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

[Russ Paielli]

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