[EM] Arrow's Theorem - The Return (again)

[Bart Ingles]

>From Arrow (1962):

Condition 1' (from p.96):  All logically possible orderings of the
alternative social states are admissible.

The original Condition 1 (from p.24) stated:  Among all the alternatives
there is a set S of three alternatives such that, for any set of
individual orderings T1, ..., Tn of the alternatives in S, there is an
admissible set of individual orderings R1, ..., Rn of all the
alternatives such that, for each individual i, x Ri y if and only if x
Ti y for x and y in S.

Condition 3 (from p.27):  Let R1, ..., Rn and R1', ..., and Rn' be two
sets of individual orderings and let C(S) and C'(S) be the corresponding
social choice functions.  If, for all individuals i and for all x and y
in a given environment S, x Ri y if and only if x Ri' y, then C(S) and
C'(S) are the same (independence of irrelevant alternatives).

Condition P (from p.96):  If x Pi y for all i, then x P y (Pareto
principal).

Condition 5 (from p.30):  The social welfare function is not to be
dictatorial (non-dictatorship).


Theorum 2 (from p.97): Conditions 1', 3, P, and 5 are inconsistent.


Clearly, approval voting does not satisfy either Condition 1 or 1'. 
Since x Ri y in Condition 3 appears to refer to *admissible* orderings,
and not to the individual orderings from which they were derived (see
original condition 1), approval voting does seem to meet Condition 3.

Bart


Alex Small wrote:
> 
> Eric Gorr said:
> > Arrow seems to be perfectly content with allowing equal rankings.
> >
> > Approval voting merely requires the user to divide the options into  two
> > different groups, but doing do does not violate Arrow's Axioms or
> > definitions regarding voter choice.
> >
> > Again, from his book:
> >
> >    "However, it may be as well to give sketches of the proofs, both to
> > show
> >     that Axiom I and II really imply all that we wish to imply about the
> > nature of orderings of alternatives and to illustrate the type of
> > reasoning to be used subsequently." (page 14)
> 
> No, no, no, no, no!
> 
> Arrow assumes that, whether voters happen to have strict preferences or
> whether they instead rank some candidates equal, voters are free to report
> ANY transitive ranking of candidates (which might include equal rankings).
>  Based on this information, i.e. every voter giving his COMPLETE
> preference order, the election method then chooses a winner.
> 
> Approval does NOT allow voters to express their complete preferences.
> Sure, if a person happens to have a lot of equal rankings so that the
> candidates fall into two groups, then Approval lets that PARTICULAR voter
> express his full preferences.  BUT, if the person sorts the candidates
> into 3 or more categories, Approval does not let them express their
> complete preferences.
> 
> Once again, Arrow basically proves that no method can satisfy all of the
> following criteria simultaneously when there are 3 or more candidates:
> 
> 1)  Non-dictatorship (plenty of methods, and any seriously proposed
> method, will satisfy this)
> 
> 2)  Pareto (plenty of methods, and any seriously proposed method, will
> satisfy this)
> 
> 3)  IIAC
> 
> 4)  The use of RANKED ballots, which allow ANY voter to indicate his
> entire preference, WHATEVER it might happen to be.
> 
> APPROVAL DOES NOT SATISFY THIS CRITERION!  If my preference is A>B>C then
> I have to make a decision about whether or not to approve B.
> 
> Can I please get some back-up here?  Somebody, anybody, please back me up
> on this?  I can't believe how difficult it is to drive home the point that
> Approval Voting is not a ranked method.
> 
> Anyway, since Approval flunks criterion #4, it is in keeping with Arrow's
> Theorem.  It flunks at least one of his criteria.
> 
> Alex