Envelope paradox

The envelope paradox is a paradox of probability. You are given two indistinguishable envelopes, each of which contains a positive sum of money. One envelope contains twice as much as the other. You may open one envelope, examine its contents, and then, without opening the other, choose which envelope to take. Your aim is to maximise your profit.

Suppose you open one envelope and find in it the amount of money A. You reason as follows:

The other envelope may contain either ˝A or 2A

Because the envelopes were indistinguishable, either amount must be equally likely.

So the expected value of the money in the other envelope is ˝(˝A + 2A) = 1ĽA.

This is greater than A, so you gain by swapping.
But you could have gone through this same chain of reasoning before you opened the envelope and deduced the same result, that you should always take the other envelope. But that's clearly nonsense.

Solution

Step 2 in the argument above is flawed. It uses the unstated assumption that the pairs of amounts (˝A, A) and (A, 2A) are equally likely, for all values of A. But there is no probability distribution with this property. (In particular, there is no uniform probability distribution on the positive real numbers.) So some values of A must be more likely than others.

A second paradox

The flaw in the original statement of the problem doesn't rule out the possibility that there is some distribution of sums in the envelopes (but not a uniform distribution, since that's impossible) that makes the paradox work.

Suppose that the envelopes contain the non-negative integer sums 2ⁿ and 2ⁿ⁺¹ with probability e(1 − e)ⁿ for some fixed e < ˝. (This variation is due to Marcus Moore.)

Now of course there's a sensible strategy which guarantees a win, which is to swap only when the envelope you open contains 1, when you know that the other must contain 2. But you can apparently do better than that, for suppose you open the envelope and find 2ⁿ for n ≥ 1. Then the other envelope contains
:2ⁿ⁻¹ with probability e(1 − e)ⁿ⁻¹
:2ⁿ⁺¹ with probability e(1 − e)ⁿ
So your expected gain if you switch is
:˝(2ⁿe(1 − e)ⁿ − 2ⁿ⁻¹e(1 − e)ⁿ⁻¹)
:= 2ⁿ⁻²e(1 − 2e)
which is positive when e < ˝, so you expect to gain if you switch.

But once again, you can go through this reasoning before you open either envelope, and deduce that you should always choose the other envelope. This conclusion is just as clearly wrong as it was in the first paradox, but now the flaw noted above doesn't apply, as we have a specified distribution.

Solution to second paradox

The distribution in the statement of the second paradox has an infinite mean, so before you open any envelope the expected gain from switching is ∞ − ∞, which is not defined. So the conclusion is not in fact justified.

Non-Probability version of paradox

The envelope paradox can be expressed in a way which does not use probabilities. The following arguments lead to conflicting conclusions:

Let the amount in the envelope you chose be X. Then by swapping, if you gain you gain X but if you lose you lose X/2. So the amount you might gain is strictly greater than the amount you might lose.
Let the amounts in the envelopes be Y and 2Y. Now by swapping, if you gain you gain Y but if you lose you also lose Y. So the amount you might gain is equal to the amount you might lose.
Special case

When the envelope that you open contains 5.23 pounds, 7.99 dollars, 6.01 euro, or another such sum ending in an odd digit, you should always switch envelopes since the unopened envelope cannot contain half the amount of the current envelope, owing to the integer nature of money. Therefore it must contain twice the amount of the current envelope.

References

Mark Wainwright, On Large Numbers, in Eureka 50 (1990) 70–73.

Raymond Smullyan, Satan, Cantor and Infinity

Aaron S. Edlin, Forward Discount Bias, Nalebuff's Envelope Puzzle, and the Siegel Paradox in Foreign Exchange, in Topics in Theoretical Economics, Volume 2, Issue 1 2002 Article 3, PDF

Friedel Bolle, The Envelope Paradox, the Siegel Paradox, and the Impossibility of Random Walks in Equity and Financial Markets, PDF

Randall Barron, The paradox of the money pump: a resolution, in Maximum Entropy and Bayesian Methods, 1988 ed J Skilling.

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