[EM] Thoughts on Arrow's Theorem and the IIAC

[fsimmons at pcc.edu]

In liberal arts mathematics text books Arrow's impossibility theorem is usually
quoted as saying that no election method can simultaneously satisfy (1)
neutrality, (2) anonymity, (3) decisiveness (4) monotonicty (5) the majority
criterion (6) the Condorcet Criterion, and (7) the Independence from Irrelevant
Alternatives Criterion (the IIAC), as though all of these requirements were
equally to blame for the incompatibility, when in reality conditions one through
six are perfectly compatible with each other, but condition seven is not even
compatible with the existence of a Condorcet cycle.

To see why the IIAC is not compatible with the existence of a Condorcet cycle,
let M be any method that satisfies the IIAC.  We will show that the only kind of
winner that there can be under M is a Condorcet Winner:

Let E be an arbitrary election that is decided by M.  Let X be the winner of
election E according to M.  Let Y be any of the other candidates.  Eliminate all
of the other candidates one by one until only X and Y remain.  According to the
IIAC, the winner is not changed at any stage of the elimination, so X is still
the winner according to M when the choice is between X and Y.  Since the choice
of Y was arbitrary, we see that M makes its winner X defeat each of the other
candidates head to head. 

Thus we see that the IIAC is a totally unreasonable requirement.  How would you
like it if somebody asked you to do something that was logically impossible, and
then complained that you were imperfect for not doing it?  It's like the
philosopher that requires god to make an immoveable object and then to move it,
because (in his opinion) a perfect being would have to be capable of both
requirements.

On the other hand there are methods that satisfy requirements one through six
along with other reasonable requirements in place of the IIAC, including (8)
independence from clones, (9) independence from Pareto dominated alternatives,
and (10) independence from non-Smith alternatives, simultaneously.

Woodall's incompatibility theorems for various combinations of his criteria are
more interesting because they spread the blame around; it's not so easy to
single out a single criterion as being unreasonable.