Condorect sub-cycle rule

[David Marsay]

Dear Markus,

      Thank you for your responses. You inspired me to look through 
the archives more thoroughly. I still don't understand the sub-cycle 
rule, though. First, do you have a reference to Tideman, MIIAC and 
GITC?

I wont attempt the original example. Here's a simpler one.

Consider options A,B,C,D, where A>B>>C>>>A, D>B>>C>>>D, and A~D, 
where more >s indicates a larger majority. Then ABCD is a cycle with 
symmetric sub-cycles ABC, DBC. Mike's sub-cycle rule that you quote 
allows to to pick either sub-cycle arbitrarily. You claim that this 
makes no difference.

The way I see it, if I start with the sub-cycle ABC I have 
A>B>>C>>>A, for which A>B is the weakest link, and the 
Condorcet-winner is B. Thus I delete A and C, leaving:
D>B, so D is the overall winner.

Now I repeat, but with the other sub-cycle. By symmetry, A is now the 
winner.

The only way around this that I can see is to look at the whole 
cycle.

I understand from a DEMOREP1 posting of 26 July 1996 that Young 
cites Condorcet as considering all cycles together. This gives B.

However, if one considers all cycles together, then a strong 
sub-cycle A>>>B>>>C>>A will effectively destroy all information of a 
lesser strength. This seems undesirable.

It seems to me that one needs to consider all sub-cycles of the same 
cycle together. This notion is given a meaning as follows:

Note that if a relation has cycles, then there exists a unique set of 
basis cycles. These are disjoint connected cycles, such that every 
cycle lies within a basis cycle.
(To construct, choose any cycle and put it into the set. For each 
remaining cycle, r, in turn, for each cycle s in the set that it 
intersects,  add s to r, and put r in the set).

Now, where the basic Condorcet procedure gives cycles, form the basis 
cycles. These correspond to Condorcet's "impossible opinions". For 
each basis cycle, find the least threshold T such that discounting 
all members of the cycle with Condorcet-plurality of less than T 
removes the cycles (creates possible opinions).

This sub-cycle rule is now rather like the one I 
posted earlier.
--------------------------------------------------
Sorry folks, but apparently I have to do this. :-(
The views expressed above are entirely those of the writer
and do not represent the views, policy or understanding of
any other person or official body.