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The person you are responding to is right actually. I drew in an ASCII art a diagram of the various game trees so it will be easier for you to see this: On Wik

The person you are responding to is right actually. I drew in an ASCII art a diagram of the various game trees so it will be easier for you to see this: On Wikipedia they created two trees one with 2A. -> Keep -> 2A 2A -> Switch -> A 0.5 -> A -> Keep -> A -> Switch -> 2A And another with 1/2A -> Keep -> A A -> Switch -> A/2 0.5 -> A/2 -> Keep -> A/2 -> Switch -> A Notice now that there is no game in which A is both 2A and A/2. 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! 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. You can't do it over information sets and strategy spaces. You can only do it over states and action spaces. But they do it. And then they combine them back into the same analysis when they say 1/2(Game tree 1 switch case) + 1/2 (Game tree 2 keep case). And you can't do that, because game tree one depends on game tree ones keep case and game tree 2 depends on its switch case! You can't tell which you are in, so you are in both - not one or the other. You can't combine the two different subgames from two different trees like you could in perfect information! You have to stick with the same game tree. And that is what the person you are mistakenly correcting is saying. Pick one or the other, you can't have both. PS: There are other errors in the problem, like that when you solve it with an MDP or the Bellman equations you actually /do/ end up with 100% Switch having an expected value of zero. 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. PPS: This exact confusion that people are talking about is actually something that happened in practice in the solving of imperfect information games! IIRC the Pluribus paper author corrected a reasoning error of this type when he introduced sub game reach solving? I might be misremembering here, since its been a while since I listened to his talk, but my fuzzy memory suggests that confusion about subgames interacting with each other was the root cause of a failure in some of the poker research for subgame solving in a blueprint abstraction.

baobabKoodaa 4y ago
> 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.

baobabKoodaa 4y ago

> 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.