Strategic Axioms (for Profile class)

Green (✓) cells mean no violation was found after exhaustive checking of (representatives of anonymous and neutral equivalence classes of) profiles with the specified numbers of candidates (c) and voters (v). Note that violations may exist for numbers of candidates and voters not yet checked! In some cases, you can click on "(ref)" for a reference stating that the method satisfies the axiom.

Red (✗) cells show the lexicographically smallest configuration (smallest number of candidates, then smallest number of voters, among configurations checked) with a violation. Click on a red cell to view the details of the violation.

Yellow (N/A) cells indicate that the voting method cannot process profiles with ties created in order to check the axiom.

MethodEarlier No HelpLater No HarmParticipation
(Single Winner)		Participation
(Weak Dominance)		Participation
(Optimist)		Participation
(Pessimist)		Strategy Proofness
(Single Winner)		Strategy Proofness
(Weak Dominance)		Strategy Proofness
(Optimist)		Strategy Proofness
(Pessimist)
Anti-PluralityN/AN/A✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-3v; 6c, 2v✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
BaldwinN/AN/A✗ 3c, 12v (checked 2c, 2-11v)✗ 3c, 9v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Baldwin PUTN/AN/A✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Banks✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-8v, 10v; 5c, 2-3v; 6c, 2v✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Beat Path✗ 3c, 5v (checked 2c, 2-11v)✗ 4c, 10v (checked 2c, 2-11v; 3c, 2-33v)✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
BenhamN/AN/A✗ 3c, 8v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Benham PUTN/AN/A✗ 3c, 8v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Benham TBN/AN/A✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
BlacksN/AN/A✗ 3c, 8v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
BordaN/AN/A✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✗ 4c, 2v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Borda-Minimax FaceoffN/AN/A✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✗ 4c, 7v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Bottom-Two-Runoff Instant RunoffN/AN/A✗ 3c, 10v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Bottom-Two-Runoff Instant Runoff PUTN/AN/A✗ 3c, 12v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
BucklinN/AN/A✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Condorcet✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-3v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-3v; 6c, 2v✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Condorcet IRVN/AN/A✗ 3c, 8v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Condorcet IRV PUTN/AN/A✗ 3c, 8v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Condorcet PluralityN/AN/A✗ 3c, 10v (checked 2c, 2-11v)✗ 3c, 8v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Consensus BuilderN/AN/A✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
CoombsN/AN/A✗ 3c, 8v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Coombs PUTN/AN/A✗ 3c, 10v (checked 2c, 2-11v)✗ 3c, 8v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Coombs TBN/AN/A✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Copeland✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 4c, 2v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Copeland-Global-BordaN/AN/A✗ 3c, 11v (checked 2c, 2-11v)✗ 3c, 9v (checked 2c, 2-11v)✗ 3c, 10v (checked 2c, 2-11v)✗ 3c, 9v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Copeland-Global-Minimax✗ 3c, 5v (checked 2c, 2-11v)✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 8v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 7v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Copeland-Local-BordaN/AN/A✗ 3c, 13v (checked 2c, 2-11v)✗ 3c, 10v (checked 2c, 2-11v)✗ 3c, 11v (checked 2c, 2-11v)✗ 3c, 11v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Copeland-Local-Minimax✗ 3c, 5v (checked 2c, 2-11v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
DaunouN/AN/A✗ 3c, 10v (checked 2c, 2-11v)✗ 3c, 8v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
DodgsonN/AN/A✗ 5c, 4v (checked 2c, 2-11v; 3c, 2-33v; 4c, 2-8v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 2v (checked 2c, 2-11v; 3c, 2-25v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
GOCHA✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Instant RunoffN/AN/A✗ 3c, 10v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Instant Runoff PUTN/AN/A✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Instant Runoff TBN/AN/A✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Iterated Removal Condorcet Loser✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-3v; 6c, 2v✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Kemeny-Young✗ 3c, 5v (checked 2c, 2-11v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-33v)✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Knockout VotingN/AN/A✗ 3c, 11v (checked 2c, 2-11v)✗ 3c, 9v (checked 2c, 2-11v)✗ 3c, 10v (checked 2c, 2-11v)✗ 3c, 9v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Leximax✗ 3c, 3v (checked 2c, 2-11v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-20v)✗ 4c, 8v (checked 2c, 2-11v; 3c, 2-20v)✗ 4c, 7v (checked 2c, 2-11v; 3c, 2-20v)✗ 4c, 7v (checked 2c, 2-11v; 3c, 2-20v)✗ 4c, 8v (checked 2c, 2-11v; 3c, 2-20v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Llull✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Loss-Trimmer Voting✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 4c, 7v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 9v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Minimax✗ 3c, 5v (checked 2c, 2-11v)✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Minimax (Support)✗ 3c, 5v (checked 2c, 2-11v)✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Most Wins, Smallest Loss✗ 3c, 3v (checked 2c, 2-11v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-20v)✗ 4c, 8v (checked 2c, 2-11v; 3c, 2-20v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-20v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-20v)✗ 4c, 7v (checked 2c, 2-11v; 3c, 2-20v)✗ 4c, 2v (checked 2c, 2-11v; 3c, 2-20v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Pareto✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v
PluralityN/AN/A✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-3v; 6c, 2v✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Plurality with Runoff PUTN/AN/A✓ 2c, 4c, 2-11v; 3c, 2-20v; 5c, 2-4v; 6c, 2v✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Positive-Negative VotingN/AN/A✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✗ 4c, 2v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Preliminary Weighted CondorcetN/AN/A✗ 3c, 18v (checked 2c, 2-11v)✗ 2c, 3v✗ 2c, 3v✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 2c, 3v✗ 2c, 3v✗ 2c, 3v
Ranked Pairs TB✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Ranked Pairs ZTN/AN/A✗ 3c, 13v (checked 2c, 2-11v)✗ 3c, 13v (checked 2c, 2-11v)✗ 3c, 13v (checked 2c, 2-11v)✗ 3c, 13v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Raynaud✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 11v (checked 2c, 2-11v)✗ 3c, 8v (checked 2c, 2-11v)✗ 3c, 9v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
River TB✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
River ZTN/AN/A✗ 3c, 13v (checked 2c, 2-11v)✗ 3c, 13v (checked 2c, 2-11v)✗ 3c, 13v (checked 2c, 2-11v)✗ 3c, 13v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Simple Stable Voting✗ 3c, 5v (checked 2c, 2-11v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 9v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 7v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 8v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Simplified BucklinN/AN/A✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 3v (checked 2c, 2-11v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 3v (checked 2c, 2-11v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 2v (checked 2c, 2-11v)✗ 4c, 2v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 2v (checked 2c, 2-11v)
Simplified Dodgson✗ 3c, 3v (checked 2c, 2-11v)✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-20v)✓ 2c, 2-11v; 3c, 2-20v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-20v)✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-20v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-20v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-20v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Slater✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Smith IRVN/AN/A✗ 3c, 8v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Smith IRV PUTN/AN/A✗ 3c, 8v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Smith-Minimax✗ 3c, 5v (checked 2c, 2-11v)✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Split Cycle✗ 3c, 5v (checked 2c, 2-11v)✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-3v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-3v; 6c, 2v✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Stable Voting✗ 3c, 5v (checked 2c, 2-11v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 9v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 7v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 7v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Strict NansonN/AN/A✗ 4c, 7v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Superior VotingN/AN/A✗ 3c, 8v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Tideman Alternative GOCHAN/AN/A✗ 3c, 9v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Tideman Alternative GOCHA PUTN/AN/A✗ 3c, 9v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Tideman Alternative Top CycleN/AN/A✗ 3c, 8v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Tideman Alternative Top Cycle PUTN/AN/A✗ 3c, 8v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Top Cycle✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-3v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-3v; 6c, 2v✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-3v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-3v; 6c, 2v✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Uncovered Set✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✗ 3c, 8v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Uncovered Set - Bordes✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Uncovered Set - Fishburn✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Uncovered Set - McKelvey✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Weak Condorcet✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Weak NansonN/AN/A✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-33v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Weighted BucklinN/AN/A✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Weighted Covering✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
WoodallN/AN/A✗ 3c, 8v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)