Strategic Axioms (for ProfileWithTies 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).

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.

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)
Banks✗ 2c, 2v✗ 2c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✓ 2c, 2-11v; 3c, 2-16v; 4c, 2-4v; 5c, 2v✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Beat Path✗ 2c, 2v✗ 2c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-17v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Condorcet✗ 2c, 2v✗ 2c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Copeland✗ 2c, 2v✗ 2c, 2v✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-17v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 4c, 2v (checked 2c, 2-11v; 3c, 2-16v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Copeland-Global-Minimax✗ 2c, 2v✗ 2c, 2v✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-17v)✗ 3c, 6v (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✗ 2c, 2v✗ 2c, 2v✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-17v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
GOCHA✗ 2c, 2v✗ 2c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2-3v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2-3v✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-16v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Iterated Removal Condorcet Loser✗ 2c, 2v✗ 2c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2-3v✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✓ 2c, 2-11v; 3c, 2-16v; 4c, 2-5v; 5c, 2v✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Kemeny-Young✗ 2c, 2v✗ 2c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-17v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Leximax✗ 2c, 2v✗ 2c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Llull✗ 2c, 2v✗ 2c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-17v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-16v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Loss-Trimmer Voting✗ 2c, 2v✗ 2c, 2v✗ 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, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Minimax✗ 2c, 2v✗ 2c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-17v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Minimax (Support)✗ 2c, 2v✗ 2c, 2v✗ 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, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Most Wins, Smallest Loss✗ 2c, 2v✗ 2c, 2v✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-17v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Pareto✗ 3c, 2v (checked 2c, 2-11v)✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✓ 2c, 2-11v; 3c, 2-16v; 4c, 2-5v; 5c, 2v✓ 2c, 2-11v; 3c, 2-16v; 4c, 2-5v; 5c, 2v✓ 2c, 2-11v; 3c, 2-16v; 4c, 2-5v; 5c, 2v
Ranked Pairs TB✗ 2c, 2v✗ 2c, 2v✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-17v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Raynaud✗ 2c, 2v✗ 2c, 2v✗ 3c, 8v (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, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
River TB✗ 2c, 2v✗ 2c, 2v✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-17v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Simple Stable Voting✗ 2c, 2v✗ 2c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-17v)✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Simplified Dodgson✗ 2c, 2v✗ 2c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-17v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Slater✗ 2c, 2v✗ 2c, 2v✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-17v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-16v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Smith-Minimax✗ 2c, 2v✗ 2c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-17v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Split Cycle✗ 2c, 2v✗ 2c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-17v)✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-17v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Stable Voting✗ 2c, 2v✗ 2c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-17v)✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Top Cycle✗ 2c, 2v✗ 2c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 2v (checked 2c, 2-11v)
Uncovered Set✗ 2c, 2v✗ 2c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✗ 4c, 4v (checked 2c, 2-11v; 3c, 2-17v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✓ 2c, 2-11v; 3c, 2-16v; 4c, 2-5v; 5c, 2v✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Uncovered Set - Bordes✗ 2c, 2v✗ 2c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✗ 3c, 4v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✓ 2c, 2-11v; 3c, 2-16v; 4c, 2-5v; 5c, 2v✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Uncovered Set - Fishburn✗ 2c, 2v✗ 2c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-17v)✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-16v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Uncovered Set - McKelvey✗ 2c, 2v✗ 2c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✗ 3c, 5v (checked 2c, 2-11v)✗ 3c, 4v (checked 2c, 2-11v)✓ 2c, 2-11v; 3c, 2-16v; 4c, 2-5v; 5c, 2v✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Weak Condorcet✗ 2c, 2v✗ 2c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✗ 3c, 6v (checked 2c, 2-11v)✗ 3c, 5v (checked 2c, 2-11v)✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-16v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)
Weighted Covering✗ 2c, 2v✗ 2c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)✗ 3c, 3v (checked 2c, 2-11v)