Why IIA?

Few pieces of work have managed to distract entire generations of social scientists. Ken Arrow’s work did just that. Initially published in 1953 and updated in 1963, Arrow’s impossibility theorem states that there is no preference aggregation rule that satisfies some desirable conditions of fairness.

One of these conditions is the requirement that social preferences between alternatives X and Y should depend only on the individual preferences between X and Y  — i.e. IIA (independent of irrelevant alternatives). Another way to put this is to say that if we add an option to the list of alternatives, the preferences between X and Y should not change. Stated as such, it is not straightforward why IIA should or should not be dumped.

Among the many reasons why IIA is considered important, one that I find particularly convincing is that this form of binary independence is equivalent to the strategy-proof requirement as defined in the Gibbard-Satterthwaite theorem (this was shown to be equivalent to the strong monotonicity assumption, see, for example, Eggers (2015)). The fact that IIA ignores big changes in profiles or ignores (essentially dismisses) the existence of perfect substitutes (McFadden 1974 — incidentally 10 years later he proposed a test with Hausmann for the IIA restriction for the case of multinomial logit), is not even that fatal in my opinion. What is far more demanding is this: if IIA holds, then we must drop the assumption that people have transitive preferences (Donald Saari is to be credited for this observation). Yet, is this enough of a reason to discard IIA? Some may say no as IIA is often credited itself with strategy-proofness. This, I argue, is simply a misinterpretation, for consider the following example (credits go again to Saari (2008)):

My true preference: A>C>B
Outcome: C>A>B but C>B is very close

Since A, my top choice, cannot win, I could instead vote strategically  B>C and thus create a cycle in the case of majority rule over triplets.

Essentially this happens because IIA and Pareto do not imply monotonicity, something I discuss in more detail in this post. This, I believe, becomes the first thing to put on a table when one is asked “Why IIA?”.