Sunday, December 26, 2010

Two envelopes sophism puzzle

I'm currently on vacation. And it allows me to return to some small topics that I postponed earlier. One of them is Two envelopes problems. This is quite interesting and difficult sophism to solve and 'get'. This morning I'm not in 100% work-ready mood so I decided to return to my list of 'Later'. Surprisingly, this time it took quite small amount of time to solve this puzzle. Actually, I'm a bit puzzled why I postponed it earlier. Maybe I should take vacations more seriously ?

I'm copying definition from wikipedia:

The basic setup: You are presented with two indistinguishable envelopes containing some money. You are further informed that one of the envelopes contains twice as much money as the other. You may select any one of the envelopes and you will receive the money in the selected envelope. When you have selected one of the envelopes at random but not yet opened it, you get the opportunity to take the other envelope instead. Should you switch to the other envelope?

The switching argument: One line of reasoning proceeds 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 is the larger amount is 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

    {1 \over 2} 2A + {1 \over 2} {A \over 2} = {5 \over 4}A
  8. This is greater than A, so I gain on average by switching.
  9. After the switch, I can denote that content by 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.
The puzzle: The puzzle is to find the flaw in the very compelling line of reasoning above.

IMO the most easily understandable flaw in that reasoning is that it adds values of A from two possible realities. And those values obviously differ. If you analyze this problem correctly then it's obvious, that switching envelopes does not make sense. So the lesson is: be careful with your variables.

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