## Tuesday, October 26, 2010

### 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?

jsalvati said...

Dear god thank you. The absurd volume of overinterpretation has been annoying the ba-jesus out of me. At least one other person can call bullshit when they see it.
I don't think I saw that LW post before, thanks for that.

Anonymous said...

First of all, the preferences you give in your example are not intransitive. Each individual's preferences are perfectly reasonable, the problem is in putting them all together to make a group decision. You don't even appreciate the question at hand, and think you are in a position to criticize people much, much smarter than you.

And this:
"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!"
You don't think there aren't hundreds of paper written about relaxing IIA? Are you that conceited that you think you can just walk up to a question that hundreds of people have studied, and throw down some brilliant objection that no one has previously thought of? Likewise, do you understand how IIA is intuitive, and an obvious place to start in designing voting rules? What if we were voting on building a police station, a firehouse, or a hospital, we decided on firehouse, but then had to vote again between firehouse and police station, and we chose police station? That would be strange, no?

Lastly, "if a transaction really is Pareto-efficient, then rights won't get in the way, because the relevant parties will waive the relevant rights". Let's say I own an object you want. Everyone knows I have zero value for it. But I still say to you, "Hey, I'll sell it to you for... five dollars?" If you don't value it over 5 dollars, we don't trade, and even though gains from trade were common knowledge, it doesn't happen. Are you like a college sophomore or something? All you set straight is how intellectually immature you are.

Silas Barta said...

@Anonymous:

1) Yes, I know what intransitive means. I meant that their aggregated preferences exhibit a generalized form of intransitivity and figured readers were capable of inferring the meaning. If you know a briefer way to characterize the kind of situation I showed showed, let me know.

2) I'm not objecting to the mathematical validity of the theorem (which would justify a critique of conceitedness) but to its practical applicability. In at least one sense, I'm obviously right -- people don't consider it fatal that a system has to use tiebreakers, but Arrow's Theorem no longer matches our intuitions when it deems this fatal.

Note that other mathematical theorems -- like the Bayes Theorem or Noisy Channel Coding Theorem, *do* have practical applicability despite the assumptions you need in order for them to work.

3) Yes, it is counterintuitive that a removal of an option would reverse the aggregate ordering. And that intuition is _wrong_, for the reasons BBB gave. To elaborate: since people pick their preferences with knowledge of the voting system, such reversals, though "strange" are necessary to capture all information about preference strength.

4) If the trade would really be a Pareto improvement, you would be willing to sell it for less than five dollars. The only thing stopping such a move is your inability to waive your right for less than \$5, which isn't a problem with rights, and therefore doesn't demonstrate how rights interfere with efficiency.

Nathan Byrd said...

You might find the following article interesting:

Donald Saari, The American Mathematical Monthly, Vol. 111, No. 5, May 2004. You can find it on gigapedia or perhaps your local library has a copy on file.

People do not generally consider the structure of voting to be as crucial as 'counting every vote' or making sure that campaigning is done fairly. It's actually the rules of the vote itself that are most important. He makes the following proposal:

"For a price, I will come to your organization before your next election. You tell me who you
want to win. After talking with members of your group, I will design a voting procedure that
involves all candidates in which your designated choice will be the sincere winner."

James Hanley said...

Sorry to come in almost a year late, but it looks to me like you've actually drawn Condorcet's paradox, not Arrow's impossiblity theorem (although, of course, they're related).

I think the argument that irrelevancy of alternatives is undesirable misses the point. Irrelevance of alternatives is an assumption of the model; it either holds or it doesn't, and "undesirability" doesn't really matter.

Finally, I don't follow the logic of arguing that people will give up weak rights in order to achieve a Pareto efficient outcome. They will only do so if they personally gain from sacrificing the right. If their is not change in their own position, it will take a side payment to get them to surrender it, no?

Silas Barta said...

Regarding your second paragraph, desirability of a criterion doesn't affect the validity of the proof, correct, but it does affect the practical relevance. And "a voting system is bad because it incorporates preference strengths" just doesn't seem all that important of a point.

Re 3rd para, if a move is Pareto-efficient, then it's still Pareto-efficient with the requisite side-payment. So to get Say's result, you do indeed have to replace "rights" with "obligations".

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