[EM] The issue of comments about Arrow's theorem
Alex Small
alex_small2002 at yahoo.com
Sun May 15 00:02:30 PDT 2005
>Arrow's result can best be reported as: Arrow proved that a few criteria
>that he likes are incompatible with eachother.
I think there's more to it than that. It might not sound like such a big deal if you think of his criteria as just some guy's pet concerns. But I think there's a greater significance.
First, his criteria don't sound terribly restrictive at first glance. They sound more like common sense than like stringent demands, or at least they did to me when I first learned of it. To find out that something so (apparently) simple is mathematically impossible is a very surprising insight.
Second, he demonstrated that one can make very general insights about election methods. You don't have to do it piece-meal, by writing down the description of a method, writing down a criterion, and seeing if the method satisfies the criterion. You can make general statements that apply to ANY method, or to a wide variety of methods.
I think it's rather powerful to realize that one can prove the general inconsistency of different criteria. It might not seem like such a big deal nowadays, now that we all know about Arrow's Theorem. But he wasn't just one guy proving that his pet criteria were incompatible. He was the first person (or at least the first widely-known person) to prove the general incompatibility of different criteria.
OK, so I hope we can agree that he's at least of historical significance. One could still legitimately ask how significant his work would be if some other general impossibility theorem had been proven before his work. I would suggest that his work is still pretty important precisely because the criteria are so seemingly simple. Admittedly, a rigorous definition of his criteria requires some care (especially for IIA), but the basic idea that removing a loser shouldn't change the outcome is a simple one. Then the details can be filled in later. So it's an easy finding to explain to beginners in the field.
Also, learning that a seemingly simple thing is impossible is always more impressive than learning that more elaborate tasks are impossible. For instance, suppose that somebody took the Condorcet Criterion plus 2 others, and proved that they are incompatible. Well, there's a hell of a lot of methods out there that don't satisfy Condorcet, including a lot of methods that some people might consider interesting for public elections, elections in private groups, or machine decision-making. On the other hand, non-dictatorship and Pareto span a much wider range of interesting election methods. Proving that this very wide range of methods can't satisfy something that's so seemingly simple makes the insight of much greater interest.
On a theoretical level, I'd make 2 other observations. The first is that most proofs that I've seen for Arrow's Theorem ultimately invoke the Condorcet paradox. That is interesting, because it shows that the consequences of Condorcet's paradox go much further than just the need for Condorcet methods to have a "backup" algorithm. Second, while I think that the Gibbard-Satterthwaite Theorem is of greater practical significance (because a lot of people complain about spoilers, lesser-of-two-evils, and other strategic aspects of voting), Arrow paved the way for that work.
Anyway, just my $0.02. I give Dr. Arrow mad props for his finding.
Alex
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