Hi,
Jameson, I did most of what I looked into. I didn't complete the Asset
methods though. I did come up with a good universal way to estimate the
candidates' utilities for each other though: Have the candidates read the
minds of the voters (the base quantities of each bloc), and multiply each
opinion by their opinion of the candidate asking the question. (I'm
assuming utilities range from 0 to 1.) So, a candidate will like the same
candidates you like, if you like that candidate. This way, B will like
C not because of the spectrum, but because most of his "goodwill" comes
from the C voters.
A likely tweak would be to have every candidate assured of liking
themselves the best.
I threw together a method to test something similar: The voters cast rated
ballots. Voting power for each candidate is determined by top ratings
(fractional if tying). "Candidates" (simulated) determine their preferences
based on the voters' ratings. Then the candidates cast full sincere
rankings for a round of Minmax, to find the winner. It turned out to be
not that great in this particular scenario being considered:
"AICMM" 86 86
----MCM 79 85 85
----MMC 77 85 85
----CMM 69 83 86
-B--MCM 32 85 85
B---CMM 26 86 86
-B--CMM 26 87 87 ...
You can see the SCWE and utility maximizer election rates are to the
right of the frequencies. This was a thousand trials, so pretty split
up.
Anyway, the Asset methods stumped me somewhat because I couldn't come
up with a deterministic way to solve the method that doesn't seem to
be contrived. For instance, it's possible that two of the three candidates
are able to transfer. Who has initiative? How do they even know if they
would like to have initiative? Maybe they'd rather do nothing. So, I
didn't attempt to write a method that might not be faithful to the idea.
Here are some methods to compare.
MCA 97 87
TTTTM-- 220 100 88
TTTT--- 194 100 86
TTTT--M 130 100 87
TTTT-M- 126 100 85
TTT-MMM 63 92 93
T-TTM-- 23 100 88 ...
This looks like mostly the same order as last time.
MAFP 95 87
TTTTM-- 158 100 88
TTTT--- 92 100 86
TTTT--M 83 100 87
TTTT-M- 80 100 88
TTT-MMM 49 94 95
-TTTM-- 22 100 86 ...
Really this could be called "MATR" because it breaks all mid-slot ties
on top ratings.
"MCARA" 96 87
BBB-MMM 99 100 86
TTTT--- 75 100 90
TTTTM-- 46 100 86
TTTTMM- 44 100 85
BBT-MMM 41 100 86
TBB-MMM 35 100 88
TTTT--M 31 100 89
TTTTMMM 29 100 84
BTB-MMM 28 100 85
BBB-CCC 28 93 94
***MCCC 26 94 93
TTTT-MM 26 100 87 ...
You were right that this would do pretty well by SCWE, but your guess
for strategy was ---TTTT (which is actually a quite rare outcome under
any method).
"MCARP" 97 87
TTTTMM- 71 100 84
BBB-MMM 70 100 87
***MCCC 62 94 95 (*=M+P+B, i.e. "A=C>B")
TTTTM-M 48 100 89
TTTTMMM 32 100 88
TTB-MMM 32 100 88
BBT-MMM 30 100 87
TTTT-MM 29 100 85
BTB-MMM 28 100 87
TBB-MMM 24 100 87 ...
This was a little higher by SCWE but your MMMTTTT guess never occurred
once in any (recorded) result for any method!
By the way, it seemed to me that there was only one difference between
MCARA and MCARP. That is, MCARP's tie finalists are only picked by TRs,
while MCARA could be based on TRs or approval. Let me know if that sounds
wrong.
Next, Majority Judgment, with ER and non ER. This was tricky to think
through. However, it seems to me that this method gives the same result
as MCA unless there is a tie on the middle slot. In that case you must
find how many votes must be removed to make each eligible candidate a
majority favorite or majority disapproved. I don't think it matters where
you take the votes from as long as it's on the correct side of the
boundary. (It's possible to find that more votes must be removed than are
even there, but I doubt these values ever determine the result.)
I can certainly share the method code I used here. Strict then ER:
MJgmtSt 96 86
TTTT--- 571 100 89
TT-T--- 65 100 87
-TTT--- 52 100 88
T-TT--- 40 100 88
TTTTCCC 32 94 92
TTTTC-C 30 94 94 ...
Not too special here. Like Bucklin but not as concentrated at the top
(Bucklin had 702 TTTT---).
MJgmtER 97 87
TTTTM-- 172 100 89
TTTT--- 120 100 88
TTTT--M 101 100 86
TTTT-M- 99 100 88
TTT-MMM 38 93 94
TT-TM-- 22 100 88
TT-T--M 17 100 87 ...
So, very similar to MCA but not as certain for some reason.
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I tried, out of curiosity, FPP where the *second* candidate wins:
WrFPP 73 72
CCCCPPP 669 93 94
PPPPPPP 254 14 14
PPPPCCC 77 90 80
Interesting that most of the time a decent candidate wins.
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DAC update: I have DAC with ER and no ER.
DACstr 96 86
TTTT--- 569 100 87
-TTT--- 61 100 85
TT-T--- 52 100 88
T-TT--- 45 100 86
TTTTCCC 36 94 93 ...
DACer 97 87
TTTTM-- 172 100 88
TTTT-M- 129 100 87
TTTT--- 116 100 88
TTTT--M 101 100 87
TTT-MMM 29 94 93 ...
Interestingly the ER version doesn't seem to have as much compression as
I expected. A bit comparable to MCA actually.
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Next, I thought I should try the ER versions of IRV. IRV earlier gave
these results:
IRV (93 and 93 stats)
88 ----CCC
44 ----CC-
39 -----CC
36 ----C-C
34 --T-CCC ...
ER-IRV(fractional) gives:
IRVerF 93 93
----MMM 37 93 94
----MCM 26 93 94
----MMC 19 92 93
----CMC 18 92 95
----CCM 17 94 95
----CMM 17 93 93
-T--MMM 16 94 94
T---MMM 15 92 94 ...
ER-IRV(whole) gives:
IRVerW 93 93
----MMM 182 94 94
T---MMM 65 93 92
--T-MMM 57 94 94
-T--MMM 50 93 94
----MMC 33 93 94
----CMM 27 93 93
----MCM 22 94 94
----MCC 19 93 94 ...
So, ER didn't make IRV any better but it did reduce the amount of
compromise. Chris had an idea, I believe, that ER-IRV(whole) was
susceptible to a straightforward push-over strategy, but I'm not sure
I see that here unless some Ms or Cs are playing that role. Since the
C voters are very frequently using that strategy (in many methods) I'm
doubtful...
Also, you can note here that the IRV methods did better wrt utility
maximizers than the other methods in this post. You can see (by scanning
for high values) that this is expected when C is losing his win odds
due to the voting patterns, and B is winning when sincere Condorcet says
it should be C.
Kevin
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