[EM] Tanking advantage of cycle proof conditions

[fsimmons at pcc.edu]

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?