Setting Arrow's Impossibility Theorem Straight

Okay, by now, you might have noticed the econ blogosphere cooing over how awesome and insightful and useful the Arrow Impossibility Theorem is: Here, here, here, here, and here (in random order).

Um, to put it mildly ... no.

First, a summary of the theorem: let's say you want to convert individual preference rankings over outcomes into a social preference ranking that faithfully reflects these individual preferences as best as possible (i.e., create a voting system). You place a few "obvious" constraints on it that it voting system should meet, and it turns out -- you can't! Boo hoo, democracy sucks. (Well, in many senses, it does ... just not for this reason.)

This issue was discussed almost exactly one year ago on LessWrong. Long story short, the result has much less practical application than you might think. The requirements it asks of an aggregation system are far too strict. For one thing, the "determinism" requirement rules out the use of randomized tie-breakers. Keep in mind, there's always the possibility of some hopeless tangle involving a preference ordering like:

Person 1: A > B > C
Person 2: B > C > A
Person 3: C > A > B

Such preferences are completely intransitive, so no method of aggregation has any hope of being faithful. Normal people react to this by saying, "Okay, in the occasional pathological case, just use some tie-breaker that's not slanted in favor of any option -- in the end, it all averages out, so no problem". But Arrow's Theorem throws up its abstract arms and says, "Gosh, how hopeless. You can never satisfactorily aggregate preferences. Look how insightful I am!"

Needless to say, "We are not impressed."

It gets better though. "Black Belt Bayesian" makes the point that the "independence of irrelevant alternatives" (IIA) requirement is undesirable in the first place. (IIA means basically, if you remove some option, it should not change the aggregated ordering of the remaining options.) Why is it undesirable? Because so-called "irrelevant alternatives" aren't. Rather, they give evidence about the relative _strengths_ of preferences and therefore SHOULD affect the aggregated preference ordering!

Why was the econ blogosphere talking about Arrow's Theorem in the first place? Because someone had asked about underappreciated ideas in economics. Well, I think it's clear by now that this one doesn't suffer from a lack of deserved appreciation.

But what's even worse is that Amartya Sen's celebrated Liberal Paradox is viewed as a corrolary to the Arrow Theorem, and is just as ridiculous. It basically says you can't *both* respect people's rights *and* achieve Pareto optimality. Now, how do you imagine that works out? Well, you cheat by equating rights with obligations -- that is, you eliminate the possibility of people waiving a right when it's infringement would make everyone -- everyone -- weakly better off.

But who cares about that case? Not me. The very reason that rights allow for Pareto-optimality is because people can trade them as necessary when they find welfare-improving opportunities! If you equate "property rights in a specific apple" with "the obligation never to trade the apple away" ... well, you kinda throw a kink in all that.

As I said a year ago, if a transaction really is Pareto-efficient, then rights won't get in the way, because the relevant parties will waive the relevant rights! (Epic tongue-twister, too.)

Reassuringly, the folks on the opposite end of the ideological spectrum from me come to the same conclusion.

So are we set straight now?