[EM] The issue of comments about Arrow's theorem
Russ Paielli
6049awj02 at sneakemail.com
Tue May 17 20:31:11 PDT 2005
Curt Siffert siffert-at-museworld.com |EMlist| wrote:
>
> 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
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