[EM] Thoughts on majority potential simulations

[Forest Simmons]

Tony,

Your comments below remind me that it would be interesting to do a
singular value analysis to find the effective dimension of the issue
space.  Just throw out the dimensions corresponding to relatively small
singular values. The eigenvectors corresponding to the larger eigenvalues
would give the natural combinations of issues that the various dominant
factions disagree on.

I also have a "poor man's singular value decomposition" based on L_1 and
L_infinity norms that might go better with the spaces that we are dealing
with, thinking along Richard's line of reasoning.

I have more thoughts along these lines, but am late for class.

Forest

On Mon, 21 May 2001, Anthony Simmons wrote:

> >> From:  Richard Moore
> >> Subject: [EM] thoughts on majority potential simulations
> 
> [...]
> 
> >> 4. I am going to use a 2-dimensional policy space. I am
> >> going to calculate utilities based on the L1 (Hamming)
> >> distance between the voter and the candidate. The
> >> rationale for L1 is that if policy X has a cost to the
> >> voter and policy Y has a cost to the voter (not
> >> necessarily monetary costs but such can be used as a
> >> tangible example), then the combined costs of both
> >> policies is a simple sum. If there are any arguments for
> >> using L2 distance please let me hear them.
> 
> Usually, if there are a whole lot of factors (as in "factor
> analysis"), they aren't independent.  For example, you'd
> imagine that if I'm morally opposed to ice cream, I'd most
> likely be opposed to frozen yogurt as well.  If you make one
> the X coordinate and one the Y coordinate, and plot the
> positions of actual moral philosophers, you'd expect to find
> a pretty steady correlation between X and Y, with most of the
> pack along a straight line through the origin and
> representing a third factor (a derived one in this case), Z,
> "moral correctness of frozen confections".
> 
> It's reasonable to consider Z as basic as X and Y, so we'd
> like to be able to rotate our graph so that this factor is
> horizontal or vertical and becomes one of the coordinate
> axes, without changing any of the relationships between the
> variables or the distances between points on the Z line; we
> wouldn't want a policy to become more or less extreme on our
> scale just because we rotated the scales.
> 
> Using the root-sum-of-squares distance makes all of this very
> clean.  Of course, you could preserve distances in other
> ways.  If you were using the the city block distance, and you
> rotated the coordinates so that (1, 1) ended up on the X
> axis, you could stretch it to (2, 0).  After all, if you're
> not using Euclidean distance, then there's no requirement
> that (1, 1) rotates onto (1.414, 0).  But when you throw out
> the way we normally measure distances, you throw out the
> underlying geometry, and we all know that the most important
> thing about any measurement is the underlying geometric
> aesthetics.
> 
> Another consideration:  In the illustration above, the data
> actually lies along one dimension, embedded in a two-
> dimensional space.  If we were to add popsicles to ice cream
> and yogurt, we would appear to have a third dimension, but
> it's spurious.  The data still lie along a single dimension.
> 
> In real life, this isn't likely to be so clear.  We'd expect
> some moral philosophers to adhere to the orthodox position
> that petty distinctions between frozen sweets are the work of
> the devil, while others might take a more complex view.
> Thus, as the number of frozen confections increases, the
> number of dimensions required to describe the actual data
> would increase, but not as fast as the number of raw
> dimensions.  For example, with five desserts, perhaps we
> could describe all positions as lying on a two-dimensional
> surface within the five-dimensional ambient space.
> 
> Using root-sum-of-squares reflects this situation nicely.
> If we want to know how far from neutral the position of
> treating all desserts equally is (that is, how far from the
> origin is the point (1, 1, ...., 1), we find that adding more
> desserts has less and less effect.  Which is what we'd want.
> We'd like the difference between plots of one and two
> desserts to be large, but the difference between 72 and 73 to
> be almost identical.  Simply creating an artificial
> disctinction between chocoloate and vanilla ice cream
> shouldn't change my fundamental position if all of the
> philosophers consider them to be of equal moral merit.
> 
> Just some thoughts.  Sorry it's so long and rambling.  I
> didn't have time to make it shorter.
> 
>     Tony
> 
> P.S.:  I just ran across this quote.  It's from the
> bibliography in "Loring Ensemble Rules", fetchable from
> http://member.aol.com/loringrbt/
> 
>     "Dimensional Analysis of Ranking Data". by Henry E. Brady
>     American Journal of Political Science. 34 (11/90).  Brady
>     found that a spatial model with 2 or 3 opinion dimensions
>     depicted a real electorate well.  Knowing a voter's
>     position on 1 issue helps predict his positions on
>     related issues in that dimension.  Knowing his 3D
>     position helps predict his ranking of candidates.
> 
> I haven't read the paper.
> 
>