As Benham pointed out (basically) my claim
that "all Condorcet methods" suffer from the DH3 pathology was
not quite right.  I should have said
"all Condorcet methods which handle a 3-cycle by eliminating the weakest 
This includes Schulze-beatpaths, Heitzig-river, Tideman-ranked-pairs, 
least-reversal-classic-Condorcet, Simpson-Kramer-minmax, my method I call the
"maxtree" method...  but not ALL Condorcet methods.

(One also can consider emthods where a vote is a rank ordering AND 
an "approval threshold" but I will not do that here.)

An example of a Condorcet order-only method which is immune to DH3 is
Rob LeGrand's BTR-IRV method; another is the "IRV restricted to the Smith set" 
method;  another is the "plurality restricted to Smith set" method.

These are all immune in the sense that it does not make strategic sense 
for  the A- and B-voters to pretend D>C, because that does not work - C still 
elected.  (They are all NOT immune in the perhaps-sillier sense that if the 
stupidly do it anyway, then you get the "dark horse" D winning.)

Of these three, which should we prefer?
None of the three are monotonic, all three suffer from "honest voting can hurt 
paradoxes, and all three can suffer from embarrassing 
"winner=loser reversal paradoxes":

Plurality paradox:
2 A>B>C
2 A>C>B
3 B>C>A
7-voter scenario where 
plurality-winner = BTR-IRV-winner = IRV-winner = Condorcet-winner = A = 

BTR-IRV paradox:
4 A>B>C
4 A>C>B
6 B>C>A
5 C>A>B
19-voter scenario where 
plurality-winner = BTR-IRV-winner = IRV-winner = A = plurality-loser = 

IRV paradox:
9        B>C>A
8        A>B>C
7        C>A>B
24-voter scenario where 
IRV-winner = A = plurality-loser = IRV-loser = BTR-IRV-loser

Benham notes IRV enjoys (and I now note BTR-IRV also enjoys, but Plurality does 
the "Dominant mutual third" property:
 If a more than a third of the voters rank (in any order) the members
 of a subset S of candidates above all others, and all the
 members of S pairwise beat all the non-members; then the
 winner must come from S.

That is because the IRV (or BTR-IRV)
election will, after eliminations and vote-transfers, reduce to
a 2-man contest between an S-member and somebody else.  (This is a weakened
form of the Condorcet or Smith Set property.)

Also BTR-IRV and IRV and Smith//IRV
are both clone-immune but plurality is not.  These facts cause
us to demerit Smith//Plurality.   

Now IRV is better than BTR-IRV in the sense it is immune to add-top failure
and enjoys "later-no-harm".  However, both these IRV advantages no longer hold
if we are speaking of Smith//IRV.  

So...  I guess BTR-IRV and Smith//IRV  both are good Condorcet methods and I 
uncovered no basis here for preferring one over the other.

But we HAVE uncovered a good reason (DH3 immunity) to prefer either to
Schulze-beatpaths, Heitzig-river, Tideman-ranked-pairs, 
least-reversal-classic-Condorcet, Simpson-Kramer-minmax, and my method I call 

Benham pointed out there are two ways to go with  IRV restricted to the Smith 
either eliminate the candidates not inthe Smith set beforehand, or do not
(i.e. just use regular IRV until have eliminated all but one Smith set member X
plus some perhaps-large number of non-Smith-set members also remain, then X 
I am not sure which of these two is better.
Ditto for BTR-IRV.  That complicates matters.

BTR-IRV also has the advantage over Schulze-beatpaths that it does not
matter whether you hew to the margins or winning-votes philosophies - BTR-IRV
is the same in either.  (With Schulze-beatpaths, you have to worry about that.)

In view of this, I should probably change CRV's recommendation
("if you insist on using a rank-order-ballot method") away from 
and toward BTR-IRV or Smith//IRV - probably the former since it is simplest to 

Warren D Smith
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