[EM] GMC compliance a mistaken standard? (was "CDTT criterion"...)
Markus Schulze
markus.schulze at alumni.tu-berlin.de
Mon Dec 29 13:22:35 PST 2008
Dear Chris Benham,
you wrote (29 Dec 2008):
> The "Generalised Majority Criterion" says in effect that
> the winner must come from Woodall's CDTT set, and is
> defined by Markus Schulze thus (October 1997):
>
> > Definition ("Generalized Majority Criterion"):
> >
> > "X >> Y" means, that a majority of the voters prefers
> > X to Y.
> >
> > "There is a majority beat-path from X to Y," means,
> > that X >> Y or there is a set of candidates
> > C[1], ..., C[n] with X >> C[1] >> ... >> C[n] >> Y.
> >
> > A method meets the "Generalized Majority
> > Criterion" (GMC) if and only if:
> > If there is a majority beat-path from A to B, but
> > no majority beat-path from B to A, then B must not
> > be elected.
>
> With full strict ranking this implies Smith, and obviously
> "Candidates permitted to win by GMC (i.e.CDTT), Random
> Candidate" is much better than plain Random Candidate.
> Nonetheless I think that compliance with GMC is a mistaken
> standard in the sense that the best methods should fail it.
>
> The GMC concept is spectacularly vulnerable to Mono-add-Plump!
>
> [Situation #1]
>
> 25: A>B
> 26: B>C
> 23: C>A
> 04: C
> 78 ballots (majority threshold = 40)
>
> B>C 51-27, C>A 53-25, A>B 48-26.
>
> All three candidates have a majority beat-path to each other,
> so GMC says that any of them are allowed to win.
>
> [Situation #2]
>
> But say we add 22 ballots that plump for C:
>
> 25: A>B
> 26: B>C
> 23: C>A
> 26: C
> 100 ballots (majority threshold = 51)
>
> B>C 51-49, C>A 75-25, A>B 48-26.
>
> Now B has majority beatpaths to each of the other candidates
> but neither of them have one back to B, so the GMC says that
> now the winner must be B.
>
> The GMC concept is also naturally vulnerable to Irrelevant
> Ballots. Suppose we now add 3 new ballots that plump for an
> extra candidate X.
>
> [Situation #3]
>
> 25: A>B
> 26: B>C
> 23: C>A
> 26: C
> 03: X
> 103 ballots (majority threshold = 52)
>
> Now B no longer has a majority-strength beat-path to C,
> so now GMC says that C (along with B) is allowed to win
> again.
>
> (BTW this whole demonstration also applies to "Majority-Defeat
> Disqualification"(MDD) and if we pretend that the C-plumping
> voters are truncating their sincere preference for B over A
> then it also applies to Eppley's "Truncation Resistance"
> and Ossipoff's SFC and GFSC criteria.)
Several versions of the "Generalized Majority Criterion" (GMC)
have been discussed at the Election Methods mailing list in
the past. Therefore, I recommend that you should use the term
"beatpath GMC" for my 1997 proposal to distinguish it from
the other proposals.
Your argumentation is incorrect. Example:
In many scientific papers, the Smith set is criticized
because the Smith set can contain Pareto-dominated
candidates. However, to these criticisms I usually
reply that the fact, that the Smith criterion doesn't
imply the Pareto criterion, is not a problem as long
as the used tie-breaker guarantees that none of these
Pareto-dominated candidates is elected. It would be
a problem only if the Smith criterion and the Pareto
criterion were incompatible.
You made the same mistake as the authors of these papers.
You didn't demonstrate that "the GMC concept is spectacularly
vulnerable to mono-add-plump". You only demonstrated that
beatpath GMC doesn't imply mono-add-plump.
However, the fact, that Schulze(winning votes) satisfies
mono-add-plump and always chooses from the CDTT set and
isn't vulnerable to irrelevant ballots, shows that these
properties are not incompatible.
In all three situations, Schulze(winning votes) chooses
candidate B. Therefore, you demonstrated neither a
"spectacular failure of mono-add-plump" nor a "vulnerability
to irrelevant ballots" for methods that satisfy beatpath GMC.
You wrote: "All three candidates have a majority beatpath
to each other, so GMC says that any of them are allowed to
win." No! Beatpath GMC doesn't say that "any of them are
allowed to win"; beatpath GMC only doesn't exclude any of
them from winning. Similarly, the Smith criterion doesn't
say that even Pareto-dominated candidates must be allowed
to win; that would have meant that the Smith criterion and
the Pareto criterion were incompatible; the Smith criterion
only doesn't imply the Pareto criterion.
Markus Schulze
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