Tuesday, December 7, 2010

What "The People" "Want", Part Two: It Gets Worse.

Last time I talked about what the people want, there were two main points.

First, there are always more than two options. The point of this is that if even in seemingly two-party systems such as that here in the US, there are so-called third parties, and the presence of third-party candidates is sometimes consequential. The big scary example is Allende in Chile in the 1970s; the less scary but more proximate example (for US readers) is Perot in the 1992 US presidential election.

Second, once there are more than two options, what the people "want" is hard to define. Using the Borda count and plurality rule voting methods, I showed how society's first choice depends on the manner in which people decide.

In the academic literature on "social choice," as it's called, the issue is one of preference aggregation and whether it's possible to aggregate preferences in a rational, fair way.

Today, I'll start with a statement of Arrow's Theorem, which answers that question with a resounding "No, it's not possible. Sorry." I'll follow with some intuition, and in future posts I'll flesh out the ideas.

Rational Preferences
To state the theorem, we need first to talk about rational preferences. A person with rational preferences has complete and transitive preferences, meaning 1) she can state which of any two options she prefers or that she's indifferent and 2) if she prefers option A to option B and she prefers option B to option C, she must also prefer option A to option C.

For example, she either prefers Bush to Clinton, Clinton to Bush, or is indifferent, and so on for every pair of candidates. Furthermore, if she prefers Bush to Clinton and Clinton to Perot, she must prefer Bush to Perot. If she preferred Perot to Bush, there would be a cycle: since she prefers Bush to Clinton and Clinton to Perot and Perot to Bush. So who does she prefer? No one. Everyone. This is, you know, a problem from a philosophical point of view.

To avoid the philosophical problem, we're going to assume everyone has rational preferences. I'll talk sometime in the future about the fact that people don't actually have rational preferences, but keep in mind that Arrow's Theorem is a sort of "this is as good as it gets" kind of result: even if people are rational, we'll still have problems, as you'll see below. (I owe this observation to Andy Rutten, a political science instructor at Stanford.)

A Statement of Arrow's Theorem
Here are five conditions on a preference aggregation rule, which is just a ranking scheme for social preferences—the people prefer A to B, B to C, etc.—that seem like they are necessary to be fair and sensible.

1) Universality (aka Unrestricted Domain): what this means is that people can have whatever preferences they have and they can still participate. This one is here because it seems rather undemocratic to say, "you're a Democrat, you can't vote" or "you're a Tory, you can't shout in parliament."

2) Completeness: the society can choose between any two alternatives or is indifferent, just like individuals.

These two are often taken to be so fundamental that formal statements and proofs take them for granted; as we'll see in the future, restricted domain plays a big role in solving the problems Arrow's Theorem raises.

3) Transitivity. If social preferences aren't transitive, there can be cycles, and that means society can't decide. If we as a society prefer Bush to Clinton to Perot to Bush, we can't pick a president.

Aside: If you're in most any democracy, you're very, very familiar with a voting method that violates transitivity, as you'll see below. Whoops. It's important to understand that this means there's a violation for some set of voter preferences, but not all sets of preferences. But, unrestricted domain means we have to consider all sets of voter preferences.

4) Unanimity (aka the weak Pareto property): if everyone prefers one thing to another, the group prefers it. Of fairly obvious importance—it would be weird if everyone favored Bush to Clinton, but we somehow elected Clinton—and hard to violate in practice, though not impossible.

5) Independence of Irrelevant Alternatives (IIA): the choice between two alternatives doesn't depend on how people feel about a third. We've seen an example of this already in Part One...see if you can figure out where.

Arrow's Theorem is the following somewhat frightening result: Any preference aggregation rule that respects universality transitivity, unanimity, and independence of irrelevant alternatives is a dictatorship.

That's right. Even assuming rational voters, you can't guarantee a group can choose between two options in a fair, rational manner without making someone a dictator—kind of a contradiction, if you think about it. And that's the point: you can't have everything you'd hope to. Now, some intuition for two of the more easily violated rules.

Intuition: Transitivity
Let's consider a rule that satisfies completeness, unrestricted domain, unanimity, and IIA: plurality rule. Plurality rule means that if more people prefer, say, Clinton to Bush than the other way around, society prefers Clinton to Bush, and, were these the only two options, would elect Clinton. Note that plurality does not mean majority—we don't need more than half, just more than the other choice.

Plurality rule automatically respects completeness, since it's always true you can come up with a social preference (there are rules where you can't), and IIA, since the social preference depends directly and only on the comparison of two alternatives. It respects unanimity for the same reason: if everyone prefers Clinton to Bush, then more people prefer Clinton to Bush, and hence the group prefers Clinton to Bush. Because it doesn't refer to any specific set of allowed preferences, plurality rule respects universality. Finally, plurality rule is clearly not a dictatorship, since it depends on the preferences of more than one person.

That leaves transitivity, and here's the classic example. Suppose a third of the people prefer Clinton to Bush to Perot, a third prefer Bush to Perot to Clinton, and a third prefer Perot to Clinton to Bush. Then, two thirds of the people prefer Clinton to Bush (combine the first and third groups), so the group prefers Clinton to Bush. For the same reason, the group prefers Bush to Perot (combine the first and second groups) and Perot to Clinton (combine the second and third groups). Whoops. A cycle. Transitivity is violated.

You might say "well, people wouldn't actually have those preferences." This is where universality comes in. People might have those preferences, and if so, the rule for making decisions should respect them. On the other hand, if we could show that people would never have certain kinds of preferences, maybe we could restrict the preferences we consider. One way to do so is by using "spatial preferences," which I'll discuss eventually.

Yikes. This is getting long. Next time, I'll demonstrate a violation of independence of irrelevant alternatives that respects the other requirements—it involves the Borda count—and give a rough proof of Arrow's Theorem. Stay tuned for the next episode.