James replying to Mike, trying to clear up some of the confusion about the immune set, the beatpath criterion, the beatpath winner set, and your majority rule definition.
Mike, you wrote: > >>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. > Markus wrote: >That's not correct. >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. > Mike, you wrote: >I was assuming that James' definition of Jobst's immune set was correct. >James defined it in a way that's the same as the set of candidates who >can >be elected without violating the Strong Beatpath Criterion. You say it's >the >Beatpath Criterion rather than the Strong Beatpath Criterion. Either >you're >wrong or James was. It doesn't matter. Jobst could clarify whether it's >you >or James who has correctly defined the immune set, but it isn't something >that I'd make an issue about. some background... Jobst wrote (May 1, 2004) "An option is *immune* here when each of its defeats is countered by a chain of stronger defeats leading back." I wrote (in my new definitions page) http://fc.antioch.edu/~james_green-armytage/vm/define.htm#immune "Immune set: The set of candidates such that every defeat against a candidate within the set is countered by a string of stronger defeats leading back to the defeating candidate. There is an immune set definition for each possible defeat strength definition, e.g. the WV-defined immune set, the margins-defined immune set, the CWP-defined immune set." You wrote http://lists.debian.org/debian-vote/2000/02/msg00016.html Beatpath Criterion (BC): If X beats Y, and if Y's strongest beatpath to X is weaker than X's defeat of Y, then Y shouldn't win. In summary, I don't think that Markus or I were incorrect in our definitions of the immune set. It seems that the immune set is equivalent to the set of candidates who can be elected according to the beatpath criterion. I believe that the error was when you (Mike) stated that the set of candidates who can be elected according to the beatpath criterion is equivalent to the beatpath (Schulze) method's winner set. Example (courtesy of Jobst, 5/1/04): A four candidate cycle between candidates A and B. There are six defeats, which are listed here in order of strength: AB>BD>CD>DA>AC>BC The immune set is {A, C}. The beatpath winner is candidate C. ... Now I'd like to re-evaluate my statement that your definition of majority rule is equivalent to the WV-defined immune set. Your definition (March 26, 2005) was as follows: >"Majority rule: >X has a majority pairwise vote (MPV) against Y if a majority vote X over >Y. >An MPV's strength is measured by the number of people who vote X over Y. >An MPV for X over Y is outdone if there's a sequence of MPVs from Y to X >consisting of MPVs that are all at least as strong as the one for X over >Y. >Violating majority rule means electing someone who has an MPV against him >that isn't outdone." To me, this seems identical to the WV-defined immune set or WV-defined beatpath, *except* for the complication caused by the distinction between a "majority pairwise vote" and a "non-majority" pairwise defeat. For example, if all pairwise defeats in a 3 candidate top cycle are "minority" beats of different strengths, then your majority rule set would probably include all candidates, whereas the WV-defined immune set would consist of a single candidate (the candidate with the weakest (as defined by WV) defeat against it. Is this right? So, it looks as though I was close in saying that the two are equivalent, but not 100% correct. Sincerely, James ---- Election-methods mailing list - see http://electorama.com/em for list info
