[EM] More issue-dimensions. Answer to an objection.
Forest Simmons
fsimmons at pcc.edu
Sat Oct 29 16:53:16 PDT 2016
Michael,
Suppose that issue space is a sphere of any dimension and the candidates
are uniformly (and fairly densely) distributed within the sphere, with one
candidate C at or very near the center of the sphere..
Let X be the candidate that you like least, and Y be the candidate that you
like best.
Suppose that intuitively you rate these candidates at 100 percent and zero
respectively, and rate all other candidates by linear interpolation based
on their distances from you:.
score(Z)= (distY - dist Z)/(distY - distX) .
If you have an estimate of your distance r from the center C, then (under
the above assumption of intuitive rating based on issue space distance) the
center candidate will have a ratingon your ballot of 1/(1+r/R) , where R
is the radius of the sphere.
So for example if r/R is 50%, i.e. you are half way out from the center to
the boundary of the distribution, your rating of the CWs will be 1/1.5, or
2/3.
In that case you should approve every candidate that you rate above 66%.
If you are way out on the periphery, then r/R will be 1, and your rating of
the CW will be 1/(1+1) = 1/2, so you should approve all candidates that you
rate at or above 50%.
Since 1/(1+r/R) is never below 50%, you should never approve a candidate
below the midrange of your ballot ratings.
When I have more time I will show how to estimate r in three different ways:
(1) On the basis of what score s1 supporters of your most despised
candidate give your favorite.
(2) On the basis of how the supporters of the "most despised of your most
despised" score your favorite, say s2..
(3) On the basis of how you score s3 the most despised of your most
despised.
It turns out that on the basis of estimates (1) and (2), your rating of the
CW will be about 1/(1+s1 - s2).
Since 1>s1>s2>0, this value can never be smaller than 1/2.
On the basis of (3), your rating of the CW will be about 1 - (s3)/2, which
can never be smaller than 1/2 since s3 is never greater than 1.
So if you have information about s1, s2, and s3, you should approve every
candidate that you rate above the minimum of (1 - .5s3) and 1/(1 + s1 - s2)
The geometry is this: Your most despised A, and the most despised B of A's
supporters, are endpoints (on the periphery of the sphere) of diameter that
contains your position (near your favorite).
Your ratings of them and their ratings of your favorite are functions of
your distance from the center as a fraction of the radius R of the sphere.
The above estimates of your rating of the CW come from working out the
details, which I will give in a future post.
Forest
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