[EM] Beatpath and SSD, Goldfish

[Markus Schulze]

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