What "The People" "Want", Part Three: Voting Rules Gone Bad

Last time, I introduced Arrow's Theorem, which states, essentially, that we can't have our voting cake and eat it too. I also showed how plurality rule fails to give us one thing we’d like, transitivity.

Today I want look at the other classic problem: the Borda count and independence of irrelevant alternatives. First, let's review a little bit.

1) We want complete and transitive group preferences. Complete means a group can say it prefers option A to option B, B to A, or it’s indifferent. Transitive means that if the group prefers A to B and B to C, it also prefers A to C.

Note that preferences can be complete but not transitive; that was the case in the Clinton-Bush-Perot example from last time.

2) There are three things we'd like out of a democratic preference aggregation rule. First, universality: we'd like it respect all sorts of individual preferences. Second, unanimity: if everyone prefers A to B, the group does, too. Third, independence of irrelevant alternatives (IIA): how people feel about C doesn’t matter for the group’s choice between A and B.

3) Arrow's Theorem says that if we have each of these things, then the preference aggregation rule doesn't aggregate preferences at all: it's a dictatorship.

The Borda Count and IIA

Recall from the previous posts that the Borda count is like golf. Everybody ranks each alternative, and each time an alternative comes up first, it gets one point; second, two points; and so on, and the group prefers alternatives with fewer points to those with more. 

As before, we take it for granted that Borda respects universality.

Borda is complete. Everybody gets a number after summing up everyone's rankings, and the number tells you the preference relation for any two alternatives. For the same reason, Borda satisfies transitivity: if the group prefers Clinton to Bush and Bush to Perot, then Clinton’s number is lower than Bush’s, which is lower than Perot’s, so Clinton’s must also be lower than Perot’s. (Remember: lower scores are better in the Borda count, like golf and cycling.)

What about unanimity? Let's say for the sake of argument everybody prefers Bush to Perot. Then, everybody ranks Bush higher than Perot, meaning everybody gives Bush fewer points. Automatically, Bush gets fewer points in total. (Don’t believe me? Try it yourself, and remind me to explain one of the prettiest things in the universe: mathematical induction.) 

On to independence of irrelevant alternatives, or IIA. (I adapted this example from Don Saari’s Basic Geometry of Voting.) Let's suppose we look one more variation on the 1992 election. This time, we'll say that 2/5 of the people prefer Clinton to Bush to Perot, 2/5 prefer Bush to Clinton to Perot, and the rest—1/5—prefer Perot to Bush to Clinton. So that the calculation is simple, we'll say there are five people total, although things will work out for any number of people.

From the first two people, Clinton two first-place votes, corresponding to one point each, or two points total so far. From the second two, he gets two second-place votes, worth a total of four points. Finally, he gets one third-place vote worth three points. His total is then nine points. 

Similarly, Bush gets eight points—it's the same as Clinton, except that the Perot voter gives him two points instead of three. Perot gets a whopping 13 points—four third-place votes and one first-place vote.

Under this arrangement the group prefers Bush to Clinton to Perot, since 8 is less than 9 is less than 13. Using the Borda count, Bush wins the election. In particular, he beats Clinton. Keep that in mind.

Now, remember what IIA means. It means that the group preference between two alternatives, say Clinton and Bush, shouldn't depend on the preferences between these two and a third alternative, say Perot. To test that, let's suppose that the first 2/5, which had preferred Clinton to Bush to Perot, now prefers Clinton to Perot to Bush. We can view this as just another set of preferences, or we can view it as this group acting strategically—they really do prefer Bush to Perot, but in an attempt to ensure Clinton wins, they rank Bush lower on their lists—but either way, the key thing is that no one's preferences over Clinton and Bush have changed. Let's see what happens.

No one has changed Clinton's position in her list, so his score is the same: 9 points.

Perot now has two second-place votes, two third-place votes, and one first-place vote, for a total of 11 points.

Finally, Bush now has two third-place votes, two first-place votes, and one second-place vote, for a total of...interesting...10 points.

Bush now loses to Clinton, even though no individual's preferences over Bush and Clinton changed. Borda violates IIA.

I've commented on the ups and downs of Borda before, in this post in response to a reader's question and this post on a Borda-like system in the Bay Area. Borda definitely has some things going for it. You can always reach a decision (because of the ranking), and it has a habit of selecting alternatives everyone at least kinda of likes. Plurality has a sort of opposite property: a candidate can win, even if most people prefer someone—even anyone—else, as long as not everybody agrees on other candidates.

Stay tuned, as always.

Feeling Blue? Try a Dose of Blue Light at New Scientist

After a bit of a lull, it's been a busy week (and day) at Nathan Explains Science. This afternoon I have a new story over at New Scientist (link below) on using a blue-light activated, algae-derived protein called Channel Rhodopsin 2 (ChR2) to cure depression in mice.

(While you're checking things out, be sure to check out the posts from earlier today and yesterday, about flying—sort of—with light, more on basic problems with voting, and the first installment of Fielding Reader Questions.)

The gist is that a bunch of neuroscientists took depressed mice and implanted ChR2 in the medial prefrontal cortices (mPFCs) of their brains. They first "depressed" them—defining depression in mice is problematic, so they actually induced "social defeat stress—by bullying them. Perfectly nice lab mice had to endure the taunts of a bully mouse for ten days straight and had to interact with the bully directly for five minutes a day, at the end of which they were submissive, avoided contact, and so forth. It was, in short, kind of sad.

But then the happy part (before the unpleasant part that follows all such studies): shining blue light via an optical fiber on the mice's mPFCs, the team activated the ChR2, which, the biophysicists in the audience will know, improves ion flow and essentially helps depressed neuron firing along. And behold: the mice weren't depressed any more. Pretty awesome.

Here's the story.

Fielding Reader Questions: The Strength of Voter Preference

Reader F. Tyler asks the following (slightly edited) question regarding the Borda count, a voting method in which people rank alternatives, higher-ranked alternatives get fewer points, and the option with the fewest points wins, just like golf (see this post and its follow-up on basic problems with voting):

"But isn't there a problem with the Borda count assuming that these preferences are of ratio interval, when they are, in fact, simply ordinal? If I'm voting for Perot in 1992, I'd MUCH prefer Bush, and HATE to see Clinton elected. Similarly, in 2000 I cast my vote for Nader, and would be ok with Gore, but Bush Jr is the worst. Right?

There isn't a consistent ratio presented. Maybe if I gave Nader a 1, Gore a 3 and Bush a 10, it would reflect my preferences, but the Borda count doesn't seem to allow that."

F. Tyler is absolutely right. First, a little background. Economists (of the academic variety) sometimes think of preferences as ordinal, i.e., they are just a ranking and nothing else. But preferences may also be cardinal, i.e., they reflect how much you prefer one option over another. Although Borda argued his method as capturing some of the latter, voters still just rank alternatives, meaning their preferences are treated as ordinal.

There are methods of voting that attempt to capture cardinal preferences, or utilities. For example, in one method, each voter has a certain number of points she can distribute among candidates. In Tyler's (2000) example, if he had 100 points, he could give Nader 70 points, Gore 23, and Bush 7, roughly reflecting the strength of his preferences (assuming his Borda-like scheme above translated into liking Nader three times as much as Gore and ten times as much as Bush).

However, even this method is problematic because of so-called interpersonal utility comparisons. The problem is, there isn't any obvious way to compare how strongly one person feels about something to another person's feelings. People try—there are those who erroneously believe you can do it by asking people how much they would pay for a certain choice over another, problematic because, despite what misguided Chicago-school acolytes and stock brokers will tell you, not everyone cares about money equally—but really, nobody knows a theoretically sound way to do it.

An interesting question is whether such a method would be susceptible to Arrow's Theorem problems, as other methods are. My guess is that it would be, but I don't have my textbooks in front of me and don't recall the answer perfectly. As always, stay tuned. In the not-too-distant future, I'll talk about arguments for and against voting rules such as the Borda count.

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.

Light "Flight" at ScienceNOW

Nifty brief on using the momentum carried by light rays to get a piece of glass to move perpendicular to the light. Puzzled? Read more here. I'll give a more thorough explanation later today.


- Posted using BlogPress from my iPhone