Tally method

The Ranked Pairs method

The transparent alternative to Schulze: duels are kept one by one, from the most decisive to the closest, and anyone can follow the calculation by hand. Almost always the same winner.

When to choose it: when you want to be able to explain the result duel by duel, or if your bylaws specify it.

Origin: Nicolaus Tideman, 1987
Variants offered: Ranked Pairs (winning votes) · Ranked Pairs (margins)
Computation: Polynomial; up to 60 options in CondorcetPHP

The principle

Lock in the most decisive victories first.

Ranked Pairs sorts all the duels won, from the most decisive to the closest, then locks them in one by one, except those that would form a cycle with the ones before: they are skipped.

The locked duels draw an order free of contradictions: the winner is the option that none of them beats, and the ranking follows.

Step by step

The same cycle, locked in duel by duel.

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 Sort the duels. Adèle beats Sophie with 75 votes, Camille beats Adèle with 65, Sophie beats Camille with 60.
2 Lock in the first two. Adèle › Sophie, then Camille › Adèle: no cycle.
3 Skip the last one. Sophie › Camille would close the cycle and contradict two more decisive victories.

1 Adèle beats Sophie 75–25 locked
2 Camille beats Adèle 65–35 locked
3 Sophie beats Camille 60–40 skipped

Result: Camille › Adèle › Sophie, as with Schulze.

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).

Ranked Pairs (winning votes)

Duels are sorted by the winner’s number of votes. The most widespread variant today.

Ranked Pairs (margins)

Duels are sorted by the vote margin, as in Tideman’s original papers.

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.

Almost all of Schulze’s guarantees, plus local independence.

Criterion	Ranked Pairs	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 Note 1	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.	Yes	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 2
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 When two duels have exactly the same strength, full independence of clones requires the Zavist and Tideman (1989) tie-breaking rule, which CondorcetPHP does not apply: it breaks ties by the weakest opposition, then processes together any duels that remain tied.
2 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. The options that no locked duel beats take first place; they are removed and the process starts again for the following ranks.
Duels of equal strength. They are separated by the weakest opposition; if they remain equal, they are examined together and any duel that would create a cycle is skipped.
No chance involved. The additional tie-breaking rule of Zavist and Tideman (1989) is not applied: rare ties remain possible.

Sources

Type	Reference
Encyclopedia	Ranked pairs Wikipedia
Documentation	Voting Methods CondorcetPHP · docs.condorcet.io
Paper	Independence of clones as a criterion for voting rules T. Nicolaus Tideman (1987) · Social Choice and Welfare 4(3)

The other methods

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