How rational are you? I once posted a test, based on ideas of the economist Maurice Allais, that most people seem to fail. Today’s test, based on ideas of the economist Dick Zeckhauser and the philosopher Richard Jeffrey, is one that almost everyone fails.
Suppose you’ve somehow found yourself in a game of Russian Roulette. Russian roulette is not, perhaps, the most rational of games to be playing in the first place, so let’s suppose you’ve been forced to play.
Question 1: At the moment, there are two bullets in the six-shooter pointed at your head. How much would you pay to remove both bullets and play with an empty chamber?
Question 2: At the moment, there are four bullets in the six-shooter. How much would you pay to remove one of them and play with a half-full chamber?
In case it’s hard for you to come up with specific numbers, let’s ask a simpler question:
The Big Question: Which would you pay more for — the right to remove two bullets out of two, or the right to remove one bullet out of four?
The question is to be answered on the assumption that you have no heirs you care about, so money has no value to you after you’re dead.
Almost everybody says they’d pay more in the first case than the second. Arguably, that means that in this scenario almost nobody is rational — because a rational person would give the same answer to both questions.
The reason for that is not immediately obvious. To understand it, you’ve got to think about four questions:
Question A: You’re playing with a six-shooter that contains two bullets. How much would you pay to remove them both? (This is the same as Question 1.)
Question B: You’re playing with a three-shooter that contains one bullet. How much would you pay to remove that bullet?
Question C: There’s a 50% chance you’ll be summarily executed and a 50% chance you’ll be forced to play Russian roulette with a three-shooter containing one bullet. How much would you pay to remove that bullet?
Question D: You’re playing with a six-shooter that contains four bullets. How much would you pay to remove one of them? (This is the same as Question 2.)
Now here comes the argument:
- In Questions A and B you are facing a 1/3 chance of death, and in each case you are offered the opportunity to escape that chance of death completely. Therefore they’re really the same question and they should have the same answer.
- In Question C, half the time you’re dead anyway. The other half the time you’re right back in Question B. So surely questions C and B should have the same answer.
- In Question D, there are three bullets that aren’t for sale. 50% of the time, one of those bullets will come up and you’re dead. The other 50% of the time, you’re playing Russian roulette with the three remaining chambers, one of which contains a bullet. Therefore Question D is exactly like Question C, and these questions should have the same answer.
Okay, then. If Questions A and B should have the same answer, and Questions B and C should have the same answer, and Questions C and D should have the same answer — then surely Questions A and D should have the same answer! But these, of course, are exactly the two questions we started with.
So. Did you pass the test? I for one did not. That leaves me (and also you, if you failed along with me) two options. Either we can maintain that there’s some flaw in the above argument — some way in which it fails to capture the “right” meaning of rationality — or we can conclude that we don’t always make good decisions, and that meditating on our failures can help us make better decisions in the future (including in situations more likely to arise than forced Russian Roulette). I am mostly in the latter camp. How about you?