Incidentally: you can also see this evidence that IRV
hurts racial minorities (as compared with plain plurality voting). I have no idea whether
that phenomenon is related to the one we shall describe here.
Cool graphics: 2-dimensional scenarios
For each of the pictures below, there are 14 candidates
(small circles).
Each pixel represents an election.
The blue pixels denote the region where the blue candidate wins an election for voters
distributed 2D-normally centered at that pixel (and similarly for other colors).
Each normal distribution is rotationally symmetric with
σx=σy=70, and each picture is 200x200 pixels.
In each picture, the coordinates of the 14 candidates
were chosen randomly from the 150x150 centered
subsquare. They are simply the first 112 random numbers output by my random generator.
Pictures produced by IEVS 3.24.
The idea behind these pictures is that candidates and voters are distributed in a 2-dimensional
"issue space" and voters prefer candidates whose stances on the two issues are close to their own,
e.g. closer in Euclidean
distance. There is no need to label the axes and the length units are arbitrary
(and moving 1 pixel away horizontally or vertically implies distance=1).
Voters prefer candidates closer to them.
All voters vote honestly.
The pictures in the top row are the hypothetical optimum voting system, which always
elects the best possible candidate
for society (maximizing summed utility, utility being a decreasing function of
voter-candidate distance).
The corresponding pictures in the bottom row are IRV (Instant Runoff Voting).
Observations:
With IRV usually only about 7 of the 14 candidates can ever win. The other 7 are
"zeroed out" and can never win, even when they are exactly located at the center of
the voter distribution.
The 7 favored candidates are usually the 7 "extremists." The 7 "centrists"
(located nearest the center of the picture) tend to be the ones IRV zeros out.
The "extremist" win-regions grow into, invade, and take over "centrist territory."
IRV also exhibits some other strange phenomena. In several cases, a candidate has a
disconnected win-region, for example the yellow region in the leftmost picture.
(Why is that yellow "island" there in the middle of the olive-green "sea"? The
yellow candidate is nowhere near. Surely the three candidates that actually are
located on the island ought to win there? But no – IRV never permits any of them to win.)
There are also several other "islands" and split win-regions in these pictures.
In this particular kind of 2D scenario (it may be proven)
Condorcet voting systems with honest voters
always produce the optimum diagram (top row),
Condorcet winners always exist,
and Condocet cycles are impossible.
Thus every pixel where the IRV color
disagrees with the Condorcet color is an example of a situation in which IRV refuses to
elect a "beats-all winner" who would defeat every opponent head-to-head.
Some IRV-defenders have contended that such behavior is very rare.
As you can see, that is not so; it can be quite common – it happens
for roughly 1/4 of the pixels
in each IRV picture.
You are looking at about 40,000 examples of this "very rare"
behavior.
Why the "random dot" regions? Each pixel in these diagrams was computed
from a random sample of
≈5000 voters from each normal distribution. In some regions of the diagrams,
the elections are "tough calls," i.e involve near-ties, and in those regions we
get "random dots." If we were to instruct the computer to work harder and use, say,
a million voters per pixel,
then that randomness would largely vanish and the boundaries of all regions would become crisp
line segments. However, that brings us to our point.
Compared to most voting systems, IRV is very vulnerable to tied
and near-tie elections, which is why a noticeable fraction of some of the IRV pictures
are random-dotty. (Think of each random-dot region yielding a chad-counting and recounting,
lawsuit and court battle over who really should have won that election.)
In contrast, at least in this kind of 2D scenario, Condorcet voting
systems and range voting are comparatively invulnerable to tied and near-tie elections
and their diagrams yield crisp boundaries and almost no stray dots, even with only 5000 voters
per pixel.
1-dimensional scenarios
Some people have objected that they consider politics to be largely "one dimensional"
from "left" to "right," not two dimensional. The below pictures were created in the same
way as the above ones except there are 5, not 14, candidates and the candidates are random
points in a 150x15 rectangle, not a 150x150 square. Again the top row in each group
shows the best winner, and the bottom row shows what IRV yields.
And here are some more:
Again, IRV often "zeros out" candidates
– although not as often as in the 2D scenarios above; 10% of them are zeroed
in the 1D scenarios here, versus about 50% in the 2D scenarios.
And when it does so, the victims again tend to be "centrists."
Again, random-dottiness and weird-shaped (including apparently
disconnected) regions are common.
(Actually, more than 10%
are zeroed out if you only look at the region of the pictures near the centerline;
that view also engenders more "tough-call" elections.)
