There exists the **Envelope Paradox**, or Two Envelopes problem for which it appears that a lot of ink has been expended in order to explain away a false argument. Copying from wikipedia,

<<The problem typically is introduced by formulating a hypothetical challenge of the following type:

*Of two indistinguishable envelopes, each containing money, one contains twice as much as the other.**The subject may pick one envelope and keep the money it contains.**Having chosen an envelope at will, but before inspecting it, the subject gets the chance to take the other envelope instead.**What is the optimal rational strategy for maximising the amount of money to be gained?*

There is no point at all in switching envelopes as the situation is symmetric. However, the story now introduces the so-called switching argument that shows that it is more beneficial to switch. The problem is to show what is wrong with this argument.>>

I really don’t understand why is there a problem to show what is wrong with the “switching argument” (you can read it in the wikipedia article), since a basic formulation of the problem shows easily that there is no increase in expected gain if one switches the envelope. I do this **in a recent post in economics.se. **

My intention was mainly to showcase how beneficial, and in fact necessary, is to treat such cases formally, avoiding vague language and seemingly logical verbal arguments: expected utility theory, set-theoretic probability and game theory are all put to work (lightly) to provide definite answers.

But I also treat the more interesting variant of the paradox, where one gets to open the envelope, learn the amount in it, and *then *decide whether to switch or not: In this case, **risk-takers and risk-neutral expected utility maximizers should always switch, as well as risk-averse people with risk-aversion lower than that implied by a logarithmic utility function. At logarithmic utility, one becomes again indifferent with respect to switching, while for more risk-averse persons, it is optimal not to switch.**

In both variants, selecting one of the envelopes is a given. So in both variants, what is in the chosen envelop is a given. What is interesting and fascinating here is that ** just the knowledge of what you got changes the optimal action you should take, **even though it does not change the actual allocation of the two prizes.