[EM] "Beatpath GMC" compliance a mistaken standard? (was "GMC compliance...")

[Chris Benham]

Marcus,
 
You wrote (29 Dec,2008):
 
"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. "


I can't see that the distinction between "allowed to win" and 
"not excluded from winning" is anything more than that between
"the glass is half full" and "the glass is half empty", so I reject your
semantic quibble. Any candidate that a criterion C doesn't exclude
from winning is (as far as C is concerned) "allowed to win".
 
"You didn't demonstrate that "the GMC concept is spectacularly
vulnerable to mono-add-plump". "
 
Well, I think I did. Perhaps you misunderstand my use of the 
word "concept".

Beatpath GMC says that the winner must come from a certain set
S, but a candidate X can fall out of  S if a relatively large number
of new ballots are added, all plumping (bullet-voting) for X.
 
Is there any other criterion with that absurd feature?

"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."

Yes, and I never meant to suggest otherwise.  In your previous post
you (referring to beatpath GMC as the "CDTT criterion") wrote:
 
"When Woodall's CDTT criterion is violated, then this
means that casting partial individual rankings could
needlessly lead to the election of a candidate B who
is not a Schwartz candidate; "needlessly" because
Woodall's CDTT criterion is compatible with the
Smith criterion, independence of clones, monotonicity,
reversal symmetry, Pareto, resolvability, etc.."

The Schwartz criterion doesn't imply beatpath GMC, so
by a "Schwartz candidate" you mean a '[presumed] sincere 
Schwartz candidate' instead of a  'voted Schwartz candidate'.
 
I don't accept that this stated aim is necessarily so desirable
partly because it isn't the case that (assuming sincere voting
and no strategic nominations) a Schwartz candidate is the
one that is mostly likely to be the SU winner (as evidenced by
my suggested "Comprehensive 3-slot Ratings Winner" criterion's
incompatibility with Condorcet).

Secondly I don't accept your suggestion that compliance with
beatpath GMC is acceptably cheap (let alone free) because it
isn't compatible with recently suggested "Smith- Comprehensive 3-slot
Ratings Winner" criterion, which I value much more.

In other words the CDTT set can fail to include the candidate that on
overwhelming common-sense (mostly positional) grounds is the strongest
candidate (e.g. C in "Situation # 2").
 
So given a method that meets what I've been recently calling "Strong
Minimal Defense"  (and so Minimal Defense and Plurality) and Schwartz
(and so fails LNHarm and meets Majority for Solid Coalitions), I consider
the addition of compliance with "beatpath GMC" a negative if without it the
method can meet "Smith- Comprehensive 3-slot Ratings Winner" (which
should be very very easy).


Chris Benham


 

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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