THE RANK-ORDER VOTES IN THE 2009 BURLINGTON MAYORAL ELECTION
-------By Juho Laatu & Warren D. Smith-----March 2009-------
Juho Laatu independently produced a summary of all 8980 valid votes
(i.e. excluding the 4 invalid ones). He wrote software that input official ballot
files, and employed these abbreviations:
K = Bob Kiss
M = Andy Montroll
N = James Simpson
H = Dan Smith
W = Kurt Wright
R = Write-in
The line "83: H>M>K" means 83 voters ranked Smith top, Montroll 2nd, Kiss 3rd,
and the others implicitly co-equal last.
---the votes---
840: W
355: K>M
326: K
271: W>H
256: W>M
234: K>M>H
200: M>K
178: M
147: W>M>H
145: M>K>H
139: M>H
125: K>H
124: H
123: K>M>H>W>N
122: W>H>M
121: K>H>M
120: K>M>H>N>W
104: W>M>H>N>K
103: K>M>N>H>W
102: W>M>H>K>N
101: W>H>M>K>N
94: M>W
93: H>W
93: M>H>K
90: H>M
90: W>H>M>N>K
88: H>K>M
83: H>M>K
80: K>H>M>N>W
80: W>K
76: W>H>N>M>K
72: M>K>H>W>N
71: K>M>W>H>N
69: M>H>K>W>N
69: W>M>K>H>N
66: K>W
62: M>H>W
61: M>K>H>N>W
60: K>M>H>W
60: M>W>H>K>N
58: H>K>M>N>W
57: M>W>H
56: M>W>K>H>N
55: K>H>M>W>N
54: W>M>H>K
53: M>H>K>N>W
52: H>W>M>K>N
52: K>M>N
52: M>H>W>K>N
52: W>H>K>M>N
51: H>K
50: H>W>M
50: M>K>W>H>N
50: W>K>M>H>N
48: K>M>W
46: M>K>W
46: W>M>K
45: W>H>M>K
45: W>K>H>M>N
43: H>M>K>W>N
42: K>M>H>N
41: K>M>N>H
41: M>H>K>W
39: K>W>M>H>N
39: M>W>H>N>K
38: H>M>K>N>W
38: M>K>H>W
37: H>M>W
37: M>W>H>K
34: M>W>K
32: H>K>M>W>N
32: M>H>W>K
31: H>M>W>K>N
31: K>W>M
30: W>H>K
29: K>H>M>W
28: H>W>M>N>K
27: H>K>M>W
27: H>M>W>K
27: H>W>K>M>N
27: M>H>N>K>W
27: M>K>N>H>W
26: H>M>K>W
26: K>H>W>M>N
25: W>N
24: K>M>W>H
24: K>N>M>H>W
24: M>K>W>H
24: W>K>M
23: K>H>N>M>W
23: M>H>W>N>K
22: M>K>H>N
21: K>H>M>N
21: K>W>H>M>N
21: W>K>H
21: W>K>M>N>H
20: M>W>K>H
19: K>N
19: W>H>K>M
19: W>M>N>H>K
18: H>M>N>K>W
17: H>W>M>K
17: W>H>K>N>M
16: H>K>W>M>N
16: H>W>K
16: K>N>M
16: W>H>N
15: H>M>N>W>K
15: K>H>N
15: K>H>W>M
15: K>W>H>M
15: W>H>N>K>M
15: W>K>M>H
15: W>M>K>H
14: H>K>N>M>W
14: K>M>W>N>H
13: H>K>M>N
13: H>M>W>N>K
13: K>H>W
13: K>M>N>W>H
13: M>H>K>N
12: H>W>K>M
12: H>W>N>M>K
12: K>N>H>M>W
12: K>W>H
12: K>W>M>H
12: K>W>M>N>H
12: M>K>N
12: M>W>N>H>K
12: W>K>H>N>M
11: H>K>W
11: M>H>N>W>K
11: M>K>N>H
11: M>K>W>N>H
11: W>N>H>M>K
11: W>R
10: H>K>W>N>M
10: R
10: W>H>M>N
10: W>N>H>K>M
9: H>N>M>K>W
9: K>H>M>N>R
9: K>H>N>M
9: K>M>H>N>R
9: M>N>H
9: M>N>K>H>W
9: W>M>K>N>H
8: H>K>W>M
8: H>M>K>N
8: K>N>H
8: K>N>M>H
8: M>H>N
8: M>N>H>K>W
8: M>N>H>W>K
8: W>N>H
7: H>K>N
7: K>N>M>W>H
7: M>N>K
7: W>M>H>N
7: W>M>N>K>H
6: H>N
6: K>H>N>M>R
6: K>H>N>W>M
6: K>N>H>M
6: K>W>H>N>M
6: K>W>N>H>M
6: M>K>H>N>R
5: H>K>M>R>N
5: H>K>N>M
5: H>M>N
5: H>N>K>M>W
5: H>N>M
5: H>W>K>N>M
5: K>H>W>N>M
5: K>W>N>M>H
5: M>R
5: M>W>H>N
5: W>H>N>M
5: W>K>H>M
5: W>N>M>H>K
4: H>N>K>M
4: H>W>N
4: K>H>M>R>N
4: K>R
4: K>R>M>H>N
4: K>W>M>N
4: M>H>W>N
4: M>H>W>R
4: M>K>N>W>H
4: M>K>W>N
4: M>N
4: N>H>K>M>W
4: W>H>R
4: W>M>R
4: W>N>M>K
4: W>N>M>K>H
3: H>M>K>R>N
3: H>M>R
3: H>N>M>K>R
3: H>W>M>N
3: H>W>N>M
3: K>H>R>M>N
3: K>M>N>H>R
3: K>N>H>W>M
3: K>N>W>M>H
3: M>K>H>W>R
3: M>K>R
3: M>W>K>N>H
3: M>W>N
3: N
3: N>K
3: N>W
3: R>K
3: R>M
3: R>W
3: W>M>N
3: W>N>R
2: H>K>M>R>W
2: H>K>N>M>R
2: H>K>R>M>N
2: H>M>K>R>W
2: H>M>K>W>R
2: H>M>N>K
2: H>N>K
2: H>N>K>W>M
2: H>N>W
2: H>N>W>M>K
2: H>W>N>K>M
2: H>W>R
2: K>H>R
2: K>M>R
2: K>M>W>R
2: K>N>H>W
2: K>N>M>R>H
2: K>N>W
2: K>W>H>N
2: M>H>K>N>R
2: M>H>N>K
2: M>H>W>N>R
2: M>H>W>R>K
2: M>K>H>R>N
2: M>N>H>W
2: M>N>K>H
2: M>N>K>W
2: M>N>W
2: M>N>W>H>K
2: M>N>W>K>H
2: M>R>K>H>N
2: N>K>M
2: N>K>M>W>H
2: R>M>H
2: R>M>H>K>N
2: R>M>H>N>K
2: R>M>N>H>K
2: W>H>M>N>R
2: W>K>H>N
2: W>K>N
2: W>K>N>H>M
2: W>M>N>K
2: W>N>K>M>H
2: W>R>M>H>K
2: W>R>M>K
1: H>K>M>N>R
1: H>K>M>W>R
1: H>K>N>W
1: H>K>W>N
1: H>K>W>N>R
1: H>M>K>N>R
1: H>M>N>K>R
1: H>M>W>N
1: H>M>W>R>N
1: H>N>K>W
1: H>N>M>K
1: H>N>M>R>K
1: H>N>M>W
1: H>N>M>W>K
1: H>N>R
1: H>N>R>M>K
1: H>N>W>K>M
1: H>R
1: H>W>K>M>R
1: K>H>M>R
1: K>H>M>R>W
1: K>H>N>R>M
1: K>H>W>N
1: K>M>H>R
1: K>M>H>W>R
1: K>M>N>R>W
1: K>M>N>W
1: K>M>R>W>N
1: K>N>H>R
1: K>N>H>R>W
1: K>N>M>H>R
1: K>N>M>W>R
1: K>N>R
1: K>N>R>M>W
1: K>N>W>H>M
1: K>R>H>M>N
1: K>R>M
1: K>R>M>H
1: K>R>W
1: K>R>W>H>N
1: K>R>W>N>H
1: K>W>H>M>R
1: K>W>N>M
1: K>W>N>M>R
1: M>H>N>W
1: M>H>N>W>R
1: M>K>H>R>W
1: M>K>N>H>R
1: M>K>N>R>H
1: M>K>N>W
1: M>K>R>W
1: M>K>R>W>N
1: M>N>H>K
1: M>N>H>K>R
1: M>N>R>H>K
1: M>N>W>K
1: M>N>W>R>K
1: M>R>H>K>N
1: M>R>N
1: M>R>N>H
1: M>W>H>R
1: M>W>N>K>H
1: M>W>R
1: M>W>R>N>K
1: N>H>K
1: N>H>M
1: N>H>M>K>W
1: N>H>W
1: N>H>W>M>K
1: N>K>H>M
1: N>K>H>M>R
1: N>K>H>W
1: N>K>M>H
1: N>K>M>H>W
1: N>K>M>W>R
1: N>K>W>M>H
1: N>M>H
1: N>M>H>W>K
1: N>M>W>H>K
1: N>M>W>K>R
1: N>R
1: N>W>K>H>M
1: R>H
1: R>H>K>W>M
1: R>K>H>N>M
1: R>K>M
1: R>K>W>M
1: R>M>K>H>N
1: R>M>K>W
1: R>W>H>N>M
1: R>W>N>M>K
1: W>H>K>M>R
1: W>H>M>R>K
1: W>H>M>R>N
1: W>H>N>M>R
1: W>H>R>M
1: W>H>R>M>N
1: W>H>R>N
1: W>K>N>M
1: W>K>N>M>H
1: W>M>H>K>R
1: W>M>K>H>R
1: W>M>K>N
1: W>M>N>H
1: W>M>R>H>K
1: W>N>H>M>R
1: W>N>H>R>M
1: W>N>K
1: W>N>K>H>M
1: W>N>M
1: W>N>M>H
1: W>R>H>M>N
1: W>R>K
1: W>R>M
1: W>R>M>H>N
1: W>R>N
1: W>H>K=M
1: M>H>N=W>K
1: K=R>W>M>H>N
1: W=R>H>M>N>K
1: W=R>M
1: W=R>M>K>H
---end of votes---
384 voter types (380 at least partly valid).
5 candidates plus write-ins (combined regarded as 6th).
8980 votes, 8976 at least partly valid.
The last 6 votes included tied-rankings among candidates not at the bottom-end of
the ordering.
Laatu's explanation of why (he thinks) his counts differ from the official counts:
4 out of the 6 votes with ties (the last 4 listed above) were considered "exhausted"
i.e. not used in the official counts.
It seems this decision was made when the calculation algorithm read the tied
part. In one of the remaining 2 ballots with ties the algorithm did not need to
read the tied parts of the ballot to compute the IRV winner & IRV round-totals.
Finally, it seems that the other remaining ballot was
not "exhausted" because one of the tied candidates was already
eliminated when the calculation reached that point!
I don't know if these differences in handling of the ties ("exhausted if met",
"ok if not met", "ok if met only after elimination of the other tied candidate"?)
should be considered a bug or a feature of the counting algorithm.
W.D. Smith & Jan Kok comment:
Actually, the official method -- discarding ballots with the top-two entries ranked equal --
!seems stupid to us. It is a holdover from the olden pre-computer days when a ballot had to be
put in one or the other pile. Since this election was counted by computer there was nothing
stopping the computer from putting HALF of the vote in BOTH piles. That, it seems to us,
would have more-accurately reflected what the voter wanted (versus just discarding her
vote entirely).
But anyhow, the bottom line is that the disagreements between the
Laatu, Official, and Univ. of Vermont counts all were so small (at most 5 votes)
that in this particular election that they were unable to affect any of our conclusions.
=========================================================================
Here is a summary of Juho Laatu's summary (by WD Smith)
also including comparisons with (WDS's deductions from) the Univ. of Vermont
counts.
Laatu and the UVM counts do not always agree, but in no case is the disagreement
more than 5 votes.
Here's Juho Laatu's vote-summary with all candidates besides W,M,K eliminated,
replacements such as "M>K becomes M>K>W" and "W>K=M becomes W" made,
and reordered to group common vote-types together:
M>K>W
1332 = 200+145+93+83+72+69+61+53+50+46+43+41+38+38+27+27+26+24+22+18+13+12+11+11+9+9+8+8+7+6+4+4+3+3+3+3+2+2+2+2+2+2+2+2+2+2+2+2+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1
M>W>K
767 = 94+62+60+57+56+52+39+37+37+34+32+31+27+23+20+15+13+12+11+8+5+4+4+3+3+2+2+2+2+2+2+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1
M (note: UVM's count had only 451)
455 = 178+139+90+9+8+5+5+5+4+3+3+2+1+1+1+1
K>M>W (note: UVM's count had 2047)
2043 = 355+234+123+121+120+103+88+80+71+60+58+55+52+48+42+41+32+29+27+24+24+23+21+16+14+14+13+13+12+9+9+9+8+7+6+6+5+5+5+4+4+4+4+3+3+2+2+2+2+2+2+2+2+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1
K>W>M
371 = 66+39+31+26+21+16+15+15+13+12+12+12+11+10+8+6+6+6+5+5+4+3+3+2+2+2+2+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1
K (note: UVM's count had 564)
568 = 326+125+51+19+15+8+7+4+3+3+2+2+1+1+1
W>M>K (note: UVM's count had 1517)
1513 = 256+147+122+104+102+101+90+76+69+54+52+50+46+45+28+19+17+15+12+11+10+9+7+7+5+5+4+4+4+3+3+3+2+2+2+2+2+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1
W>K>M (note: UVM's count had 496)
495 = 80+52+50+45+30+27+24+21+21+19+17+16+15+15+12+12+10+5+5+2+2+2+2+2+1+1+1+1+1+1+1+1+1
W (note: UVM's count had 1284)
1289 = 840+271+93+25+16+11+8+4+4+3+3+3+2+2+1+1+1+1
total
8833 = 1332+767+2043+371+495+1513+1289+568+455
Note: there were also 147=124+10+6+3+1+1+1+1 votes
which gave no preference amongst K,M,W. They were:
124: H
10: R
6: H>N
3: N
1: H>N>R
1: H>R
1: N>R
1: R>H
larger total
8980 = 8833 + 147
====================================================
VARIOUS VOTING SYSTEMS USING THE (FULL or ABBREVIATED) LAATU VOTE COUNTS:
All of these voting methods (since they are Condorcet methods) elect MONTROLL
using either the *full* or *abbreviated* (both work) Laatu counts:
Nanson-Baldwin
Black
Raynaud
Schulze
Simpson-Kramer
Tideman ranked pairs
Copeland
Small
Also electing MONTROLL using the *full* Laatu counts:
Borda
Dodgson
Arrow-Raynaud
Warning these counts may be slightly off (but it should not be enough to matter):
Counts of 1st choices: H(1306) - K(2586) - M(2063) - N(35) - R(40) - W(2954)
Counts of 2nd choices: H(2107) - K(1404) - M(2637) - N(307) - R(50) - W(994)
Counts of 3rd choices: H(1806) - K(962) - M(1398) - N(659) - R(42) - W(722)
Probably also Approval and Range voting both would have
elected MONTROLL (because it is very rare that range/approval winners
differ from Condorcet winners and there are theoretical reasons for that)
but we cannot be sure since we only have rank-order ballot data.
Some approval-variants using *full* Laatu counts (which probably are
not terribly realistic as guesses for what plain-approval would do):
Approval of top-1-only (i.e. plurality) elects WRIGHT (Kiss 2nd).
Approval of top-1 and half-approval of 2nd choice elects WRIGHT (Montroll 2nd).
Approval of top-1 and (2/3)-approval of 2nd choice elects MONTROLL.
Approval of top-2 elects MONTROLL.
Approval of top-2 and half-approval of 3rd choice elects MONTROLL.
Approval of top-3 elects MONTROLL.
These methods elect MONTROLL using the {M,K,W}-only Laatu counts with
"M" regarded as half M>W>K and half M>K>W, etc:
Coombs
AntiPlurality (which in this case is top-two approval)
Bucklin
Dodgson
Borda (here equivalent to approval of top and half-approval of 2nd-choice)
Every Condorcet method
Elects WRIGHT:
Plurality (which is top-one approval)
Elects KISS using the {M,K,W}-only Laatu counts (regarding
"M" as half M>W>K and half M>K>W, or not, either way):
IRV.
It thus appears IRV and Plurality are uniquely bad performers in this election --
pretty much every other voting method mankind ever invented would elect MONTROLL.
Here are the {M,K,W}-only Laatu vote counts:
1332: M>K>W
767: M>W>K
455: M
2043: K>M>W
371: K>W>M
568: K
1513: W>M>K
495: W>K>M
1289: W
which yield this defeats matrix:
Canddt K M W
K * 3477 4314
M 4067 * 4597
W 4064 3668 *
But UVM found the following slightly different matrix:
Canddt K M W
K * 3478 4314
M 4067 * 4597
W 4064 3668 *
[Either way Montroll is a beats-all winner.]
corresponding to these counts
1332: M>K>W
767: M>W>K
451: M
2047: K>M>W
371: K>W>M
564: K
1517: W>M>K
496: W>K>M
1284: W
Using the Laatu {W,K,M}-only vote counts the IRV election proceeds:
Canddts Rnd#1 Rnd#2
K 2982 4314 (wins)
W 3297 4064
M 2554
which if the 495 plus 258 of the 1289 W-votes are switched to K is
1332: M>K>W
767: M>W>K
455: M
2043: K>M>W
371: K>W>M
1342: K
1513: W>M>K
0: W>K>M
1031: W
yielding this IRV election
K 3755 3755
M 2554 4067 (wins)
W 2544
which is slightly different from the official IRV election:
Canddts Rnd#1 Rnd#2
K 2981 4313 (wins)
W 3294 4061
M 2554
while the UVM counts yield:
Canddts Rnd#1 Rnd#2
K 2982 4314 (wins)
W 3297 4064
M 2550
Here are the {M,K,W}-only Laatu vote counts with "M" regarded
as half M>W>K and half M>K>W, etc (note: all these counts *doubled*):
3119: M>K>W
1989: M>W>K
4654: K>M>W
1310: K>W>M
4315: W>M>K
2279: W>K>M
This is the pairwise matrix they yield (with *doubled* entries):
K M W
K * 8243 9083
M 9423 * 9762
W 8583 7904 *
The *full* pairwise matrix (but, unlike above matrices, now counting
voters who do not express a preference for A vs B as half A>B and half B>A):
Canddt H K M N R W
H * 4305.5 3702.5 6916 7460 4399
K 4674.5 * 4195 6826 7506.5 4615
M 5277.5 4785 * 7328 7767 4954.5
N 2064 2154 1652 * 6076.5 2507.5
R 1520 1473.5 1213 2903.5 * 1540
W 4581 4365 4025.5 6472.5 7440 *
The *full* pairwise matrix (usual count this time, BUT wins only):
Canddt H K M N R W
H * 0 0 5573 6057 0
K 3946 * 0 5517 6149 4314
M 4573 4067 * 6267 6658 4597
N 0 0 0 * 3338 0
R 0 0 0 0 * 0
W 3975 0 0 5274 6063 *
(also showing losses):
Option H K M N R W
H * 3577 2998 5573 6057 3793
K 3946 * 3477 5517 6149 4314
M 4573 4067 * 6267 6658 4597
N 721 845 591 * 3338 1309
R 117 116 104 165 * 163
W 3975 4064 3668 5274 6063 *
Note that M defeated every opponent pairwise, and the "big three"
{M,K,W} pairwise-defeated everybody not in the Big Three.
KEENER EIGENVECTOR VOTING:
M := matrix(6,6, [
1, 3578, 2999, 5574, 6058, 3794,
3947, 1, 3478, 5518, 6150, 4315,
4574, 4068, 1, 6268, 6659, 4598,
722, 846, 592, 1, 3339, 1310,
118, 117, 105, 166, 1, 164,
3976, 4065,3669, 5275, 6064, 1]);
eigenvectors(M);
???
M := matrix(3,3, [
1, 3478, 4315,
4068, 1, 4598,
4065, 3669, 1] );
eigenvectors(M);
All 3 eigenvectors real. The one with greatest eigenvalue 8041.7 is
(0.563441903756, 0.626340182887, 0.467281589294)
which anoints M (MONTROLL) the winner with 0.626 ???
Reversed ballots:
1332: M>K>W
767: M>W>K
455: M
2043: K>M>W
371: K>W>M
568: K
1513: W>M>K
495: W>K>M
1289: W
W wins (plur) 3886.5 = 568/2 + 2043 + 455/2 + 1332
K = 3152 = 767 + 455/2 + 1513 + 1289/2
M = 1794.5 = 371 + 568/2 + 495 + 1289/2
W is a winner with every method in sight (Borda, all Condorcets, IRV) with reversed ballots
reversed IRV:
W 3886.5 4541.5 (wins)
K 3152 4291.5
M 1794.5
Possible interesting future calculation: try Sinkhorn & Keener-eigenvector.
Full IRV election proceeds:
eliminate N(35)
eliminate R(37)
eliminate H(1317)
eliminate M(2554)
eliminate W(4060)
winner K(4313)
If the 840 W-only votes artificially removed then
eliminate N(35)
eliminate R(37)
eliminate H(1317)
eliminate W(2454)
eliminate K(3476)
winner M(4064)
The STATED SECOND PREFERENCES OF W-TOP VOTERS ABOUT M-versus-K WERE
1510: M
495: K
*after* all candidates besides {W,M,K} eliminated. This is a 3.05:1 ratio.
If however we assess this *before* eliminating anybody, then the
STATED SECOND PREFERENCES OF W-TOP VOTERS ABOUT M-versus-K INSTEAD WERE
1340: M
430: K
This is a 3.12:1 ratio.
REALISTIC APPROVAL VOTING (DEDUCTION):
Stephen Unger noted the voter were allowed to rank only SOME of the candidates.
If rank K only: assume they would with approval voting, approve K only (among the Big Three K,M,W).
If rank K and M only: assume they would approve both K and M but not W.
If rank all three (K,M, and W) then assume they would approve their top 2 with some
frequency X (0N
3: N
1: H>N>R
1: H>R
1: N>R
1: R>H
K ONLY RANKED:
326: K
125: K>H
51: H>K
19: K>N
15: K>H>N
8: K>N>H
7: H>K>N
4: K>R
3: N>K
3: R>K
2: H>N>K
2: K>H>R
1: K>N>H>R
1: K>N>R
1: N>H>K
M ONLY RANKED:
178: M
139: M>H
90: H>M
9: M>N>H
8: M>H>N
5: H>M>N
5: H>N>M
5: M>R
4: M>N
3: H>M>R
3: R>M
2: R>M>H
1: M>R>N
1: M>R>N>H
1: N>H>M
1: N>M>H
W ONLY RANKED:
840: W
271: W>H
93: H>W
25: W>N
16: W>H>N
11: W>R
8: W>N>H
4: H>W>N
4: W>H>R
3: N>W
3: R>W
3: W>N>R
2: H>N>W
2: H>W>R
1: N>H>W
1: W>H>R>N
1: W>R>N
K&M BUT NOT W RANKED:
355: K>M
234: K>M>H
200: M>K
145: M>K>H
121: K>H>M
93: M>H>K
88: H>K>M
83: H>M>K
52: K>M>N
42: K>M>H>N
41: K>M>N>H
22: M>K>H>N
21: K>H>M>N
16: K>N>M
13: H>K>M>N
13: M>H>K>N
12: M>K>N
11: M>K>N>H
9: K>H>M>N>R
9: K>H>N>M
9: K>M>H>N>R
8: H>M>K>N
8: K>N>M>H
7: M>N>K
6: K>H>N>M>R
6: K>N>H>M
6: M>K>H>N>R
5: H>K>M>R>N
5: H>K>N>M
4: H>N>K>M
4: K>H>M>R>N
4: K>R>M>H>N
3: H>M>K>R>N
3: H>N>M>K>R
3: K>H>R>M>N
3: K>M>N>H>R
3: M>K>R
2: H>K>N>M>R
2: H>K>R>M>N
2: H>M>N>K
2: K>M>R
2: K>N>M>R>H
2: M>H>K>N>R
2: M>H>N>K
2: M>K>H>R>N
2: M>N>K>H
2: M>R>K>H>N
2: N>K>M
2: R>M>H>K>N
2: R>M>H>N>K
2: R>M>N>H>K
1: H>K>M>N>R
1: H>M>K>N>R
1: H>M>N>K>R
1: H>N>M>K
1: H>N>M>R>K
1: H>N>R>M>K
1: K>H>M>R
1: K>H>N>R>M
1: K>M>H>R
1: K>N>M>H>R
1: K>R>H>M>N
1: K>R>M
1: K>R>M>H
1: M>K>N>H>R
1: M>K>N>R>H
1: M>N>H>K
1: M>N>H>K>R
1: M>N>R>H>K
1: M>R>H>K>N
1: N>K>H>M
1: N>K>H>M>R
1: N>K>M>H
1: R>K>H>N>M
1: R>K>M
1: R>M>K>H>N
K&W BUT NOT M RANKED:
80: W>K
66: K>W
30: W>H>K
21: W>K>H
16: H>W>K
13: K>H>W
12: K>W>H
11: H>K>W
2: K>N>H>W
2: K>N>W
2: K>W>H>N
2: W>K>H>N
2: W>K>N
1: H>K>N>W
1: H>K>W>N
1: H>K>W>N>R
1: H>N>K>W
1: K>H>W>N
1: K>N>H>R>W
1: K>R>W
1: K>R>W>H>N
1: K>R>W>N>H
1: N>K>H>W
1: W>N>K
1: W>R>K
M&W BUT NOT W RANKED:
256: W>M
147: W>M>H
122: W>H>M
94: M>W
62: M>H>W
57: M>W>H
50: H>W>M
37: H>M>W
10: W>H>M>N
7: W>M>H>N
5: M>W>H>N
5: W>H>N>M
4: M>H>W>N
4: M>H>W>R
4: W>M>R
3: H>W>M>N
3: H>W>N>M
3: M>W>N
3: W>M>N
2: M>H>W>N>R
2: M>N>H>W
2: M>N>W
2: W>H>M>N>R
1: H>M>W>N
1: H>M>W>R>N
1: H>N>M>W
1: M>H>N>W
1: M>H>N>W>R
1: M>W>H>R
1: M>W>R
1: R>W>H>N>M
1: W=R>M
1: W>H>M>R>N
1: W>H>N>M>R
1: W>H>R>M
1: W>H>R>M>N
1: W>M>N>H
1: W>N>H>M>R
1: W>N>H>R>M
1: W>N>M
1: W>N>M>H
1: W>R>H>M>N
1: W>R>M
1: W>R>M>H>N
ALL THREE RANKED:
123: K>M>H>W>N
120: K>M>H>N>W
104: W>M>H>N>K
103: K>M>N>H>W
102: W>M>H>K>N
101: W>H>M>K>N
90: W>H>M>N>K
80: K>H>M>N>W
76: W>H>N>M>K
72: M>K>H>W>N
71: K>M>W>H>N
69: M>H>K>W>N
69: W>M>K>H>N
61: M>K>H>N>W
60: K>M>H>W
60: M>W>H>K>N
58: H>K>M>N>W
56: M>W>K>H>N
55: K>H>M>W>N
54: W>M>H>K
53: M>H>K>N>W
52: H>W>M>K>N
52: M>H>W>K>N
52: W>H>K>M>N
50: M>K>W>H>N
50: W>K>M>H>N
48: K>M>W
46: M>K>W
46: W>M>K
45: W>H>M>K
45: W>K>H>M>N
43: H>M>K>W>N
41: M>H>K>W
39: K>W>M>H>N
39: M>W>H>N>K
38: H>M>K>N>W
38: M>K>H>W
37: M>W>H>K
34: M>W>K
32: H>K>M>W>N
32: M>H>W>K
31: H>M>W>K>N
31: K>W>M
29: K>H>M>W
28: H>W>M>N>K
27: H>K>M>W
27: H>M>W>K
27: H>W>K>M>N
27: M>H>N>K>W
27: M>K>N>H>W
26: H>M>K>W
26: K>H>W>M>N
24: K>M>W>H
24: K>N>M>H>W
24: M>K>W>H
24: W>K>M
23: K>H>N>M>W
23: M>H>W>N>K
21: K>W>H>M>N
21: W>K>M>N>H
20: M>W>K>H
19: W>H>K>M
19: W>M>N>H>K
18: H>M>N>K>W
17: H>W>M>K
17: W>H>K>N>M
16: H>K>W>M>N
15: H>M>N>W>K
15: K>H>W>M
15: K>W>H>M
15: W>H>N>K>M
15: W>K>M>H
15: W>M>K>H
14: H>K>N>M>W
14: K>M>W>N>H
13: H>M>W>N>K
13: K>M>N>W>H
12: H>W>K>M
12: H>W>N>M>K
12: K>N>H>M>W
12: K>W>M>H
12: K>W>M>N>H
12: M>W>N>H>K
12: W>K>H>N>M
11: M>H>N>W>K
11: M>K>W>N>H
11: W>N>H>M>K
10: H>K>W>N>M
10: W>N>H>K>M
9: H>N>M>K>W
9: M>N>K>H>W
9: W>M>K>N>H
8: H>K>W>M
8: M>N>H>K>W
8: M>N>H>W>K
7: K>N>M>W>H
7: W>M>N>K>H
6: K>H>N>W>M
6: K>W>H>N>M
6: K>W>N>H>M
5: H>N>K>M>W
5: H>W>K>N>M
5: K>H>W>N>M
5: K>W>N>M>H
5: W>K>H>M
5: W>N>M>H>K
4: K>W>M>N
4: M>K>N>W>H
4: M>K>W>N
4: N>H>K>M>W
4: W>N>M>K
4: W>N>M>K>H
3: K>N>H>W>M
3: K>N>W>M>H
3: M>K>H>W>R
3: M>W>K>N>H
2: H>K>M>R>W
2: H>M>K>R>W
2: H>M>K>W>R
2: H>N>K>W>M
2: H>N>W>M>K
2: H>W>N>K>M
2: K>M>W>R
2: M>H>W>R>K
2: M>N>K>W
2: M>N>W>H>K
2: M>N>W>K>H
2: N>K>M>W>H
2: W>K>N>H>M
2: W>M>N>K
2: W>N>K>M>H
2: W>R>M>H>K
2: W>R>M>K
1: H>K>M>W>R
1: H>N>M>W>K
1: H>N>W>K>M
1: H>W>K>M>R
1: K=R>W>M>H>N
1: K>H>M>R>W
1: K>M>H>W>R
1: K>M>N>R>W
1: K>M>N>W
1: K>M>R>W>N
1: K>N>M>W>R
1: K>N>R>M>W
1: K>N>W>H>M
1: K>W>H>M>R
1: K>W>N>M
1: K>W>N>M>R
1: M>H>N=W>K
1: M>K>H>R>W
1: M>K>N>W
1: M>K>R>W
1: M>K>R>W>N
1: M>N>W>K
1: M>N>W>R>K
1: M>W>N>K>H
1: M>W>R>N>K
1: N>H>M>K>W
1: N>H>W>M>K
1: N>K>M>H>W
1: N>K>M>W>R
1: N>K>W>M>H
1: N>M>H>W>K
1: N>M>W>H>K
1: N>M>W>K>R
1: N>W>K>H>M
1: R>H>K>W>M
1: R>K>W>M
1: R>M>K>W
1: R>W>N>M>K
1: W=R>H>M>N>K
1: W=R>M>K>H
1: W>H>K=M
1: W>H>K>M>R
1: W>H>M>R>K
1: W>K>N>M
1: W>K>N>M>H
1: W>M>H>K>R
1: W>M>K>H>R
1: W>M>K>N
1: W>M>R>H>K
1: W>N>K>H>M
CALCULATION PROCEEDS
I am removing a "W>K=M" vote from the KMW set and placing it in the W set.
Then here are the counts of vots ranking various subsets of {K,M,W}:
emptyset=147 = 124+10+6+3+1+1+1+1
k=568
m=455
w=1289
km=1720
kw=271
mw=905
kmw=3625
approvals if the all-three-ranked votes are ignored:
k=2559=568+1720+271
m=3080=455+1720+905
w=2465=1289+271+905
Looking at the 3625 votes ranking all three of {K,M,W} we find these 6=3! counts:
kmw=962
kwm=253
mwk=488
mkw=693
wkm=342
wmk=887
approvals if fraction 1-X of the 3-ranked votes approve top-1 and X approve top-2:
k=2559+962+253+X*(693+342);
m=3080+488+693+X*(962+887);
w=2465+342+887+X*(253+488);
simplify to get final result:
k = 3774 + 1035 X
m = 4261 + 1849 X
w = 3694 + 741 X
MONTROLL is the approval-voting winner for EVERY X value with 0<=X<=1.
This election featured 1481 = 840 + 326 + 178 + 124 + 10 + 3
"bullet style" votes (ranking exactly one candidate) out of 8980 valid ballots (16.5%).
It also had
1931=355+271+256+200+139+125+94+93+90+80+66+51+25+19+19+11+6+5+4+4+3+3+3+3+3+1+1+1
balots ranking exactly two (21.5%)
YET ANOTHER APPROVAL-VOTING SIMULATION (by Juho Laatu):
In the web page you referred to "big
three". There might actually be four.
I made 100 simulations where voters
approved random number of candidates
that they marked on their ballot
(from 1 (first) to length of the
vote). (Equals were ignored. All
voters that voted the same way also
approved the same candidates.)
The result of this limited test
was that Montroll won in 72.8% of
the simulations, Smith won 14.3% and
Kiss won 12.8%. Wright never won.
The assumptions were purely technical
and did not take into account any
psychological implications or the real
life political set-up. Maybe they were
quite reasonable though. With these
assumptions Montroll seems to be the
most probable winner. Wright didn't
win in any of the simulations. Smith
was also a strong candidate, about as
strong as Kiss. Simpson got plenty of
approvals (more that what the number
of his first preference votes would
indicate) although he didn't win any
simulations.
--end.