The “Two Envelope Paradox” is an intriguing puzzle that some prefer to solve using the logic of Bayesian probability. Can we avoid Bayes with a classic solution or a realistic experiment?
The Classic Two Envelope Paradox
In the two envelope paradox, you are asked to choose one of two envelopes, each containing a check for some amount of money that is greater than zero. One is worth twice as much as the other, but you have no way of knowing which.
You select one envelope.
If you are asked whether you want to switch before opening the envelope, you might say “There’s no reason to switch,” but you are told that Mathematics says you should switch. Why?
Let’s say that your envelope is worth ‘A’ dollars (‘$A’). The other envelope has an equal chance of having the value of either $A/2 or 2*$A. So the expected value in the other envelope is ($A/2 + 2*$A)/2 = 1.25*$A, or 1.25 times the value of your envelope. Now that you know that the value of the other envelope is mathematically greater, don’t you want to trade envelopes?
Of course, once you switch envelopes, a similar argument says that you now have ‘$B’, but the other envelope’s expected value is 1.25*$B, or 1.25 times the value of your new envelope. Will you continue switching forever?
Why is the Two Envelope Problem a Math Paradox?
Two factors contribute to calling the “Two Envelope” problem a paradox. First, it does run counter to one’s intuition.
From the standpoint of mathematics, an unintuitive problem that does not resolve to a single solution may often be labelled a “paradox.”
On the other hand, sometimes finding that a problem cannot be resolved is itself an important result. Alan Turing’s “Turing Machine,” for example, showed that Hilbert’s Decision Problem was impossible to solve.