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 defeats. 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. Mike Ossipoff _________________________________________________________________ Get your FREE download of MSN Explorer at http://explorer.msn.com/intl.asp.
