[BBB]

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Dear, PTB. You state that the place where argument 1 breaks down is at the part where it assumes 50% probablility. However, unless the probabilities now take on specific values, switching will still be either +EV or -EV.

If this is true, then that leaves a bit of a strange situation. I could switch, and then after I did, switch again on the basis of the exact same reasoning, and then again, and again, and in the end I'd have to be rich.

This is basically stating what has been stated before, argument 1 would create +EV situations from both ends of the switch.

How do you reason your perspective deals with this problem?

@BBB, thank you as well for your elaborate explanation. As far as I can see my reply to PTB is also valid as a response to your post and I ask you the same thing.

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NaobisDad,

I'm not quite clear on what your question is... If your strategy is to always switch, no matter what amount is in your envelope, then this strategy will always be neutral EV. This is because, if the amount that you see will not affect your decision, then switching after looking is the same as switching without looking, and argument 2 applies. Once you see the amount in the envelope, if you can infer any information based on that amount as to the chances that the amount you see is N versus those that it is 2N, then switching may become +EV or -EV, depending on what the amount tells you.

Suppose we change the problem slightly. Let's say you're presented with two envelopes, except that all you're told is that one envelope contains mroe than the other (not necessarily twice as much). You open one and you see $100. Should you switch? What if you see $1 million dollars?

If you apply similar logic to argument 1 in the N-2N case, you could conclude that you should always switch, no matter what amount you see. After all, if you see $100, switching can only cost you up to $100, but it could earn you thousands, millions, or more. If you see $1 million, switching could cost you a million, but it could gain you billions or trillions.

But actually, a strategy of switching no matter what amount we see is clearly neutral EV (prior to opening the envelope), just like in the N-2N problem. But in this case, it's harder to jump to the not-necessarily-true conclusion of argument 1, becuase it's easier to see that automatically figuring that the other envelope will contain more money than ours on average is arbitrary and not necessarily correct.

If you were actually presented with this new problem, and you found $100 in your envelop, then if you think there's even a small chance that the other envelope contains thousands of dollars or more, then of course you should switch. However, if you know that there's no way this guy could afford to give you even $200, then I would keep the $100. Similar reasoning should be used if you find $10 million dollars (although personally, that would be enough money that if I felt that on average, the other envelope would contain $20 million dollars, but if there was a significant chance that it contained much much less than $10 million, then I would probably decide that $10 million would be life-changing money for me, and I wouldn't risk the switch, even though switching would be significantly +EV if looked at strictly in terms of dollar amounts (but if I were already a billionaire, then I would switch)).