Here's a four slot method that takes advantage of the impossibility of beat cycles under certain conditions:.
Use range style ballots with four levels: 0, 1, 2, and 3. (1) First eliminate all candidates that are pairwise defeated by a ratio greater than 3/1. (2) Then eliminate all of the candidates that are pairwise defeated by a ratio greater than 2/1 based on only those comparisons that involve an extreme rating, i.e. 3 beats a 0, 1, or 2, while 1, 2, or 3 beats a 0, but don't count a 2 as beating a 1, since neither 1 not 2 is an extreme rating on our four slot ballot. (3) Finally, eliminate all of the candidates that are pairwise defeated by any ratio greater than 1/1 on the basis of comparisons that involve a rating difference of at least two, i.e. 3 vs. 0 or 1, and 2 or 3 vs. 0, while considering 3 vs. 2, 2 vs. 1, and 1 vs. 0 to be too weak for this final elimination decision that is based on a mere 1/1 defeat ratio cutoff. The candidate that remains is the winner. If there is a pairwise tie in step three, use the middle two levels to resolve it, which is the same as electing the tied candidate with the greatest number of ratings strictly above one. None of the three elimination steps can eliminate all of the candidates because the elimination conditions are cycle proof. Furthermore, (with the tie breaker in place) the third step will eliminate all of the remaining candidates except one. Notice that the ballot comparisons get progressively stronger as we go from step one to step three, while the defeat ratio requirements get weaker, (from 3/1 to 2/1 to 1/1) but stay strong enough at each step to prevent cycles. Isn't that cool? ---- Election-Methods mailing list - see http://electorama.com/em for list info
