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