Two envelopes problem

You seem to misunderstand the point of that constraint. Correction is not an error or a "step too far," it's just insufficient.

You can arrive at the correct conclusion either by pinpointing exactly where the original argument is incorrect, or you can come up with a completely different argument that does not have an error.

The puzzle challenges you to pinpoint the error because coming up with the correct solution is trivial (and the puzzle is deliberately set up this way).

This is not a "nonsense goal." If this came up in real mathematical research - two papers coming to contradictory conclusions - and no one could find where either paper went wrong, we would have a real paradox on our hands.

> Ah this is a different version of the problem. There are quite a few listed on the wikipedia page.

The vast majority of the Wikipedia article concerns variants that do not involve opening the first envelope.

The first definition of the problem states "before inspecting it".

The first section, "Introduction", also describes the problem similarly: "before they open it". This section - which defines the problem - does not introduce any variants which involve opening the first envelope.

The next section, "Simple resolution", describes a resolution to the variant described in the introduction (where the first envelope is not opened).

The next section, "Other simple resolutions", also discusses the same variant (where the first envelope is not opened). At the end of this section there is a remark "We could actually open our envelope before deciding on switching or not and the above formula would still give us the correct expected return", but that is the extent to which the first-envelope-is-opened variant is discussed there.

I could go on. My point is that the discussion very clearly revolves around the variant where the first envelope is not opened, and you have to go down a pretty deep rabbit hole in order to find a variant where the first envelope is opened.

> I believe the version you are talking about is just a simple equivocation.

Yes, just a simple equivocation, with a Wikipedia page that could be printed as a book. Does anything strike you as odd about that?

> Since you seem to be making some authority arguments just know that I have a PhD in ML / stats and was top 30 in my country in olympiad level competitive maths so I DO know what I do and in this case with absolute certainty.

Let's review what you're claiming with absolute certainty. This is the first claim that I'm contesting:

> they say "one envelope contains 2A and the other A/2".

When pointed out that they actually don't say this, you clarified your point:

> the "writing" doesn't say it but the written formula at bullet point 7 says it. That's the mistake, the formula is wrong and does not describe reality.

The second part of that quote is correct: bullet point 7 has the mistake, the formula is wrong and does not describe reality.

The first part of that quote is incorrect: the written formula at bullet point 7 does not claim that one envelope contains 2A and the other envelope contains A/2. It claims that the first envelope contains A and the second envelope has a 50% probability of containing 2A and a 50% probability of containing A/2.

You correctly identify where the mistake is, but you misidentify what the mistake is.

If you read bullet point 6, it is very clear: "the other envelope contains 2A with probability 1/2 and A/2 with probability 1/2". Notice that bullet point 6 does not claim "One envelope contains 2A and the other envelope contains A/2".

To illustrate my point, consider a simple coin toss game. If the coin comes up heads, you win $2. If it comes up tails, you win $0.50. How might we construct the EV calculation for this coin toss?

0.5 * $2 + 0.5 * $0.50

Notice how both parts of the formula refer to the same coin. They don't refer to different coins. You have one coin that you flip, and depending on how that one coin lands, there is some probability that you get $2, and some probability that you get $0.50. If you looked at that formula, you wouldn't think that the first part refers to different coin that the second part.

Similarly, here we have one envelope (similar to having one coin) and we have uncertainty about the what the envelope contains (similar to having uncertainty about how the coin will land). We have 50% chance that the envelope will contain 2A, and we have 50% chance that the envelope will contain A/2. Thus, we arrive at the (incorrect) formula:

0.5 * 2A + 0.5 * A/2

It's very obvious that both parts of the formula refer to the same envelope, just like in the coin toss formula both parts of the formula refer to the same coin. To be extremely clear, I am not claiming that the formula is correct. I am claiming that both parts of the formula refer to the same envelope, not to different envelopes. To be specific, I am refuting this claim that you made:

> they say "one envelope contains 2*A and the other A/2".

They don't say that.

Furthermore, you make an additional incorrect claim:

> The right formula is changing either the 0.5A by A or the 2A by A.

Here you are just randomly changing the formula in order to make it output 0EV (which is the correct answer). Just because you get a correct answer does not mean your computation was correct. In this case it is incorrect, because A was defined badly. In order to fix the formula you would have to fix the definition of A. If you keep the incorrect definition of A and just shuffle symbols around until you get the correct result, your computation is still incorrect.

> You can’t reflect the probability to be 2A AND 1/2A if A represents the value of an envelope.

Yes, that's right. That's exactly what I said: "A is the amount in one of the envelopes, so if A=50 then either E1 is 50 or E2 is 50, and the other E is 25 or 100. But under no circumstances can E1=100 and E2=25 at the same time."

Okay. lets start with the easy analysis; the version of the game that terminate after only one switch.

I[null] = {0.5: A, 0.5: 2A}

We get the expected change in EV for switching like this:

    (
        # The expected gain of switching if we are in subgame A
        (2A-A) * 0.5
    
        +

        # The expected loss of switching if we are in subgame 2A
        (A-2A) * 0.5
    )

See how we had to consider two different subgames? We didn't know whether we had A or 2A. We only knew we had I[null].

You had to consider the benefit of switching from A to 2A and the cost of switching from 2A to A. You had to consider the factual reality you were in, but also the counterfactual reality you were not in.

Here is the reality of the first part of the game tree:

   ChanceNode(0.5)
  /           \ 
 A            2A

In a perfect information game, A and 2A are disconnected because they are in two different subtrees, but what watch what happens when we convert from the perfect information view to the imperfect information view of that world that we are actually dealing with:

        0.5
      /    \
    I[nil] I[nil]

We have two information sets now as the branches. One is actually A, but when we do reasoning about it we need to counterfactually consider it the other parts of the information set.

                      0.5
      /                                    \
    {Factually A, Counterfactually 2A}     {Factually 2A, Counterfactually A}

So I hope you're starting to realize that A and 2A are fundamentally connected. You can't reason about A without including 2A. You can't reason about 2A without including A. This is really really important to one of several reason that their analysis is wrong. You don't know which branch you are in. So you can't condition on being in A, like they do in the wikipedia article.

Look at the trouble they run into when they /do/ condition on A. I just showed you these counterfactuals exist, but when they condition they still exist. It treats the situation as if you can just deal with A and just deal with 2A. So they get a 2A case and a A/2 case both of which have their own counterfactuals. Notice what just happened when they did that.

In the A/2 case since we are in imperfect information there is a counterfactual associated with it. What is that counterfactual? It is A! So they don't just have A/2 they also have counterfactual A.

And now look at the other case. 2A has a counterfactual associated with it too. What is it? A.

So they have this:

[F: A, CF: 2A], [F: A/2 CF: A]

And what I'm trying to point out to you is that they just declared A=A/2 and 2A=A, because they neglected the counterfactual relationships.

You can't condition on A in the subgame; you don't have perfect information - you don't have A. You have I[null]. Even when you get told 60 you still have I[null].

/A/ isn't 60 and it can't be because you don't know which subgame you are in. A isn't given. If you knew A, if it was possible to know A, you wouldn't be in an imperfect information game. A is defined in point one yes - but it is also used in point seven and when it is used there we get the logical contradiction.