[EM] strange 2-dim. fun example
Jobst Heitzig
heitzig-j at web.de
Tue May 17 15:30:15 PDT 2005
Hi folks!
Here's some fun example showing that, even with a 2-dimensional model
and only 4 candidates, classical voting methods can give completely
different winners and orders...
In the example, 100 voters from all over the globe elect the earth's
capital from the following candidates (which build an almost regular
tetrahedron):
CANDIDATES:
candidate phi theta
| | | place
| | | |
a 145 -38 Melbourne
b -123 38 San Francisco
c -58 -35 Buenos Aires
d 53 30 Persepolis
In this, phi and theta are angular coordinates with respect to Greenwich
and to the equator, in degrees.
The voters' preferences are simply by distance, and all approve the
nearest two cities:
VOTERS:
voters
| preferences (according to distance)
| and approval cutoffs:
| | phi theta
| | | | place
| | | | |
4 a>b>>c>d 178 -17 Fidji Islans
3 a>b>>d>c 147 -9 Port Moresby
8 a>c>>b>d 173 -44 Christchurch
5 a>d>>b>c 134 -23 Alice Springs
6 a>d>>c>b 78 -39 St. Paul Island
7 b>a>>c>d -159 22 Honolulu
8 b>c>>d>a -83 23 Havana
8 b>d>>c>a -113 53 Edmonton
5 c>a>>b>d -68 -50 Piedrabuena
3 c>a>>d>b -27 -76 Halley Bay
8 c>b>>a>d -91 0 Galapagos Islands
3 c>b>>d>a -60 -3 Manaus
4 c>d>>a>b 18 -34 Cape Town
1 c>d>>b>a -28 -20 some small island east of Rio de Janeiro
6 d>a>>b>c 114 23 Hong Kong
5 d>a>>c>b 78 4 Male
5 d>b>>a>c 69 42 Tashkent
3 d>b>>c>a 38 56 Moscow
8 d>c>>b>a 4 7 Lagos
---
100
Now let us see what different methods do with this information:
ANALYSIS:
------------------------------
c b d a
------------------------------
1st ranks 24 23 27 26 --> plurality order: d>a>c>b
2nd ranks 24 26 24 26
------------------------------
approval 48 49 51 52 --> approval order: a>d>b>c
------------------------------
3rd ranks 33 33 17 17
4th ranks 19 18 32 31 --> "skating" order: a>d>b>c
------------------------------
Borda score 153 154 146 147 --> Borda order: b>c>a>d
------------------------------
above c - 49 49 49
above b 51 - 46 49
above d 51 54 - 49 --> transitive defeats,
above a 51 51 51 - minmax order: c>b>a>d
------------------------------
Copeland score 3 2 1 0 --> Copeland order: c>b>d>a
------------------------------
IRV 24 23 27 26
31 35 34
51 49 --> IRV order: d>a>c>b
------------------------------
random ballot and DFC
winning probabilities:
.24 .23 .27 .26
------------------------------
If I didn't mess the numbers up, the classical methods
Approval/Borda/Condorcet/Plurality elect A/B/C/D, respectively.
As it turns out, each of the cities can get each rank from 1 to 4
depending on the method, but there is not a single defeat circle which
would indicate any problems...
Aren't such examples a good reason to always consider more than just one
piece of the information (like approval or ranks or defeats or direct
support) and instead combine them to get the whole picture?
Yours, Jobst
More information about the Election-Methods
mailing list