[EM] The issue of comments about Arrow's theorem

[Curt Siffert]

On May 17, 2005, at 8:31 PM, Russ Paielli wrote:

> If I am not mistaken, Arrow's theorem says that you can't satisfy both 
> the Condorcet criterion *and* the independence of irrelevant 
> alternatives (IIA). Should that bother us? I think it should bother us 
> at least a bit. I am bothered by the fact that eliminating a losing 
> candidate can change the winner. Like failure of monotonicity, it 
> suggests a certain irrationality.

My understanding of IIA is that it can never replace one singular 
Condorcet Winner with another.  It can only lead from a Condorcet 
Winner to a multi-member Smith Set.  It's also obvious that the 
introduction of a new candidate does not actually *change* the ordering 
of an individual voter's preferences with respect to the pre-existing 
candidates.  The new candidate merely gets inserted.  And in the 
already described case, failing IIA exposes a confused electorate, when 
that confusion - still present - was hidden beforehand.

This is enough to prove that IIA is an unreliable criterion.  
abd at lomaxdesign.com had a great response describing this in more 
detail.

These vote-method criteria are supposed to test the reliability of the 
vote methods.  Seems to me that IIA instead tests the certainty of the 
electorate.

Here's another thought that illustrates to me why IIA is messed up.  
Think of it in reverse.  You have an election that ends up with a 
multi-member Smith Set, one that most tie-breaking procedures would 
settle by awarding "A" with the victory.  They'll do this by 
eliminating other candidates from the Smith Set.  But it's possible to 
instead start with eliminating a candidate that doesn't make sense, and 
then recount the ballots to find a Condorcet Winner that doesn't make 
sense.  Now, it wouldn't make sense to do that.  But IIA tells us that 
it should.  It's a stupid criterion, at least if it's interpreted to 
mean that the Condorcet Winner is the rightful winner, and that the IIA 
failure leading to a different winner is evidence of vote method 
*failure*.

So, it's obvious that there are some cases where the failure of IIA 
shouldn't bother us at all.  Adding the new candidates merely expose a 
confused electorate that was already confused before the introduction 
of the new candidate (even though the confusion was masked by the lack 
of choices).  And, it's also difficult to prove that that isn't always 
the case.

It seems like IIA could instead be constructively used to test whether 
an electorate is confused even when they seem to be certain.  In other 
words, just because someone is a Condorcet Winner, maybe they're still 
not the best choice yet.  Introducing some more arbitrary candidates 
might show that the electorate requires more debate before they really 
feel settled about their decision.

If it's true that a Condorcet Winner may not actually be the best 
winner (even leaving aside "strength of preference" ratings), due to 
the confused electorate, and that the IIAC helps expose that when it's 
true, then maybe I should conclude that an election with a Condorcet 
Winner isn't the democratic perfection that I've believed it was over 
the past many months.  It seems to me that failing IIAC is sometimes a 
feature, not a bug, which shakes my perception of the Condorcet 
Criterion as a hard requirement of a valid voting method.

Curt