EM list: The Beatpath Criterion (BC) has been defined here. I'd like to point out that it can be also defined in terms of cycles instead of beatpaths: No one should win who has a pairwise defeat that isn't the weakest defeat in some cycle. (The strength of B's defeat by A is measured by how many people ranked A over B). *** When BC is worded that way, it's immediately obvious that the cycle Condorcet methdos meet BC. I've defined those methods as methods that solve circular ties by dropping defeats, and which won't drop a defeat unless it's the weakest defeat in some cycle. *** I should add that BC was defined before the Cycle Condorcet methods were defined. BC was defined because it is the essence of the defensive strategy criteria. It's easy to show why BC-compliance implies compliance with all 5 defensive strategy criteria. And it's also obvious why compliance with BC makes it easy to ensure that a candidate won't win if a majority don't want him to. Easy in the sense that they can do it without drastic measures like voting someone less-liked over someone more-liked. They can give him a majority defeat, and deny him a majority victory, ensuring that his defeat isn't the weakest defeat in any cycle. And if X majority-beats Y, and X is in the sincere Smith set, and Y isn't, then any cycle containing X & Y has to include a non-majority link, since no one outside the sincere Smith set can majority-beat a member of the sincere Smith set if no one votes unfelt preferences. So in that instance, Y's defeat can't be dropped, and no one needs strategy to achieve that result. Mike Ossipoff ______________________________________________________ Get Your Private, Free Email at http://www.hotmail.com