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.