Kathy Dopp wrote:
From: Jameson Quinn <jameson.qu...@gmail.com>
To: EM <election-methods@lists.electorama.com>
Here's a toy model where the math is easy and you can get some interesting
results.
-Voters are distributed evenly from [-1, 1] along the ideology dimension.
-Candidates are represented by an ordered pair (i,q) where i is an ideology
from -1 to 1 and q is a quality from 0 to 2.
Such a one-dimensional ideology dimension grossly over-simplifies
IMO.In reality, people do not line up along a simple one ideology
dimension.
It seems fairly simple to extend the toy model to multiple dimensions if
you wish. Say you have n dimensions and a quality dimension. Then set
the utility for a candidate c to a voter v equal to c's quality value
plus the Euclidean distance between the two - or use another norm if you
want to experiment.
Political scientists tend to oversimplify, beginning with
Anthony Downs. The mathematics could take into account more than one
issue position or dimension when using spatial geometry to model how
close voters and candidates are to each other. It's on my to-do list
to write up a far more logically coherent way of using spatial
analysis of positions of voters and candidates that would essentially
unify much of the field of voting behavior research -- although
political scientists seem to enjoy carrying on the same debates
endlessly rather than deriving new theory on what they agree on.
Condensing reality down to one ideological dimension, even adding one
quality dimension, grossly distorts the more complex picture of
reality. A unidimensional model cannot even accurately model how
three different persons, say candidates, stand on two different issues
relative to each other or to voters. I think Downs basic approach
makes sense only if his mathematics is repaired to respond to the
multi-dimensional nature of the real world.
There have been attempts to find out the number of dimensions in opinion
space by looking at dimensions alone. Most of these have used principal
components analysis to align the greatest variety along one axis, the
greatest among the remaining to another, and so on.
To my knowledge, these have generally found somewhere between one and
two dimensions. However, at least for the examples I gave in my reply to
David's post, these have been done in countries that use FPTP, and
FPTP's relative failure to handle more than two parties could be
affecting the way in which people consider their opinions, squeezing
those opinions to fit along a line and thus reducing the dimensionality.
Also, if one relaxes the requirement that each axis should "mean"
something (e.g. left-vs-right, centralized-vs-decentralized,
pragmatic-vs-idealist), then metric multidimensional scaling would work
better than PCA (I think). It would be interesting to take some
political data, such as survey responses or legislature voting records,
define a distance between these (Euclidean, for instance, or Hamming in
the case of aye/nay), and then try to reconstruct a lower-dimensional
space focusing only on making the model distances as close to the
reported distances as possible.
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