Opiniatrety

Two Envelopes: Theory Refined, Trouble Encountered

[The set-up for the Two-Envelope Problem is here.]

The Chronological Ordering Principle relied on the idea that, if a process with a finite number of possible outcomes takes place after a process with an infinite number of possible outcomes, the outcome of the finite process can't affect the outcome of the infinite process. So, if a strategy yields the best expected utility (EU) no matter what happened in the finite process, that strategy should be followed. The conditional EUs will be depend on the probability distribution of the finite process, but there's no way that can cause trouble.

The COP can't handle the Advance Coin Flip, because in the Advance Coin Flip the finite process (the coin flip) takes place before the infinite process (the St. Petersburg bet). But it seems as though finite process doesn't affect the outcome of the infinite process anyway. The real point is whether the infinite process is independent of the finite process, not whether it takes place first.

So let's try modifying the Chronological Ordering Principle to state that the outcome of the infinite process is independent of the outcome of the finite process, no matter the order in which they occur.

(What makes processes independent? I'm not sure, really. In fact, I'm not even sure how to define processes--you might be able to gerrymander them in all sorts of tricksy ways. At the end of the post we'll see how this might cause trouble.)

Anyway:

(Independent Process Principle) Suppose that probability space is partitioned by performing process Q, which has an infinite number of outcomes, and process R, which has a finite number of outcomes, and that the outcomes of Q are independent of the outcomes of R. Let P_Q be the partition induced by the outcomes of Q alone--that is, two ultimate outcomes are in the same cell of P_Q iff they result from the same outcome of Q. Suppose that, in every cell C of P_Q, strategy S yields a higher EU than strategy T given that you're in C. Then strategy S is preferred over strategy T.

The IPP yields the right answer for the Advance Coin Flip; take R to be the advance coin flip, and Q to be the St. Petersburg; no matter how Q comes out, your EU is higher (probabilizing over R) if you take the coin flip.

In the Traditional Two-Envelope Problem, can you take Q to be the process of writing down the number that you actually get and R to be the process of deciding whether you get the bigger number or the smaller number (x as opposed to 2x)? I think it's pretty clear that you can't. The way the problem is set up, the pair [x, 2x] is chosen first, and then you're given one of the two envelopes. So, if you try to set up Q as an infinitary process such that the outcome of Q is the number you get in your envelope, it's pretty clear that the outcomes of Q are not independent of the finite process of deciding which envelope you get. So the IPP is not applicable.

(We can reframe the traditional two-envelope problem as follows:

(R) one angel flips a coin to decide whether you get the bigger or the smaller envelope;
(Q) then another angel, not knowing the outcome of (R), randomly picks the x such that x is in the smaller envelope and 2x is in the bigger;
and then a third angel hands you the envelope and gives you the option of switching.

Then Q is clearly independent of R, you can apply the IPP to P_Q even though the COP doesn't apply, and the IPP tells you it doesn't matter whether you switch.)

But when we try to apply the IPP to the St. Petersburg two-envelope problem, we run up against what might be a gerrymandered process. The St. P two envelope problem can be framed either of the following ways:

(1) First (R) they decide whether to give you the red envelope or the blue envelope (1/2 chance of either). Then (Q) they run a St. Petersburg for the red envelope and a St. Petersburg for the blue envelope.

(2) First (R) they decide whether to give you the red envelope or the blue envelope (1/2 chance of either). Then (Q) they run a St. Petersburg for the envelope they're going to give you and a St. Petersburg for the one they're not going to give you.

Either (1) or (2) could be used to describe the same process. If (1) is the description at issue, then the IPP tells you that it doesn't matter whether you switch. The cells of P_Q are of the form [outcome of red St. P, outcome of blue St. P], and in each such cell the EU of switching and the EU of standing pat (probabilized over R) are the same.
But if (2) is the description at issue, the IPP yields no verdict. The cells of P_Q are of the form [outcome of the St. P you have, outcome of the St. P you don't]. Obviously, in some of those cells the EU of switching is positive, and in some the EU is negative.
Nor do I think that it's obvious that, in (2), Q is not independent of R.

Now, I'm not sure this is fatal. The IPP might best be phrased existentially--"If there is some way of partitioning probability space such that it is partitioned by process Q and process R" etc. "and that in every cell C of P_Q, strategy S yields a higher EU than strategy T given that you're in C. Then choose S over T." Then the possibility of framing the St.P2NV as in (1) would mean that the IPP says it doesn't matter whether you switch--and (2) doesn't yield a conflicting verdict. One problem is that I don't know whether it might be possible to come up with a problem that can be framed in two ways so as to yield a conflicting verdict. (Any proof might depend on a rigorous definition of "independent process.")

In fact, there's something even more worrying here. Take again the case in which they run a St. P for the one you're going to get, and another for the one you're not going to get. There's just no coin flip. It seems as though you have to describe this as in (2). And then, as above, the IPP doesn't yield an answer, though it should tell you (I think) that it doesn't matter whether you switch.

[AFTERTHOUGHT: Actually, I'm leaning towards the view that the utility of switching is undefined rather than zero. I'm certainly leaning toward that thought in the Traditional Two-Envelope problem. Maybe Brian's new problem will shed some light.]

Now, you could try to decompose this into two concurrent processes: one process that yields an unordered pair [outcome of the two St Ps], and one process on which, given that, you have a 50/50 chance of getting the bigger or smaller (if there's a difference). But it seems to me that the first process really--really really--wouldn't be independent of the second.

Suggestions?

Posted by Matt Weiner at January 30, 2004 12:21 PM