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> I drew in an ASCII art a diagram of the various game trees so it will be easier for you to see this: This game tree only makes sense if you change the defini

> I drew in an ASCII art a diagram of the various game trees so it will be easier for you to see this: This game tree only makes sense if you change the definition of A. In particular, you need to change the definition to "A = 1/3 of the total amount of money in the envelopes". If you define A like this, then your game tree is correct, and it corresponds to the "simple resolution" described on the Wikipedia page (where they use x instead of A, for clarity). This is an unsatisfactory answer, because it provides an "alternate path" to the correct answer, without demonstrating which step went wrong in the "incorrect path" that leads to the wrong answer. It is not particularly difficult to construct these "alternate paths" that lead to the correct answer. The difficulty of the puzzle is in demonstrating what goes wrong in the incorrect path, as stated on the Wikipedia page: The puzzle is to find the flaw in the very compelling line of reasoning above. This includes determining exactly why and under what conditions that step is not correct, in order to be sure not to make this mistake in a more complicated situation where the misstep may not be so obvious. In short, the problem is to solve the paradox. Thus, in particular, the puzzle is not solved by the very simple task of finding another way to calculate the probabilities that does not lead to a contradiction. > Notice now that there is no game in which A is both 2A and A/2. I have no idea what you are referring to here. I never claimed that A = 2A in any circumstance (there is nonzero amount of money in the envelopes). > If you don't think this is the error, you're not thinking in imperfect information terms. You're thinking in perfect information terms. This sort of splitting action is allowed in perfect information games! In imperfect information games, you are in multiple subgames at the same time. A has to be the same A in both subgames, because you can't discriminate between which game you are in. So you're not allowed to do this. It is wrong! I agree with you that the crux of the issue is in the definition of A. In particular, "mixing" "two different A's" within the same computation, even though they sort of mean different things. This is a fuzzy explanation that gives a feeling "something about this explains the issue", but it's not a complete explanation that precisely pinpoints what is wrong. > This gets at the core of what the question is trying to make people notice. You can't do this with imperfect information. You can only do it with perfect information. This is false. We can easily construct imperfect information games similar to this one, but tweak a few things in such a way that a "naively constructed EV calculation" yields the correct results. > And that is what the person you are mistakenly correcting is saying. Pick one or the other, you can't have both. Nope! I was very clear about which very specific claims I was refuting. They were unrelated to the topics you are bringing up here. I posted a long, clarifying comment to a sibling comment here, you can look that up if you want to refute specific claims. > So if you spotted some other error, that doesn't mean that the person you are correcting is wrong in their correction. There are multiple issues with the way the wikipedia article approaches the conclusion. I didn't say "I spotted error X in Wikipedia which explains Two Envelope problem", I said "I spotted error X in this other poster's Hacker News comment". Please read the very specific comments I made about very specific claims made by the person.

JoshCole 4y ago
> This game tree only makes sense if you change the definition of A. Switching away from using A to avoid ambiguity. > This is a fuzzy explanation that gives a feeling "something about this explains the issue", but it's not a complete explanation that precisely pinpoints what is wrong. Yes it does, actually. Well, to me. I'll go more in depth so it less fuzzy for you. The issue is that when you handle imperfect information correctly you have counterfactuals implied by your state-oriented reasoning, because you can't actually escape into perfect information world - you're not in it. When you combine the state-derived probabilities the - I don't know the word for this, so forgive me, but the - superposition of the two counterfactuals implied by those state-derived probabilities end up occupying the same space. They exist, because they are implied by the imperfect information game. They just aren't writing them out in the equations on wikipedia. Edit: My 'spatial' probabilistic reasoning in my head is explained here, but I show you the identity rule that proves this in the next section. > I never claimed that A = 2A You aren't, but the counterfactual terms that get realized into the same position, but which weren't written out, claim that Z = 2Z and that Z = Z/2. The two cases were under two different games. So their implied counterfactual components don't agree with each other and we should never have been allowed to combine them. We agree on this! You're just more focused on the they come from different games part and not noticing the other part of what I'm telling you: you can equate the counterfactual components that they don't show between the two games, because there is an identity implied by being under imperfect information. Look at the set relationship closely across the two subgames and you'll realize that the set is defined to be equal to itself across all subgames where the set is used. That is quite literally one of the logical contradictions. This is where they claim that 2=1. The sets aren't equal to each other after their operations. So by the identity implied by the set, they've claimed 2=1. And yet when I try to claim this is a core aspect you throw out this... > This is false. We can easily construct imperfect information games similar to this one, but tweak a few things in such a way that a "naively constructed EV calculation" yields the correct results. This is a really dumb nitpick with very poor support. If you're going to be this fallacious, you might as well go all out. Why not say I'm wrong because first graders learning to add numbers aren't learning how to handle imperfect information subgames? After all, your argument for me being wrong is that the pedagogical value of problems is independent of the subproblems they contain. > Nope! I was very clear about which very specific claims I was refuting. You have a more subtle point than I thought you were making. I'll check out your other post. > I didn't say "I spotted error X in Wikipedia which explains Two Envelope problem", I said "I spotted error X in this other poster's Hacker News comment". Please read the very specific comments I made about very specific claims made by the person. Well, he does have another error; EV isn't defined over all policy choices. It is a problem on many of the wikipedia solutions too. He does see it - its why he mentioned other people talking about infinite series - but his correction attempt was more like "halt here because of type error" which is something that I as a programmer can respect.

JoshCole 4y ago

> This game tree only makes sense if you change the definition of A. Switching away from using A to avoid ambiguity. > This is a fuzzy explanation that gives a feeling "something about this explains the issue", but it's not a complete explanation that precisely pinpoints what is wrong. Yes it does, actually. Well, to me. I'll go more in depth so it less fuzzy for you. The issue is that when you handle imperfect information correctly you have counterfactuals implied by your state-oriented reasoning, because you can't actually escape into perfect information world - you're not in it. When you combine the state-derived probabilities the - I don't know the word for this, so forgive me, but the - superposition of the two counterfactuals implied by those state-derived probabilities end up occupying the same space. They exist, because they are implied by the imperfect information game. They just aren't writing them out in the equations on wikipedia. Edit: My 'spatial' probabilistic reasoning in my head is explained here, but I show you the identity rule that proves this in the next section. > I never claimed that A = 2A You aren't, but the counterfactual terms that get realized into the same position, but which weren't written out, claim that Z = 2Z and that Z = Z/2. The two cases were under two different games. So their implied counterfactual components don't agree with each other and we should never have been allowed to combine them. We agree on this! You're just more focused on the they come from different games part and not noticing the other part of what I'm telling you: you can equate the counterfactual components that they don't show between the two games, because there is an identity implied by being under imperfect information. Look at the set relationship closely across the two subgames and you'll realize that the set is defined to be equal to itself across all subgames where the set is used. That is quite literally one of the logical contradictions. This is where they claim that 2=1. The sets aren't equal to each other after their operations. So by the identity implied by the set, they've claimed 2=1. And yet when I try to claim this is a core aspect you throw out this... > This is false. We can easily construct imperfect information games similar to this one, but tweak a few things in such a way that a "naively constructed EV calculation" yields the correct results. This is a really dumb nitpick with very poor support. If you're going to be this fallacious, you might as well go all out. Why not say I'm wrong because first graders learning to add numbers aren't learning how to handle imperfect information subgames? After all, your argument for me being wrong is that the pedagogical value of problems is independent of the subproblems they contain. > Nope! I was very clear about which very specific claims I was refuting. You have a more subtle point than I thought you were making. I'll check out your other post. > I didn't say "I spotted error X in Wikipedia which explains Two Envelope problem", I said "I spotted error X in this other poster's Hacker News comment". Please read the very specific comments I made about very specific claims made by the person. Well, he does have another error; EV isn't defined over all policy choices. It is a problem on many of the wikipedia solutions too. He does see it - its why he mentioned other people talking about infinite series - but his correction attempt was more like "halt here because of type error" which is something that I as a programmer can respect.