Warren suggested trying range ballots and seeking an approval equilbrium winner X such that X is the approval winner when each candidate rated above X is approved, each candidate rated below X is not approved, and each candidate rated equal to X (including X itself) gets half approval. This doesn't seem quite right when X has max or min range rating. It seems like you would want to give full or no approval, respectively, in these cases. Also, I would like to point out that in the case of no equal rankings X would be an equilibrium winner (according to Warren's rule) iff the max pairwise opposition against X was less than 50 percent of the ballots. Ordinal ballots are sufficient for this. I would like to suggest an alternative: instead of giving half approval to candidates ranked equal with X, give them full approval if X is above midrange, no approval if X is below midrange, and half approval only if X is right at midrange. Ordinal ballots with approval cutoff could be adapted to this variant. For a variant that requires range ballots consider the following: Give candidates that are rated equal with X partial approval equal to the percentage of the way X lies between the min range and max range values, i.e. the normalized range score for X. This would agree with the rule in my previous suggestion at the top, bottom, and middle, while interpolating the other positions naturally. If there is no equilibrium candidate, then the winner should be the one closest to equilibrium, i.e. the one for which the maximum difference in approval of Y and X is minimum when X is the approval cutoff, i.e. the sitting duck in one of the three variants above. If there are several equilibria, then the strongest one in the above sense should be chosen. Forest
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