There could be an error here, but it appears to me that CSSD & PC are both monotonic. CSSD is Cloneproof SSD, which is equivalent to BeatpathWinner. In BeatpathWinner, if I raise candidate X's position in my ranking, without changing the order of the others, I might increase X's defeat of someone, or I might decrease his defeat by someone, but that's the only way that I affect any defeats. If I decrease his defeat by someone, then I'm decreasing one or more beatpaths to him. If I increase his defeat of someone, then I'm increasing one or more beatpaths from him. Those are the only ways that I'm affecting beatpath strengths. Comparing X to some other candidate Y, then, I might be increasing X's strongest beatpath to Y, or weakening Y's strongest beatpath to X. In no way could that make Y have a beatpath win against X that Y didn't already have. So, by upranking X, without changing the orders of the other candidates in my ranking, I can't make X not win, if he'd previously won. With PC (Minimax), by upranking X without changing the order of the other candidates, I might decrease defeats of X, or I might be increasing defeats of other candidates. That won't increase X's strongest defeat, and it won't lessen someone else's strongest defeat. So if X's strongest defeat was less than anyone else's strongest defeat, before I upranked X, then it must still be. I don't call those proofs, and their hasty wording might not be as explicit as it should. But you know what I mean, and it could be worded more explicitly. Aren't those arguments valid & convincing? Any general definition of Monotonicity will be expected to result in a requirement that upranking someone without changing the order of the other candidates mustn't make him lose. Even though we don't have a general definition yet, it will have to be like that, in order to be accepted. MIke Ossipoff _________________________________________________________________ MSN Photos is the easiest way to share and print your photos: http://photos.msn.com/support/worldwide.aspx
