Archive for August, 2007

Interesting paradox, not resolved yet, give it a try:

The setup: Let’s say you are given two indistinguishable
envelopes, each of which contains a positive sum of money. One envelope
contains twice as much as the other. You may pick one envelope and keep
whatever amount it contains. You pick one envelope at random but before
you open it you’re offered the possibility to take the other envelope
instead.

The switching argument: Now, suppose you reason as follows::

1. I denote by A the amount in my selected envelope
2. The probability that A is the smaller amount is 1/2, and that it’s the larger also 1/2
3. The other envelope may contain either 2A or A/2
4. If A is the smaller amount the other envelope contains 2A
5. If A is the larger amount the other envelope contains A/2
6. Thus, the other envelope contains 2A with probability 1/2 and A/2 with probability 1/2
7. So the expected value of the money in the other envelope is

   

8. This is greater than A, so I gain on average by swapping
9. After the switch I can denote that content B and reason in exactly the same manner as above
10. I will conclude that the most rational thing to do is to swap back again
11. To be rational I will thus end up swapping envelopes indefinitely
12. As it seems more rational to open just any envelope than to swap indefinitely we have a contradiction

From Wikipedia.