3 Candidate Condorcet Simulations
Mike Ossipoff
dfb at bbs.cruzio.com
Sun Nov 24 01:26:53 PST 1996
First of all, it's difficult to comment on Demorep's simulation
results, because there have been very few times when I've been
able to decypher his diagrams or tables, and this isn't one of
them.
Secondly, a simulation would ordinarily be done to find out
something--it would have a stated initial purpose, and a
conclusion, regarding that purpose, as part of its report.
But my main objection is that it's a most unrealistic simulation
if it assumes that people are equally likely to vote any
of the possible 8 rankings (equal rankings not used). I
repeat that the number of possible rankings, with truncations
but not equal rankings, is 8, and not 15.
There's a better way to do a simulation. I haven't talked
about this before, because I've never felt that a simulation
would tell us anything important. I can suggest 1 use for
a simulation: Estimating the liklihood of natural circular
ties, with a given number of canddiates, voters, & issue
dimensions. It would be interesting, but not necessary, and
I haven't brought it up because it would distract us from
more relevant business.
Anyway, if you're going to do a simulation, it would be more
realiatic if it's a "spatial study", as we've mentioned earlier.
Here's what a spatial study is (I won't be offended if anyone
corrects me about some aspect of this):
Using a random number generator, or other randomizing device,
like a table of random numbers, you randomly place the voters
& the candidates in N-dimensional space. Each dimension is
an issue. One's co-ordinate in a particular dimension is one's
position on the issue to which that dimension corresponds.
Then, having done that, it's easy to make out each voter's ranking,
based on hir distance to each canddiate. How far is a voter
from a candidate? Determine the difference between the voter's
co-ordinate & the candidate's co-ordinate in that dimension, and
square that difference. Sum that squared difference between that
candidate & that voter in each of the N dimensions. The sum of
those squared differences is what's used as the distance between
the voter & the candidade. Actually, the actual distance through
the N-dimensional space is equal to the _square root_ of that sum,
but we can just use the sum.
So the candidate with whom a voter has the lowest distance sum
is the 1st choice of that voter, etc.
***
Someone might suggest using a normal distribution to weight
the probability of a voter or candidate having a certain
distance from the mean along a particular dimension axis. No,
don't do that, because it would just mean that the middle
candidate would be the big winner, by Plurality & by
every other method, and you'd be less likely to find
natural circular ties. Maybe someone could show that, as the
number of voters & candidates is incraeased, the liklihood
of a natural circular tie approaches zero when the candidates'
& voters are given normal distributions along each issue
dimension. So such a study would be a waste of time. We
already know what happens if the voters are concentrated
at the middle: It doesn't matter what method is used.
Steve pointed out how a double-peaked distribution would
be a likely natural result of strategy. Maybe someone could
work out a weighting scheme for a double-peaked distribution,
but I'd just start with random distribution.
How many dimensions to use? I'd suggest 3. Voters surely don't
go by more issue dimensions than that. And 3-dimensional issue
space matches the dimensionality of physical space. More dimensions
than 3 wouldn't make sense intuitively.
How many voters? You might want to try it a few times with _lots_
of voters, just for fun, to see what happens, but that isn't
realistic, because, when the distribution is random, it produces
a very uniform distribution. It seems to me that I once determined
that, at least in 1 dimension, a uniform distribution would give
the election to the middle of 3 candidates every time. I don't
know; it was some time ago. Anyway, so a _big_ random sample
isn't desirable, since it would look more uniform than random,
and might regularly favor the middle. And who says the voters
would be uniformly distributed in a real election. So don't
use too many voters. I don't know--10? 20? Fewer than 10?
That's how I'd do it, for the most interesting & varied
results, and probably the most useful results, it seems to
me.
How many candidates? Well it seems like the number of candidates
should always be less than the number of voters, as much less
as possible. So maybe start out with just 3 candidates, & then
go up to more, always keeping the number of voters somewhat
more than the number of candidates. Realistically, I'd expect
one would want the number of candidates to be much less than
the number of voters, but that's difficult to achieve without
having too many voters.
***
Is it worth all this trouble just to find out how likely
natural circular ties are under various conditions? I doubt
it. That's why I've never brought this up.
***
Mike
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