[EM] Toy election model: 2D IQ (ideology/quality) model

[Kristofer Munsterhjelm]

Kathy Dopp wrote:
>> From: Jameson Quinn <jameson.quinn at gmail.com>
>> To: EM <election-methods at 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.