a candidate no voter ranks above the winner still flips Schulze's result.

Where Ranked Robin and STAR fit

  • Ranked Robin (Equal Vote) is essentially Copeland + a margins tiebreak — a pragmatic choice: cycles are rare, so the simple win-loss count plus a sum-of-margins fallback is usually plenty. (Consensus Choice uses a different fallback, "Most Wins, Smallest Loss" — same family, different cycle rule.) See Ranked Robin (RCV-RR / Copeland).
  • STAR is not a Condorcet method and doesn't try to resolve cycles at all. Its score-then-runoff just produces a winner, which can differ from the Condorcet winner (BV2156 (STAR's miss)). The trade: these ranked methods capture pure majority preference but ignore intensity; STAR captures intensity (how much, not just which) at the cost of strict Condorcet guarantees. Neither is "the" right answer — it's a values choice.

How often do cycles even happen?

Rarely, but not never. Cycles get likelier in small, sharply three-way-divided races and much rarer as the electorate grows and preferences spread along a spectrum. As with the runoff-reversal frequency caveat: any rate you quote depends on the voter model, so state the assumptions rather than a bare number.

Try it yourself / verify

Both profiles on this page ship as runnable YAMLs in method_comparisons/cycle_resolution. The repo tool prints every rule at once:

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

It reports the margins, the Smith set, and Copeland / Minimax / Ranked Pairs / Schulze / Split Cycle / Stable Voting side by side, all computed by pref_voting (cross-check engine). The LH engine itself runs only the Copeland column — that's Ranked Robin. For a quick manual run without the repo, paste a count:A>B>C block into LeGrand's calculator, which reports Minimax, Ranked Pairs, Schulze, Copeland and more side by side.

Learn more (external)

Draft — open question to refine later: decide whether this lives here (Condorcet-family folder) or graduates to a topics/ hub if we add more cross-method deep-dives. (The "add runnable YAMLs for the two worked profiles" item is done — see method_comparisons/cycle_resolution.)