[EM] Explicit & complete--Why CSSD is monotonic
nkklrp at hotmail.com
Tue Jan 15 19:53:10 PST 2002
My previous letter showing why BeatpathWinner, Cloneproof SSD,
& PC are monotonic was hurried, and not as complete & explicit as
it should have been. So now I'm saying it in a more complete way:
Say the method is BeatpathWinner, and you uprank X without changing
the order of the other candidates in your ranking. How can that
When you rank X over some other candidate Y, whom you'd previously
ranked over X, there are several ways that can affect a beatpath:
You could reverse X's defeat by Y. That could eliminate a beatpath
to X, or it could create a beatpath from X.
That couldn't cause X to no longer have a beatpath win
over another candidate.
Of course it could also make X the BeatsAll winner, or stop Y from
being BeatsAll winner. That couldn't make X lose.
Or, if X already beats Y, then newly ranking X over Y increases
the magnitude of the XY defeat, and could thereby strengthen a
beatpath from X to another candidate (which could be Y).
Of, if Y beats X, but your XY reversal doesn't reverse that defeat,
the fact that you're no longer voting Y over X weakens the magnitude
of that defeat, and thereby could weaken a beatpath to X.
That, too, can't cause X to no longer have a beatpath win against
Can you tell me some way in which upranking X without changing the
order of the other candidates could make another candidate the
BeatsAll winner, stop X from being BeatsAll winner, or increase the
strength of a beatpath to X, or decrease the strength of a beatpath
from X? Can you tell me how upranking X could cause X to no longer
have a beatpath win against another candidate? If it can't cause
those results then BeatpathWinner is monotonic. Cloneproof SSD is
equivalent to BeatpathWinner, and so it's monotonic.
If anyone has an example in which BeatpathWinner or CSSD is nonmonotonic,
then will you post it?
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