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Secondary method

The Dodgson Quick method

Lewis Carroll’s idea: elect the option that would need the least shuffling to beat everyone. Intuitive, but without proven guarantees comparable to Schulze’s.

When to choose it: if your bylaws specify it, or if this reading of the result speaks to your voters.

Origin
Charles Dodgson (Lewis Carroll), 1876; approximation by McCabe-Dansted, Pritchard and Slinko, 2008
Variants offered
Dodgson Quick
Computation
Polynomial, in n² for n options (exact Dodgson: NP-hard)
The principle

Measuring the distance to the Condorcet winner.

Dodgson elects the option that would become a Condorcet winner with the fewest swaps on the ballots. This exact calculation is NP-hard (Bartholdi, Tovey and Trick, 1989).

Dodgson Quick estimates it simply: for each duel lost, at least half the gap has to be swung. The score adds up these half-gaps, rounded up; the lowest wins.

Step by step

The same cycle, measured in swaps.

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 Camille. Loses to Sophie by 20 votes: score 10.
  2. 2 Adèle. Loses to Camille by 30 votes: score 15.
  3. 3 Sophie. Loses to Adèle by 50 votes: score 25.
Score (lowest wins)
  1. Camille beaten by Sophie, by 20 votes 10
  2. Adèle beaten by Camille, by 30 votes 15
  3. Sophie beaten by Adèle, by 50 votes 25
Result: Camille › Adèle › Sophie, the same order as Schulze on this example.
Properties

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

No published table covers Dodgson Quick: only the cells established by its construction or by a general result are filled in.

Criterion Dodgson Quick First-past-the-post
Majority An option ranked first by more than half of the voters wins. Yes Note 1 Yes
Majority loser An option ranked last by more than half of the voters never wins. Not established No
Mutual majority If a majority ranks a group of options ahead of all the others, the winner belongs to that group. Not established No
Condorcet winner The option that beats each of the others in a duel is always elected. Yes Note 2 No
Condorcet loser The option beaten by each of the others in a duel is never elected. Not established No
Smith criterion The winner belongs to the smallest group of options that beat every option outside it. Not established No
Independence of clones Adding an option very similar to another does not change the outcome among the others. Not established No
Monotonicity Ranking the winner higher can never make them lose. Not established Yes
Local independence Removing the option that finished last (or first) in the result does not change the order of the others. Not established No
Reversal symmetry If every ballot is reversed, the former winner no longer wins. Not established No
Participation Voting sincerely can never leave a voter worse off than abstaining. No Note 3 Yes
Consistency If an option wins in two separate groups, it wins when they are combined. No Note 3 Yes
Later-no-harm Ranking an additional choice can never hurt those already ranked ahead of it. No Note 3 Yes Note 4
Resolvability A final tie becomes unlikely as the number of voters grows. Not established 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 A direct consequence of the Condorcet criterion: an option ranked first by a majority beats every other option in a duel.
  2. 2 Guaranteed by construction: CondorcetPHP always places the Condorcet winner first.
  3. 3 No method that satisfies the Condorcet criterion can satisfy this criterion: this is a general result.
  4. 4 Trivially: first-past-the-post never looks beyond the first choice.
All the methods, criterion by criterion
In Condorcet.Vote

Our implementation choices.

  • The Condorcet winner first. When one exists, CondorcetPHP always puts it in first place.
  • A deliberate approximation. Exact Dodgson is not offered: its computation can become impractical.
  • A limit worth knowing. The score looks only at defeats: an option beaten by all the others can rank ahead of an option that loses only one duel.
Sources

Further reading.

Encyclopedia Dodgson’s method Wikipedia
Documentation Voting Methods CondorcetPHP · docs.condorcet.io
Paper Approximability of Dodgson’s rule John C. McCabe-Dansted, Geoffrey Pritchard, Arkadii Slinko (2008) · Social Choice and Welfare 31(2)

The other methods

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