[EM] "Possible Approval Winner" set/criterion (was "Juho--Margins fails Plurality. WV passes.")

Juho juho4880 at yahoo.co.uk
Sun Mar 11 08:49:05 PDT 2007

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  

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.


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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