[EM] Does the 'Independence of Irrelevant Alternatives Criterion' Imply a Condorcet Winner ?

[Forest Simmons]

On Mon, 29 Mar 2004, Marcos C. Ribeiro wrote:

> Hi Democracy Friends.
>
> As long as I know, the 'Independence of Irrelevant Alternatives Criterion' may be explained as: If an election is held and a winner is declared, this winning candidate should remain the winner in any recalculation of votes as a result of one or more of the losing candidates dropping out.
>
> To me, this criteria implies that any election should have a Condorcet winner, because we might drop out all of the defeated candidates, except 1 of them, and this would establish a pairwise dispute between the winner and only one of the defeated candidates, any one of them.
>
> However, the absence of a Condorcet winner can not be considered a problem with an election method, but, instead, it only reflects ambiguities in the electors' preferences. If what I wrote is correct, this means that the 'Independence of Irrelevant Alternatives' is not a valid criteria to evaluate any voting method.
>
> I'm afraid I am not understanding correctly the 'Independence of Irrelevant Alternatives Criterion' , so I ask you to correct me if this is happening.
>
> I also ask confirmation if Arrow used this criteria to prove his famous impossibility theorem.
>
> Thank you.
>
> Marcos C. Ribeiro
> Belo Horizonte - Minas Gerais - Brasil.
>

Yes, Arrow did use the IIA criterion, and yes, most folks here agree that
it is the criterion that is too strict, and therefore should be relaxed in
one way or another.  For example, the winner should come from the Smith
set, and (at very least) the method should have "clone independence."

Any time there is a "Condorcet cycle" of the type A beats B beats C beats
A, no matter who the winner is, eliminating the candidate that the winner
beats, turns the winner into a loser.

Some folks believe that this is a defect of Condorcet methods.  But
Condorcet cycles are a fact of life, whether or not my favorite method
detects them.

So at some fundamental level no method really, truly satisfies the IIAC.
However, some methods like Approval satisfy it technically when it is
expressed in terms of ballots.

Suppose in the above example that A is the Approval winner and that B
withdraws from the contest.  Then the approval ballots will still say that
A beats C even though, if the voters had a chance to vote for A or C with
B out of the contest, they would choose C.

How is this possible?  Ballots that approved only B would approve nobody
after B's withdrawal, while ballots that disapproved both A and C before
the withdrawal of B, would disapprove nobody after the withdrawal.

In other words, the ballots would not be realistic reflections of the
voter wishes for a two way contest between A and C.

You probably already figured this out, but perhaps it is worth putting in
writing one more time for new lurkers.

Forest