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

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@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 > > > > > > > > >