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 | Independence of Pareto-Dominated Alternatives | Independence of Smith-Dominated Alternatives | Immunity to Spoilers | Stability for Winners |
|---|---|---|---|---|
| Banks | ✗ 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 | ✓ 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 | ✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v | ✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v |
| Condorcet | ✗ 3c, 2v (checked 2c, 2-11v) | ✗ 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 |
| Copeland | ✗ 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 |
| Copeland-Global-Minimax | ✗ 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 |
| Copeland-Local-Minimax | ✗ 3c, 2v (checked 2c, 2-11v) | ✓ 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) |
| GOCHA | ✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-17v) | ✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2-3v | ✗ 5c, 3v (checked 2c, 2-11v; 3c, 2-17v; 4c, 2-5v) | ✗ 5c, 3v (checked 2c, 2-11v; 3c, 2-17v; 4c, 2-5v) |
| Iterated Removal Condorcet Loser | ✗ 3c, 2v (checked 2c, 2-11v) | ✗ 3c, 2v (checked 2c, 2-11v) | ✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2-3v | ✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2-3v |
| Kemeny-Young | ✗ 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) |
| Leximax | ✗ 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 | ✗ 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 | ✓ 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 | ✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-17v) | ✗ 4c, 3v (checked 2c, 2-11v; 3c, 2-17v) |
| Minimax | ✓ 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 | ✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v |
| Minimax (Support) | ✓ 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 | ✗ 4c, 5v (checked 2c, 2-11v; 3c, 2-17v) |
| Most Wins, Smallest Loss | ✗ 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 |
| Pareto | ✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 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 |
| Ranked Pairs TB | ✗ 3c, 2v (checked 2c, 2-11v) | ✓ 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) |
| Raynaud | ✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2-3v | ✗ 4c, 3v (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) |
| River TB | ✗ 3c, 2v (checked 2c, 2-11v) | ✓ 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) |
| Simple Stable Voting | ✗ 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 | ✗ 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 | ✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v |
| Slater | ✗ 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) |
| 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 | ✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v |
| Split Cycle | ✗ 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 | ✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v |
| Stable Voting | ✗ 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 | ✗ 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 |
| 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 |
| Uncovered Set - Bordes | ✗ 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 | ✓ 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 |
| Uncovered Set - McKelvey | ✗ 4c, 4v (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 | ✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v |
| Weak Condorcet | ✗ 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, 2v | ✓ 2c, 2-11v; 3c, 2-17v; 4c, 2-5v; 5c, 2v |
| Weighted Covering | ✗ 3c, 3v (checked 2c, 2-11v) | ✗ 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 |