Tyler Cowen asks which economic ideas are hardest to popularize. Arnold Kling nominates the Arrow Impossibility Theorem. Tyler responds with an attempt to popularize it. Alex Tabarrok weighs in with another. Here’s my own attempt:
Every day, Alice, Bob and Charlie split a pizza with one topping — anchovies, mushrooms or pepperoni. Their preference orderings change from day to day — some days Alice is in the mood for mushrooms, other days the very thought of mushrooms makes her queasy. Every day, they have to call in their first, second and third choice pizza orders. (The pizza delivery place insists that you specify your second and third choices in case they run out of something.) So Alice, Bob and Charlie need a method for translating their individual preferences to a pizza order.
Now as it happens, this past Tuesday, their preferences ran as follows:
|First Choice||Anchovies||Mushrooms||Pepperoni||Second Choice||Mushrooms||Pepperoni||Anchovies||Third Choice||Pepperoni||Anchovies||Mushrooms|
I’m not going to tell you in any detail what system these three were using to determine their order, but I will tell you that on Tuesday they reported Anchovies as their first choice.
On Wednesday, Alice ranked Anchovies over Pepperoni again. (I don’t remember whether she ranked them first and second, or first and third or second and third — but that doesn’t matter anyhow.) I don’t remember anything about Bob’s or Charlie’s rankings. But I don’t have to remember anything to know this: On Tuesday, only Alice ranked Anchovies over Pepperoni, whereas on Wednesday, Alice plus possibly some others ranked Anchovies over Pepperoni. So if the pizza order ranked Anchovies above Pepperoni on Tuesday, then it should certainly have ranked Anchovies over Pepperoni on Wednesday, no?
Any other outcome would have seemed unreasonable to Alice, Bob and Charlie, so when they designed their system, they designed it with the following feature:
- If we list Anchovies over Pepperoni on our order on one day, and if none of the people who prefer Anchovies over Pepperoni change their minds about that the next day, then we should list Anchovies over Pepperoni the next day as well.
Because this was built into their system (and because they’d listed Anchovies over Pepperoni on Tuesday when only Alice had that preference), they always listed Anchovies over Pepperoni on any day when Alice preferred Anchovies to Pepperoni. Alice, in other words, was sort of an “Anchovy/Pepperoni dicatator”.
Now on Thursday, Alice was in an Anchovies/Pepperoni/Mushroom sort of mood. I don’t remember much about Bob’s or Charlie’s moods except that they both favored Pepperoni over Mushrooms. Since everyone preferred Pepperoni to Mushrooms, of course Pepperoni was listed higher than Mushrooms in the pizza order. Once again, any other outcome would have seemed unreasonable to Alice, Bob and Charlie, so they’d designed their system with this feature also:
- Whenever we unanimously prefer Topping X to Topping Y, Topping X should rank higher than Topping Y on our order.
In fact, though I don’t remember everything about Thursday’s pizza order, I can figure it out. First, Alice (the Anchovy/Pepperoni dictator) preferred Anchovies to Pepperoni, so of course Anchovies ranked higher than Pepperoni. Second, everyone preferred Pepperoni to Mushrooms, so Pepperoni ranked higher than Mushrooms. And using my IQ test skills, I can figure out that Anchovies must have ranked higher than Mushrooms.
By the same logic — On any day when Alice prefers Anchovies/Pepperoni/Mushrooms in that order, and everyone else prefers Pepperoni to Mushrooms, Anchovies must rank higher than Mushrooms.
Now here’s another design feature these three agreed on:
- Our preferences about Pepperoni should not affect the relative ranking of Anchovies and Mushrooms.
Therefore, the green boldfaced statement above should remain true if we drop all the Pepperoni-related assumptions. In other words, On any day when Alice prefers Anchovies to Mushrooms, Anchovies must rank higher than Mushrooms. That is, Alice was not just an Anchovy-Pepperoni dictator. She was an Anchovy-Mushroom dicatator as well.
On Friday, Alice’s preferences ran Mushrooms/Anchovy/Pepperoni, while the other two both preferred Mushrooms to Anchovies. Since they all preferred Mushrooms to Anchovies, Mushrooms came out higher than Anchovies on the pizza order. Since Alice was an Anchovy/Pepperoni dictator, Anchovies came out higher than Pepperoni. Our IQ test skills tell us that Mushrooms came out higher than Pepperoni. And the same would be true on any day when Alice preferred Mushrooms/Anchovies/Pepperoni and everyone else preferred Mushrooms to Anchovies. But the ranking of Mushrooms vs. Pepperoni was designed to be unaffected by how anyone cared about Anchovies, so the Anchovy-related information can’t be relevant. This tells us that on any day when Alice prefers Mushrooms to Pepperoni, Mushrooms rank higher than Pepperoni. She’s not just an Anchovy/Pepperoni dictator and an Anchovy/Mushroom dictator; she’s a Mushroom/Pepperoni dictator also.
What we’ve discovered is that any Anchovy/Pepperoni dictator is also an Anchovy/Mushroom dictator and a Mushroom/Pepperoni dictator. Interchanging the names of the toppings, we could as easily have discovered that any Anchovy/Mushroom dictator (e.g. Alice!) is also an Anchovy/Pepperoni and a Pepperoni/Mushroom dictator — and so on until every pair of toppings appears. In other words, Alice is an absolute dictator. All of her preferences are fully reflected in the pizza order, every single day.
Now to get this ball rolling, I had to assume that Anchovies came out on top on Tuesday. But if Mushrooms had come out on top, I could have proved that Bob is an absolute dictator, and if Pepperoni had come out on top, I could have proved that Charlie is an absolute dictator. Regardless of what happened Tuesday, someone must be an absolute dictator.
And what if there had been more than three voters or more than three toppings? Then the argument gets more complicated to keep track of, but not more complicated in spirit. The conclusion remains the same — if you have a system that translates a collection of individual preference orderings into a single “social preference ordering”, and if that system is designed to have certain features that strike many people as reasonable, then the system anoints a dictator.
Arnold? Tyler? How did I do?