Forest Simmons, responding to questions from Mike Ossipff, wrote (19 Nov
2011):
> 4. How does it do by FBC? And by the criteria that bother some
> people here about MMPO (Kevin's MMPO bad-example) and MDDTR
(Mono-Add-Plump)?
I think it satisfies the FBC.
Forest's definition of the method being asked about:
Here’s my current favorite deterministic proposal: Ballots are Range
Style, say three slot for simplicity.
When the ballots are collected, the pairwise win/loss/tie relations are
determined among the candidates.
The covering relations are also determined. Candidate X covers
candidate Y if X
beats Y as well as every candidate that Y beats. In other words row X
of the
win/loss/tie matrix dominates row Y.
Then starting with the candidates with the lowest Range scores, they are
disqualified one by one until one of the remaining candidates X covers
any other
candidates that might remain. Elect X.
Forest,
Doesn't this method meet the Condorcet criterion? Compliance with
Condorcet is incompatible with FBC, so
why do you think it satisfies FBC?
http://lists.electorama.com/pipermail/election-methods-electorama.com/2005-June/016410.html
Hello,
This is an attempt to demonstrate that Condorcet and FBC are incompatible.
I modified Woodall's proof that Condorcet and LNHarm are incompatible.
(Douglas R. Woodall, "Monotonicity of single-seat preferential
election rules",
Discrete Applied Mathematics 77 (1997), pages 86 and 87.)
I've suggested before that in order to satisfy FBC, it must be the case
that increasing the votes for A over B in the pairwise matrix can never
increase the probability that the winner comes from {a,b}; that is, it
must
not move the win from some other candidate C to A. This is necessary
because
if sometimes it were possible to move the win from C to A by increasing
v[a,b], the voter with the preference order B>A>C would have incentive to
reverse B and A in his ranking (and equal ranking would be inadequate).
I won't presently try to argue that this requirement can't be avoided
somehow.
I'm sure it can't be avoided when the method's result is determined solely
from the pairwise matrix.
Suppose a method satisfies this property, and also Condorcet. Consider
this
scenario:
a=b 3
a=c 3
b=c 3
a>c 2
b>a 2
c>b 2
There is an A>C>B>A cycle, and the scenario is "symmetrical," as based on
the submitted rankings, the candidates can't be differentiated. This means
that an anonymous and neutral method has to elect each candidate with
33.33%
probability.
Now suppose the a=b voters change their vote to a>b (thereby
increasing v[a,b]).
This would turn A into the Condorcet winner, who would have to win
with 100%
probability due to Condorcet.
But the probability that the winner comes from {a,b} has increased
from 66.67%
to 100%, so the first property is violated.
Thus the first property and Condorcet are incompatible, and I contend
that FBC
requires the first property.
Thoughts?
Kevin Venzke
Chris Benham
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