Access to issue space positions is the election methods designer's dream,
but direct access is impractical if not impossible.

However, those familiar with applications of Whitney's embedding theorem
know that there are indirect methods of accessing a space, namely through
one-to-one bi-continuous transformations (i.e. embeddings) of the space
into some coordinate space of sufficiently high dimension (twice the
dimension of the space plus one is sufficient, according to Whitney's
theorem).

[Delay coordinates in chaos theory come to mind as a good example of
an application of Whitney's theorem.]

In the context of election methods, suppose that the issue space is three
dimensional, i.e. there are three major issue axes along which all of the
various issues tend to align themselves.  [Any deviations into a fourth
dimension would be relatively small compared to the variation along the
three principal axes.]

Then according to Whitney's embedding theorem, it would take no more than
a 2*3+1=7 dimensional coordinate system to get a faithful image of this
issue space.

If there are seven or more (non clone) candidates, then the ratings of the
candidates on CR ballots could well serve as such a coordinate system for
locating the images of the voters in the image space. [The marked ballots
are the "images" of the voters who marked them.]

In other words, if the issue space is three dimensional, and there are
seven or more candidates, then the greater the similarity between two
marked ballots, the closer their two voters are positioned in issue
space.

[The converse is true even if there are fewer than Whitney's required
number of coordinates; the closer two voters are positioned in issue
space, the more similar their ballots.]

For a one or two dimensional issue space, a 2*2+1=5 dimensional coordinate
system would suffice, so five non-clone candidates would suffice to
discern the shape of the space ... in particular to discern whether the
space was one or two dimensional.

A good example is Adam's recent far-left, left, centrist, right, far-right
election ballot summary in a posting challenging the IRV supporters to
justify IRV's choice.  Even if the names of the candidates didn't give it
away, the one dimensional shape of the issue space could be easily
inferred from the ballots.

Next time (if there is even a particle of interest)... "How to Take
Advantage of the Correspondence Between Ballot Space and Issue Space When
Designing and Testing Election Methods."

Forest

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