Nondictatorial & Nonmanipulable axioms (was Re: York's

[Steve Eppley]

Bruce Anderson wrote:
[snip]
>according to every definition I have seen here (or elsewhere),
>Condorcet voting method does not fail the non-dictatorial axiom;
>instead, Condorcet voting method fails the non-manipulable axiom. 
[snip]

Are you talking Arrow, Gibbard-Satterthwaite, or both?  Which of 
Arrow's 5 axioms does Condorcet violate?

Another Saari (Mike, in the eVote maillist) claims that, by Arrow's
theorem, ranked ballot methods like Condorcet must be "dictatorial".
I'd certainly appreciate learning the truth.

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.  
And that the purpose of this limitation is to prevent voters from
manipulating the outcome by strategically misrepresenting their
ratings.  

Bruce, are you saying that Condorcet violates A4 and not A5? 

- -

Arrow's 5 axioms:
A1. (Collective rationality)  The social preference function 
satisfies the following two assumptions:

   (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.  

A2. (Unrestricted domain)  Every individual preference relation that 
satisfies the two assumptions in A1 is admissible.

A3. (Pareto principle)  If every person in the group likes
alternative x more than y, then x is preferred to y in the social
preference order.  

A4. (Independence)  If R is a profile of individual preferences 
over some set of alternatives that includes x and y, 
if G(R,{x,y}) = (x is liked more than y), 
and if R' is another preference profile such that each person's 
preference between x and y is the same in R' as in R, 
then G(R',{x,y}) = (x is liked more than y).  

Note: G(R,O) --> O is a social choice function that takes individual
preferences and selects outcomes in O as the social outcome.

A5. (Nondictatorship)  No one person is decisive for every pair of 
outcomes.
(I.e., no one can get his/her way if opposed by everyone else.)

Arrow's Impossibility Theorem: If there are at least three possible
outcomes, then voting methods which satisfy A1 through A4 violate A5.

--Steve