Wednesday, November 05, 2003

The Devil in the Details

Chris Bertram of Crooked Timber poses the following little gem of a thought problem:
You are in hell and facing an eternity of torment, but the devil offers you a way out, which you can take once and only once at any time from now on. Today, if you ask him to, the devil will toss a fair coin once and if it comes up heads you are free (but if tails then you face eternal torment with no possibility of reprieve). You don’t have to play today, though, because tomorrow the devil will make the deal slightly more favourable to you (and you know this): he’ll toss the coin twice but just one head will free you. The day after, the offer will improve further: 3 tosses with just one head needed. And so on (4 tosses, 5 tosses, ….1000 tosses …) for the rest of time if needed. So, given that the devil will give you better odds on every day after this one, but that you want to escape from hell some time, when should accept his offer?
I briefly thought this problem was going to be a paradox with no answer. But then I proved to myself that there is, in fact, a definite answer, so long as one discounts the future relative to the present, as human beings generally do. (More on that below.)

Suppose that d is your one-day discount rate, which tells you how much you value utility tomorrow relative to utility today. E.g., if d = 0.95, then tomorrow’s happiness is worth 95% of today’s happiness, from today’s perspective. Then if the following condition holds, you should take the bet on day N rather than waiting one more day:
(0.5)^N < (1 – d)/(1 – 0.5d)
(Here's the proof, if you really want it.) For instance, if d = .95, then the right-hand side of the condition is 0.095. On day 1, the left-hand side is 0.5, which is greater than 0.095, so you wait. On day 2, the left-hand side is 0.25, which is greater than 0.095, so you still wait. On day 3, the left-hand side is 0.125, so you still wait. But on day 4, the left-hand side is 0.0625, which is less than 0.095. Therefore, you should take the bet on the fourth day.

The more you value the future, the longer you should wait. If d = 0.99, then you should wait until day 6. If d = 0.999, you should wait until day 9. If d = 0.9999, you should wait until day 13.

Tyler Cowen, admitting he hasn’t gone through the math, guesses that you’d wait a very, very long (possibly infinite) time. But as you can see from the above, even if your valuation of the future is extremely close (0.9999) to your valuation of the present, you’d still take the bet within a couple of weeks. One of the commenters on Chris Bertram’s original post says, “The rational answer to this question is obviously that it depends on your discount factor, your relative utility for being in hell or not, and the extent to which you trust the devil.” But it turns out that your relative utility for being in hell or not – i.e., the disutility of torment – actually doesn’t matter. (I was surprised by this; the parameter representing how bad hell is canceled out of my calculations.)

There is a way to squeeze a paradox into the problem. If you do not discount the future at all relative to the present, then the condition above never holds, meaning you’d stay in hell forever. This outcome is mentioned by a couple of Chris’s commenters, and one notes that this is allegedly a paradox of rational choice: “The punch line was that if you were completely rational in the economic sense of the term – maximizing your expected utility – you’d spend forever in hell.” But it seems to me that this is not really a paradox brought on by rational choice, but by the general weirdness of infinity. In the calculations that produce the condition above, you have to divide by (1 – d) at some point, which means dividing by zero if you don’t discount the future. So you can’t really use the condition above. When you look at the original comparisons that led to the condition, it turns out that you get infinite terms on both sides, meaning that you’re indifferent between taking the bet and waiting at any point in time. This happens because no matter what you do, there’s a chance you’ll be in hell forever, and that’s an infinite amount of suffering. The probabilities don’t matter, because anything times infinity is still infinity. In any case, the correct conclusion is not that the rational non-discounter must remain in hell forever, but that he’s indifferent between doing so and taking the bet.

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