Monotonicity 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.

MethodMonotonicityWeak Positive Responsiveness
Anti-Plurality✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v
Baldwin✗ 3c, 5v (checked 2c, 2-11v)✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v
Baldwin PUT✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)
Banks✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✗ 3c, 4v (checked 2c, 2-11v)
Beat Path✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v (ref)✗ 4c, 9v (checked 2c, 2-11v; 3c, 2-33v)
Benham✗ 3c, 9v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)
Benham PUT✗ 3c, 9v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)
Benham TB✗ 3c, 8v (checked 2c, 2-11v)✗ 3c, 8v (checked 2c, 2-11v)
Blacks✓ 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
Borda✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v (ref)✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v (ref)
Borda-Minimax Faceoff✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)
Bottom-Two-Runoff Instant Runoff✗ 3c, 12v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)
Bottom-Two-Runoff Instant Runoff PUT✗ 3c, 12v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)
Bucklin✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v (ref)✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v
Condorcet✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✗ 3c, 2v (checked 2c, 2-11v)
Condorcet IRV✗ 3c, 9v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)
Condorcet IRV PUT✗ 3c, 9v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)
Condorcet Plurality✓ 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
Consensus Builder✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v
Coombs✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)
Coombs PUT✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)
Coombs TB✗ 3c, 7v (checked 2c, 2-11v)✗ 3c, 8v (checked 2c, 2-11v)
Copeland✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v (ref)✗ 3c, 5v (checked 2c, 2-11v)
Copeland-Global-Borda✓ 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
Copeland-Global-Minimax✓ 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
Copeland-Local-Borda✓ 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
Copeland-Local-Minimax✓ 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
Daunou✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-33v)
Dodgson✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-9v; 5c, 2-4v; 6c, 2v
GOCHA✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✗ 3c, 5v (checked 2c, 2-11v)
Instant Runoff✗ 3c, 9v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)
Instant Runoff PUT✗ 3c, 9v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)
Instant Runoff TB✗ 3c, 8v (checked 2c, 2-11v)✗ 3c, 8v (checked 2c, 2-11v)
Iterated Removal Condorcet Loser✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Kemeny-Young✓ 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
Knockout Voting✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 7v (checked 2c, 2-11v; 3c, 2-33v)
Leximax✓ 2c, 2-11v; 3c, 2-20v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-20v; 4c, 2-10v; 5c, 2-4v; 6c, 2v
Llull✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-3v; 6c, 2v✗ 3c, 5v (checked 2c, 2-11v)
Loss-Trimmer Voting✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)
Minimax✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v (ref)✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v (ref)
Minimax (Support)✓ 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
Most Wins, Smallest Loss✓ 2c, 2-11v; 3c, 2-20v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-20v; 4c, 2-10v; 5c, 2-4v; 6c, 2v
Pareto✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✗ 2c, 4v
Plurality✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v (ref)✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v
Plurality with Runoff PUT✗ 3c, 9v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)
Positive-Negative Voting✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v✓ 2c, 4c, 2-11v; 3c, 2-33v; 5c, 2-3v; 6c, 2v
Preliminary Weighted Condorcet✗ 2c, 3v✗ 2c, 2v
Ranked Pairs TB✓ 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
Ranked Pairs ZT✓ 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
Raynaud✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)
River TB✓ 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
River ZT✓ 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
Simple Stable Voting✗ 4c, 8v (checked 2c, 2-11v; 3c, 2-33v)✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v
Simplified Bucklin✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✗ 3c, 4v (checked 2c, 2-11v)
Simplified Dodgson✓ 2c, 2-11v; 3c, 2-20v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✓ 2c, 2-11v; 3c, 2-20v; 4c, 2-10v; 5c, 2-4v; 6c, 2v
Slater✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-3v; 6c, 2v (ref)✗ 3c, 5v (checked 2c, 2-11v)
Smith IRV✗ 3c, 9v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)
Smith IRV PUT✗ 3c, 9v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)
Smith-Minimax✓ 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
Split Cycle✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-3v; 6c, 2v (ref)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)
Stable Voting✗ 4c, 8v (checked 2c, 2-11v; 3c, 2-33v)✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v
Strict Nanson✗ 3c, 5v (checked 2c, 2-11v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)
Superior Voting✗ 3c, 8v (checked 2c, 2-11v)✗ 3c, 7v (checked 2c, 2-11v)
Tideman Alternative GOCHA✗ 3c, 9v (checked 2c, 2-11v)✗ 3c, 9v (checked 2c, 2-11v)
Tideman Alternative GOCHA PUT✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Tideman Alternative Top Cycle✗ 3c, 9v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)
Tideman Alternative Top Cycle PUT✗ 3c, 9v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)
Top Cycle✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-3v; 6c, 2v✗ 3c, 2v (checked 2c, 2-11v)
Uncovered Set✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✗ 3c, 4v (checked 2c, 2-11v)
Uncovered Set - Bordes✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✗ 3c, 4v (checked 2c, 2-11v)
Uncovered Set - Fishburn✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✗ 3c, 5v (checked 2c, 2-11v)
Uncovered Set - McKelvey✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✗ 3c, 4v (checked 2c, 2-11v)
Weak Condorcet✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✗ 3c, 5v (checked 2c, 2-11v)
Weak Nanson✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-33v)✗ 4c, 6v (checked 2c, 2-11v; 3c, 2-33v)
Weighted Bucklin✓ 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
Weighted Covering✓ 2c, 2-11v; 3c, 2-33v; 4c, 2-10v; 5c, 2-4v; 6c, 2v✗ 3c, 4v (checked 2c, 2-11v)
Woodall✗ 3c, 9v (checked 2c, 2-11v)✗ 3c, 6v (checked 2c, 2-11v)