Dear Kevin, you wrote (7 Sep 2003): > Can anyone refer me to an explanation of why Schwartz > Sequential Dropping is the same thing as "elect the > candidate whose beatpaths to all other candidates are > stronger than the beatpaths back"? The reasoning for > the Schwartz limitation is clear to me (non-Schwartz > members are not useful beatpath nodes), but I am fuzzy > on how discarding weakest defeats obtains the beatpath > winner.
The strength of a beatpath is the strength of its weakest link. Let's say that "p[X,Y]" with X<>Y is the strength of the strongest beatpath from candidate X to candidate Y. Then p[A,B] > p[B,A] simply means that in the "Schwartz Sequential Dropping" heuristic the weakest link in the strongest beatpath from candidate B to candidate A is dropped before the weakest link in the strongest beatpath from candidate A to candidate B can be dropped. As soon as the weakest link in the strongest beatpath from candidate B to candidate A is dropped, candidate B has no chain to candidate A anymore while candidate A still has a chain to candidate B so that candidate A eliminates candidate B. Therefore, in the "Schwartz Sequential Dropping" heuristic candidate B is eliminated before candidate A can be eliminated. Markus Schulze ---- Election-methods mailing list - see http://electorama.com/em for list info
