## Same IRV 3-candidate paradox probabilities from different random number generator

 Election Example REM Prob. Dirichlet Prob. Quas-1D Prob. QRSTUVWXYZ 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):

PhenomenonREMDirichletQuas 1D
Participation failure W∪X20.6285%10.2593%19.1666%
Nonmonotonicity U∪V15.2304%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" failure2.4464%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 winning5.5827%4.0798%11.1111%
X: Would be strategic mistake for more voters of some single type to come16.2296%7.2917%8.0555%
T: Plurality and IRV winners differ24.4660%12.3264%24.9999%
S: Condorcet cycle8.7740%6.2500%0.0000%
Both kinds of participation failure simultaneously W∩X1.1837%1.1122%0.0000%
Both kinds of nonmonotonicity simultaneously U∩V1.8733%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:

PhenomenonREMDirichletQuas 1D
Participation failure W∪X69.3394%65.0222%49.1112%
Nonmonotonicity U∪V35.8572%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" failure4.5151%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 winning7.8425%14.8899%16.8891%
X: Would be strategic mistake for more voters of some single type to come66.3352%59.1548%32.2222%
T: Plurality and IRV winners differ100.0000%100.0000%100.0000%
S: Condorcet cycle18.6197%25.3522%0.0000%