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

Chris Benham cbenhamau at yahoo.com.au
Mon Dec 29 10:43:03 PST 2008

The  "Generalised Majority Criterion" says in effect that the winner
must come from Woodall's CDTT set, and is defined by Marcus 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!

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. 

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 51-27,   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.

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 trucating their sincere preference for B
over A then it also applies to Eppley's "Truncation Resistance" and Ossipoff's SFC and
GFSC criteria.)

If  the method uses 3-slot ratings ballots and we assume that the voted 3-slot ratings are
sincere, then the GMC can bar the plainly highest SU candidate from winning as evidenced
by its incompatibility with my recently suggested  "Smith-Comprehensive 3-slot Ratings
Winner" criterion:

*If no voter expresses more than three preference-levels and the ballot 
rules allow the expression of 3 preference-levels when there are 3 (or 
more) candidates, then (interpreting candidates that are voted above one
or more candidates and below none as "top-rated", those voted above
one or more candidates but below all the top-rated candidates as 
"middle-rated" and those not voted above any other candidate and below
at least one other candidate as "bottom-rated", and interpreting above-
bottom rating as approval) it must not be possible for candidate X to
win if there is some candidate Y which has a beat-path to X and  
simultaneously higher Top-Ratings and Approval scores and a lower  
Maximum Approval-Opposition score.*


25: A>B
26: B>C 
23: C>A
26: C

TR scores:      C49,   B26,    A25
App. scores:   C75,   B51,    A48
MAO scores: C25,   B49,    A52

That criterion says that C  must win here. GMC says only B can win.

Frankly I think any method needs a much better excuse than any that Winning Votes
can offer for not electing C here. As I discuss in another recent post, any method
that doesn't elect C here must be vulnerable to Push-over. So another reason not
to be in love with GMC is that it is incompatible with "Pushover Invulnerability".


As I hope some may have guessed from the spectacular failure of Mono-add-Plump, the GMC 
concept is grossly unfair to truncators.  And Winning Votes  (as a GMC complying method) is
unfair to truncators. 

Say the 26C "we're just here to elect C and don't care about any other candidate" voters use a 
random-fill strategy, each tossing a fair coin to decide between voting C>B or C>A; then even if as
few as 4 of them vote C>A they will elect C. Their chance making C the decisive winner is  99.9956% 
(according to an online calculator http://stattrek.com/Tables/Binomial.aspx  ).

I have some sympathy with the idea of giving up something so as to counter order-reversing buriers,
but not with the idea that electing a CW is obviously so wonderful that when there is no voted CW
we must guess that there is a "sincere CW" and if we can infer that that can only (assuming no voters
are order-reversing) be X then we must elect X.

Chris Benham

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