I think there is a well known paradox concerning this envrlope bussiness.

]]>Proof:

(Left as an exercise for the reader.)

I have several envelopes containing different amounts of money. I will randomly pick one of them, see the amount that it contains and will give it closed to player A. Then I toss a coin and if I get tails, I will get an empty envelope and put half the amount of player's A envelope, while if I get heads, will put double. I will then give this envelope (closed) to player B.

Then I will invite each of you privately and ask you to decide whether you will swap envelopes or not. If you both agree in swapping, you will do so, otherwise you will keep your initial envelopes.

A and B agree with the procedure and then A asks B to reveal his amount, so that they get an idea on what to propose to the logician. They both see that B has $100. Right afterwards, each of them must meet the logician in private to announce their decision. Which decision ensures the biggest expected gain for players A and B separately? Explain your answer.]]>