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Comment by not2b | Hacker News Reader

The right resolution is: suppose the two amounts are $50 and $100. If the envelope you have contains $50 and you switch, your gain is $50. If it contains $100 and you switch, your gain is -$50. So your expected gain from switching is (0.5)(50) + (0.5)(-50) which is zero.

The way that they fool you is by saying that you either double your money or you halve it, leading you to think that your expected ratio is 2.5/2. But that isn't correct. That ratio doesn't mean anything. That's the wrong step.

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Yep! You're in both subgames at once. You're playing both the A state and the 2A state at the same time, because the information set I you are in contains both of them.

This is the key difference between perfect information and imperfect information. In perfect information you can do the assumption of just one subgame and it works because you know what game you are in - its the one you are in. So you are playing over states and actions. In imperfect information you have to assume you are in both at the same time - so you can't assume one subgame, because you don't know which one you are in. You are playing over information sets - every state contained within that information set, not just one state.

I have a feeling that saying it like this, in the general way, is a lot more confusing to people than your phrasing. So thanks for sharing your phrasing.

I think you nailed it. If the question were posed as "Two envelopes with $A and $(A+1)" then there would be no talk of ratios and no "paradox" would ever emerge.

Exactly. The other envelope might be the good one or the bad one. The "paradox" is basically just sophistry.

Oh, I just realized what I missed. You can use the ratio, but if you do you need to take the geometric mean, since it is a multiplying factor. The geometric mean of 2 and 0.5 is sqrt(2 * 0.5) or 1. So the expected ratio is 1.

That’s a different problem though. In the original problem you don’t know the two amounts ahead of time. All you see is $50 in your envelope - you don’t know if the other one is $100 or $25 and you’re back to square one.

But the two choices aren't equally likely, because someone has chosen the amounts in the envelopes through a non-random process. You could make inferences based on what you expect the person to spend on prices, but it's no longer a simple mathematical problem at that point.

Which is why the problem as stated doesn't let you look in the envelope before deciding whether to switch or not. Then it's obvious switching makes no difference.

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