A problem with the standard solution of the two envelopes paradox

[Fredrik]

I completely fail to see what it is that you state is missing from the explanation in, say, Devlin, that is actually missing. All of the things you claim are required for a full explanation of the paradox are in that linked article.

I have explained it lots of times. I don't know what else I can say. Devlin never mentions that if it's possible to calculate the expected gain as a function of the amount in the first envelope, the result of the calculation is the solution of a different problem.

If you get the probabilities correct, then it tells you the correct answer in *every* situation, well, the *only* situation that there is.

I guess you can say that, but this "only situation there is" is not the situation that was specified in the problem we're working with! Read post #1 again. We don't know the prior distribution!

It really isn't hard: if you use the correct probabilities then you get the correct answer.

Yes, the correct answer to the wrong problem.

The explanation of the paradox is that the probabilities that are shoved in are nonsense.

No it isn't. The first step of the explanation is to realize that the probabilities in the calculation of E(B|A=a)-E(A|A=a) depend on the prior distribution. The second and final step is to realize that that means that E(B|A=a)-E(A|A=a) will tell us what the correct decision is in a situation where both a and the prior distribution is known, but not in the situation that was specified in the problem!

Devlin doesn't do the second step. Everything he does after the first step is irrelevant.

No, it isn't. The 'first one' was that you asserted that the expected gain from swapping is zero.

Unless Hurkyl has told you privately that he meant something other than what he said, you are wrong. He quoted two sencences of mine and said that he can't follow "either of these assertions". Each sentence is one assertion. The 'first one' is that Devlin claims that the cause of the paradox is the confusion of prior and posterior probabilities.

And stop saying that I have asserted that E(B|A=a)-E(A|A=a) is zero! I have never said that.

 

[matt grime]

Given your misuse of terminology, it is hard to decide what you have asserted is the problem. See post 25 where you contradict yourself.

 

[Fredrik]

No, I don't. And what misuse of terminology? Yes, I wrote E(B) when I should have written E(B|A=a), but that's a minor detail. It can't have caused any of your confusion.

However, there are a few things in #25 that reveals that at the time I had not fully understood what the cause of the paradox is. It didn't become completely clear to me until I wrote #28.

For example, I said in #25 that I don't know if we should be able to show that E(B|A=a)=a. At the time, I didn't know. (Now I know we can't). I just knew that if we can't, then there's something wrong with Devlin's argument. (That's what made me consider the possibility that maybe we should be able to show that E(B|A=a)=a after all). It took me a while to figure out what was missing from his argument.

 

[EnumaElish]

That whole thing about the probabilities not really being 1/2 is completely irrelevant. It adds nothing to his understanding of the resolution of the paradox.

Don't your observations 1 and 2 in #25 two roundabout (or overly general) ways of saying that p=1/2 doesn't work?

 

[Hurkyl]

In #29, I quoted two separate passages, each of them comtaining an assertion (this implies that). I meant I didn't follow either passage.

(I mentioned my edit in because wouldn't be right to have corrected the typo unannounced)



As for what "resolves" the paradox, I consider pointing out the flaw in the argument a resolution. The flaw in the envelope paradox is (in my notation) that it uses the value P(A=2B) when it's supposed to use P(A=2B|A=a).

Anything beyond merely pointing out the flawed step in the argument is simply a bonus.