Ordered Envelopes

Suppose a man hands us an envelope containing a check for some amount of money, and then he holds up another envelope and informs us that it contains a check for either twice or half as much, and that the two possibilities are equally probable, and asks if we would like to switch. Should we switch?

The question is underspecified. Consider, for example, the following three different cases.

Case 1:  The man wrote a check for a certain amount of money, X, and placed it in one envelope, and wrote a check for twice that amount, 2X, and placed it in the other envelope.  Then he shuffled the envelopes and randomly selected one of them to hand to us first, and then offers us the chance to switch to the other envelope.

Case 2:  The man wrote a check for a certain amount of money, X, and placed it in the first envelope, and then wrote checks for 2X and X/2, and placed them in two other envelopes.  Then he shuffled those two other envelopes, and randomly threw one of them away. Then he hands us the first envelope (containing X), and offers us the chance to switch to the remaining envelope.

Case 3:  This is the same as Case 2, except the man first offers us the envelope that contains either 2X or X/2, and then offers to let us switch to the envelope with X.

In each of these three cases the man is perfectly correct in saying that the second envelope he offers to us is equally likely to have twice or half as much money as the first. (This is true both from the standpoint of his knowledge and from the standpoint of the frequency of outcomes in repeated trials under the stated conditions.) However, knowing what the man knows about how the envelopes were prepared, the expected benefit of switching in the three cases is different, as summarized below, where “E” signifies the expected value.

We, as the recipients, are not told how the envelopes were prepared. The man knows, but he has merely informed us that the second offered envelope is equally likely to have twice or half as much as the first. This condition is satisfied in all three cases. However, the ratio ϕ = E₂/E₁ of the expectations for the two envelopes in the three cases is 1, 5/4, and 4/5 respectively. Hence, the man (who knows how the envelopes were prepared) knows that in the first case the expectations are equal, and in the second case the expectations favor switching, and in the third case the expectations favor not switching.

More generally, the man could prepare the envelopes by first placing 2X and X in two envelopes, and placing qX and 2qX in two other envelopes. Then he places the pairs (retaining knowledge of the ordering of each pair) in two boxes and shuffles the boxes. He randomly selects a box and takes the two envelopes (in order) as the first and second envelopes. Thus we have the situation

The two rows each have probability 0.5, so there is an equal probability that the second envelope contains half or twice as much as the first, but the ratio of expectations is

For sufficiently large q the ratio approaches 2, whereas for sufficiently small q the ratio approaches 1/2. This shows that the man can prepare the envelopes in such a way that the ratio of expectations for the two envelopes is anywhere from 1/2 to 2, while satisfying the condition that the second envelope is equally likely to have half or twice as much as the first. Our previous three cases correspond to q = 1, 2, and 1/2 respectively.

This puzzle superficially resembles the famous “Two Envelope Problem”, but in our scenario the envelopes are ordered and need not be symmetrical, because they can be prepared differently. Also, since the discussion mentions “switching”, people sometimes jump to the conclusion that this puzzle is related to the “Monty Hall Problem”, but it is actually quite different, as there is no initial choice on the part of the recipient, and the envelopes have a defined probability of being double or half. The only similarity is that the answer is indeterminate and depends on the conditional actions of the man (like the host in Monte Hall) that are unknown to the recipient.

One can also cite a connection with the famous (albeit silly) “wine-and-water paradox”, and naïve applications of the so-called principle of indifference. Recall that this “paradox” concerns a jug containing a mixture of wine and water in the ratio x (= wine/water) between 1/3 and 3, and asks for the probability that the ratio is greater than 1. Obviously the question is under-specified, because no distribution is defined for the mixture. Nevertheless, students who have heard about the “principle of indifference” may be tempted to reason that the ratio wine/water can be assumed to be uniformly distributed from 1/3 to 3, and hence the probability is (3 – 1)/(3 – 1/3) = 3/4. Of course, one could just as well assume that the reciprocal ratio of water/wine is uniformly distributed, in which case the probability is (1 – 1/3)/(3 – 1/3) = 1/4. There are infinitely many other possible distributions, and the answer depends entirely on what distribution we assume. (Typically the “paradox” ask for the probability that the ratio of wine/water is greater than 2, in which case the answers are 3/8 and 1/16, which helps to disguise fallacy of the reasoning.)

Likewise, for our envelopes, we might stipulate that the man prepares the envelopes with some particular value of q (in terms of Case 4), but the recipient doesn’t know the value of q. On this basis, what is the expected ratio of expectations? The recipient might naively suppose, by a facile application of the principle of indifference, that the value of q is uniformly distributed between 0 and infinity, so the expected ratio is

However, with no less justification the recipient could suppose that the reciprocal values p = 1/q are uniformly distributed. Noting that ϕ(p) = (p+2)/(2p+1), we have

If we had some reason to suppose that the range of possible q values extends only between, say, 1/3 and 3, or more generally between 1/n and n, the assumption of uniformity over q implies

whereas on the same range the assumption of uniformity over p (= 1/q) implies

Of course, these are not the only two possibilities. Instead of assuming uniformity over q⁺¹ or q^-1, we might more generally evaluate the expectation of the expectations (for the ratio of expected values) with q^s where the exponent s is assumed to be uniformly distributed from –1 to +1. But even this distribution leads to asymmetric results. We reach symmetry by noting that the man could just as well have reversed the order of the envelopes from Case 4, leading to

If it is stipulated that the man used some fixed value of q and either Case 4 or Case 5 with equal probability, then the expectation for each envelope is (3/4)(q+1)X and hence we have ϕ = 1. This emphasizes the fact that, if we know the ratio of the amounts of money in two envelopes up to reciprocation, and the envelopes are unordered, then clearly the expectations are equal (more or less implicit in the definition of “unordered”), but if the envelopes are ordered, such as being presented to us in order by the man, then the expectations are indeterminate without stipulating that the envelopes were prepared symmetrically.

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