Tyler Cowen is of course one of the primary reasons to be grateful that you live in the age of the Internet. But none of us is infallible, and I believe Tyler has stumbled in his account of Arrow’s Theorem. His example:
Let’s say you had two people on a desert island, John and Tom, and John wants jazz music on the radio and Tom wants rap. Furthermore any decision procedure must be consistent, in the sense of applying the same algorithm to other decisions. In this set-up (with a further assumption), there is only dictatorship, namely the rule that either “Tom gets his way” or “John gets his way.”
Not true. A rule (or, in Arrow’s language, a social welfare function) has to prescribe a choice not just today, but every day, even as Tom’s and John’s preferences might change from one day to another. So there are in fact 16 possible rules. One is “Tom always gets his way.” Another is “John always gets his way.” Another is “Always turn the radio to jazz”, which seems pretty unreasonable since it prescribes jazz even on days when Tom and John both prefer rap. Yet another is:
- If Tom and John agree, do whatever they agree on. If they disagree, turn the radio to jazz.
That last rule is particularly interesting because it satisfies every one of Arrow’s “reasonableness” criteria without anointing a dictator. What Arrow’s theorem says is that no non-dictatorial rule can meet all of those criteria.
Hold on a minute. I just gave you an example of a rule that meets all of Arrow’s criteria, and then told you that according to Arrow there is no such rule. What gives? What gives is the reason why Tyler’s example is irrelevant: Arrow’s theorem applies only when there are at least three options. With two voters and two options, the theorem fails and everything is copacetic.
In my own recent attempt to explain Arrow’s theorem, I assumed three voters and three options. It would have been simpler (and therefore better) to emulate Tyler by assuming only two voters (say Ann and Bob) arguing over three options (say Anchovies, Mushrooms and Pepperoni). Then you’re up against the fact that your rule must tell you what to do on days when their preferences run like this:
|First Choice||Anchovies||Mushrooms||Second Choice||Mushrooms||Pepperoni||Third Choice||Pepperoni||Anchovies|
On those days, you must either grant Alice a smidgen of dictatorial power by ranking Anchovies over Pepperoni even though she’s the only voter with that preference, or grant Bob a smidgen of dictatorial power by ranking Pepperoni over Anchovies. Once you’ve granted (say) Alice that smidgen of dictatorial power, Arrow’s argument demonstrates that — in order to satisfy his reasonableness criteria — you’ve got to grant her a bigger smidgen by ranking Anchovies over Pepperoni on any day when she has that preference. And then in order to continue satisfying his criteria, you’ve got to grant her yet another smidgen by ranking Anchovies over Mushrooms on any day when she has that preference. And then you’ve got to grant her another and another until finally you’ve made her an absolute dictator.
The details of this “argument by increasing smidgens” are in my earlier post, where you can just ignore the third voter (Charlie) to keep things a little simpler.
But Tom and John, living on Tyler’s island and facing only two choices, are exempt from all this and therefore an irrelevant diversion.