A few observations about cycles and subgroups -- The very simple circular tie- N1 ABC N2 BCA N3 CAB Assume the N amounts produce A>B>C>A. The AB, BC and CA combinations are each in 2 of the 3 vote groups. By reversing the groups there is N4 CBA N5 ACB N6 BAC The AB/BA, BC/CB, CA/AC combinations are now in 4 of the 6 vote groups. In a larger example each choice (A, B, C) might be 2 or more subchoices-- such as A might be a D/E subgroup (especially simple clone subgroups); B might be a F/G/H subgroup, etc. It would appear that with 3 or more choices ALL election methods fail to meet one or more test standards (aka criteria) due to possible cycles and subgroups. What to do ??? How many test standards (if not ALL of them) regarding cycles and subgroups are irrelevant in getting a majority rule winner (for single winner elections) ??? How many test standards are in effect chicken-egg type standards ??? That is, certain cycle/subgroup data produces the test standard. Such test standard then causes other cycle/subgroup data to fail the test standard. I note again for the simple 3 choice case (with 3 or 6 vote groups), that if no choice gets a first choice majority and is not a Condorcet winner, then one of the choices will get a majority if the first plus second choice votes are combined (ignoring ties and truncated votes). The same tiebreaker can be used if there are 4 or more choices with no first choice majority and no Condorcet winner.
