The paradox is interesting to discover and discuss errors in reasoning, but we should agree from the start that any proof that one should always switch (or that

The paradox is interesting to discover and discuss errors in reasoning, but we should agree from the start that any proof that one should always switch (or that any behaviour could increase the expected gain) is flawed. This can be easily proven by: 1. Taking at face value that choosing an envelope is random. 2. Adding a second player that is given the other envelope. In that scenario, any reasoning that leads one player to always switch would apply to the other player too. Both players would swap their envelopes. Clearly, the expected value for both cannot go up. Thus any reasoning that mandates switching /has/ to be wrong. (If you have issue with adding another player, instead assume a parallel universe where the player chose the other envelope.) The error in reasoning is usually to assume that the envelope contains X and then reasoning about X/2 and 2X. In reality, the envelopes always contain X and 2X. When you actually use the parallel universe version, clearly one has X and the otehr has 2X. Nobody has X/2 (nor 4X, nor any other expected value your theory may come up with).