12 Nov 2009

Democracy Is More Important Than Avoiding Paradox

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I can’t stop reading Tyler! He knows so much, and his views are so similar to mine, and so when we disagree… Aaaaaaaa! I can’t stand it!!

While discussing a voting procedure that one of his readers had asked about, Tyler said:

The main question to get out of your head is whether or not range voting satisfies Arrow’s Impossibility Theorem. (In fact it doesn’t, most forms of range voting violate the independence of irrelevant alternatives, but don’t worry about that!). There’s no major reason why a democratic system should follow all of Arrow’s axioms as defined across universal domain, which means you have to rule out the very possibility of paradoxes. Can anyone do that? No, not even when you’re deciding which book to read next. (But should you stop reading? No.) We do, however, care if the system can:

1. Deliver decent economic growth and an acceptable level of civil liberties.

2. Build consensus and legitimacy going forward, and

3. Toss out the truly bad politicians.[Emphasis in Tyler’s original.]

If you don’t know what Arrow’s Theorem is, here’s a link [.pdf] but I’ll give you the thumbnail sketch. (Note that I actually have a proof of Arrow’s Theorem that I wrote up for undergrad students in my Game Theory class, so please don’t say in the comments that I’ve misunderstood the theorem. However, I admit that my discussion of the historical context of Arrow and his mission may be apocryphal; I’m simply repeating what an NYU professor told us in class one time. I haven’t read this in a book.)

OK so Arrow wanted to inject some rigor into the analysis of different social decision problems. Basically, if you have a collection of people with different preference rankings over possible outcomes (like distribution of wealth, whether women have to wear veils, whether Tom Palmer should have blogging privileges, etc.) then how do you aggregate those diverse preference rankings into one collective Social Welfare Ordering? More casually, how do you take everyone’s unique utility function and generate a social utility function? How do you know which “state of the world” is both feasible and achieves the highest level of “social well being”?

Economists had known for some time (e.g. the work of Condorcet) that there were problems with things like majority-rule voting. In fact for any proposed system, economists had found undesirable attributes. (E.g. with majority voting you can get cycling. If you have just 3 candidates and you use a two-stage election, the order can matter.)

OK so Arrow just wanted to rule out all the stupid voting procedures–the ones plagued by cycling etc.–so economists could focus on the remaining set of sensible ones, in order to decide which they liked best, which were most consistent with liberal values of tolerance etc.

Arrow came up with a bunch of axioms that, on the surface, seem pretty innocuous and all but one of them seem perfectly sensible for a voting system we can believe in. For example, one of the axioms says that if every single individual in society thinks outcome X is better than outcome Y, then the “social welfare ordering” had darn well better agree that outcome X is better than outcome Y. The other axioms are not as simple and self-evidently desirable, but they’re pretty innocuous as I say.

But guess what? Arrow found to his surprise–and again I’m just relying on what the NYU guy said, maybe he was embellishing and Arrow actually had a hunch to guide his axiom choice–that the set of aggregating rules (“voting procedures” if you will) that satisfied his axioms was empty! In other words, if you found a voting system that satisfied 3 of the axioms, it would necessarily violate the 4th. (BTW some expositions describe it as 5 axioms, where 1 is “universal domain,” but the way I learned it universal domain–meaning we don’t put any restrictions on the type of preferences people can have–was just assumed as part of the original problem, so that’s why I think Arrow’s Theorem only uses 4 axioms.)

Now that you have that background, you will understand my middle-aged-angry-man comment to Tyler’s post:

Tyler wrote:

There’s no major reason why a democratic system should follow all of Arrow’s axioms as defined across universal domain…

I think the major reason is, “The axioms all sounded perfectly innocuous and reasonable when Arrow first dreamed them up, since his original intent was just to rule out all the self-evidently undesirable voting procedures and focus on the sensible ones.”

And then when Arrow realized he had just ruled out every possible voting procedure, people moved the goalposts.

If it were called Arrow’s Reasonability Criteria, I think social democrats would be citing it all over the place to justify their desired reforms, just like they use Pareto Optimality. But since Arrow’s axioms should have been a game ender, out the window they go.

Now as far as Tyler’s analogy with book reading: To my knowledge, no one has come up with a proof showing that very sensible rules to use in book selection will contradict each other. If I’m understanding him, Tyler seems to want me to prove, “I can avoid paradox in my rules of book selection.” But that’s not what’s going on with voting. The issue isn’t that I’m demanding someone to justify voting. No, I’m demanding that proponents of voting explain why they are ignoring Arrow’s demonstration that the rules violate quite sensible features.

Last point: This really isn’t about voting. I’d have to think about it more carefully, but I think Tyler could come back and say, “OK then champ, please explain to us why you continue to espouse the wonders of a private property order, when its ‘rules’ necessarily violate Arrow’s Theorem?”

Assuming that’s correct, then perhaps it’s close to what Tyler was saying about books. But as you know, I like to err on the side of criticizing Tyler.

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