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The Schulze method

The most robust Condorcet method: it settles cycles by the strength of chains of victories, resists clones and makes ties unlikely.

When to choose it: by default, unless your bylaws require another method. Keep winning votes, recommended by its author; prefer margins if your bylaws reason in terms of vote gaps.

Origin
Markus Schulze, 1997; published in 2011
Variants offered
Schulze (winning votes) · Schulze (margins)
Computation
Polynomial, in n³ for n options
The principle

Compare chains of victories, not just duels.

If Camille beats Adèle and Adèle beats Sophie, Camille has a chain of victories leading to Sophie, even if she loses their direct duel. A chain is only as strong as its weakest link.

For each pair of options, Schulze compares the strongest chain in each direction: the option whose chain prevails ranks ahead. The resulting ranking can never go round in circles.

Step by step

The cycle of Camille, Adèle and Sophie, settled by Schulze.

The 100 ballots of the example
  • Camille , then Adèle , then Sophie 40 voters
  • Adèle , then Sophie , then Camille 35 voters
  • Sophie , then Camille , then Adèle 25 voters
  1. 1 Measure each victory. Camille beats Adèle with 65 votes, Adèle beats Sophie with 75, Sophie beats Camille with 60.
  2. 2 Find the strongest chains. From Camille to Sophie, the chain Camille › Adèle › Sophie is worth 65; in the other direction, the direct duel is worth 60.
  3. 3 Compare and rank. Camille prevails over both her rivals (65 to 60), then Adèle over Sophie (75 to 60).
Strongest chain, from row to column
Camille Adèle Sophie
Camille 65 65
Adèle 60 75
Sophie 60 60

In color, the chain that beats the reverse chain.

Result: Camille › Adèle › Sophie. Only the least decisive victory, Sophie’s over Camille, gives way.
The variants

Winning votes or margins: two ways to measure a victory.

Same calculation; only the strength given to each victory changes. Without ties on the ballots, both variants always give the same result; they can diverge when voters rank options as tied, or leave some out (those then count as tied, last).

Schulze (winning votes)

Default

The strength of a victory is the number of voters who prefer the winner. The variant recommended by Markus Schulze.

Schulze (margins)

The strength of a victory is the vote margin between winner and loser.

See a vote where the two variants diverge
  • 45 × Camille › Adèle = Sophie
  • 11 × Adèle › Camille = Sophie
  • 15 × Adèle › Sophie › Camille
  • 29 × Sophie › Adèle › Camille

45 voters rank Camille first, with Adèle and Sophie tied behind; 11 rank Adèle first, with the other two tied. With winning votes, the weakest victory is Sophie’s (29 votes): it gives way, and Adèle wins. With margins, it is Camille’s victory over Sophie, by a single vote: Sophie wins.

Adèle beats Camille
55–45
Camille beats Sophie
45–44
Sophie beats Adèle
29–26
Winning votes
Adèle › Camille › Sophie
Margins
Sophie › Adèle › Camille
Properties

What the method guarantees, compared with first-past-the-post.

A rare set of guarantees. Like every Condorcet method, it gives up only the criteria that none can satisfy.

Criterion Schulze First-past-the-post
Majority An option ranked first by more than half of the voters wins. Yes Yes
Majority loser An option ranked last by more than half of the voters never wins. Yes No
Mutual majority If a majority ranks a group of options ahead of all the others, the winner belongs to that group. Yes No
Condorcet winner The option that beats each of the others in a duel is always elected. Yes No
Condorcet loser The option beaten by each of the others in a duel is never elected. Yes No
Smith criterion The winner belongs to the smallest group of options that beat every option outside it. Yes No
Independence of clones Adding an option very similar to another does not change the outcome among the others. Yes No
Monotonicity Ranking the winner higher can never make them lose. Yes Yes
Local independence Removing the option that finished last (or first) in the result does not change the order of the others. No No
Reversal symmetry If every ballot is reversed, the former winner no longer wins. Yes No
Participation Voting sincerely can never leave a voter worse off than abstaining. No Yes
Consistency If an option wins in two separate groups, it wins when they are combined. No Yes
Later-no-harm Ranking an additional choice can never hurt those already ranked ahead of it. No Yes Note 1
Resolvability A final tie becomes unlikely as the number of voters grows. Yes Yes
Fast computation The result is computed in polynomial time, even with many options. Yes Yes
  • ✓ always satisfied
  • ✕ can be violated in some votes
  • ? no published source establishes it
  1. 1 Trivially: first-past-the-post never looks beyond the first choice.
All the methods, criterion by criterion
In Condorcet.Vote

Our implementation choices.

  • A complete ranking. CondorcetPHP puts first the options that no other dominates, removes them and starts again, down to the last.
  • No drawing of lots. Two options with exactly equal chains share the rank: the random tie-break suggested by Markus Schulze is not applied.
  • Partial ballots. Unranked options count behind the others, tied with each other. This is what can make the variants diverge.
Sources

Further reading.

Encyclopedia Schulze method Wikipedia
Documentation Voting Methods CondorcetPHP · docs.condorcet.io
Paper A new monotonic, clone-independent, reversal symmetric, and Condorcet-consistent single-winner election method Markus Schulze (2011) · Social Choice and Welfare 36(2)

The other methods

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