The Two-Envelope Paradox

Two envelopes each contain a positive amount of money; one contains exactly twice the other. You pick one envelope at random and find it contains $x$ dollars. The naive argument says: the other envelope is equally likely to contain $2x$ or $x/2$, so the expected value of switching is $(1/2)(2x) + (1/2)(x/2) = 5x/4 > x$, and you should always switch — but this leads to the absurd conclusion that you should switch back and forth indefinitely. (a) Identify the precise flaw in the naive argument. (b) Suppose the smaller amount $S$ is drawn from a known proper prior distribution with $E[S] = \mu < \infty$. Show that the unconditional expected gain from switching is zero. (c) Explain why conditional on observing $x$, it *can* be rational to switch for some values of $x$ and not others.