Two players, A and B, are going to play a game. A perfect logician explains the terms:
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.
Swapping maximizes expected gain for player A and minimizes it for B.
(Left as an exercise for the reader.)
For any amount M that A receives, there's 50%-50% probability that B gets either 2M or M/2, so on average A would gain from swapping and by symmetry B would lose. But what if we apply the same logic starting from B? How does the reference to B's amount ($100) change our reasoning?
Well, B could either lose 25 or gain 50 by swapping, so it would be good for him to swap, too.
I think there is a well known paradox concerning this envrlope bussiness.
Here lies the reader who will never open this book. He is forever dead.
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment
The correct answer is the one that muxdemux submitted but I am not sure about the reasoning for B. Why wouldn't he want to swap?
I think swapping provides mutual benefits to both the users. Hypothetically superposition can also be reached under special conditions. Totoring Services
Last edited by mnuelreyes (2013-02-28 02:53:11)
Complex Numbers Rocks