QRSTUVWXYZ | Election Example | REM Prob. | Dirichlet Prob. | Quas-1D Prob. |
---|---|---|---|---|

0000000000 | ABC= 0, ACB= 2, BAC= 0, BCA= 1, CAB= 0, CBA= 0 | 69.0764% | 82.6389% | 61.1113% |

0001000000 | ABC= 0, ACB= 7, BAC=12, BCA= 0, CAB= 6, CBA= 0 | 6.3360% | 3.3758% | 11.3889% |

0001000100 | ABC=15, ACB= 0, BAC=20, BCA= 0, CAB=10, CBA= 0 | 7.8076% | 2.7826% | 8.0555% |

0001010100 | ABC=20, ACB= 0, BAC=16, BCA=14, CAB=15, CBA= 0 | 2.8818% | 0.9163% | 0.0000% |

0101000000 | ABC=23, ACB=29, BAC=17, BCA=40, CAB= 6, CBA= 0 | 0.0164% | 0.0096% | 0.0000% |

0101000100 | ABC=18, ACB= 0, BAC= 8, BCA=15, CAB=10, CBA= 0 | 1.5332% | 1.0079% | 0.0000% |

0101010100 | ABC=26, ACB= 0, BAC=16, BCA=23, CAB=18, CBA= 0 | 0.1903% | 0.0675% | 0.0000% |

1000000010 | ABC= 0, ACB=13, BAC= 0, BCA=12, CAB= 6, CBA= 0 | 0.0000% | 0.6970% | 6.9999% |

1000001010 | ABC= 0, ACB=19, BAC= 0, BCA=18, CAB=12, CBA= 0 | 0.0000% | 0.1087% | 2.3750% |

1000100010 | ABC= 0, ACB=25, BAC= 6, BCA=18, CAB=18, CBA= 0 | 0.7542% | 0.2211% | 0.0000% |

1000101010 | ABC= 0, ACB=22, BAC= 0, BCA=21, CAB=16, CBA= 0 | 1.4849% | 0.8829% | 4.5138% |

1001000010 | ABC= 0, ACB=13, BAC= 4, BCA=14, CAB=10, CBA= 0 | 0.0000% | 0.2796% | 1.3333% |

1001001010 | ABC= 0, ACB=15, BAC= 0, BCA=20, CAB=10, CBA= 0 | 0.0000% | 0.0866% | 1.7917% |

1001100010 | ABC= 0, ACB=13, BAC= 6, BCA=12, CAB=12, CBA= 0 | 0.5816% | 0.1912% | 0.0000% |

1001101010 | ABC= 0, ACB=25, BAC= 0, BCA=30, CAB=20, CBA= 0 | 0.5637% | 0.4842% | 2.4306% |

1010000000 | ABC= 6, ACB= 7, BAC= 0, BCA=12, CAB= 6, CBA= 0 | 0.0000% | 0.7006% | 0.0000% |

1010000001 | ABC=13, ACB= 0, BAC= 0, BCA=12, CAB= 6, CBA= 0 | 0.0000% | 0.8732% | 0.0000% |

1010001000 | ABC= 9, ACB=10, BAC= 0, BCA=18, CAB=12, CBA= 0 | 0.0000% | 0.0901% | 0.0000% |

1010001001 | ABC=19, ACB= 0, BAC= 0, BCA=18, CAB=12, CBA= 0 | 0.0000% | 0.0938% | 0.0000% |

1010100000 | ABC=12, ACB=13, BAC= 6, BCA=18, CAB=18, CBA= 0 | 1.3645% | 0.1487% | 0.0000% |

1010100001 | ABC=25, ACB= 0, BAC= 6, BCA=18, CAB=11, CBA= 7 | 0.6750% | 0.1497% | 0.0000% |

1010101000 | ABC=11, ACB=11, BAC= 0, BCA=21, CAB=16, CBA= 0 | 1.5123% | 0.5507% | 0.0000% |

1010101001 | ABC=22, ACB= 0, BAC= 0, BCA=21, CAB=16, CBA= 0 | 0.6668% | 0.5181% | 0.0000% |

1011000000 | ABC= 5, ACB=11, BAC= 3, BCA=18, CAB=10, CBA= 0 | 0.0000% | 0.4084% | 0.0000% |

1011000100 | ABC= 8, ACB= 8, BAC= 3, BCA=18, CAB=10, CBA= 0 | 0.0000% | 0.3064% | 0.0000% |

1011001000 | ABC= 5, ACB=12, BAC= 0, BCA=23, CAB=11, CBA= 0 | 0.0000% | 0.0677% | 0.0000% |

1011001100 | ABC= 9, ACB= 8, BAC= 0, BCA=22, CAB=11, CBA= 0 | 0.0000% | 0.0990% | 0.0000% |

1011010100 | ABC=10, ACB=10, BAC= 6, BCA=24, CAB=15, CBA= 0 | 0.0000% | 0.0672% | 0.0000% |

1011011100 | ABC=10, ACB=10, BAC= 0, BCA=30, CAB=15, CBA= 0 | 0.0000% | 0.0618% | 0.0000% |

1011100000 | ABC=11, ACB=17, BAC= 9, BCA=24, CAB=22, CBA= 0 | 0.5675% | 0.0468% | 0.0000% |

1011100100 | ABC=14, ACB=14, BAC= 9, BCA=24, CAB=22, CBA= 0 | 0.9166% | 0.0898% | 0.0000% |

1011101000 | ABC=11, ACB=17, BAC= 0, BCA=33, CAB=22, CBA= 0 | 0.1713% | 0.0848% | 0.0000% |

1011101100 | ABC=14, ACB=14, BAC= 0, BCA=33, CAB=22, CBA= 0 | 0.3933% | 0.2041% | 0.0000% |

1011110100 | ABC=15, ACB=15, BAC=11, BCA=29, CAB=25, CBA= 0 | 0.9354% | 0.1556% | 0.0000% |

1011111100 | ABC=15, ACB=15, BAC= 0, BCA=40, CAB=25, CBA= 0 | 0.4667% | 0.3326% | 0.0000% |

1111000101 | ABC=16, ACB= 0, BAC= 3, BCA=18, CAB=10, CBA= 0 | 0.0000% | 0.5566% | 0.0000% |

1111001101 | ABC=17, ACB= 0, BAC= 0, BCA=22, CAB=11, CBA= 0 | 0.0000% | 0.0738% | 0.0000% |

1111010101 | ABC=20, ACB= 0, BAC= 6, BCA=24, CAB=15, CBA= 0 | 0.0000% | 0.0823% | 0.0000% |

1111011101 | ABC=20, ACB= 0, BAC= 0, BCA=30, CAB=15, CBA= 0 | 0.0000% | 0.0347% | 0.0000% |

1111100101 | ABC=28, ACB= 0, BAC= 9, BCA=24, CAB=16, CBA= 6 | 0.4807% | 0.0665% | 0.0000% |

1111101101 | ABC=28, ACB= 0, BAC= 0, BCA=33, CAB=22, CBA= 0 | 0.1528% | 0.1374% | 0.0000% |

1111110101 | ABC=30, ACB= 0, BAC=11, BCA=29, CAB=20, CBA= 5 | 0.3003% | 0.0807% | 0.0000% |

1111111101 | ABC=30, ACB= 0, BAC= 0, BCA=40, CAB=25, CBA= 0 | 0.1708% | 0.1689% | 0.0000% |

Below is a smaller table of some of the most-requested pathology-probability information (All of the numbers in the below tables are derivable by adding up appropriate sets of numbers from the above master table):

Phenomenon | REM | Dirichlet | Quas 1D |
---|---|---|---|

Participation failure W∪X | 20.6285% | 10.2593% | 19.1666% |

Nonmonotonicity U∪V | 15.2304% | 5.7435% | 6.9444% |

V: ("less is more" nonmonotonicity) | 4.9454% | 1.9676% | 0.0000% |

U: ("more is less" nonmonotonicity) | 12.1583% | 4.5138% | 6.9444% |

Y: Condorcet winner eliminated ("thwarted majority") | 3.3843% | 2.9514% | 19.4443% |

Z: Reversal "winner=loser" failure | 2.4464% | 2.8356% | 0.0000% |

R: All scoring rules agree B wins, but IRV says A wins (failure of "sniff test") | 2.8446% | 2.2859% | 0.0000% |

W: Abstention failure: deleting A-bottom voters stops A from winning | 5.5827% | 4.0798% | 11.1111% |

X: Would be strategic mistake for more voters of some single type to come | 16.2296% | 7.2917% | 8.0555% |

T: Plurality and IRV winners differ | 24.4660% | 12.3264% | 24.9999% |

S: Condorcet cycle | 8.7740% | 6.2500% | 0.0000% |

Q∪R∪U∪V∪W∪X∪Y∪Z ("total paradox probability") | 24.5877% | 13.9854% | 27.4998% |

Both kinds of participation failure simultaneously W∩X | 1.1837% | 1.1122% | 0.0000% |

Both kinds of nonmonotonicity simultaneously U∩V | 1.8733% | 0.7379% | 0.0000% |

Q∪V: Betraying B makes either B or C win (where either way the betrayers prefer that to A winning) | 15.2304% | 10.1852% | 19.4443% |

Q: Loser drop-out paradox: If B drops out, that switches the winner from A to C. Also (which happens in exactly the same set of elections) "Favorite betrayal"; voters with favorite B, by betraying B, make C win (whom they prefer as the "lesser evil" over current winner A) | 12.1583% | 9.2014% | 19.4443% |

And below is the same table, but *restricted* to elections in which the IRV process
*matters*, i.e. in which the IRV and plain-plurality winners differ.
(Warning: The error bars are approximately twice as wide as in the tables above.)
This almost always makes pathologies substantially more likely:

Phenomenon | REM | Dirichlet | Quas 1D |
---|---|---|---|

Participation failure W∪X | 69.3394% | 65.0222% | 49.1112% |

Nonmonotonicity U∪V | 35.8572% | 26.5475% | 9.7224% |

V: ("less is more" nonmonotonicity) | 20.2132% | 15.9624% | 0.0000% |

U: ("more is less" nonmonotonicity) | 23.3006% | 16.5711% | 9.7224% |

Y: Condorcet winner eliminated ("thwarted majority") | 4.6809% | 8.4505% | 22.2222% |

Z: Reversal "winner=loser" failure | 4.5151% | 9.7418% | 0.0000% |

R: All scoring rules agree B wins, but IRV says A wins (failure of "sniff test") | 11.6268% | 18.5445% | 0.0000% |

W: Abstention failure: deleting A-bottom voters stops A from winning | 7.8425% | 14.8899% | 16.8891% |

X: Would be strategic mistake for more voters of some single type to come | 66.3352% | 59.1548% | 32.2222% |

T: Plurality and IRV winners differ | 100.0000% | 100.0000% | 100.0000% |

S: Condorcet cycle | 18.6197% | 25.3522% | 0.0000% |

Q∪R∪U∪V∪W∪X∪Y∪Z ("total paradox probability") | 74.1030% | 72.6134% | 54.4444% |

Both kinds of participation failure simultaneously W∩X | 4.8383% | 9.0226% | 0.0000% |

Both kinds of nonmonotonicity simultaneously U∩V | 7.6566% | 5.9860% | 0.0000% |

Q∪V: Betraying B makes either B or C win (where either way the betrayers prefer that to A winning) | 35.8572% | 41.7838% | 22.2222% |

Q: Loser drop-out paradox: If B drops out, that switches the winner from A to C. Also (which happens in exactly the same set of elections) "Favorite betrayal"; voters with favorite B, by betraying B, make C win (whom they prefer as the "lesser evil" over current winner A) | 23.3006% | 33.8027% | 22.2222% |