Suppose that the voters are willing and able to fill out Cardinal Ratings ballots in which each each candidate is rated on a scale of zero to 100. A pairwise matrix M can be filled out for each completed ballot as follows: The entry in the i_th row and j_th column M(i,j) is a one or a zero depending on whether or not this ballot has a higher rating for candidate i than for candidate j . The sum of all of these pairwise matrices can be used to find out if there is a "beats all winner" similar to the Condorcet Winner based on ranked ballots. Suppose there is no beats all winner (BAW). Then on each ballot divide all of the ratings by three and round to the nearest whole number to get a set of cruder ratings. Calculate the pairwise matrices based on these cruder ratings. If there is no BAW on the basis of these cruder ratings divide them by three and round again, etc. A BAW will be found before this process is repeated five times, because division by three and rounding four times will convert any number between zero and 100 into either a zero or a one. At that stage there will be a BAW, the candidate with the most ones. If it takes all four steps to reach a BAW, then the BAW winner is the same as the Approval winner where the approved candidates are the ones whose original scores were above 40 : 100 -> 33 -> 11 -> 4 -> 1 41 -> 14 -> 5 -> 2 -> 1 40 -> 13 -> 4 -> 1 -> 0 You can safely rate your lesser evil at 41 under this method. Note that this method is summable if you fill out four pairwise matrices for each ballot along with an approval count. The method comes very close to satisfying the Condorcet Criterion since any CW would almost surely turn out to be a BAW in the first step. It is also pretty obvious that the method satisfies the Favorite Betrayal Criterion. The method can be adapted to the zero to fifteen range ballot below by dividing the ratings by two and dropping the remainder at each stage. Equivalently, successively strike out the ratings columns from right to left. No more than three steps are necessary since the method reduces to Approval when only the first column remains. Forest Range Ballot for a Scale of Zero to Fifteen: Jose Blaze (8) (4) (2) (1) Sheila M. (8) (4) (2) (1) Jana P. Q. (8) (4) (2) (1) A. Ron Bla (8) (4) (2) (1) J. Q. Sten (8) (4) (2) (1) Sajh Amlkj (8) (4) (2) (1) Each candidate's score on this ballot is the sum of the digits shaded (with a number two pencil) to the right of the candidate's name (or alias). Any combination of shaded and unshaded digits produces a validly marked ballot. If none of the digits are shaded, then the score is zero for that candidate.
