I'm resending the message I sent to Kristofer because I think it's generally interesting. I redid the formula for an STV-like Range-based proportional system, and it's actually simpler than my previous (totally broken) formula. When electing candidate A, just multiply all the ballots by <meta http-equiv="content-type" content="text/html; charset=utf-8"><span class="Apple-style-span" style="font-size: large;">1-</span><span class="Apple-style-span" style="font-size: large;"><span class="Apple-style-span" style="font-size: small; "><span class="Apple-style-span" style="font-size: large; ">r</span>(<span class="Apple-style-span" style="font-size: x-small; ">A)</span></span>D/S</span><span class="Apple-style-span" style="font-size: x-small;">(A)</span> (unless it's negative), where<br>
<div><span class="Apple-style-span" style="font-size: large; ">r</span>(<span class="Apple-style-span" style="font-size: x-small; ">A)</span> = ballot score for candidate A</div><div><span class="Apple-style-span" style="font-size: large; ">D </span>= Droop quota</div>
<div><span class="Apple-style-span" style="font-size: large; ">S</span><span class="Apple-style-span" style="font-size: x-small; ">(A) </span>= Sum of squares of scores for candidate A</div><div><br></div><div>(detail - you need to either normalize the scores to [0,1], or multiply the droop quota by the top allowed score)</div>
<div><br></div><div>Note that my (still unpublished) summability trick can summably give results which are the same as this method with a high probability. (I developed the summability trick for approval ballots, but to use it for Range, you just divvy up the ballot into approval-style "slices" which approve all candidates which score higher than a given total. For instance, in range(3), A3 B1 C0 would be equivalent to 2 (A) ballots and one (AB) ballot.)</div>
<div><br></div><div>Kristofer - have you been able to get results for this formula? If you send me the source code, I can try myself.</div><div><br></div><div>JQ</div><div><br></div><div><div class="gmail_quote">---------- Forwarded message ----------<br>
From: <b class="gmail_sendername">Jameson Quinn</b> <span dir="ltr"><<a href="mailto:jameson.quinn@gmail.com">jameson.quinn@gmail.com</a>></span><br>Date: 2010/6/8<br>Subject: Re: [EM] Preliminary Range PAV results<br>
To: Kristofer Munsterhjelm <<a href="mailto:km-elmet@broadpark.no">km-elmet@broadpark.no</a>><br><br><br><br><br><div class="gmail_quote">2010/6/8 Kristofer Munsterhjelm <span dir="ltr"><<a href="mailto:km-elmet@broadpark.no" target="_blank">km-elmet@broadpark.no</a>></span><br>
<blockquote class="gmail_quote" style="border-left:1px solid rgb(204, 204, 204);margin:0pt 0pt 0pt 0.8ex;padding-left:1ex"><div class="im">
<div>Jameson Quinn wrote:<br>
<blockquote class="gmail_quote" style="border-left:1px solid rgb(204, 204, 204);margin:0pt 0pt 0pt 0.8ex;padding-left:1ex">
OK, I think there's a bug in this formula. Can you try max(0,(R/M-D(rN/R))/N)? M is the maximum Range vote (1, 99, 100, whatever); the minimum is 0.<br>
</blockquote></div>
<br></div><div class="im">
If you have<br>
10: A (0.9) > B (0.5) > C (0.1) <- set a<br>
5: B (0.7) > C (0.3) > A (0.2) <- set b<br>
<br>
and you're calculating the weighting for set a, is N 10 or is it 15?<br></div></blockquote><div><br>Thanks for giving me a concrete example. It inspired me to work the formula out again from the top. I'm sorry, I should not try to do it in my head.<br>
<br>Normalize all scores to [0,1] for notational simplicity.<br><br>Define S as the sum of the squares of the scores. S/R is the average score, weighted by support. For instance, if there are two groups x and y, which rate the given candidate at s and 0, then the weight of group y is 0, so S/R = s, independent of the group sizes.<br>
</div></div><br>Formula:<br><b>w = max(0,1-rD/S)</b><br>D->0, w->1, check.<br>r->0, w->1, check<br>for any p, D->pR and r->S/R, w ->1-p, check.<br><br>This is the formula I want. The only problem is that it might fail to take a full droop quota, even though one exists, because it is forbidden from going negative. This can only happen if D > S, or, more intuitively, D/R > S/R, that is, the ratio of the droop quota to the range total is above the average supporter's support.<br>
<br>In your example, for candidate A, set a, 2 candidates to be elected,<br>R=10<br>S=10*.81+5*.04=8.3<br>D=15/(2+1)=5<br>r=0.9<br>factor = .458. rescored a group A total, 4.120482<br><br>For voter set b, the rescored A total is .879518<br>
<br>New A total = 5 = R-D<br><font color="#888888"><br>JQ<br></font><br>
ps. Maybe you should share your source code on Sourceforge or Google Code or the like, so we could add our own systems?
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