You have to have some prior distribution over the possible amounts of money in the envelopes.
The expected value of switching is equal to 1⁄2xP(I have the envelope with more money | I opened an envelope containing x dollars) + 2xP(I have the envelope with less money | I opened an envelope containing x dollars). This means that, once you know what x is, if you think that you have less than a 2⁄3 chance of having the envelope with more money, you should switch.
According to what I read, if your prior is such that there is no finite X for which you would decide that you have less than a 2⁄3 chance of having the envelope with more money, then your expected value for x, before you learned what it was, was infinite—and if you were expecting an infinite amount of money, then of course any finite value is disappointing. (Note that having a proper prior doesn’t keep you from expecting an infinite value for x). So it doesn’t matter what you actually find in the envelope, once you find out what it is, you should switch—but there’s no reason to switch before you open at least one envelope.
rstarkov wrote a nice discussion piece on the two envelopes problem: Solving the two envelopes problem. thomblake commented that the error most people make with this problem is treating the amounts of money in the envelopes as fixed values when calculating the expectation.
I’ve read a pretty good resolution to the two envelopes problem.
You have to have some prior distribution over the possible amounts of money in the envelopes.
The expected value of switching is equal to 1⁄2xP(I have the envelope with more money | I opened an envelope containing x dollars) + 2xP(I have the envelope with less money | I opened an envelope containing x dollars). This means that, once you know what x is, if you think that you have less than a 2⁄3 chance of having the envelope with more money, you should switch.
According to what I read, if your prior is such that there is no finite X for which you would decide that you have less than a 2⁄3 chance of having the envelope with more money, then your expected value for x, before you learned what it was, was infinite—and if you were expecting an infinite amount of money, then of course any finite value is disappointing. (Note that having a proper prior doesn’t keep you from expecting an infinite value for x). So it doesn’t matter what you actually find in the envelope, once you find out what it is, you should switch—but there’s no reason to switch before you open at least one envelope.
rstarkov wrote a nice discussion piece on the two envelopes problem: Solving the two envelopes problem. thomblake commented that the error most people make with this problem is treating the amounts of money in the envelopes as fixed values when calculating the expectation.