Forest,
25: A>B
26: B>C
23: C>A
26: C
I don't like any method that fails to elect C here, unless like IRV it
has the property
that a Mutual Dominant Third (MDT) winner can't be successfully buried
to elect a
non-MDT winner.
If these rankings are from sincere 3-slot ratings ballots, then C is the
big SU winner.
Also the truncating C supporters don't have to do much burying to elect C.
In common with Winning Votes, your suggested method of using MinMax and
weighing
the defeats by the winner's approval (ranking) opposition to the loser
elects B:
A>B 23, B>C 25, C>A 52.
34: A>B
17: C>A
49: B
Here A is a MDT winner, but if the B truncators change to B>C then
AWP(ranking)
elects B:
A>B 17, B>C 34, C>A 49.
I think the idea that the CW should always be elected but it is
sometimes ok to elect
from outside the Smith set is a bit philosophically weird, and not easy
to sell.
AWP(ranking) needs a lot of information that isn't just in the gross
pairwise matrix, which
could count as a practical disadvantage compared to some other pairwise
methods.
But AWP(ranking) may have sufficient strategy-resistant qualities to
justify it as not a bad
method, but I'm not a fan.
The Approval-Weighted Pairwise method that James Green-Armytage
originally envisaged
allowed voters to rank among unapproved candidates. That version fails
mono-raise.
31: A>>B
04: A>>C
32: B>>C
33: C>>A
B>C 32, C>A 33, A>B 35. C's defeat is the weakest so C wins.
Now say the 4 A>>C ballots change to C>>A.
31: A>>C
32: B>>C
37: C>>A
A>B 31, B>C 32, C>A 37. Now B wins.
Chris Benham
Forest Simmons wrote (21 April 2010):
I don't know if Juho is still cheering for MinMax as a public proposal. I used to be against it because of its clone dependence,
but now that I realize that measuring defeat strength by AWP (Approval Weighted Pairwise) solves that problem, I'm starting to warm
up more to the idea.
MinMax elects the candidate that suffers no defeats if there is one, else it
elects the one whose maximum strength defeat is minimal.
There are various ways of measuring defeat strength. James Green Armytage has advocated one called AWP as making Condorcet methods
less vulnerable to strategic manipulation.
If all ranked candidates on a ballot are considered approved, then the AWP strength of a defeat of B by A is the number of ballots on
which A is ranked but B is not.
Then more recently I was reading a paper by Joaquin Pérez in which he shows that MinMax is the only commonly known Condorcet method
that satisfies the following weak form of Participation:
If A wins and then another ballot with A ranked unique first is added to the
count, A still wins.
Beatpath, River, Ranked Pairs, etc. fail this weak participation criterion, but
they do satisfy this even weaker version:
If A wins and then another ballot with only A ranked is added to the count,
then A still wins.
Proof: First add a ballot in which no candidate is ranked. The above mentioned methods allow this, and it doesn't affect their outcome
since no mention of absolute majority is made in any of them. Then raise A while leaving the other candidates unranked. This cannot hurt
A since all of the above mentioned methods are monotone.
Knowing that Beatpath satisfies the weaker version but not the weak version may be an inducement for voters to bullet vote candidate A to
make sure that they avoid the no show paradox. But MinMax is free of this temptation; they wouldn't have to truncate the other candidates.
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