[EM] issue space revisited

[Alex Small]

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."

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?

> 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.



Alex