[EM] issue space revisited

[Forest Simmons]

On Thu, 14 Aug 2003, Alex Small wrote:

> Forest Simmons said:
> > 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).
>
> Could you explain a little more of Whitney's theorem?  I get the idea of
> what you're saying:  If we have n-dimensional issue space, having CR
> ballots with 2n+1 candidates (assume sufficient resolution on the ballots)
> would be enough to infer information on the distribution of voters in
> issue space.  However, I'd like to know more about the mathematical tool
> that justifies this statement.  I don't know what you mean by "accessing a
> space."

The key idea of Whitney's theorem is this: in Euclidean 2*n+1 dimensional
space, two n dimensional hyperplanes in general position will not
intersect each other. In other words, you wouldn't expect two randomly
chosen n dimensional hyperplanes to intersect each other in a 2*n+1
dimensional coordinate space.

For example, in three space you don't expect two randomly chosen lines to
intersect each other; you expect them to be skew lines.

If you take a simple closed curve in three space and give it an exact half
twist, it will form a figure eight, which is topologically distinct from a
simple closed curve, but a slight perturbation will remove the self
intersection.  The two intersecting tangent lines become skew.

RP2 (real projective two space) and the Klein bottle are two examples of
closed two dimensional manifolds that cannot be embedded (i.e. represented
without "self intersection") in three dimensional space.

[They both contain moebius bands, so they are non-orientable, and
therefore cannot separate space into inside and outside components, but
every closed manifold embedded in three-space must so separate it,
according to the generalized Jordan curve theorem.]

Whitney's theorem says that if we take our best three dimensional
representations of these two manifolds into five dimensional space, then
we can get rid of the self intersections as easily as we did for the
figure eight.

Note that a two dimensional plane in five dimensional coordinate space is
determined by three equations with five unknowns, leaving two degrees of
freedom.  The intersection of two such planes would be determined by six
equations with only five unknowns.  So you don't expect a common solution,
i.e. you don't expect the planes to intersect.


>
> For 1D issue space, 3 candidates would suffice, because we'd see that
> every voter is of the type A>B>C, C>B>A, B>A>C, or B>C>A.  However, 2
> candidates would not suffice, because all we'd really know is that
> everybody likes one or the other of them, which will always be true
> regardless of how complicated the issue space might be.
>
> What would be the necessary level of resolution on the CR ballots?  2n+1
> slots for 2n+1 candidates?  4n+2?

If we want to use Whitney's theorem rigorously (and non trivially) then we
must have infinite resolution ballots and infinitely many voters, since
every space with dimension greater than zero has infinitely many points.


In practice, however, the number of required slots grows slower than the
number of required candidates because the possible ballot combinations
grow so rapidly, and the distinct combinations distinguish the voters into
factions and sub-factions, which serve to outline the skeleton of issue
space.

To flesh it out, we connect the dots with line segments (i.e. 1-simplices)
and fill in the higher dimensional simplices to get a triangulation that
approximates our issue space.  The higher the resolution and the greater
the number of voters, the finer the mesh of the triangulation and the
better the approximation to the real issue space manifold.

>
> > 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."
>
> Count me as interested.

Great! As soon as I get through grading summer course finals, I'll start
another installment.

Forest