The "Possible Approval Winner" criterion looks actually quite natural
in the sense that it compares the results to what Approval voting
could have achieved.
The definition of the criterion contains a function that can be used
to evaluate the candidates (also for other uses) - the possibility
and strength of an approval win. This function can be modified to
support also cardinal ratings.
In the first example there is only one entry (11: A>B) that can vary
when checking the Approval levels. B can be either approved or not.
In the case of cardinal ratings values could be 1.0 for A, 0.0 for C
and anything between 0.0001 and 0.9999 for B. Or without
normalization the values could be any values between 0.0. and 1.0 as
long as value(A) > value(B) > value (C). With the cardinal ratings
version it is possible to check what the "original utility values"
leading to this group of voters voting A>B could have been (and if
the outcome is achievable in some cardinal ratings based method, e.g.
max average rating).
The max average rating test is actually almost as easy to make as the
PAW test. Note that my description of the cardinal ratings for
candidate B had a slightly different philosophy. It maintained the
ranking order of the candidates, which makes direct mapping from the
cardinal values to ordinal values possible. The results are very
similar to those of the approval variant but the cardinal utility
values help making a more direct comparison with the "original
utilities" of the voters.
Now, what is the value of these comparisons when evaluating the
different Condorcet methods. These measures could be used quite
straight forward in evaluating the performance of the Condorcet
methods if one thinks that the target of the voting method is to
maximise the approval of the winner or to seek the best average
utility. This need not be the case in all Condorcet elections (but is
one option). There are several utility functions that the Condorcet
completion methods could approximate. The Condorcet criterion itself
is majority oriented. Minmax method minimises the strength of
interest to change the selected winner to one of the other
candidates. Approval and cardinal ratings have somewhat different
targets than the majority oriented Condorcet criterion and some of
the common completion methods, but why not if those targets are what
is needed (or if they bring other needed benefits like strategy
resistance).
I find it often useful to link different methods and criteria to
something more tangible like concrete real life compatible examples
or to some target utility functions (as in the discussion above). One
key reason for this is that human intuition easily fails when dealing
with the cyclic structures (that are very typical cases when studying
the Condorcet methods). In this case it seems that PAW and
corresponding cardinal utility criterion lead to different targets/
utility than e.g. the minmax(margins) "required additional votes to
become the Condorcet winner" philosophy. Maybe the philosophy of PAW
is to respect clear majority decisions (Condorcet criterion) but go
closer to the Approval/cardinal ratings style evaluation when the
majority opinion is not clear. You may have different targets in your
mind but for me this was the easiest interpretation.
Juho
P.S. One example.
1: A>B
1: C
Here B could be an Approval winner (tie) but not a max average rating
winner in the "ranking maintaining style" that was discussed above
(since the rating of B must be marginally smaller than the rating of
A in the first ballot).
On Mar 7, 2007, at 16:28 , Chris Benham wrote:
Juho wrote (March7, 2007):
The definition of plurality criterion is a bit confusing. (I don't
claim that the name and content and intention are very natural
either :-).)
- http://wiki.electorama.com/wiki/Plurality_criterion talks about
candidates "given any preference"
- Chris refers to "above-bottom preference votes" below
If the number of ballots ranking A as the first preference is
greater than the number
of ballots on which another candidate B is given any preference,
then B must not be elected.
Electowiki definition could read: "If the number of voters ranking A
as the first preference is greater than the number of voters ranking
another candidate B higher than last preference, then B must not be
elected".
Yes it could and to me it in effect does (provided "last" means
"last or equal-last") The criterion come
from Douglas Woodall who economises on axioms so doesn't use one
that says that with three candidates
A,B,C a ballot marked A>B>C must always be regarded as exactly the
same thing as A>B truncates. He
assumes that truncation is allowed but above bottom equal-ranking
isn't.
A similar criterion of mine is the "Possible Approval Winner"
criterion:
"Assuming that voters make some approval distinction among the
candidates but none among those
they equal-rank (and that approval is consistent with ranking) the
winner must come from the set of
possible approval winners".
This assumes that a voter makes some preference distinction among
the candidates, and that truncated
candidates are equal-ranked bottom and so never approved.
Looking at a profile it is very easy to test for: considering each
candidate X in turn, pretend that the
voters have (subject to how the criterion specifies) placed their
approval cutoffs/thresholds in the way
most favourable for X, i.e. just below X on ballots that rank X
above bottom and on the other ballots
just below the top ranked candidate/s, and if that makes X the
(pretend) approval winner then X is
in the PAW set and so permitted to win by the PAW criterion.
11: A>B
07: B
12: C
So in this example A is out of the PAW set because in applying the
test A cannot be more approved
than C.
IMO, methods that use ranked ballots with no option to specify an
approval cutoff and rank among
unapproved candidates should elect from the intersection of the PAW
set and the Uncovered set
One of Woodall's "impossibility theorems" states that is
impossible to have all three of Condorcet,
Plurality and Mono-add-Top. MinMax(Margins) meets Condorcet and
Mono-add-Top.
Winning Votes also fails the Possible Approval Winner (PAW)
criterion, as shown by this interesting
example from Kevin Venzke:
35 A
10 A=B
30 B>C
25 C
A>B 35-30, B>C 40-25, C>A 55-45
Both Winning Votes and Margins elect B, but B is outside the PAW set
{A,C}.
Applying the test to B, we get possible approval scores of A45,
B40, C25.
ASM(Ranking) and DMC(Ranking) and Smith//Approval(Ranking) all meet
the Definite
Majority(Ranking) criterion which implies compliance with PAW. The
DM(R) set is
{C}, because interpreting ranking (above bottom or equal-bottom) as
approval, both
A and B are pairwise beaten by more approved candidates.
Chris Benham
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