[EM] explicit & complete--why PC is monotonic too
nkklrp at hotmail.com
Tue Jan 15 20:05:50 PST 2002
Say the method is PC, and that you uprank X, without changing the
order in which you rank the other candidates. Say a number of other
people, voting the same as you, do the same.
For some Y over whom you now rank X, but whom you'd previously
ranked over X:
You may have reversed a YX defeat. That could stop Y from being
BeatsAll winner, or it could make X BeatsAll winner, if Y was his
only defeater. But it couldn't make Y BeatsAll winner or stop
X from being BeatsAll winner.
Reversing a YX defeat, giving Y a new defeat, could give Y a defeat
that's stronger than is previous strongest defeat, which could have
the result that Y is no longer the candidate whose greatest defeat
is the least.
By taking away a defeat that X had, that could be his strongest defeat,
and now his strongest defeat is the least of all the candidates'
strongest defeats, making X the winner.
But reversing a YX defeat couldn't stop X from being the winner.
But maybe X already beat Y, and by voting X over Y, you merely
strengten the XY defeat. The only way that could affect the outcome
is if it is or becomes Y's strongest defeat, and causes Y's strongest
defeat to no longer be the least of all the candidates' strongest
If Y previously beat X, and still does, after the change, then,
by no longer voting Y over X, you've weaked a defeat of X. The only
way that could affect the outcome would be if the YX defeat was
X's strongest defeat. Either that defeat is still X's strongest defeat,
or now another defeat is X's strongest defeat, but either way,
X's strongest defeat is weaker than it was before the change.
That, too, can't cause X to no longer have the weakest of all the
candidates' strongest defeats.
Unless I've missed something--and let me know if I have--it certainly
seems that you can't make X lose by upranking him without changing
the order of the other candidates in your ranking. PC is monotonic.
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