Two Envelope Paradox
If I place two envelopes in front of you and tell you that both envelopes have some amount of money while one of them has twice as much as another. You have a choice to pick either one. Once you picked, you are given another chance to either keep the picked one or switch to another. There are two ways of reasoning you could employ. One way to think is that both envelopes has equal opportunity of having larger amount of money so that the one has been picked is as good as another. This way, either keeping or switching would make no difference to you. Another way is to argue that the opportunity of the second envelope having either twice as much or half of the first envelope is 50%. Assuming the amount of money inside the first envelope is $100, then the expected total value inside the second envelope is 50%*($200) + 50%*($50) = $125 which is greater than what you have on hand. Therefore, you should switch.
The problem now becomes which way of reasoning is correct one. It seems that both make sense while one or both of them must be wrong. To me, as an outsider of your reasoning/decision making process, the outcome is the same, i.e. you picked one at the end regardless which way of thinking you employed. Either one you picked is no better or worse than another. In other words, your decision/will has no meaning or your decision/will is equivalent to no decision/will.
Somehow I feel that the Two Envelope Paradox is related to Bell’s inequality which may lead to an insufficient prove.
1 Comments:
I wonder if the relationship between the envelopes and Bell's Theorem have anything to do with this thought experiment, from David Harrison's lucid discussion of Bell's inequality:
"Imagine we take a coin and carefully saw it in half so that one piece is a 'heads' and the other is a 'tails.' We put each half in a separate envelope and carry them to different rooms. If we open one of the envelopes and see a heads, we know that the other envelope contains a tails. This correlation 'experiment' corresponds to spin measurements when both polarisers have the same orientation."
Your post doesn't seem to point to a problem with reality as classical physics describes it (ontology), but to a flaw in the nature of knowledge itself (epistemology). After you have taken one envelope and then switched it for the other one, the first envelope still has an expected total value of $125. Of course, if you are optimistic (and logical) you know the envelope you hold in your hands must also have an expected total of $125.
But you might look at the problem this way: you are told that both envelopes have some amount of money while one of them has twice as much as another; and of course you realize that is logically the same as saying one has HALF as much the other. You assume $100. The expected total value in the second envelope after you have taken the first is now 50% * ($100) + 50% * ($50) = $75. It was your optimism that led you to expect a greater payoff if you switch.
When you balance optimism with a pessimistic estimation, you end up with $125 + $75 = $200, or the real POSSIBILITY that either of the envelopes may contain the expected $100, assuming $100 was the amount you expect in the first place. Remember, you multiplied the estimated amounts by a 50% POSSIBILITY.
Of course, reality hasn't been the least affected by your optimism or pessimism. Only your expectations changed.
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