On May 9, 2010, at 9:51 PM, robert bristow-johnson wrote:
thanks, Juho, for this summary.
On May 9, 2010, at 2:24 PM, Juho wrote:
All classical Condorcet methods can handle equal rankings and their
impact has been analyzed quite well.
Usually the discussion focuses on how to measure the strength of
the pairwise preferences. This is the next step after the matrix
has been populated. The two most common approaches are margins and
winning votes.
- Let's define AB = number of votes that rank A above B
- In margins the strength of pairwise comparison of a A against B
is: AB - BA
- In winning votes the strength of pairwise comparison of a A
against B is: AB if AB>BA and 0 otherwise
- Note that in margins ties could be measured either as 0:0 or as
0.5:0.5 since the strength of the pairwise comparison will stay the
same in both approaches
The most common (/classical) Condorcet methods give always the same
winner if there are three candidates and all votes are fully
ranked. If there is no Condorcet winner then the candidate with
smallest defeat will win. The strengths of the defeats (and
therefore also the end result) may differ in margins and winning
votes if there are equal rankings.
do you mean that marginal defeat strength can differ if equal
rankings are allowed compared to if equal rankings are not allowed?
that, of course is true. but if you mean that marginal defeat
strength is different where equal ranks are counted (as votes for
both candidates) vs. if they are not counted (for both candidates),
then i think i disagree. that's one reason why i think that
marginal defeat strength is a more salient measure than simply the
number of winning votes in each pairing (as a metric to be used in
resolving a cycle).
I was thinking about results given by margins vs. given by winning
votes with the same ballots. Here's one example that has no Condorcet
winner.
49: A>B>C
48: B>C>A
03: C>A>B
Both approaches to measuring the strength of pairwise comparisons
agree (as always with full rankings wthout equalities) that A wins (in
typical Condorcet methods that "ignore" the weakest defeat in the case
of a three candidate loop).
But if the A supporters use equal rankings...
49: A>B=C
48: B>C>A
03: C>A>B
... then winning votes gives victory to C while in margins A still wins.
There is a preference loop in the opinions of the society where A>B,
B>C and C>A. According to margins the strengths of the defeats of A, B
and C changed from (2, 4, 94) (= (48+3-49, 49+3-48, 49+48-3) ) to (2,
4, 45) and according to winning votes from (51, 52, 97) (= (48+3,
49+3, 49+48) ) to (51, 52, 48).
(I agree that margins is a more natural measure of preference strength
than winning votes.)
Juho
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r b-j [email protected]
"Imagination is more important than knowledge."
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