<br><br><div class="gmail_quote">2011/10/19 Andy Jennings <span dir="ltr"><<a href="mailto:elections@jenningsstory.com">elections@jenningsstory.com</a>></span><br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
<div>Good points, Ross and Jameson.</div><div><br></div><div>Section 4.3 of my dissertation (<a href="http://ajennings.net/dissertation.pdf" target="_blank">http://ajennings.net/dissertation.pdf</a>) talks about this very thing.</div>
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<br></div><div>The Ac-Bc rule was proposed by David Gale (<a href="http://en.wikipedia.org/wiki/David_Gale" target="_blank">http://en.wikipedia.org/wiki/David_Gale</a>) before his passing, to Balinski and Laraki directly. I proposed a rule very similar to Jameson's <span style="font-family:arial, sans-serif;font-size:13px;background-color:rgb(255, 255, 255)">(Ac - Bc)/(Mc + |Ac - Bc|). </span><span style="font-family:arial, sans-serif;font-size:13px;background-color:rgb(255, 255, 255)">Mine was </span><span style="font-family:arial, sans-serif;font-size:13px;background-color:rgb(255, 255, 255)">(Ac - Bc)/(2*Mc). Both are continuous everywhere (assuming fractional voters), even at the junction where the median changes from one grade to another. (See graphs in the pdf.) </span></div>
</blockquote><div><br></div><div>"Andy's" rule is simpler to state. "My" rule makes straighter lines, reduces to something more sensible in the case of approval, and is a bit better for stating the error on a poll. The two rules always give the same winner, so it doesn't really matter.</div>
<div> </div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;"><div><span style="font-family:arial, sans-serif;font-size:13px;background-color:rgb(255, 255, 255)">Also in my dissertation, I gave examples for the other two rules where a small change in the voter profile could cause a candidate to fall multiple rankings.</span></div>
<div><br></div><div>On the other hand, Balinski and Laraki's rule is constant with respect to either Ac or Bc almost everywhere. I think this might make it a little more resistant to strategic voting. </div></blockquote>
<div><br></div><div>I've explored this a bit - not rigorously, just enough to sharpen my intuition - and it does not seem to be true.</div><div> </div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
<div> Plus, <span style="font-family:arial, sans-serif;font-size:13px;background-color:rgb(255, 255, 255)">the remove-one-median-rating-at-a-time method has a certain simplicity and elegance to it, especially for very small electorates, even if it gets a little convoluted for large ones.</span></div>
<div><br></div><font color="#888888"><div>~ Andy</div></font><div><div></div><div class="h5"><div><br></div><br><br><div class="gmail_quote">On Wed, Oct 19, 2011 at 9:12 AM, Jameson Quinn <span dir="ltr"><<a href="mailto:jameson.quinn@gmail.com" target="_blank">jameson.quinn@gmail.com</a>></span> wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div>Great suggestion. I've been thinking along those lines, but I hadn't expressed it as clearly.</div><div>
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</div>And now that Ross has given me this idea, I can make it even simpler. Ross's suggested process is of course equivalent to, and harder to explain than, using (number above median grade)-(number below median grade) as a score. The only disadvantage of my version is that it could give negative numbers. But almost all people over the age of 10 (and a lot of people under that age) can handle negative numbers just fine, so I think that's OK.<div>
<br></div><div>This tiebreaker process is good. It will also tend to agree with the MJ one, as long as the tied candidates have approximately the same number of votes at the median grade - which will generally be true for two candidates whose strengths are similar enough to tie the median grade in the first place.</div>
<div><br></div><div>Here's another "tiebreaker" which I've developed. The advantage is that it gives a single real-number grade to each candidate, thus avoiding the issue of "ties" in the first place. I call it "Continuous Majority Judgment" or CMJ.</div>
<div><br></div><div>Rc= Median rating for candidate c (expressed numerically; thus, letter grades would be converted to grade-point-average numbers, etc.)</div><div>Mc= Number of median ratings for candidate c</div><div>
Ac= Number of ratings above median for candidate c</div>
<div>Bc= Number of ratings below median for candidate c</div><div>|x| = standard notation for absolute value of x</div><div><br></div><div>CMJ rating for c = Rc + ((Ac - Bc)/(Mc + |Ac - Bc|))</div><div><br></div><div>For approval (that is, binary ratings), the CMJ rating works out to be equal to the fraction of 1s, as you'd expect. Note that the adjustment factor is always in the range of -0.5 to 0.5, because the difference |Ac - Bc| can never be greater than Mc or it wouldn't be the median.</div>
<div><br>I prefer either of these methods to the MJ method - not for results, but for simplicity. (Ac - Bc) is simplest to explain, while CMJ is simplest to compare candidates / post results. All three of them should give the same results in almost all cases. But Balinski and Laraki preferred the remove-one-median-rating-at-a-time method because they could prove more theorems about it, and they wrote the MJ book, so until I write my own book about it I'm fine with promoting their method.<br>
<br>JQ</div><div><br></div><div><br><div class="gmail_quote">2011/10/19 Ross Hyman <span dir="ltr"><<a href="mailto:rahyman@sbcglobal.net" target="_blank">rahyman@sbcglobal.net</a>></span><br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
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<table cellspacing="0" cellpadding="0" border="0"><tbody><tr><td valign="top" style="font:inherit">It seems to me that there is a simpler way to compare candidates with the same median grade in majority judgement voting than the method described in the Wikipedia page for majority judgement. Why isn't this simpler way used? <br>
<br>Every voter grades every candidate. Elect the candidate with the highest median grade (the highest grade for which more than 50% of voters grade the candidate equal to or higher than that grade.) If there are two or more candidates with the same highest median grade, elect the candidate with the highest score of those with the highest median grade. A candidate's score is equal to the the number of voters that grade the candidate higher than the median grade plus the number of voters that grade to candidate equal to or higher than the median grade. This is equivalent to giving one point to
each candidate for each voter who grades the candidate its median grade and two points for each voter who grades the candidate higher than its median grade. Motivation: voters who vote median grade instead of something lower should increase the score for the candidate by the same amount as voters who vote above the median grade instead of equal to the median grade. With this scoring, going from less than median to median increases the candidate score by one point and going from median to higher than median also increases the candidate score by one point.<br>
<br>Example using same example from Wikipedia's majority judgement entry:<br>26% of voters grade Nashville as Excellent and 42% of voters grade Nashville as Good. Nashville's median grade is Good and its score is 26+26+42 = 94<br>
15% of voters grade Chattanooga as Excellent and 43% of voters grade Chattanooga as Good. Chattanooga's median grade is Good and its
score is 15+15+43 = 73.<br>Nashville wins.<br><br><br><br><br><br><br><br></td></tr></tbody></table><br></div></div>----<br>
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