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

Chris Benham chrisjbenham at optusnet.com.au
Wed Mar 7 06:28:28 PST 2007

```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 :-).)
>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

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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