Alternatives to Borda Count
Forest Simmons
fsimmons at pcc.edu
Fri Jan 26 12:54:46 PST 2001
Thanks to everyone for their insights on the Borda Count.
I do not intend to propose the use of the Borda Count (or any other method
that converts rankings to ratings) as a practical election method. But
when someone (like Craig L.) proposes a hypothetical situation in terms of
rankings, and asks how a ratings system like CR or AV would perform in
that situation, instead of just telling him, "Sorry, CR and AV don't take
that kind of input," it would be nice to be able to make a reasonable
conversion from the ranked ballots to CR, for example.
I agree with Mike that on the average we should assume that the data is
symmetrically distributed about the median (unless some psychometrician
comes up with irrefutable evidence to the contrary, which is not likely).
Here's the way I look at it: Start with an hundred random candidates at
one end of a football field. Pose a question about civil rights, feminism,
labor, taxes, pollution, homelessness, landmine bans, gun control,
globaloney, or some other issue. Those who impress you with their answers
move one yard towards the opposite goalpost. Repeat the process until at
least one of them reaches the 100 yard line. (Ask some easy questions if
necessary.)
This experiment is about as close as you could every come to a social
science set-up for the Central Limit Theorem, which predicts that at the
end of a typical experiment of this kind, the 100 candidates will be
approximately normally distributed along the football field.
The normal distribution has a symmetrical, bell shaped probability density
function, with the bulk of the probability concentrated near the mean,
which is identical with the median and mode in such distributions.
In particular we would expect the candidates to be clustered around the
median.
The Borda Count does not reflect this reality. It assumes that the
candidates would be more or less evenly distributed from one end of the
football field to the other.
So the Borda Count introduces a distortion that should be taken into
consideration when inferring Cardinal Ratings from Ordinal Rankings.
In particular, the Borda Count should not be used in the higher resolution
version of PAV (as I did in one example recently) without filtering the
numbers to compensate for the distortion.
Now here's another way to think about this.
A two winner election has just finished. Which of the following would make
you happier? (1) You find out that your favorite (at the 100 yd line) and
your mid ranked (at the 50 yd line) candidates are in the winners circle.
OR ... (2) You find out that the two candidates closest to the upper
quartile (near the 60 yd line) are in the winners circle.
Wait! Shouldn't they be near the 75 yd line? No; although they are half
way in the rankings between the median and the top, we must remember that
the candidates are clustered near the median on the football field, so
these two guys shouldn't be much past the sixty yard line in a typical
case.
So which would you rather have, the two who impressed you with a combined
total of 120 answers, or the two whose combined total was 150 ?
This analysis gives some insight into some other aspects of rankings. For
example, suppose someone were to suggest a test case where 24% of a large
population ranks the candidates A>B>C>D>E>F>G>H . While it is possible
that 24% could agree on the extremes without getting together and swearing
in blood that they will all abide by this order, it is unrealistic to say
they they would all subscribe to the exact same order for the candidates
clustered around the median.
To make the simulation more realistic, the 24% with this ranking should be
broken up into 24 smaller groups corresponding to the 24 permutations of
CDEF.
In general it is slightly easier to deal with a bell curve centered at
zero. The ballots should be written in the simplest way for the public to
understand, and the conversion to the simplest formulae for processing the
ballots can be done later.
Does this analysis help clarify things?
Forest
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