[EM] Beatpath(Unc, wv), a version of Beatpath that always picks from the Uncovered set
Jobst Heitzig
heitzig-j at web.de
Wed Mar 14 07:07:22 PDT 2007
Dear Forest,
you wrote:
> Let N be the number of ballots. In Beatpath(Unc, wv) the strength of
> a pairwise victory of X over Y is N+1 if X covers Y, else it is the
> number of ballots on which X is ranked above Y.
I fear a "Beatpath" or "River" method based on this definition of
strength will not be monontonic. Let's assume the defeats
X>B>Y>A>B>C>A>X>C>Y>X with winning votes of the following relative
sizes:
over...
wv A B C X Y
of...
A - 1 4
B - 1 5
C 1 - 2
X 5 3 -
Y 5 1 -
Since every defeat is on a 3-cycle, the covering relation is empty and
the beatpath strengths are of these relative sizes:
beatpath over...
strength A B C X Y
of...
A - * * 4 *
B * - * 4 *
C * * - 2 *
X 5 5 3 - 5
Y * * * 4 -
Hence your Beatpath version will elect X.
(By the way, River would lock in the defeats X>B>Y>A and X>C, and Ranked
Pairs would in addition lock in B>C>Y and C>A, both therefore also
electing X.)
Now assume that weak defeat Y>X is reversed into a weak defeat X>Y, with
winning votes like these:
over...
wv A B C X Y
of...
A - 1 4
B - 1 5
C 1 - 2
X 5 3 - 1!
Y 5 -
Then C covers Y and X covers B, hence the defeat strengths and beatpath
strengths are now of these relative sizes:
defeat over...
strength A B C X Y
of...
A - 1 4
B - 1 5
C 1 - 9!
X 9! 3 - 1
Y 5 -
beatpath over...
strength A B C X Y
of...
A - * 3 4 *
B * - 3 4 *
C 5 4 - 4! 9
X 5 9 3 - 5
Y * * 3 4 -
That is, X has a weaker beatpath against C than C has against X. The new
winner is C, proving that the method is not monotonic!
Using your definition of defeat strength with River instead would lock
in C>Y>A>X>B, also electing C.
However, using it with Ranked Pairs would lock in C>Y and X>B, then
B>Y>A, then X>C>A and B>C, giving the same social order as before in
this example.
So, could it be in this case the Ranked Pairs cycle-resolution method
works better than the Beatpath and River cycle-resolution methods??
Yours, Jobst
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