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

[Bart Ingles]

I should add that the original Condition 1 (included below) was used in
the original 1951 theorum, but not in the 1962 version.  The later
version replaced it with the stronger Condition 1'. 

I mainly included the older Condition 1 because it seemed to settle the
question of whether x Rn y in Condition 3 referred to an individual's
complete private orderings, or orderings on an actual ballot.  It seems
to be the latter, implying that approval voting satisfies at least
Arrow's definition of IIAC.


Bart Ingles wrote:
> 
> >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