Arrow's Impossibility Theorem and IIA (Signifying Nothing: how does saying nothing at all become so loud?)

Tuesday, 18 November 2003

Arrow's Impossibility Theorem and IIA

Steve Verdon explains Arrow’s Impossibility Theorem, one of the most important theorems of economics and “rat choice” political science. Unfortunately, he doesn’t explain one of the key assumptions—the independence of irrelevant alternatives or IIA assumption—in much detail, which is a shame because I’ve never found a good explanation of it that doesn’t talk about the colors of buses. (It’s often called the “red bus, blue bus” problem for that very reason—that’s the classic example used to explain IIA, which leads most people to correctly ask, “but what if my problem has nothing to do with buses?”) Despite that (very insignificant) shortcoming, it’s an interesting post.

Incidentally, IIA is important not just because of Arrow’s theorem, but also because it underlies a methodological debate over the appropriate estimator for regression equations that include a categorical dependent variable—such as when you want to come up with estimates of the effects of individual attributes on vote choice among three or more options. The preferred estimator in many situations, multinomial logit (which has the nice property of being tractable via standard numerical minimization techniques like Newton-Raphson), has the property that IIA must not be violated for its estimates to be valid, but nobody ever has been able to demonstrate (to my satisfaction, at least) that IIA is actually violated in practice. This is an important question because there are a gazillion other estimators out there that don’t suffer from the IIA assumption, but can’t be estimated directly (for example, see this implementation of multinomial probit, which doesn’t have the IIA problem but is fragile without including choice-specific variables, or mixed logit).