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.