[EM] Expected distance of winners, in n dimensions--I complicated it unnecessarily..

[Michael Ossipoff]

Oops! I've been talking about a situation in which the range of distance of
the candidates from you (but not counting the unwinnable ones you least
like).

That assumes that the range of voters can be judged by the range of
candidates.

I don't really believe that candidates & voters have the same distribution.
That's why I don't agree with the candidate merit mean Approval cutoff.

But it's unavoidable to at least assume that the range of candidates
indicates the range of voters, because what else is there to judge that by?

So anyway, I've been using R1 to stand for the distance to the nearest
candidate, and R2 to stand for the distance to the most distant candidate.

What's wrong with that? Well, the only reason to consider the range of
candidates is to estimate the range of voters. So what, really, should R1
be? Zero, because I'm not outside the range of voters, and the range of
_voters_ is what I really mean. The most distant (winnable) candidate is
needed to estimate how far the voters extend from me. But there's no
distance to the nearest voters from me.

That greatly simplifies the formula that I wrote for expected distance of
winner, in n dimensions. (I inferred it from the formulas for the distance
in 1, 2, & 3 dimensions).

What I posted was:

Expected R =

(n/(n+1)) (R2^(n+1) - R1^(n+1) )/(R2^n - R1^n)

But R1 should be 0, and so that can be written:

(n/(n+1) R.

So, for 1 dimension, (1/2) R
For 2 dimensions, (2/3) R
For 3 dimensions, (3/4) R

...etc.

I suggest that the expected distance (demerit) of the winner should be the
Approval cutoff, when the election is 0-info, and when there isn't reason
to believe that voters & candidates have the same distribution, so that
uniform voter-distribution is the best assumption. I suggest that that's
the case.

Evidently the dimensionality of our voter-&-candidate-space is 1, or is
best approximated by 1. But that might not be so in an authentic political
system.

Michael Ossipoff
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