[EM] "Beatpath GMC" compliance a mistaken standard? (was "GMC compliance...")
Markus Schulze
markus.schulze at alumni.tu-berlin.de
Thu Jan 8 08:46:46 PST 2009
Dear Chris Benham,
you wrote (29 Dec 2008):
> 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.)
I wrote (29 Dec 2008):
> 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.
You wrote (8 Jan 2009):
> 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".
Statement #1: Criterion X does not imply criterion Y.
Statement #2: Criterion X and criterion Y are incompatible.
Statement #1 does not imply statement #2. But in your
29 Dec 2008 mail, you mistakenly assume that statement #1
implies statement #2.
Example:
X = Smith criterion.
Y = Pareto criterion.
Then statement #1 is true and statement #2 is false.
The fact, that statement #1 does not imply statement #2,
is not "semantic quibble".
You proved only that beatpath GMC does not imply mono-add-plump;
but then you mistakenly concluded that this means that beatpath
GMC and mono-add-plump were incompatible ("spectacularly
vulnerable to mono-add-plump", "spectacular failure of
mono-add-plump"). However, the fact, that Schulze(winning votes)
satisfies beatpath GMC and mono-add-plump, demonstrates that
these two criteria are not incompatible.
Example:
The Smith criterion does not imply the Pareto criterion; that
means that it can happen that a Pareto-dominated candidate is
not excluded from winning by the Smith criterion. However,
this doesn't mean that the Smith criterion implies that even
Pareto-dominated candidates must be allowed to win.
************
You wrote (8 Jan 2009):
> 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?
Your argumentation doesn't make any sense.
Example:
The fact, that the Borda method satisfies monotonicity
and violates independence of clones, demonstrates that
monotonicity doesn't imply independence of clones. You
rejected beatpath GMC because it doesn't imply mono-add-plump.
But with the same logic, you could reject the "concept"
of monotonicity for the "absurd feature" of not implying
independence of clones or for being "spectacularly
vulnerable" to clones.
The fact, that Schulze(winning votes) satisfies beatpath GMC
and mono-add-plump, demonstrates that these two criteria are
not incompatible. But you claim that already the fact, that
beatpath GMC doesn't imply mono-add-plump, was an "absurd
feature" of beatpath GMC.
Markus Schulze
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