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<P>Marcus,</P>
<P> </P>
<P>You wrote (29 Dec,2008):</P>
<P> </P>
<P>"You wrote: "All three candidates have a majority beatpath<BR>to each other, so GMC says that any of them are allowed to<BR>win." No! Beatpath GMC doesn't say that "any of them are<BR>allowed to win"; beatpath GMC only doesn't exclude any of<BR>them from winning. "<BR><BR></P>
<P>I can't see that the distinction between "allowed to win" and </P>
<P>"not excluded from winning" is anything more than that between</P>
<P>"the glass is half full" and "the glass is half empty", so I reject your</P>
<P>semantic quibble. Any candidate that a criterion C doesn't exclude</P>
<P>from winning is (as far as C is concerned) "allowed to win".</P>
<P> </P>
<P>"You didn't demonstrate that "the GMC concept is spectacularly<BR>vulnerable to mono-add-plump". "</P>
<P> </P>
<P>Well, I think I did. Perhaps you misunderstand my use of the </P>
<P>word "concept".<BR></P>
<P>Beatpath GMC says that the winner must come from a certain set</P>
<P>S, but a candidate X can fall out of S if a relatively large number</P>
<P>of new ballots are added, all plumping (bullet-voting) for X.</P>
<P> </P>
<P>Is there any other criterion with that absurd feature?</P>
<P><BR>"However, the fact, that Schulze(winning votes) satisfies<BR>mono-add-plump and always chooses from the CDTT set and<BR>isn't vulnerable to irrelevant ballots, shows that these<BR>properties are not incompatible."<BR></P>
<P>Yes, and I never meant to suggest otherwise. In your previous post</P>
<P>you (referring to beatpath GMC as the "CDTT criterion") wrote:</P>
<P> </P>
<P>"When Woodall's CDTT criterion is violated, then this<BR>means that casting partial individual rankings could<BR>needlessly lead to the election of a candidate B who<BR>is not a Schwartz candidate; "needlessly" because<BR>Woodall's CDTT criterion is compatible with the<BR>Smith criterion, independence of clones, monotonicity,<BR>reversal symmetry, Pareto, resolvability, etc.."<BR></P>
<P>The Schwartz criterion doesn't imply beatpath GMC, so</P>
<P>by a "Schwartz candidate" you mean a '[presumed] sincere </P>
<P>Schwartz candidate' instead of a 'voted Schwartz candidate'.</P>
<P> </P>
<P>I don't accept that this stated aim is necessarily so desirable</P>
<P>partly because it isn't the case that (assuming sincere voting</P>
<P>and no strategic nominations) a Schwartz candidate is the</P>
<P>one that is mostly likely to be the SU winner (as evidenced by</P>
<P>my suggested "Comprehensive 3-slot Ratings Winner" criterion's</P>
<P>incompatibility with Condorcet).</P>
<P><BR>Secondly I don't accept your suggestion that compliance with</P>
<P>beatpath GMC is acceptably cheap (let alone free) because it<BR>isn't compatible with recently suggested "Smith- Comprehensive 3-slot</P>
<P>Ratings Winner" criterion, which I value much more.<BR></P>
<P>In other words the CDTT set can fail to include the candidate that on<BR>overwhelming common-sense (mostly positional) grounds is the strongest</P>
<P>candidate (e.g. C in "Situation # 2").</P>
<P> </P>
<P>So given a method that meets what I've been recently calling "Strong<BR>Minimal Defense" (and so Minimal Defense and Plurality) and Schwartz</P>
<P>(and so fails LNHarm and meets Majority for Solid Coalitions), I consider</P>
<P>the addition of compliance with "beatpath GMC" a negative if without it the</P>
<P>method can meet "Smith- Comprehensive 3-slot Ratings Winner" (which</P>
<P>should be very very easy).</P>
<P><BR><BR>Chris Benham</P>
<P><BR><BR> </P>
<P><BR>Dear Chris Benham,<BR><BR>you wrote (29 Dec 2008):<BR><BR>><I> The "Generalised Majority Criterion" says in effect that<BR></I>><I> the winner must come from Woodall's CDTT set, and is<BR></I>><I> defined by Markus Schulze thus (October 1997):<BR></I>><I><BR></I>><I> > Definition ("Generalized Majority Criterion"):<BR></I>><I> ><BR></I>><I> > "X >> Y" means, that a majority of the voters prefers</P></I>><I> > X to Y.<BR></I>><I> ><BR></I>><I> > "There is a majority beat-path from X to Y," means,<BR></I>><I> > that X >> Y or there is a set of candidates<BR></I>><I> > C[1], ..., C[n] with X >> C[1] >> ... >> C[n] >> Y.<BR></I>><I> ><BR></I>><I> > A method meets the "Generalized Majority<BR></I>><I> > Criterion" (GMC) if and
only if:<BR></I>><I> > If there is a majority beat-path from A to B, but<BR></I>><I> > no majority beat-path from B to A, then B must not<BR></I>><I> > be elected.<BR></I>><I><BR></I>><I> With full strict ranking this implies Smith, and obviously <BR></I>><I> "Candidates permitted to win by GMC (i.e.CDTT), Random<BR></I>><I> Candidate" is much better than plain Random Candidate.<BR></I>><I> Nonetheless I think that compliance with GMC is a mistaken<BR></I>><I> standard in the sense that the best methods should fail it.<BR></I>><I><BR></I>><I> The GMC concept is spectacularly vulnerable to Mono-add-Plump!<BR></I>><I><BR></I>><I> [Situation #1]<BR></I>><I><BR></I>><I> 25: A>B<BR></I>><I> 26: B>C<BR></I>><I> 23: C>A<BR></I>><I> 04: C<BR></I>><I> 78 ballots (majority threshold = 40)<BR></I>><I><BR></I>><I> B>C
51-27, C>A 53-25, A>B 48-26.<BR></I>><I><BR></I>><I> All three candidates have a majority beat-path to each other,<BR></I>><I> so GMC says that any of them are allowed to win.<BR></I>><I><BR></I>><I> [Situation #2]<BR></I>><I><BR></I>><I> But say we add 22 ballots that plump for C:<BR></I>><I><BR></I>><I> 25: A>B<BR></I>><I> 26: B>C<BR></I>><I> 23: C>A<BR></I>><I> 26: C<BR></I>><I> 100 ballots (majority threshold = 51)<BR></I>><I><BR></I>><I> B>C 51-49, C>A 75-25, A>B 48-26.<BR></I>><I><BR></I>><I> Now B has majority beatpaths to each of the other candidates<BR></I>><I> but neither of them have one back to B, so the GMC says that<BR></I>><I> now the winner must be B.<BR></I>><I><BR></I>><I> The GMC concept is also naturally vulnerable to Irrelevant<BR></I>><I> Ballots. Suppose we now add 3 new ballots that plump for
an<BR></I>><I> extra candidate X.<BR></I>><I><BR></I>><I> [Situation #3]<BR></I>><I><BR></I>><I> 25: A>B<BR></I>><I> 26: B>C<BR></I>><I> 23: C>A<BR></I>><I> 26: C<BR></I>><I> 03: X<BR></I>><I> 103 ballots (majority threshold = 52)<BR></I>><I><BR></I>><I> Now B no longer has a majority-strength beat-path to C,<BR></I>><I> so now GMC says that C (along with B) is allowed to win<BR></I>><I> again.<BR></I>><I><BR></I>><I> (BTW this whole demonstration also applies to "Majority-Defeat<BR></I>><I> Disqualification"(MDD) and if we pretend that the C-plumping<BR></I>><I> voters are truncating their sincere preference for B over A<BR></I>><I> then it also applies to Eppley's "Truncation Resistance"<BR></I>><I> and Ossipoff's SFC and GFSC criteria.)<BR></I><BR>Several versions of the "Generalized Majority Criterion" (GMC)<BR>have been discussed at the Election Methods mailing list in<BR>the
past. Therefore, I recommend that you should use the term<BR>"beatpath GMC" for my 1997 proposal to distinguish it from<BR>the other proposals.<BR><BR>Your argumentation is incorrect. Example:<BR><BR> In many scientific papers, the Smith set is criticized<BR> because the Smith set can contain Pareto-dominated<BR> candidates. However, to these criticisms I usually<BR> reply that the fact, that the Smith criterion doesn't<BR> imply the Pareto criterion, is not a problem as long<BR> as the used tie-breaker guarantees that none of these<BR> Pareto-dominated candidates is elected. It would be<BR> a problem only if the Smith criterion and the Pareto<BR> criterion were incompatible.<BR><BR>You made the same mistake as the authors of these papers.<BR>You didn't demonstrate that "the GMC concept is spectacularly<BR>vulnerable to mono-add-plump". You only
demonstrated that<BR>beatpath GMC doesn't imply mono-add-plump.<BR><BR>However, the fact, that Schulze(winning votes) satisfies<BR>mono-add-plump and always chooses from the CDTT set and<BR>isn't vulnerable to irrelevant ballots, shows that these<BR>properties are not incompatible.<BR><BR>In all three situations, Schulze(winning votes) chooses<BR>candidate B. Therefore, you demonstrated neither a<BR>"spectacular failure of mono-add-plump" nor a "vulnerability<BR>to irrelevant ballots" for methods that satisfy beatpath GMC.<BR><BR>You wrote: "All three candidates have a majority beatpath<BR>to each other, so GMC says that any of them are allowed to<BR>win." No! Beatpath GMC doesn't say that "any of them are<BR>allowed to win"; beatpath GMC only doesn't exclude any of<BR>them from winning. Similarly, the Smith criterion doesn't<BR>say that even Pareto-dominated candidates must be allowed<BR>to win; that would have meant that the Smith criterion and<BR>the
Pareto criterion were incompatible; the Smith criterion<BR>only doesn't imply the Pareto criterion.<BR><BR>Markus Schulze<BR></DIV></div><br>
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