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. [EMAIL PROTECTED] 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.
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