Can ignorance be bliss?
There is allegedly a tradition of issuing a blank cartridge to one (randomly chosen) member of each firing squad, so that no shooter knows for certain that he contributed to a death. Let’s assume that tradition really exists and let’s assume that it exists because the shooters want it. Does that prove that shooters (at least in some instances) value ignorance?
Not necessarily. It might just mean that each shooter prefers a 5/6 chance of firing a real bullet over a 100% chance of firing a real bullet. That’s not the same thing as preferring to be ignorant.
So here’s the key experiment. Offer the shooters a choice:
Under Policy X, one (randomly chosen but unidentified) shooter has a blank. Everyone fires, and nobody ever knows who fired the blank.
Under Policy Y, one randomly chosen shooter is excused from the firing squad before the triggers get pulled. Everyone else fires real bullets.
Under either policy, you (a member of the firing squad) wake up on execution morning with a 5/6 chance of firing a live bullet into a live person. If that probability is all you care about, you’ll be indifferent between the policies. But if you want to guarantee your own ignorance — so you’ll never be in a position of knowing you fired a real bullet — then you’ll prefer Policy X.
Now let’s play a game. I’m going to put 89 red balls, 10 black balls, and one white ball in an urn. The 89 red balls are all labeled “YOU LOSE”. As for the rest — it’s your choice. I can label them all “one million dollars”, or I can label the blacks “five million dollars” and the white “YOU LOSE”. After they’re labeled, you draw a ball and you get the corresponding prize.
Question: Have I given you enough information to make a choice? Not if you’re a lover of ignorance! In that case, you’ll want to know which protocol I plan to follow:
Under Protocol X, you draw a ball, observe it, and win your prize.
Under Protocol Y, you’re blindfolded when you draw the ball. I tell you (honestly) what your prize is, but I don’t tell you the color of the ball.
Now suppose you’ve chosen to label the white ball “YOU LOSE”. Under Protocol X, if that white ball comes up, you’re going to majorly kick yourself. Under Protocol Y, if the white ball comes up, you’re going to be told you lost, and you’re going to be able to walk away thinking “well, it was probably a red ball anyway, so this wasn’t my fault”. With Protocol B, you get your guaranteed ignorance.
Question 1: If the reds are all labeled “one million dollars”, what labels do you want on the blacks and on the white?
Question 2: If the reds are all labeled “YOU LOSE”, what labels do you want on the blacks and on the white?
These are equivalent to Questions 1 and 2 in our first post on this subject — except for one thing. Those questions were worded in such a way as to suggest that we’ll be using Protocol Y. In question 2, if you draw a ball that says “YOU LOSE”, all you know is that you lose. You don’t know whether it’s a red ball (which was going to say “YOU LOSE” anyway) or the white ball (which only says “YOU LOSE” because of the choice you made).
According to the way economists usually think about rationality — according, that is, to the axioms written down by von Neumann and Morgenstern — any rational person will answer Question 1 and Question 2 the same way. That’s because the usual axioms only allow you to care about outcomes and the probabilities of those outcomes. They don’t allow you to care about things like staying ignorant.
Okay, then, suppose we tweak the axioms so you are allowed to prefer ignorance. Does that rescue you from the irrationality charge? Answer: It depends. If we’re using Protocol Y, then yes, you are now allowed to give different answers to the two questions — because Protocol Y offers you a shot at blissful ignorance when you choose the risky option in Question 2 but not in Question 1. That’s an important difference. Importantly different questions can merit importantly different answers.
On the other hand, if we’re using Protocol X, you can’t get off so easily. Under Protocol X, you don’t get the blessing of ignorance in either case, so there’s no remaining excuse for inconsistent answers.
Whew. This was going to be my last post on this subject, but a) I’m not done and b) this seems more than enough to digest for now. So I’ll come back to this once more after this has had a few days to sink in.