Have any of you ever heard of the two-envelope paradox? It’s a paradox so important that Wikipedia manages to have two articles on it: Two envelopes problem and Envelopes paradox. The only thing that puzzles me about it is that I’m having trouble seeing how it’s a paradox â€” unlike, say, the Monty Hall problem, the naive answer is the correct one.

1. I always thought that the “naive” answer is the correct answer to the “Monty Hall” problem. AFTER one knows that the car is not behind door #1, the chance of it being behind oneof the other two doors is 50%. New information changes the probabilitiies. Probabilities are always not natural phnoemena: sometimes they simply reflect our state of knowledge. After all, as a matter of “fact”, the “chance” that the car is behind door #2 is 100% if it is in fact behind door #2. Otherwise, the chance is 0%.

2. I always thought that the “naive” answer is the correct answer to the “Monty Hall” problem. AFTER one knows that the car is not behind door #1, the chance of it being behind oneof the other two doors is 50%. New information changes the probabilitiies. Probabilities are always not natural phnoemena: sometimes they simply reflect our state of knowledge. After all, as a matter of “fact”, the “chance” that the car is behind door #2 is 100% if it is in fact behind door #2. Otherwise, the chance is 0%.

As for the envelope “paradox”, this is one of many similar descriptions, such as the “wine/water” paradox. They all follow from the “Principle of Indifference”, which is a convention whereby equal probabilities are (arbitrarily) assigned to mutually exclusive, exhaustive uncertain outcomes. An exposition and explanation can be found in Gillies, “Philosophical Theories of Probability”, pp. 37ff.

3. You don’t need an indifference principle to get the two envelope paradox. Christian and List at the London School of Economics have pointed out (they might not have been the first ones) that even if you assume a non-uniform prior distribution, the paradox can still arise. Let’s assume that either envelope can contain \$2^n, for n at least 0. Let’s make the probability of the pair (2^n,2^(n+1)) be the same as the probability of the pair (2^(n+1),2^n), and make them each proportional to (2/3)^n. Then, given the value of the left envelope, you can see that the expected value of the right envelope is always greater, and vice versa.

Interestingly, if you choose the function differently (say that one has exactly sqrt(x^2-1) if the other has x), you can make the expectation of the other envelope always lower than the one you know about.

4. To my understanding the issue here is that attempts to make a decision based on expectation values are doomed to failure if the other assumptions we make imply that the underlying probability distribution has no expectation value. As distributions with fat tails are pretty popular in the finance world these days I wonder if there is a nice example of this ‘paradox’ that could be constructed for a real financial model involving a distribution with no expectation value.

5. As far as I understand, this is a paradox only if you subscribe to the bayesian probability theory, right? I am not familiar with that, but I find it very strange that just because we’re not certain of the value in an envelope, we are entitled to assign probabilities to it (as if the values can change while we’re making our decision). It also bothers me that if you choose both envelopes, the sum of the expected values falls short of 3A.

6. A small comment in response to Kenny Easwaran — “Christian and List” is one person, Christian List, a political scientist who gave our Computing Science Department a very interesting seminar on social choice theory a couple of years back.

7. Oops, I meant Franz Dietrich and Christian List – that explains why I couldn’t remember the first names!

Also, I’m interested in a bunch of these puzzles where expected values don’t make the decision for us, either because they’re both infinite, or because one is undefined or something. Alan Hajek has written a bunch on several of these, including Pascal’s Wager and his own Pasadena Paradox.

8. Is the paradox solely when the expected value for the distribution of envelope contents infinite (in which case it’s just a variant of the St. Petersburg paradox)? The stuff I’ve read seems to imply that the paradox holds more generally, and points to a puzzle in subjective probability in general.