On the fictitious television game show, “Marginal Analysis for Everyone,” the host subjects contestants to unusual tests of mental skill. On one, a contestant may choose one of two identical envelopes labeled A and B, each of which contains an unknown amount of money. The host reveals, though, that one envelope contains twice as much as the other. After choosing A, the host suggests that the contestant might want to switch.
“Switching is clearly advantageous,” intones the host persuasively. “Suppose you have amount x in your Envelope A. Then B must contain either x/2 (with probability 0.5) or 2 x (also with probability 0.5). Thus, the expected value of switching is 1.25 x. In fact now that I think about it, I’ll only let you switch if you give me a 10% cut of your winnings. What do you say? You’ll still be ahead. ”
“No deal,” replies the contestant. “But I’ll be happy to switch for free. In fact, I’ll even let you choose which envelope I get. I won’t even charge you anything!”
What is wrong with the host’s analysis?