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My claim is no one can give clear, commonsense answers to both of the following questions without leading to absurd conclusions:

(1) Is the probability that you have the higher amount 50%?

(2) If you have a 50% chance of losing some amount and a 50% chance of winning twice that amount, is it a positive expected value bet?


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Well, the answer to (2) is yes, of course.

I think question (1) is what drives the paradox, and a key point to the entire world of probability. Without any extraneous information, it is meaningless, once we see $100 in an envelope, to assign a 50-50 chance, or a 2-1 chance, or any specific probability to the likelihood that N is $50 versus that that N is $200. So any argument which makes such a claim lacks foundation. If we have information (in this case, we can make judgements based on our life experience), then and only then can we assign probabilities. And the better the information we have is, the more likely it is that our probabilties will lead us to the correct envelope.

For example, suppose I tell you that I'm thinking of a ballplayer, and I don't tell you who, what team he plays for, or what type of baseball league he's in (or even if he's in a league), and I ask you to give the probability that he'll hit a grand slam tonight. If you knew absolutely nothing about baseball, and you didn't know what a grand slam was, you would presumably infer whatever you could, and give your best estimate. But to blindly guess 50% (or any other number) without thinking would be meaningless (and presumably someone who did know baseball would come up with a probability lower than 50%).

Basically, what I'm saying is that any probability is simply an estimate based on some nonzero amount of information. Any random number based on zero information, however, cannot be fairly called a probability.

So my answer to question (1) would be based on factors such as the amount in the envelope, what the envelopes looked like, how they were presented to me, and by whom. It would also be based on my life experience. But without the complete picture, all I can say is that if I am able to learn anything useful by seeing the amount in the envelope, then and only then could I give a meaningful answer to the question. Otherwise, I cannot tell you what the probability is that I have chosen the smaller amount, and consequently I cannot give you a meaningful answer as to what the EV of switching would be. (I can still tell you that applying a strategy of deciding to switch before choosing and opening an envelope would be a neutral EV strategy.)

So, regarding the original statement of the paradox, I would say that neither argument 1 or 2 is correct, and that the answer to the question, "After you view the envelope, is it +EV to switch?" is "it depends". (Argument 2 is correct when it states that always switching is a neutral EV strategy, but that does not necessarily mean that switching is neutral EV once you view the contents of either envelope. Argument 1 is simply bogus, since it arbitrarily assumes a 50-50 probability with no foundation.)

Finally, to relate to poker, clearly, to make better estimates of hand probabilities, it is best to gain as much information as possible during prior deals. Gaining as much information as possible, and then making the proper decisions about how to play a hand (i.e., those that maximize EV) based on this information, is perfect poker, IMO.