[EM] Sincere Zero Info Range

[fsimmons at pcc.edu]

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: election-methods at lists.electorama.com,

> 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.
>