January 25, 2004

An Easy Envelope Problem

In the previous post, we saw that the two-envelope paradox arose from unrestricted application of this principle:

Take a partition P of the probability space. Suppose that, for every cell C of the partition, strategy S yields a greater EU [expected utility] than strategy T given that you're in cell C. Then strategy S is preferred over strategy T.

The principle can be proved to maximize EU whenever P is a finite partition, but it just can't be applied to every case in which P is an infinite partition.

But it seems like there are some cases in which you do want to apply the principle over an infinite partition. The problem is to come up with a principle that gets the right answer in these cases without yielding paradoxical or wrong results.

Take what I'll call the Extra Coin-Flip (the case Brian mentions in his first comment here:

God runs a St. Petersburg bet and puts the result in an envelope. An angel comes down and hands you the envelope and says, "You can take this envelope, and you'll get the number of utility units there are in the envelope. Or you do the following: Give me a util, and I'll flip a coin. If it comes up heads, you get what's in the envelope plus four utils; tails, you just get what's in the envelope, and lose your util."

Here, you get what's in the envelope no matter what happens. The only question is whether you're also going to spend a util to get a 1/2 chance of four utils. If you're an EU-maximizer, it seems obvious that you should, no matter what number is in the envelope.

But when I said "No matter what number is in the envelope," I introduced an infinite partition. And the moral of the two-envelope paradox was supposed to be that the decision principle breaks down when you're faced with infinite partitions. Except in the Extra Coin-Flip it seems as though the principle shouldn't break down--because the outcome of the St. Petersburg is just irrelevant to the gains or losses you can expect from taking the coin flip.

This case, like the original two-envelope and the St. Petersburg two-envelope, involve more than one indeterministic process:
In the original two-envelope problem, you have, first, the process of God choosing the number x such that x and 2x go into the envelopes, and second, the process of the angel choosing one of the envelopes to give to you.
In the St. Pete's two-envelope problem, you have, first and second, the process of God running the two St. Pete's and putting the results in the envelopes, and third, the process of the angel choosing one of the envelopes to give to you.
In the Extra Coin-Flip, you have first, the process of God running the St. Pete's, and second, the process of the angel flipping the coin.

In the Extra Coin-Flip, it seems possible to disentangle the payoffs of the coin-flip from the payoffs of the St. Petersburg. So you can evaluate the coin-flip independently, without worrying which of the infinitely many possible payoffs the St. P had. The challenge is to characterize this intuitive judgment somehow.

Next, my first stab at a solution.

Posted by Matt Weiner at January 25, 2004 03:30 PM