Approval voting is ranked voting (was Re: Attachment CRITTBL1)

[Steve Eppley]

On 5/25/96 (long ago) Bruce Anderson wrote:
-snip-
>D.	APPROVAL  VOTING
>Approval voting requires an additional input beyond just a ranking
>from each voter.  That is, a valid Approval ballot is a valid
>(ranked) ballot, as defined here, because ties are allowed in
>voter's rankings.  Thus, any voter who believes that the
>alternatives can be put into one of two categories (either approved
>or non-approved) can always cast a valid ranked ballot as well as a
>valid Approval ballot.  Conversely, though, a valid ranked ballot
>does not generally correspond to a valid Approval ballot because it
>contains insufficient information--the additional input of where in
>a voter's ranking that voter separates his or her approved
>alternatives from the non-approved ones is also needed from each
>voter. 
-snip-

I think it's misleading to compare approval ballots and ranked 
ballots using different outcome sets.  Elections where none of the 
choices might win are different from elections where (at least) one
choice must win.  Why not use the same outcome set when comparing the 
ballot types?

If the "None" choice is on the ballot (in other words, if Arrow's
"completeness" axiom is satisfied or if "overcompleteness" is
implemented) then the info Bruce said would be missing from rankings
is not missing:   a > b > ... > NONE > c > d > ...

In my view, Approval voting is a ranked ballot system which violates
Arrow's "Universal Domain" axiom in the following ways:
  1) Only three distinct rank positions are allowed.
  2) The status quo "None" choice must be ranked 2nd.
So approval ballots correspond to:
   a=b=c=...  >  NONE=d=e=f=...  >  g=h=i=...
The voter indicates a choice is ranked 1st by voting "Yes" on it and
indicates a choice is ranked 3rd by voting "No" on it.  Unvoted
choices are treated as if they were ranked 2nd.  (The usual tally 
will award +1 point to each 1st-ranked choice and -1 point to each 
3rd-ranked choice.)

Bruce's comments about the voter believing alternatives are approved 
or disapproved omitted the meaning of the two terms.  They are pairwise
rankings comparing the non-None alternatives with the None
alternative:  i>NONE or i=NONE or NONE>i 

I'm not sure what Bruce meant when he said a voter who believes the
alternatives can be categorized "approved" or "disapproved" can
always cast a valid ranked ballot as well as an approval ballot. 
Perhaps by "valid" Bruce meant a "legal" ballot which would not be
disqualified.  But such a ballot would not in general be a valid
expression of the voter's real preferences, if it was constructed
using only the info contained in the voter's approval ballot.  So 
if Bruce meant only that a nondisqualifiable ranked ballot could be 
constructed, that meaning would not support his conclusion since a 
nondisqualifiable approval ballot can easily be constructed from the 
voter's rankings (or from zilch) for any voter.

It appears there's a symmetry of missing information: rankings which
don't include None lack info which might be useful--especially when
None is a possible outcome--for constructing an approval ballot
which reflects the voter's true preferences, as Bruce pointed out.
(More precisely, though, they're not an accurate expression of the
voter's preference order because of the constraints that violate 
Arrow's "Unrestricted Domain" axiom.)  Similarly, approval ballots 
lack info which might be useful for constructing a rank order which
reflects the voter's true preferences, but Bruce implied this isn't 
true by arguing there's an asymmetry.  

It might be helpful in further exploration along these lines to
consider approval ballots and ranked ballots when the outcome set is
the same for both:  If None is a possible outcome then it's on the
ballot for both types.  And if None isn't a possible outcome, compare
approval ballots to rankings by using an approval method which will
select at least one choice (even if it has fewer Yeses than Nos). 

---Steve     (Steve Eppley    seppley at alumni.caltech.edu)