[EM] contd, your method proposals

[Markus Schulze]

Dear Mike,

you wrote (21 Apr 2005):
> By the way, though someone has probably already mentioned
> this, the immune set, the set of candidates who could be
> elected without violating the Strong Beatpath Criterion
> (SBC) is the same as the winner set of BeatpathWinner.
> SBC is based on Steve's BC.

That's not correct.

Suppose d(z)[X,Y] is the strength of the pairwise
defeat "X vs. Y" when the strength is measured by "z"
(e.g. "z" = "margins", "z" = "winning votes").

Suppose p(z)[X,Y] is the strength of the strongest path
from candidate X to candidate Y when the strength of a
pairwise defeat is measured by "z".

Then Steve's beatpath criterion says:

   If d(z)[A,B] > p(z)[B,A] then candidate B must be
   elected with zero probability.

On the other side, the Schulze method says:

   If p(z)[A,B] > p(z)[B,A] then candidate B is not
   a potential Schulze winner.

The set of potential Schulze winners is a (not necessarily
strict) subset of the set of those candidates who can be
elected according to Steve's beatpath criterion.

What Jobst Heitzig calls "the immune set" is the set of
those candidates who can be elected according to Steve's
beatpath criterion. On the other side, "the winner set of
BeatpathWinner" is the set of potential Schulze winners.

Markus Schulze