Consistent Cycles

[Mike Ositoff]

Hi--

I'd like to add something to what I said last night about
consistent cycles. It's true that each voter will not vote
at least one of the pair-orderings in the cycle, since each
voter's expressed preferences are assumed to be transitive
and he votes a ranking.

I'd said that he could decline to vote more than 1 of the
pair-orderings if he truncates, and that therefore the
sum, over all the pair orderings of the cycle, of the
percentage of the voters not voting that pair ordering
might be more than 1, and will have to be at least 1.

What I left out was the fact that another way the voter can
decline to vote a pair-ordering in the cycle would be to
vote the _opposite_ pair ordering. That will always happen
to some extent, of course. But either way, the voter will
decline to vote at least one of the cycle's pair orderings,
and that sum I named in the previous paragraph will therefore
always have to be at least 1, if the pairwise preference vote
totals in the cycle are to be consistent.

***

I want to add that my Tideman bad-example can of course be
made consistent if we say that there are 100 voters, and if
its majority defeats are 60, 61, & 62.   
65, 66, & 67 would be closer to the limit.

***

My example in my previous posting was intended to find out
for sure how Goldfish works. In that example, all of the
methods we like meet Beatpath GMC. Condorcet(EM) (Smith & plain)
meets both GMC versions in that example. I'm not saying that
means it's better.

Though Schulze's Beatpath GMC compliance seems to follow
from its definition, now I don't know for sure if Goldfish,
Norms procedure or SD can ever fail both GMC versions. Or
Tideman or Improved Tideman (Simultaneous Dropping).

***

Mike