<div dir="ltr"><div><div><div><div><div><div>Forest--<br><br></div>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).<br><br></div>With that information, and with n issue-dimensions, I guess things can get a bit complicated.<br><br></div>Of course this is just my preliminary reply to your posting, I having just now found it.<br><br></div>This is more complicated a situation than what I was looking at, but this topic is interesting to me, even when it gets complicated.<br><br></div>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.<br><br></div>Michael Ossipoff<br></div><div class="gmail_extra"><br><div class="gmail_quote">On Sat, Oct 29, 2016 at 7:53 PM, Forest Simmons <span dir="ltr"><<a href="mailto:fsimmons@pcc.edu" target="_blank">fsimmons@pcc.edu</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="ltr"><div><div><div><div><div><div><div><div><div><div><div><div><div><div><div><div><div><div><div><div><br></div>Michael,<br><br></div>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..<br><br></div>Let X be the candidate that you like least, and Y be the candidate that you like best. <br><br>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:.<br><br></div>score(Z)= (distY - dist Z)/(distY - distX) .<br><br></div>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.<br></div><br></div>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.<br><br></div>In that case you should approve every candidate that you rate above 66%.<br><br></div>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%. <br><br></div>Since 1/(1+r/R) is never below 50%, you should never approve a candidate below the midrange of your ballot ratings.<br><br></div>When I have more time I will show how to estimate r in three different ways:<br><br></div>(1) On the basis of what score s1 supporters of your most despised candidate give your favorite.<br><br></div>(2) On the basis of how the supporters of the "most despised of your most despised" score your favorite, say s2..<br><br></div>(3) On the basis of how you score s3 the most despised of your most despised.<br><br></div>It turns out that on the basis of estimates (1) and (2), your rating of the CW will be about 1/(1+s1 - s2).<br><br></div><div>Since 1>s1>s2>0, this value can never be smaller than 1/2.<br></div><div><br></div>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.<br><br></div>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)<br><br></div>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).<br><br></div><div>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.<br><br></div><div>The above estimates of your rating of the CW come from working out the details, which I will give in a future post.<span class="HOEnZb"><font color="#888888"><br><br></font></span></div><span class="HOEnZb"><font color="#888888"><div>Forest<br></div><div><br></div><br></font></span></div></div>
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