## January 25, 2004

### My First Shot at the Two-Envelope Problem

[Previous posts: 1, 2.]

Perhaps this is more like my second shot, as it's paraphrased from my comment on this thread of Brian's. Anyway:

We know that the Dominance Principle cannot be applied without restriction to processes that have an infinite number of possible outcomes. But we would also like to apply it in the presence of some such processes.

In the previous post, discussing the Extra Coin Flip, I argued that it was possible to disentangle the outcomes of the coin flip (two possibilities) from the outcomes of the St. Petersburg (infinitely many possibilities). So it should be OK to argue from "No matter what the St. Petersburg gives me, I expect to gain by taking the coin flip" to "I should take the coin flip" in this case. (All this is vague talk, and I reserve the right to take it back at any time.)

Here's one way in which we can be sure that the outcomes of the coin flip don't interfere with the outcomes of the St. Petersburg: The coin flip comes later. My first attempt at formulating a decision principle relies on that:

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

In the case of the Extra Coin Flip [Sherlock Holmes, call your office!], Q is the St. Petersburg and R is the coin flip. No matter what the outcome of the St. Petersburg, you've got a higher EU from taking the coin flip than not. Say the St. Petersburg gave you 4 utils; if the flip comes up heads, you get 8 utils; tails, you get 3 utils; the EU is 5.5 utils, better than the 4 utils you get from standing pat. Obviously it's the same no matter what the St. Petersburg. So the Chronological Ordering Principle says you should take the flip; good.

In the first two-envelope case, Q is God's choosing a rational number x and writing down x and 2x in the envelopes, and R is the angel's randomly giving you one. So the cells of PQ are determined by the values of x. Each cell consists of two equally likely outcomes. In one of those outcomes you have x and the angel has 2x, in the other you have 2x and the angel has x. In none of these cells do you expect to gain by switching envelopes. No matter what cell you're in, the EU of switching is 3x/2, and the EU of standing pat is 3x/2. So the Chronological Ordering Principle says it doesn't matter what you do; good.

In the St. Petersburg two-envelope case, Q is God's running the two St. Petersburgs and writing down the results in the two envelopes, and R is the angel's randomly giving you one. The cells of PQ are determined by the outcomes of the St. Petersburgs, say 2^m and 2^n. Each cell consists of two equally likely outcomes; in one you have 2^m and the angel has 2^n, in the other it's the other way around. In none of these cells do you expect to gain by switching envelopes. The EU of switching is (2^m + 2^n)/2, and so is the EU of standing pat. So the Chronological Ordering Principle says it doesn't matter what you do; good.

The Chronological Ordering Principle is three for three so far. Some selection bias is at work here, natch, since I designed it to deal with these three cases. Later* I'll talk about some cases it doesn't deal with so well, but first I think I'll post a bit about another variant of the two-envelope problem that it does handle.

*Even Josh Marshall doesn't say "More on this later" as much as I do!

Posted by Matt Weiner at January 25, 2004 03:49 PM