On May 14, 2005, at 9:07 PM, Russ Paielli wrote:
The importance of IIAC is a matter of individual preference, of course, but it is a perfectly reasonable criterion. If I offer a group the choice of chocolate or vanilla ice cream, and they choose chocolate, why should the additional choice of strawberry cause them to switch their choice from chocolate to vanilla?
I think that's an overly simplistic definition of IIAC, and a very good example of how the definition of IIAC is abused to convince people that Arrow's Theorem has more destructive power than it does.
Imagine this group of people:
1) Slightly less than half is crazy about chocolate, likes strawberry okay, and hates vanilla.
2) The rest like vanilla slightly more than chocolate, but likes both. However, some of them love strawberry (first choice), and some hate strawberry (last choice).
In the choice between chocolate and vanilla, vanilla wins. Introduce strawberry, and the lukewarm edge of vanilla is exposed - the greater utility of chocolate ends up winning.
Do you see? That's the secret flaw of IIAC right there: Sometimes a Condorcet Winner is not the candidate with the greatest utility. "Failing" IIAC can make a greater utility candidate win. In this example, that's actually a good thing. And if in some cases, failing IIAC is a good thing, then it isn't exactly a reliable criterion.
Curt
That's one way to look at it. Let me propose another.
I think Arrow's theorem *is* important. As the second law of thermodynamics states a fundamental physical constraint, so Arrow's theorem states a fundamental mathematical constraint. The second law of thermodynamics is not of much use in comparing the efficiencies of various automobiles, but it provides a limit to the thermodynamic efficiency of *any* automobile. Similarly, Arrow's theorem is not very useful for comparing ordinal election methods, but it clarifies a fundamental limitation of all possible ordinal methods.
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. So I think IIA is significant, but I put less value on it than on CC -- otherwise I'd give up on ordinal methods altogether.
While Arrow's theorem is not useful for comparing ordinal election methods, it *is* a point against ordinal methods in general as opposed to non-ordinal methods. An advocate of Approval, or even plurality, could make use of it to oppose IRV or even Condorcet. And he'd have a point.
--Russ ---- Election-methods mailing list - see http://electorama.com/em for list info
