Tag Archives: Game Theory

The Bottle Imp and the Limits of Logic

Logic! It’s fantastically useful stuff. Use it all the time for sorting out your options, thinking up plans, and generally making your life easier. There’s some very real limits to it, though, and whether an idea checks out logically doesn’t always have anything to do with its relevance to the real world. Here’s the test: can this idea be used to predict what will happen?

vintage photo of a dog in a top hat and white tie suit
Very dapper, but will this dog hunt? (Also proof that dogs have been putting up with the human impulse to dress them in clothes for a very long time.)

There’s a Robert Louis Stevenson story called The Bottle Imp. Go read it in this collection, here, if you like. No plot spoilers, but I will be discussing the premise of the story, so if you want to read it before we get to that, do. The main idea of the story is this: there’s a bottle that contains an evil imp. It can grant any wish except to prolong the bottle owner’s life, and if you die with the bottle in your possession, you go straight to Hell. The only way you can get rid of the bottle is to sell it to someone for less than you paid for it. Here’s where it gets interesting.

A display of ancient glass bottles of various murky clear shades
Some 17th Century glass bottles in a museum. Probably not full of evil, though.

Truth and Consequences

Let’s play a game, and think about the Bottle Imp problem logically. Eventually, there’s an ultimate loser: someone stuck with the bottle who bought it for a single penny, and they can’t sell it. So, following that, the next person up, who sold it to them, bought it for two cents, and must have known that they wouldn’t be able to sell it to someone for one cent. There must have been someone above them who got it for three, but should have known that they couldn’t sell it for two, because the person who got it for two would have to convince someone to take it for one, which nobody would ever do. Theoretically, nobody should ever take the bottle for any price, because the problem of not being able to sell it for a cent should cascade up the chain of prospective bottle owners. This is, of course, assuming that everyone involved is thinking logically (and whenever you hear that phrase, you should also assume that this perfectly spherical, frictionless dog hunts perfectly spherical, frictionless partridges in a vacuum).

The trouble here is that real people just aren’t rational actors, any more than real hunting dogs are spherical and frictionless. Realistically, everybody in the chain, down to perilously close to the bottom, is probably going to think “eh, I’ve got plenty of time, and I’m sure I can find some sucker to sell the bottle to” – and, in the main, they’d probably be right. The existence of the whole idea of gambling in general testifies to the idea that people – real people – generally do a terrible job of thinking logically and rationally. If the odds could really be in your favor in the long term, casinos wouldn’t exist.

a shack by a creek, with a man and a dog sitting outside
Las Vegas in 1895, before the gambling industry really took off.
a view of the Vegas strip in 2011. Reasonably recent.
A view of the Las Vegas strip. Practically a monument to the irrationality of humans. Keep feeding those one armed bandits, guys…

Sometimes, especially when dealing with real human behavior in the real world, logic does a truly wretched job of predicting real-world outcomes and decisions. There’s a distinction between logic, and actual utility. Most of the time, logic is very useful, but sometimes, especially when you’re dealing with questions of real human behavior, not so much.