[[Published in Analysis 62:155-57, 2002.]]
The Two-Envelope Paradox: I am presented with two envelopes A and B, and told that one contains twice as much money as the other. I am given envelope A, and offered the options of keeping envelope A or switching to B. What should I do?
I reason: (1) For any x, if I knew that A contained x, then the odds are even that B contains either 2x or x/2, so the expected amount in B would be 5x/4. So (2) for all x, if I knew that A contained x, I would have an expected gain in switching to B. So (3) I should switch to B. But this seems clearly wrong, as my information about A and B is symmetrical.
It is widely accepted that one's reasoning in this scenario should depend on one's initial probability distribution for the amounts in the envelopes. It is also widely accepted that if this distribution has a finite expected value, (1) and (2) are false, so the reasoning goes wrong. However, it is well-known that there are certain distributions with infinite expected value such that although (1) is false, (2) is still true. For such distributions, the paradox seems to arise as strongly as ever, and there is no widespread agreement on how to handle it.[*]
*[[For the existence of paradoxical distributions with infinite expected values, see Broome 1994. Chalmers 1994 gives a proof that the paradox does not arise for finite distributions, and notes the existence of infinite paradoxical distributions. I now think that the discussion there does not get to the bottom of the residual "paradox"; the current paper is a corrective.]]
To help diagnose the residual "paradox", I offer the following variant, obtained by mixing and matching two well-known scenarios:
The St. Petersburg Two-Envelope Paradox: I am presented with two envelopes, A and B. I am told that each of them contains an amount determined by the following procedure, performed separately for each envelope: a coin was flipped until it came up heads, and if it came up heads on the nth trial, 2^n is put into the envelope. This procedure is performed separately for each envelope. I am given envelope A, and offered the options of keeping A or switching to B. What should I do?
I reason: (0) Before opening the envelopes, the expected value in each is infinite. (1) For any x, if I knew that A contained x, then the expected value in B would still be infinite. So (2) for all x, if I knew that A contained x, I would have an expected gain in switching to B. So (3) I should switch to B. But this seems clearly wrong, as my information about A and B is symmetrical.
What is going on here? Obviously the problem lies in the step from (2) to (3). When a distribution over finite amounts has an infinite expected value, any specific result will be disappointing. It will then always be in one's interest to do things over, if given the opportunity. But it does not follow from this that before knowing the result, it is in one's interest to do things over. The general moral can be put decision-theoretically or probabilistically.
Decision-theoretically, the moral is that dominance reasoning is not generally valid. Say one has two choices A and B, and a parameter C whose value is unknown, on which the utility of A or B may depend. An unrestricted dominance principle says that if, given any specific value of C, A is preferable to B, then A is preferable to B overall. The St. Petersburg two-envelope paradox (taking C as the value in envelope A, and A and B as the respective choices) shows us that this principle is false.
Probabilistically, the moral is that one cannot discharge certain universally quantified claims about conditional expected values. An initially plausible principle holds that given any two unknown quantities C and D, then if E(D|C=x) > 0 for all x, then E(D) > 0. The St. Petersburg two-envelope paradox (taking C = A, and D = B-A, where A and B are the amounts in the two envelopes, and allowing infinite positive and negative expected values) shows that this principle is false.
What applies to the St. Petersburg two-envelope paradox applies equally to the original two-envelope paradox. In both cases, the reasoning from (2) to (3) tacitly relies on a principle such as those above. These principles are false, so the reasoning is invalid.
Why do the principles fail in these cases? In both cases, the overall expected gain from switching (B-A) can be represented as the sum of an infinite series in which the sums of both the positive and the negative terms diverge. Such a series has an undefined sum, but can always be partitioned into a new series with only positive terms (each of which is a sum of terms in the original series) and also into a new series with only negative terms. (In the envelope cases, one obtains such new series via partitions that hold fixed the value of A and the value of B respectively.) When this instability under partitioning is present, it is to be expected that dominance reasoning and the principles above break down.
These principles may be true under certain restrictions: e.g., a restriction to cases involving only finite expected values and absolutely convergent series (setting aside Newcomb-style cases, in which C is not independent of one's decision between A and B). There may be weaker restrictions that suffice, since there are plausibly some cases involving infinite expected values where dominance reasoning is valid. Articulating the weakest possible restrictions here is an important open question; but for now, what matters is that the principles do not hold in general.
The upshot is a disjunctive diagnosis of the two-envelope paradox. The expected value of the amount in the envelopes is either finite or infinite. If it is finite, then (1) and (2) are false: the paradoxical reasoning results from illicitly ignoring probability distributions. If it is infinite, then the step from (2) to (3) is invalid: the paradoxical reasoning relies on dominance principles or probabilistic principles that are false in the infinite case.[*]
*[[Apart from the illustration provided by the St. Petersburg two-envelope paradox, most of the issues discussed here have been discussed elsewhere: Dreier (forthcoming) discusses the breakdown of dominance reasoning; Norton (1998) discusses matters in the vicinity of the false probabilistic principle, and Clark and Shackel (2000) discuss the role of absolute convergence. Thanks to Michael Clark and Nicholas Shackel for comments and to Jamie Dreier for discussion.]]
Broome, J. 1995. The 2-Envelope Paradox. Analysis 55:6-11.
Chalmers, D.J. 1994. The two-envelope paradox: A complete analysis? [consc.net/papers/envelope.html]
Clark, M. & Shackel, N. 2000. The two-envelope paradox. Mind 109:415-42.
Dreier, J. (forthcoming). Boundless good.
Norton, J.D. 1998. When the sum of our expectations fails us: The
exchange paradox. Pacific Philosophical Quarterly 79:34-58.