[EM] More issue-dimensions. Answer to an objection.

[Michael Ossipoff]

Forest--

That's a strategy for when you have some information about who or where the
CW is, where the voter-median is, or some information about other voters'
ratings (...which could be available from the previous election).

With that information, and with n issue-dimensions, I guess things can get
a bit complicated.

Of course this is just my preliminary reply to your posting, I having just
now found it.

This is more complicated a situation than what I was looking at, but this
topic is interesting to me, even when it gets complicated.

I was just assuming 0-info, no knowledge about the other voters, the CW, or
the voter-median (other than what can be inferred about the approximate
merit-range of what the voters want, inferred from the range of
candidate-merits). So I was making it simpler. But I'm interested in every
aspect of Approval-strategy, including the not-0-info situation you discuss.

Michael Ossipoff

On Sat, Oct 29, 2016 at 7:53 PM, Forest Simmons <fsimmons at pcc.edu> wrote:

>
> 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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>
>
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