Another Probability Problem

My student Luc Attlan suggested this problem to me: 

You are told that 2 envelopes contain a positive amount of money, and that one envelope contains twice as much as the other. You can pick either envelope, so you pick envelope 1. Before opening it, you are asked if you would like to switch to envelope 2. Unlike the “3 doors” problem though, no new information is gained at this point. It would, therefore, seem impossible that switching would make any difference in the expected outcome. However, the following reasoning would suggest that in fact, the expected amount in the other envelope is higher: if you assume envelope 1 (“E1”) contains x dollars, you know that E2 contains either 2x or x/2, and that each possibility is 50 percent likely. Therefore, you could say that switching has an expected value of 0.5(2x) + 0.5(x/2) = 1.25x . This means that if you are risk neutral, you should always switch. Is this correct?

TSU

Comments

[Lawrence Solum]

Let's assume that on envelope contains $1 and the other contains $2. There are two states of affairs that are possible. Either I have one dollar, and if I switch I will gain a dollar, or I have two dollars, and if I switch I will lose a dollar. Assumeing the two states are equiprobable, I have a .5 chance of gaining one dollar with an expected value of $0.50 and a .5 chance of losing a dollar, with an expected value of -$0.50. The sum of the expected values is $0.00.

[Answer, I think]

No, because the expected value of the envelope you hold is also 1.25x. Without switching, it's expected value is .5(2x) + .5(x/2) or 1.25x too.

[José Pacheco]

Before the first pick, each envelope was worth 1.5x [2x(.5)+x(.5)=(1.5)x]

Therefore, what you have after that pick is 1.5x, because you don't know yet if it was the high or the low value. Both envelopes are the same, so the new pick is not adding value. You could do a zillion switches and the value will always be the same: 1.5x.
As Solum replied, the switching does not add value, is neutral.

[Lewis Sage]

Here's an equivalent way to put it. Switching has an expected value of 1.5x ... and so does not switching. Rational people are indifferent between equal values.

Good article