One afterthought: Of all the cardinal ratings methods for vvarious values of p, the only one that satisfies the Favorite Betrayal Criterion (FBC) is the case of p=infinity, i.e. where the max absolute rating is limited, or equivalently, the scores are limited to some finite range, i.e. the case with which we are most familiar.
----- Original Message ----- From: Date: Friday, September 2, 2011 5:48 pm Subject: Sincere Zero Info Range To: [email protected], > Range voting is cardinal ratings with certain constraints on the > possible ratings, namely that they have to fall within a certain > interval or "range" of values, and usually limited to whole > number values. > Ignoring the whole number requirement, we could specify a > constraint for an equivalent method by simply limiting the > maximum of the absolute values of the ballot scores. Call this > "method infinity." > We could get another (non-equivalent) system by limiting the sum > of the absolute values of the scores. Call this "method one." > Yet another system is obtained by limiting the sum of the > squared values of the scores. Call this method two. > Other methods are obtained by limiting the sum of the p powers > of the absolute values of the scores. In thise scheme method two > corresponds to p=2, and methods infinity and one, respectively, > are the limits of method p as p approaches infinity or one. > Suppose that there are three candidates. Then graphically the > constraints for the three respective methods corresponding to p > equal to infinity, one, and two, turn out to be a cube, an > octahedron, and a ball with a perfectly spherical boundary, > respectively.The optimal strategies for methods infinity and one > generically involve ballots represented by corners of the cube > and octahedron, respectively. > In the case of method infinity, this means that all scores on a > strategically voted ballot will be at the extremes of the > allowed range, i.e. method infinity is strategically equivalent > to Approval. > In the case of method one, the corners represent the ballots > that concentrate the entire max sum value in one candidate, and > since negative scores are allowed, this method is strategically > equivalent to the method that allows you to vote for one > candidate or against one candidate but not both. I don't think > anybody has studied this method (Kevin has studied a different > method that allows you to vote for one candidate and against > another.), but in the case of only three candidates it is the > same as Approval. > The unit ball for method two has no corners or bulges (which all > other values of p involve), so the strategy is not so obvious. > But if Samuel Merrill is right, then in the zero information > case, the optimum strategy for method two is to vote > appropriately normalized sincere utilities. The appropriate > normalization is accomplished by subtracting the mean sincere > utility from the other utilities, and then dividing all of them > by their standard deviation. > In practice, the subtraction part is not necessary, because > adding the same constant to all of the ratings on the same > ballot makes no difference in the final outcome of a cardinal > ratings election. Note that this fact is the basis of one way > of seeing why methods infinity and one are strategically > equivalent in the case of only three candidates. > ---- Election-Methods mailing list - see http://electorama.com/em for list info
