[EM] Why SSD is a Cycle Condorcet method
MIKE OSSIPOFF
nkklrp at hotmail.com
Thu Feb 24 19:50:54 PST 2000
SSD is a Cycle Condorcet method if it can be shown that
any defeat among members of the current Schwartz set
(I'll just call it the Schwartz set) is in a cycle.
That's the same as saying that no member of the Schwartz set
has an unreturned defeat, meaning that if A beats B, then B
must have a beatpath to A.
Saying that if B is a member of the Schwartz set then B
doesn't have an unreturned defeat is the same as
saying that if B has an unreturned defeat then
B isn't in the Schwartz set. That's what I'll demonstrate:
***
The premise, then, is that A has a beatpath to B, but B doesn't
have a beatpath to A. It's desired to show that B isn't in
the Schwartz set.
Let S1 be the set of candidates including A and the candidates
who have a beatpath to A. Let S2 be the set of all the other
candidates in the election.
By the definition of a beatpath, if a candidate beats A or someone
who has a beatpath to A, then that candidate has a beatpath to A.
That means that if a candidate _doesn't_ have a beatpath to A,
then he doesn't beat A or anyone who has a beatpath to A.
That means that no S2 member beats any S1 member. That means
that S1 is an unbeaten set. S2 is not an unbeaten set, since
A beats B.
In fact, any set containing B but not containing A is not an
unbeaten set, because A beats B.
Now, say some set contains both A & B, and is an unbeaten set.
A subset of that set that leaves out B and any other S2 members
is also an unbeaten set, since no S2 member beats any S1 member.
Since that unbeaten set containing A & B therefore contains a
smaller unbeaten set, then the set containing A & B can't be
a "small unbeaten set", and therefore can't be part of the
Schwartz set. No set containing A & B can be part of the Schwartz
set. Since no set containing B & not A can be an unbeaten set
and no set containing A & B can be a small unbeaten set, then
no set containing B can be a small unbeaten set. So no set
containing B can be part of the Schwartz set. That's what
was intended to be proven.
Since a candidate with an unreturned defeat can't be in the
Schwartz set then any candidate in the Schwartz set must not
have an unreturned defeat. That means that any defeat
in the Schwartz set must be in a cycle. That's what I originally
said I'd prove.
That means that SSD never drops a candidate unless that candidate
is the weakest defeat in a cycle (since SSD only drops the
weakest defeat in the Schwartz set). So SSD is a Cycle Condorcet
method.
***
Mike Ossipoff
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