# [EM] Does the 'Independence of Irrelevant Alternatives Criterion' Imply a

Alex Small asmall at physics.ucsb.edu
Thu Apr 1 22:17:01 PST 2004

```Adam said:
> This is not a rigorous proof, since I did not provide a rigorous
> justification why C should win the pairwise contest (although it is
> obvious).  But this example suffices to show, in my opinion, that no
> reasonable method will ever pass IIA.

I think we could take your argument and make it into a weak statement of
Arrow's Theorem.  I'll basically restate you post, so I'm not in any way
suggesting this is original.  But you offered your post in the spirit of a
heuristic argument, and I want to point out that if your argument is
dressed up in slightly more formal attire it's actually a full-blown
theorem that's (almost) as good as Arrow's Theorem.

The part that you cast doubt on is the justification of why C should win a
pairwise contest.  There's actually a theorem called May's Theorem.  It
says that when there are 2 candidates a pairwise vote is the only method
that is monotonic and treats all candidates as well as all voters equally.

So, armed with that reasonable assumption, let's just state all of this in
pretty language:

Assume that our election method takes as its inputs the individual voters'
transitive preferences (to distinguish it from things like approval
voting). and that the method is capable of choosing a winner regardless of
the number of candidates.  (With occasional ties, but we won't worry about
that for now.)

Theorem:  No method satisfying the above assumptions can simultaneously
satisfy the following 2 properties:
1)  When a candidate is removed the winner remains unchanged unless the
candidate was the winner.
2)  When there are only 2 candidates the outcome is decided by a pairwise
comparison.  (This implies non-dictatorship, but not Pareto, because
Pareto applies to any number of candidates.  It is, however, consistent
with Pareto.)

Proof:  Suppose we have 3 candidates, and that the pairwise results are a
cycle A>B>C>A.  Without loss of generality we assume that A wins.  If we
drop B, a losing candidate, then it reduces to a comparison of C and A,
and C wins.  So A can't win if we're still satisfying IIA.  However, since
A was chosen without loss of generality it follows that none of the
candidates can be chosen.

To my mind, this is a pretty substantial result for public elections.
Weakening the second criterion to simple Pareto may expand the theorem's
applicability to Byzantine committee systems, where the decision isn't
always a pairwise vote (e.g. some designated officer decides unless 2/3
over-rule him), but it adds very little to the study of public elections.

However, before I bash Arrow's result too much, we should remember that
many brilliant results are simple to derive once somebody does it, and the
real insight is simply asking the question in the first place, and showing