RangeVoting.org - Same IRV 3-candidate paradox probabilities from different random number generator

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

`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%

Return to main page