The Bayesian Approach to the Two Envelope Problem
The technical issue is that, as a “prior probability”, each envelope has a 50% likelihood of having the larger or smaller amount. However, once you select an envelope (and peek inside), the “posterior probability” that the value is smaller or larger depends on the earlier distribution of funds as well as the amount contained in the envelope.
In other words, Bayes requires you to have some idea of the range of values that could be placed into the envelope. Then, if you were pleasantly surprised that the cheque is large, you’d be better off sticking with it. If you’re disappointed with the small amount, you’d switch.
As well, Bayes would say that the classic argument cannot hold, because there is no probability distribution that fits for an infinite range of possible values.
A Classic Solution to the Two Envelope Paradox
Let’s return to the problem’s classic situation. You select an envelope, but you do not peek: should you switch?
Rather than comparing the envelope in your hand to the one not taken, consider how the scenario started. Someone allocated some amount of money, ‘$M’, and wrote two checks. One check has 2*$M/3; the other has $M/3. Therefore the first check is worth twice the value of the second; the total is $M; and the expected value in each envelope is (2*$M/3 + $M/3)/2 = $M/2.
Regardless of which envelope you choose, the expected value is $M/2; switching to the other envelope is futile.
This is likely to agree with your first intuition about the Two Envelope problem.
Could the Two Envelope Paradox be a Realistic Experiment?
The Two Envelope problem could reflect a different, but potentially realistic, experiment with an infinitely growing return.
Let’s say that a hidden controller writes the first checks for $A(1) and $A(1)/2 and seals them into separate envelopes. She also writes a log of the checks that are “in play.”
The test subject selects one, then switches, and sends the rejected envelope back to the controller. The controller opens the envelope, reads that cheque and discards it. She then flips a fair coin to determine whether the next cheque should be half or double of what the test subject is holding. That cheque goes into an envelope, which is sent to the test subject.
Thus, the test subject can repeatedly switch, or even send the same envelope back for changes. The controller ensures the truth of the condition, “One envelope is worth twice as much as the other,” every time that the test subject has a choice of envelopes.
If the test subject repeatedly “rejects” envelopes, then on average across many experiments, the cheques should increase in value.