Good points, Ross and Jameson. Section 4.3 of my dissertation (http://ajennings.net/dissertation.pdf) talks about this very thing.
The Ac-Bc rule was proposed by David Gale ( http://en.wikipedia.org/wiki/David_Gale) before his passing, to Balinski and Laraki directly. I proposed a rule very similar to Jameson's (Ac - Bc)/(Mc + |Ac - Bc|). Mine was (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.) 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. 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. Plus, 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. ~ Andy On Wed, Oct 19, 2011 at 9:12 AM, Jameson Quinn <[email protected]>wrote: > Great suggestion. I've been thinking along those lines, but I hadn't > expressed it as clearly. > > 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. > > 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. > > 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. > > Rc= Median rating for candidate c (expressed numerically; thus, letter > grades would be converted to grade-point-average numbers, etc.) > Mc= Number of median ratings for candidate c > Ac= Number of ratings above median for candidate c > Bc= Number of ratings below median for candidate c > |x| = standard notation for absolute value of x > > CMJ rating for c = Rc + ((Ac - Bc)/(Mc + |Ac - Bc|)) > > 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. > > 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. > > JQ > > > 2011/10/19 Ross Hyman <[email protected]> > >> 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? >> >> 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. >> >> Example using same example from Wikipedia's majority judgement entry: >> 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 >> 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. >> Nashville wins. >> >> >> >> >> >> >> >> >> ---- >> Election-Methods mailing list - see http://electorama.com/em for list >> info >> >> > > ---- > Election-Methods mailing list - see http://electorama.com/em for list info > >
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