Conundrum: Two Envelopes
Tuesday, April 10th, 2007I overheard this once on a train and was never able to figure it out. Maybe someone here can help me.
Imagine I have two envelopes and I tell you truthfully that both contain money and that one envelope contains twice as much money as the other. I offer you your choice of envelope and you choose one of them without opening it.
Now I ask you if you would like to switch envelopes. You chose yours randomly, so it’s a 50/50 chance whether the other envelope contains half as much or twice as much. So, if the amount you now have is x, there’s a 50 percent chance that switching would get you 2x and a 50 percent chance it will get you x/2. You have twice as much to gain as you have to lose, regardless of how much is in your envelope, so it makes sense mathematically to switch envelopes.
But of course, this is ridiculous, since you have no new information about the two envelopes than you had before. Once you’ve made that switch, by the same logic, you should want to switch again. This much seems obvious. So where’s the flaw in the math above?
By the way, I consulted our good friend Wikipedia before posting this, and it was little help. It just mumbled something about Bayesian Decision Theory and said the problem would be easy if I were a mathematician. It then went on to pose a harder problem in which you can look inside one of the envelopes, and an even harder problem that was way over my head at 5:30 am. Thanks, Wikipedia.