On 04/30/2012 11:11 PM, Paul Kislanko wrote:
I always thought the “Condorcet is like a round-robin athletic
tournament” analogy was weak, because individual voters don’t get to go
through the round-robin and make their pairwise preferences explicit.
(As a voter, I’d find a “better/worse” pairwise choice for all pairs
easier than filling out a ranked ballot, but that just may be because
I’ve been making pairwise choices between the /ophthalmologist’s//
/lenses since I was six.) N x (N-1) “A or B” choices is an easier way to
fill out a ballot than “rank A1,A2,A3…” so no matter what method you use
to translate my ranked ballot into pairwise comparisons I have no way to
know if you counted my A<>B preferences the way I would have.
If your preferences are transitive, you don't even need N * (N-1) - O(n
log n) will suffice. Just reduce from computer sorting by having the
sorting algorithm ask you whether you prefer A to B whenever it would do
a comparison between A and B :-)
Now, I don’t think it’s a coincidence that JUST looking at PM^2 gives
the same winner (E) as Schultze does, since it’s counting the x->y->z
chains, giving extra credit to x >> z based upon x’s wins over
alternatives that themselves have {}->z wins, and that’s explicitly part
of the motivation for Schultze.
But *if* that is equivalent to Schultze (I’ll leave that test to people
who know better than I how it works) I find it more cosmetically
appealing than the Schultze definition.
I don't think it is equivalent to Schulze, because Schulze considers
paths of lengths up to the number of candidates. Instead, it sounds like
PM^2 would pick an uncovered candidate (rather like Copeland, which is
also used in sports).
If I'm right, then the Condorcet matrix corresponding to
40 D>B>C>A
30 A>B>C>D
30 C>A>D>B
should elect someone other than D. River, RP, and Schulze all elect D,
but D is covered by A.
There’s no “eliminate candidate based upon…” which has always rubbed me
the wrong way – too IRVish. All ballots and all alternatives are
directly involved in the final count.
One can describe Schulze without having to refer to eliminations, too. I
think this explanation is correct (if it isn't, Schulze, correct me):
- Candidate X beats Y if more voters prefer X to Y than vice versa. The
magnitude of this direct victory is the number of voters who prefer X to Y.
- X indirectly beats Y by a magnitude of no less than p if there exists
a sequence of candidates beginning in X and ending in Y so that every
candidate beats the one next in the sequence by at least magnitude p.
- The magnitude of X's indirect victory over Y is equal to the greatest
value of p for which the above is true. If no such sequence exists no
matter p, the magnitude of X's indirect victory is zero.
- X is a winner (or a tied winner) if no other candidate has a greater
magnitude of indirect victory against X than X has against that other
candidate.
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