[EM] Sincere Zero Info Range

[fsimmons at pcc.edu]

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.
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