2011/10/19 Andy Jennings <[email protected]> > 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.) >
"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. > 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. > I've explored this a bit - not rigorously, just enough to sharpen my intuition - and it does not seem to be true. > 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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