This article was also published on LessWrong
Suppose you are presented with a game. You are given a red and a blue envelope with some money in each. You are allowed to ask an independent party to open both envelopes, and tell you the ratio of blue:red amounts (but not the actual amounts). If you do, the game master replaces the envelopes, and the amounts inside are chosen by him using the same algorithm as before.
You ask the independent observer to check the amounts a million times, and find that half the time the ratio is 2 (blue has twice as much as red), and half the time it's 0.5 (red has twice as much as blue). At this point, the game master discloses that in fact, the way he chooses the amounts mathematically guarantees that these probabilities hold.
Which envelope should you pick to maximize your expected wealth?
It may seem surprising, but with this set-up, the game master can choose to make either red or blue have a higher expected amount of money in it, or make the two the same. Asking the independent party as described above will not help you establish which is which. This is the surprising part and is, in my opinion, the crux of the two envelopes problem.
This is not quite how the two envelopes problem is usually presented, but this is the presentation I arrived at after contemplating the original puzzle. The original puzzle prescribes a specific strategy that the game master follows, makes the envelopes indistinguishable, and provides a paradoxical argument which is obviously false, but it's not so obvious where it goes wrong.
Note that for simplicity, let's assume that money is a real quantity and can be subdivided indefinitely. This avoids the problem of odd amounts like $1.03 not being exactly divisible by two.
The flawed argument
The flawed argument goes as follows. Let's call the amount in the blue envelope B, and in red R. You have confirmed that half the time, B is equal to 2R, and half the time it's R/2. This is a fact. Surely then the expected value of B is (2R * 50% + R/2 * 50%), which simplifies to 1.25R. In other words, the blue envelope has a higher expected amount of money given the evidence we have.
But notice that the situation is completely symmetric. From the information you have, it's also obvious that half the time, R is 2B, and half the time it's B/2. So by the same argument the expected value of R is 1.25B. Uh-oh. The expected value of both envelopes is higher than the other?...
Game master strategies
Let's muddy up the water a little by considering the strategies the game master can use to pick the amounts for each envelope.
Pick an amount X between $1 and $1000 randomly. Throw a fair die. If you get an odd number, put X into the red envelope and 2X into blue. Otherwise put X into blue and 2X into red.
Pick an amount X between $1 and $1000 randomly. Put this into the red envelope. Throw a fair die. If you get an odd number, put 2X into blue, and if it's even, put X/2 into blue.
The difference between these strategies is fairly subtle. I hope it's sufficiently obvious that the "ratio condition" (B = 2R half the time and R = 2B the other half) is true for both strategies. However, suppose we have two people take part in this game, one always picking the red envelope and the other always picking the blue envelope. After a million repetitions of this game, with the first strategy, the two guys will have won almost exactly the same amounts in total. After a million repetitions with the second strategy, the total amount won by the blue guy will be 25% higher than the total amount won by the red guy!
Now observe that strategy 2 can be trivially inverted to favour the red envelope instead of the blue one. The player can ask an independent observer for ratios (as described in the introduction) all he wants, but this information will not allow him to distinguish between these three scenarios (strategy 1, strategy 2 and strategy 2 inverted). It's obviously impossible to figure out which envelope has a higher expected winnings from this information!
What's going on here?
I hope I've convinced you by now that the information about the likelihood of the ratios does not tell you which envelope is better. But what exactly is the flaw in the original argument?
Let's formalize the puzzle a bit. We have two random variables, R and B. We are permitted to ask someone to sample each one and compute the ratio of the samples, r/b, and disclose it to us. Let's define a random variable called RB whose samples are produced by sampling R and B and computing their ratio. We know that RB can take two values, 2 and 0.5, with equal probability. Let's also define BR, which is the opposite ratio: that of a sample of B to a sample of R. BR can also take two values, 2 and 0.5, with equal probability.
The flawed argument is simply that the expected value of RB, E(RB), is 1.25, which is greater than 1, and therefore E(R) > E(B). The flawed argument continues that E(BR) is 1.25 too, therefore E(B) > E(R), leading to a contradiction. What's the flaw?
The expected value of RB, E(RB), really is 1.25. The puzzle gets that one right. E(BR) is also 1.25. The flaw in the argument is simply that it assumes E(X/Y) > 1 implies that E(X) > E(Y). This implication seems to hold intuitively, but human intuition is notoriously bad at probabilities. It is easy to prove that this implication is false, by considering a simple counter-example courtesy of VincentYu.
Consider two independent random variables, X and Y. X can take values 20 and 60, while Y can take values 2 and 100, both with equal probability. To calculate the expected value of X/Y, one can enumerate all possible combinations, multiplying each by its probability. The four possible combinations of X and Y are 20/2, 20/100, 60/2 and 60/100. Each combination is 25% likely. Hence E(X/Y) is 10.2. This is greater than 1, so the if the implication were to hold, E(X) should be greater than E(Y). But E(X) is (20+60)/2 = 40, while E(Y) is (2+100)/2 = 51. Hence, the implication E(X/Y) > 1 => E(X) > E(Y) does not hold in general.
So there you have it. The proposed argument relies on an implication which seems true intuitively, but turns out to be false under scrutiny. Mystery solved?... Almost.
Imprecise language's contribution to the puzzle
The argument concerning the original, indistinguishable envelopes, is phrased like this: "(1) I denote by A the amount in my selected envelope. (2) The other envelope may contain either 2A or A/2, with a 50% probability each. (3) So the expected value of the money in the other envelope is 1.25A. (4) Hence, the other envelope is expected to have more dollars."
Depending on how pedantic you are, you might say that the statement made in the third sentence is strictly false, or that it is too ambiguous to be strictly false, or that at least one interpretation is true. The expected value 1.25A is "of the amount of money contained in the other envelope expressed in terms of the amount of money in this envelope". It is not "of the amount of money in the other envelope expressed in dollars". Hence the last sentence does not follow, and if the statements were made in full and with complete accuracy, the fact that it does not follow is a little bit more obvious.
In closing, I would say this puzzle is hard because "in terms of this envelope" and "in terms of dollars" are typically equivalent enough in everyday life, but when it comes to expected values, this equivalence breaks down rather counter-intuitively.