The solution to yesterday’s rationality test:
This one is much much simpler (and much less infuriating) than some of our earlier rationality puzzles (e.g. here and especially here), but it has a good pedigree, having come to me from my student Tallis Moore, who found it in a paper of Armen Alchian, who attibutes it to the Nobel prizewinner Harry Markowitz.
Several commenters got it exactly right, but whenever possible, I prefer an explanation that invokes cats and dogs. So: Suppose I give you a choice between A) a cat, B) a dog, and C) a coin flip to determine which pet you’ll get:
It’s perfectly rational to prefer the cat to the dog, and perfectly rational to prefer the dog to the cat, but (according to the traditional definition of rationality) quite indefensible to prefer the coin flip to either.
After all, if you like dogs and cats equally, then all three choices are equally good. If you like dogs more than cats, then you’ll prefer the dog to the coin flip. If you like cats more than dogs, you’ll prefer the cat to the coin flip. That seems to cover all the possibilities, and in no case does the coin flip come out on top.
Now replace the cat with yesterday’s Urn A, which contains two red balls (worth $1000 each) and 1998 black balls (worth $0). Replace the dog with Urn B, which contains twenty blue balls (worth $100 each) and 1980 black balls. Which do you prefer: Urn A, Urn B, or a coin flip to choose between the urns? As with the cats and dogs, it’s hard to justify the coin flip.
But Urn C is a coin flip between Urns A and B. Here’s why:
- Mixing Urns A and B together and then picking a ball is equivalent to flipping a coin to choose between Urns A and B — either way you’ve got a 50/50 chance of a random ball from A and a 50/50 chance of a random ball from B.
- After you’ve mixed up the balls, you might as well remove exactly half the reds, half the blues and half the blacks — this has zero effect, after all, on any of the relevant probabilities.
- But mixing Urns A and B, and then taking out half the balls of each color, produces exactly: Urn C.
So Urn C, like the coin flip, is the irrational choice. You can prefer A, you can prefer B, and you can like all three urns equally, but you can’t prefer C.
Of course some people do prefer C, which, as always with these things, leaves room for multiple interpretations. Maybe economists have too crabbed a definition of irrationality — or maybe there’s a lot of irrationality out there, which economists can strive to cure. What’s your take on this?