Nondictatorial & Nonmanipulable axioms (was Re: York's

[Steve Eppley]

Bruce Anderson wrote:
>Steve Eppley wrote:
>> My understanding is that axiom A4 (below) limits the voting method
>> to using only ranking info from the voters the way Condorcet does,
>> ignoring any rating (a.k.a. weighting, a.k.a. intensity) info.
>
>No, it's A2 that does this.

Here's A2 again (slightly restated to eliminate the reference to A1):

 A2. (Unrestricted domain)  Every individual preference relation
 that satisfies the following two assumptions is admissible.
    (Completeness)  For every pair of outcomes o1 and o2,
                    either o1 is liked at least as much as o2, or o2
                    is liked at least as much as o1.  
 
    (Transitivity)  For any three outcomes o1, o2, and o3, 
                    if o1 is liked at least as much as o2,
                    and o2 is liked at least as much as o3, 
                    then o1 is liked at least as much as o3.  

You wrote that A2 is what limits the tally algorithm to using
ranking info only (prohibits more detailed voter ratings of the
choices).  I don't see this.

If I vote {A=100, B=60, C=0}, for example, then A >= B, B >= C, and 
A >= C.  So the first assumption appears to be satisfied.

And the weighted ballot appears to satisfy the transitivity 
assumption as well:  A >= B, B >= C, and A >= C.

How could a weighted ballot with a typical tally algorithm (like 
simply summing preferences) fail either of these two assumptions?

>> And that the purpose of this limitation is to prevent voters from
>> manipulating the outcome by strategically misrepresenting their
>> ratings.  
>
>That's Gibbard-Satterthwaite, not Arrow.

But here's the paragraph in Ordeshook which follows A4:

  "Put simply, A4 requires that if the set of feasible outcomes is
  restricted to the pair x and y, if x and y are the only two outcomes
  that the group can consider, then the social preference between x
  and y depends only on individual preferences over {x,y} and not on
  individual preferences over any larger set, including sets of
  outcomes that have not even been offered to the public for
  evaluation.  In effect, this condition prohibits expressions of
  intensity of preferences over the set {x,y} by referring to some
  alternative not under consideration.  Although this might not seem
  distasteful, outcomes can be manipulated, as we argue later, by
  persons strategically misrepresenting their true intensities."

I'm not fluent in this terminology.  Doesn't the prohibition on
expressing intensities imply ranked ballots only?  And doesn't
Ordeshook clearly say that removal of this Arrow axiom would allow
manipulation by strategic misrepresentation of intensities? 

--Steve