As Benham pointed out (basically) my claim http://RangeVoting.org/WinningVotes.html 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 defeat." 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 gets elected. (They are all NOT immune in the perhaps-sillier sense that if the voters 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 you" 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 = plurality-loser 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 = BTR-IRV-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 not) 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 have 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 MaxTree. Benham pointed out there are two ways to go with IRV restricted to the Smith set: 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 wins). 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 Schulze-beatpaths and toward BTR-IRV or Smith//IRV - probably the former since it is simplest to describe? Warren D Smith http://RangeVoting.org ---- election-methods mailing list - see http://electorama.com/em for list info