Probability riddle: 2 envelopes, switch or keep?

Question

You are shown two envelopes and told that they both contain money, one twice as much as the other. You select one of them at random - it contains $100. Now you are given the choice to either keep the $100 or the contents of the other envelope instead.

Question: is it better for you to switch, or better not to switch?

This is a case of nonintuitive probability similar to the Unfinished Game Problem Jeff recently mentioned, and the good old Monty Hall problem, but the solution and its explanation are rather different. Like Jeff, I'm asking you to try and reason it out yourself rather than looking it up.

Please only write answers with explanations why you think so, and look at the already present answers to see if there's already one with the same reasoning.

I'll wait one day and then either choose the answer with the best correct explanation or provide one myself.

Answer 1

I'm tempted to answer "switch" because that is the non-intuitive answer to the Monty Hall problem. However, in that case, the newly opened door actually changes the probability because the Host is giving you information (even though most people don't see that).

In this case, though, there is no new information that I can see except the amount. And that simply doesn't tell you whether the opened envelope offers 2x or .5x relative to the other. Thus, there is no virtue to switching or to not-switching: either has equal probability of providing the maximum payment.

Update: Lasevk and Roy have it. My answer was wrong due to not considering the quantities involved in the payoff.

Answer 2

You stand to lose 50 and gain 100.

Switch.

Answer 3

It is better for you to switch, which is of course a paradox. The reason for this paradox is that the distribution is such that the expected gain is infinite.

The question itself is misleading (which is a good thing for a riddle), in that it is not said, but assumed, that the probability of getting twice the value of X is the same the probability of getting half the value of X, for ANY X. The only distributions where this is possible have infinite expected value, which is why this is impossible in real life.

An easier to understand version would be: You are given only one envelope, and are told the expected value of the money inside is infinite. You open it once, see 100$, and are then asked if you want to try again with a different value? Since the expected value is infinite, you will always want to "roll again" no matter what you got.