[EM] methods based on cycle proof conditions

[fsimmons at pcc.edu]

A couple of other possibilities for methods based on cycle proof conditions:

I.  BDR or "Bucklin Done Right:"

Use 4 levels, say, zero through three.  First eliminate all candidates defeated
pairwise with a defeat ratio of 3 to 1.  Then collapse the top two levels, and
eliminate all candidates that suffer a defeat ratio of 2 to 1.  If any
candidates are left, among these elect the one with the greatest number of
positive ratings.

II.  SSCPE or "Six Slot Cycle Proof Elimination"

Use six levels, zero through five.  First eliminate all candidates with a
pairwise defeat ratio of five to one.  Then allowing only those ballot
comparisons with strength 2 or greater (i.e. the preferred candidate is rated at
least two levels above the other), eliminate all candidates with a defeat ratio
of two to one.  Then allowing only comparisons with strength three or greater,
eliminate all candidates beaten with a defeat ratio of one to one, i.e. all
defeated candidates.  If there are two or more undefeated candidates, elect the
one with the greatest number of positive ratings.

III.  SPE or Strong Preference Elimination:

Use 2n levels.  First eliminate all of the candidates that are defeated when the
only ballot preferences counted are of strength n or greater, i.e. the rating of
the preferred one is at least n levels greater than the rating of the other.  If
there are two or more unbeaten candidates, collapse the bottom three levels to
zero, decrement the other levels by two, and decrement 2n to 2(n-1) and repeat
the process recursively. 

The idea of SPE is that the most important eliminations are done by strong
preferences, and weaker preferences are invoked only to break ties.  More than
one tie breaker step is needed only to ensure the technical compliance with
Pareto.  Random ballot could be used as a tie breaker just as well where ever
voters are not allergic to it.