[EM] The voter median candidate generalized to multidimensional issue spaces.
Forest Simmons
fsimmons at pcc.edu
Wed Jan 8 17:42:32 PST 2003
Suppose that (in a certain election) candidate X is preferred over any
other candidate Y on any issue Z by some majority (depending on Y and Z).
Such a candidate would seem like a logical choice for winner of the
election, if there were such a candidate.
How could we locate such a candidate if there happened to be one?
Otherwise, how might one find the candidate closest to this ideal?
Here's an idea along these lines:
Form the matrix whose entry in row i and column j is the rank or rating of
candidate j by voter i.
Let V0 be the average of all the row vectors of the resulting matrix.
Subtract V0 from all of the rows of this matrix, and call this new matrix
A. Let B be the transpose of A.
Let V1, V2, etc. be normalized eigenvectors of the matrix product B*A, in
descending order of the magnitudes of the eigenvalues (until the
eigenvalues are too small to represent anything other than roundoff
error).
V0 is the center of gravity vector, and the other V's are the
principal axes of rotation, i.e. those axes (through the center of
gravity) about which the rigid body (with unit masses positioned by the
row vectors) can rotate without wobble.
Statistically, the eigenvectors give the uncorrelated directions in voter
space (the row space of A offset by the cg vector V0). They may be thought
of as the voter space shadows of the decorrelated issues, in order of
importance to the voters.
Sort (by numerical value) the components of the vector given by the
product A*V1 to find the median value c1.
Sort (by numerical value) the components of the vector given by the
product A*V2 to find the median value c2.
Etc.
Let X be the vector V0+c1*V1+c2*V2+...
Then X is the voter space position of the ideal candidate we are looking
for. In other words, if voter space were identical to issue space, then a
candidate positioned at X would be unbeaten on any of the (uncorrelated)
issues by any of the other candidates.
What do we do in the almost sure case that no candidate vector (image in
voter space) points in the direction of X ?
Here are two suggestions out of many possibilities:
(1) Find the candidate vector (image) that has the largest dot product
with X.
(2) Look at the components of X, and award the win to the candidate
corresponding to the largest component. In other words, let the
components of X order the candidates from winner to loser.
After experimenting I prefer the second approach for reasons which I will
explain if this becomes a hot topic.
Remark. I'm sure that this method doesn't satisfy the Local IIAC, because
each candidate, no matter how marginal in support contributes something to
the issue space profile. In other words, even the losing candidates help
to plumb the depths of issue space.
Any method that makes intentional, intelligent use of the issue space
information inherent in the distribution of votes in voter space should
not be concerned with satisfying any version of the IIAC, because the
voter response to each candidate helps outline the shape of issue space.
Sometimes I think that the best way to look at Arrow's Paradox is that the
IIAC was just one of those "good" ideas that didn't pan out. It was based
on a lack of imagination; nobody thought of using the profiles of the
losing candidates to help determine the relative positions of stronger
candidates in issue space.
On the other hand, intelligent issue space methods should automatically
get rid of clone problems, because clones just give redundant information
about the shape of issue space. The singular value decomposition
decorrelates whatever residual information there might be when the
redundancy is factored out, just as it finds uncorrelated linear
combinations of random variables in taxonomy problems associated with
measuring and recording various dimensions and features of related plants,
for example.
[In that context the SVD brings out the distinguishing features for
classification of related species. Similar uses are made in image
processing. If nostril width is directly proportional to ear lobe
thickness, the SVD will find that out automatically and count these two
measurements as only one dimension.]
Thanks for your patience in wading through these rather technical musings.
Forest
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