Interesting Mathematical Paradox?

[jason1990]

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Right. I'm sweeping it under the rug with the word "probably". If the situation were like with your brother then seeing the amount in the envelope does give you more information. I'm not exactly sure what information it might give in general.

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Seeing the amount in the envelope gives you very specific information: the amount in the envelope. In the situation with my brother, I knew how to deal with that information. In general, you may not know how to deal with it. You may not know how it affects the probability of having the smaller envelope. But that does not imply that the information is independent of having the smaller envelope. To assume so is to apply a variation of the indifference principle.

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However, here's the sense in which I think my statement has validity. Suppose many different people are offered such envelopes. Each of them wagers $100 that they have the smaller envelope. Each of them ignores the amount they see in their envelope and continues their $100 bet. At 2-1 odds they will on average make money on their $100 bets. But they will not on average make money from the Envelope Switch. This I think is at the heart of the psychological conundrum.

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Of course, if they ignore the amount, then it is as if they never opened the envelope at all. The $100-wagerers will make money without opening, and the Envelope Switchers will not. But this does not mean that the information they ignored did not affect their conditional probabilities.

I think your observations here are correct and I think they demonstrate nicely the mistake people make when they fail to acknowledge that the amount they are wagering (half their chosen envelope) is a random variable tied to the result of their bet. But I do not see how that is at the heart of the psychological conundrum. You said,

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Someone says, before I open the envelope, surely I have a 50% chance of switching to the larger. It just seems unreal that seeing amount A in the envelope should change that when it doesn't really give me any new information about whether it's the smaller amount.

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It seems that this is the heart of the psychological conundrum. It should not seem unreal that seeing A causes some sort of change. It definitely changes the experiment. At the very least, it removes the symmetry from the experiment, as I mentioned in my previous post. You may not know how this change affects the probability of having the smaller envelope, but that does not mean you can assume that it does not change it at all.