Hi,
Well, yesterday I spent a lot of time studying generally how things were
working. One notable thing I observed was that in a 1D centrist scenario,
Approval and Range would overselect the centrist (elect him even when he
is not best) while truncated Condorcet methods would underselect him.
The obvious explanation for the latter is that the voters were not
"approving" him, but since ApprPoll uses the same cutoff, that doesn't
work: What seems to be the case is that, when voters truncate at
(approval poll-based) expectation, respecting sub-majority defeats isn't
just less important than respecting majority defeats, it seems to be
harmful.
So I spent a lot of time thinking about what methods could be a "Bucklin
killer" or "DAC killer" (as these methods pick the centrist in the correct
scenarios). I implemented my old MDDA and MAMPO methods and was promptly
disappointed. I added in MAFP (Bucklin where you tie-break based on the
previous rank's count) and also newly named MARO1 and MARO2, which are
more promising.
"MARO" stands for "Majority Approval Runoff." The idea is if the top two
approval candidates have majority approval, there is a pairwise contest.
(As a real method you might limit qualification to the top three on first
preferences, to avoid the obvious way to abuse this method.) With MARO1
the contest is decided on the ballot as truncated. With MARO2 all
preferences are used, so either a second round is being held, or voters
are allowed to rank beneath the approval cutoff.
(I had also, earlier, implemented Chris' new IBIFA method, which will
show up below as well.)
After this I got to working on my simulations that consider
competitiveness of nomination, which seem to change the game a bit and
make me a bit more optimistic that the method to beat may not be Bucklin.
The idea is that usually candidates in reality are nominated in an attempt
to be competitive. Most randomly generated candidate allocations are not
very competitive. As I would prefer to simulate realistic elections, it
seems reasonable to try removing scenarios where one candidate wins a
high percentage of the time under a given method.
The stats I have put together are for one- and two-dimensional random
scenarios, limited to the most viable candidate winning within 65%, 55%,
and 45% of the elections in that scenario.
To begin with I'll note that the likelihood of a scenario being
competitive doesn't vary all that much from method to method, and it's
not clear to me what variation means anyway. Generally the truncated
Condorcet methods admit the most scenarios as competitive, and Range
and Approval the fewest, sometimes even fewer than the method that
magically picks the best winner.
Just a few stats on this: Of all 2D scenarios within the 65% threshold
for *some* method, MMPO is in this threshold 79% of the time, MMWV 71%,
IRV 63%, Bucklin 55%, Approval and Range 46-50%, magic BEST method 47%.
So, here are the average distances for 1D and 2D regarding only the
<65% scenarios for that method.
Dims Rank Method Avg Dist Avg Norm Dist
1 1 BEST 51.43 0.00
1 2 MMstrict 52.26 1.79
1 3 MARO2 52.26 1.79
1 4 MAP 52.26 1.79
1 5 MDDA 52.48 2.06
1 6 Bucklin 52.49 2.05
1 7 DAC 52.50 2.14
1 8 MAMPO 52.58 2.41
1 9 MARO1 52.65 2.63
1 10 MAFP 52.66 2.69
1 11 CdlA 52.83 4.25
1 12 IBIFA 52.84 4.25
1 13 QR 52.98 4.32
1 14 RangeNS 53.15 3.07
1 15 MMWV 53.16 5.07
1 16 C//A 53.17 5.07
1 17 SMDTR 53.21 5.21
1 18 DSC 53.31 3.99
1 19 2sMMPO 53.49 2.83
1 20 ApprPoll 53.49 2.83
1 21 MMmarg 53.91 7.06
1 22 IRV 54.10 6.76
1 23 Raynaud 54.60 9.38
1 24 MMPO 54.61 9.37
1 25 SPST 54.61 6.91
1 26 ApprZIS 54.70 3.01
1 27 VFA 55.81 9.42
1 28 WORST 77.76 100.00
2 1 BEST 114.46 0.00
2 2 RangeNS 116.09 3.71
2 3 MMstrict 116.23 5.64
2 4 MARO2 116.30 5.36
2 5 MARO1 116.37 5.62
2 6 MAFP 116.38 5.69
2 7 DSC 116.41 6.50
2 8 MDDA 116.47 5.51
2 9 DAC 116.48 5.29
2 10 MAMPO 116.49 5.70
2 11 Bucklin 116.50 5.33
2 12 IBIFA 116.54 6.48
2 13 MAP 116.59 7.07
2 14 QR 116.66 6.75
2 15 C//A 116.69 7.11
2 16 CdlA 116.69 6.89
2 17 MMWV 116.72 7.40
2 18 SMDTR 116.78 7.44
2 19 MMmarg 116.79 7.80
2 20 ApprPoll 116.82 5.61
2 21 IRV 116.83 7.46
2 22 ApprZIS 117.12 5.26
2 23 Raynaud 117.18 9.00
2 24 SPST 117.26 8.57
2 25 VFA 117.60 9.32
2 26 MMPO 118.15 11.45
2 27 2sMMPO 120.03 14.68
2 28 WORST 149.97 100.00
Perhaps not that surprising. It shows that the new methods I added can
be competitive with Bucklin. Approval has fallen in the rankings quite
a bit in the 2D case.
So, let's look at "very close" races where the most viable candidate is
winning within 55% of the elections:
Dims Rank Method Avg Dist Avg Norm Dist
1 1 BEST 50.35 0.00
1 2 MMstrict 51.30 1.66
1 3 MARO2 51.30 1.66
1 4 MAP 51.30 1.66
1 5 MAFP 51.93 2.81
1 6 MARO1 52.00 2.78
1 7 DAC 52.06 2.22
1 8 MAMPO 52.09 2.50
1 9 QR 52.12 4.70
1 10 DSC 52.12 4.04
1 11 IBIFA 52.21 4.48
1 12 CdlA 52.22 4.48
1 13 Bucklin 52.29 2.13
1 14 MDDA 52.32 2.16
1 15 MMWV 52.38 5.46
1 16 C//A 52.38 5.47
1 17 SMDTR 52.39 5.63
1 18 RangeNS 52.61 2.95
1 19 MMmarg 53.41 7.91
1 20 IRV 53.49 7.56
1 21 SPST 53.64 7.35
1 22 MMPO 54.27 10.71
1 23 Raynaud 54.32 10.72
1 24 ApprPoll 54.33 3.21
1 25 2sMMPO 54.33 3.21
1 26 ApprZIS 55.11 3.24
1 27 VFA 55.52 10.41
1 28 WORST 77.96 100.00
2 1 BEST 114.86 0.00
2 2 SMDTR 117.04 8.15
2 3 IBIFA 117.12 6.98
2 4 C//A 117.21 7.64
2 5 DSC 117.25 6.89
2 6 CdlA 117.27 7.41
2 7 MMWV 117.29 7.95
2 8 MARO2 117.30 5.74
2 9 MMstrict 117.36 6.24
2 10 MAP 117.45 8.08
2 11 MAFP 117.47 6.21
2 12 MMmarg 117.48 8.37
2 13 MARO1 117.49 6.09
2 14 QR 117.51 7.44
2 15 MAMPO 117.59 6.17
2 16 RangeNS 117.61 3.82
2 17 Bucklin 117.65 5.80
2 18 DAC 117.66 5.79
2 19 SPST 117.67 9.20
2 20 IRV 117.72 8.31
2 21 Raynaud 117.86 9.64
2 22 MDDA 117.88 5.92
2 23 VFA 118.01 10.11
2 24 ApprZIS 118.29 5.71
2 25 ApprPoll 118.52 6.03
2 26 MMPO 118.77 12.36
2 27 2sMMPO 121.42 16.58
2 28 WORST 149.01 100.00
The top two methods in 2D are both Chris's!
Lastly, let's look at scenarios that are so competitive under the method
that the leading candidate only wins within 45% of the time:
Dims Rank Method Avg Dist Avg Norm Dist
1 1 WORST NULL NULL
1 2 BEST 47.96 0.00
1 3 MMstrict 48.19 1.20
1 4 MARO2 48.19 1.20
1 5 MAP 48.19 1.20
1 6 RangeNS 48.48 1.49
1 7 Bucklin 48.85 1.65
1 8 MDDA 48.91 1.73
1 9 DAC 49.36 2.14
1 10 MAMPO 49.44 2.52
1 11 DSC 49.84 4.04
1 12 MARO1 49.90 3.04
1 13 MAFP 49.94 3.11
1 14 CdlA 49.98 5.01
1 15 QR 50.03 5.40
1 16 IBIFA 50.07 5.02
1 17 MMWV 50.83 6.38
1 18 C//A 50.84 6.41
1 19 SMDTR 51.12 6.66
1 20 SPST 51.87 8.16
1 21 IRV 52.53 9.61
1 22 MMmarg 52.57 9.73
1 23 MMPO 54.13 13.07
1 24 Raynaud 54.15 12.95
1 25 VFA 54.81 12.21
1 26 ApprZIS 56.12 3.87
1 27 ApprPoll 56.49 4.27
1 28 2sMMPO 56.49 4.27
2 1 SMDTR 117.11 9.94
2 2 BEST 117.65 0.00
2 3 IBIFA 118.39 8.36
2 4 MMWV 118.46 9.25
2 5 Raynaud 118.53 11.35
2 6 MMstrict 118.63 7.40
2 7 MMmarg 118.70 9.92
2 8 C//A 118.79 8.87
2 9 DSC 118.80 8.14
2 10 VFA 118.86 11.73
2 11 RangeNS 118.90 4.08
2 12 MAFP 119.02 7.35
2 13 QR 119.06 8.92
2 14 CdlA 119.11 8.64
2 15 SPST 119.34 11.14
2 16 IRV 119.44 9.60
2 17 MDDA 119.44 6.11
2 18 MARO1 119.46 6.87
2 19 MAMPO 119.74 6.73
2 20 MARO2 119.84 6.37
2 21 DAC 119.95 6.39
2 22 MMPO 120.28 13.94
2 23 Bucklin 120.41 6.00
2 24 ApprPoll 120.62 6.48
2 25 ApprZIS 121.09 6.38
2 26 MAP 121.83 11.04
2 27 2sMMPO 123.42 19.28
2 28 WORST 148.68 100.00
A couple of oddities here are that there were no 1D scenarios where the
WORST candidate was that unclear. (In general we are down to having
about 60-300 scenarios to look at for a given method, though Range and
the Approvals have only 16-32 of these in the 1D case.) Also, the SMD,TR
method actually beat BEST in the 2D case.
What I still want to do is analyze what these scenarios look like.
I am concerned about the fact that a scenario can be competitive without
being realistic, if only due to the fact that the position could be
dominated by another strategy for both sides. For example, an FPP election
nominating an extreme left and an extreme right candidate could be
competitive, but it's not likely to occur, because it would be at least
as effective for either to nominate a candidate closer to the median.
(Incidentally I didn't include FPP or Antiplurality or some similar
methods in this run, partly because I had trouble wrapping my mind around
the idea of assuming sincere voting but intelligent nomination.)
Kevin Venzke
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