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

[Michael Ossipoff]

In the 2nd-to-last paragraph of my previous post, I left out a when-clause.
That paragraph should read:

I suggest that the expected distance (demerit) of the winner should be the
Approval cutoff, when there is no top-set or bottom-set, and when the
election is 0-info, and when there isn't reason to believe that voters &
candidates have the same distribution, so that a uniform voter-distribution
is the best assumption.

Michael Ossipoff


On Wed, Nov 2, 2016 at 12:38 AM, Michael Ossipoff <email9648742 at gmail.com>
wrote:

> 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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>
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