Here's the strongest argument that I can think of in defense of giving candidates who are rated precisely at the approval cutoff level exactly half approval:
The problem is that the CR values are discrete. Suppose that the allowed values are the whole numbers between zero and ten. In this case the CR value "six" really means six plus or minus one half, since voters cannot "vote in the cracks" so to speak. If we assume that half of the ratings of "6" reflect preferences in the upper range of this interval (between 6 and 6.5) and the other half in the lower half of the interval, then it makes perfect sense to give half approval for a candidate with a CR of 6 when the approval cutoff is precisely 6. This brings me to a proposal that I have been toying with for a long time. When the CR resolution is small or when there are apt to be a significant percentage of voters with approval cutoffs near the same candidates (as in the case of voters copying "candidate cards" or party voting guides, not to mention typical EM list examples), a small shift in the approval cutoff can make a dramatic change in the distribution of total approval among the candidates. This discontinuity is the source of most (if not all) of the erratic behavior observed in some of the methods that focus on refining the approval cutoff. [In the case of "Auntie" it is the only discontinuity. Other methods have have additional discontinuities. GIA has a discontinuity, for example, that stems from going (in one step) from three to two candidates with equal weight (in the approval cutoff calculation).] We can overcome this main source of discontinuity by considering ratings on CR ballots to be intervals of uniformly distributed values, rather than discrete point values. However, I propose that the two extreme values retain their point status for two reasons: (1) They do not contribute to the type of discontinuity we are considering since no approval cutoff moves past either of them. (2) In the methods under consideration it is important to respect the wishes of voters voting only at the extremes. As an example, suppose that we humor the sociologists by going with a seven slot CR ballot. The two extreme slots would count as point values of zero and 100 percent, respectively. The other five slots would be intervals of width 20% centered at the values 10%, 30%, 50%, 70%, and 90%. Suppose that the approval cutoff is computed to be 70% on some ballot where one of the candidates has been rated "slightly above average" corresponding to "somewhere in the interval between 60% and 80%." As discussed above, this ballot should contribute a fraction of one half to the approval of that candidate. Now suppose that in a similar case the approval cutoff turned out to be 72% with nothing else changed. Wouldn't it make sense to have this ballot contribute a fraction of 8/20 to the approval of that candidate? In general, if the cutoff c is a fraction p/q of the way from the top of an interval to the bottom of the interval, then any candidate rated in that slot should get a contribution of p/q from that ballot. In more detail ... if the interval in question is [a,b], and cutoff c is a number in this interval, then the fraction p/q is given by p/q = (b-c)/(b-a) . It is easy to see that the value of p/q varies from zero to one as c moves from top to bottom of the interval, with 1/2 at the center of the interval. In the other example above we have a = 60%, b = 80%, and c = 72%, so p/q = (80-72)/(80-60) = 8/20 . In the rare case that an approval cutoff c turns out to be precisely at the upper or lower extreme of the CR range (one of the two point values zero or 100 Percent), then the extreme rated candidates should still get either full or no approval, depending on which extreme because this convention does not violate continuity, and is probably consistent with most voter wishes. Forest ---- Election-methods mailing list - see http://electorama.com/em for list info