You’re stuck in a motel because of a business meeting. It’s 6 o’ the clock post meridian, and you’re hungry. The town in which you must linger is small and the dining choices stunted. It’s either a hamburger at Wendy’s, or a hamburger at McDonald’s. Wendy’s is better, you think, because their soft-serve frozen drink is called a “Frosty” and McDonald’s isn’t.
The phone book says Burger King (hamburger) is an option, and they have hats; plus the creepy guy they had in their ads in oddly intriguing. Burger King wins over Wendy’s. Yet your dear old mother was a McDonald’s lady from way back. So, if you had to choose—and you do—you would pick McDonald’s over Burger King.
Thus far, our situation in the head-to-head competitions is this:
W > MC,
BK > W,
MC > BK.
If we were to string these out into one line, we’d get this:
BK > W > MC > BK.
This is so because, from the first line, W > MC, and from the third line M > BK; but from the second, you decided BK > W. There is no joy to be found. Your decision is intransitive, irreparably so: no matter what you decide, you’ll be violating one of your preferences. You’ll be accepting another, of course: you’ll be happy and sad simultaneously.
You may think that they solution lies in looking outside the (such as it is) food quality of these dining establishments—perhaps one is twice as far as the other, and you’re low on gas—but this is not so. Because even factoring them in, it is still possible that the ordering remains the same. (You can always, of course, change your order.)
It’s not the system that broken, it’s that the utopia of deciding upon the optimal vote is impossible. Not always, mind; but sometimes, and more often than you would like. Intransitive systems are not that rare. Think of the childhood game of rock, paper, scissors (to which there is devoted at least one society). There is no best, or ideal solution that guarantees a winner in this game. Neither expected value, game theory, nor any other branch of mathematics can help here. But there may be help in multitudes.
Turns out that you have two comrades with you, similarly exiled: Bob and Randolph. And you, knowing the forlorn solution, have decided to reorder your preferences to BK > W > MC (you figure your mother will never know). You call up Bob and Randy and ask them theirs. Here’s how they respond:
You: BK > W > MC,
Bob: W > MC > BK,
Randy: MC > BK > W.
For first place, each restaurant has one vote. Again, no joy. But how about if we count the votes with what’s called the Borda system, after the chevalier, incongruously named the sailor? This assigns an arbitrary number of points for first place, fewer points for second, and so on, to produce a weighted count.
It’s easy to see—if not, look harder—that no matter how many points you pick for first, second, and third place, each dining establishment will have an equal weight for first place. Thus, the only way to decide is by somebody taking on the role of dictating where you’ll eat. And when he does—it may even be you that does the dictating—somebody is going to be unhappy: somebody is going to get their last place choice.
Similar dispiriting results can be found in any election in which there are more than two choices. Kenneth Arrow quantified one such result in a well known impossibility theorem, which said that if a voting system had three or more choices, and that it met certain eminently sensible desiderata, there always existed a “dictator”, by which he meant a person whose ordered preferences are substituted for all other voters’. This dictator might be an accidental and not a manipulative or domineering one.
The complexities of Arrow’s theorem are interesting, but in an important sense, they are beside the point. For we can prove that there will always be an unhappy voter or voters in nearly any election. We can show authoritatively, that is, that the human state is imperfectible, as measured by satisfaction.
Take any vote in which the choices are two or more and there are two or more voters. Then, it frequently occurs that unanimity is absent. And that’s it. The people on the losing end of the vote go away unhappy necessarily. This result is so banal that we scarcely notice it, but its consequences are deep and fundamental.
The unhappiness of the losers may be temporary, and may even be assuaged by “enlightening” them such that they change their vote (if only in their mind). But the unhappiness is there and it is real, and it may be enduring.