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> It would be boring and unsatisfying to accept a simple answer that explains how to get the "correct answer" without explaining what made it a paradox in the f

> It would be boring and unsatisfying to accept a simple answer that explains how to get the "correct answer" without explaining what made it a paradox in the first place Well, this is just a place where they do the math wrong. We can quibble about why, but it isn't a paradox. We both agree they do it wrong and we even have an overlap in the reason why - we both think they aren't comparing them relative to all the subgames they are in and we both think that this is required to reason correctly. So lets say we get to the correct relative EV of 0. If both are worth the same, well, why not always switch? They are the same EV, right? IMO this is the real paradox. We have a false equivocation. EV(policy, game) is not equal to EV(envelopes), because for the policy `ALWAYS_SWITCH` we get the logical contradiction that undefined = EV(envelope) or if you take the limit with a discount under a different formalism that 0 = EV(envelope) When you tackle this problem starting there rather than at the other error the entire algorithm changes, because we need to modify the problem so that we can calculate EV with respect to policy. The algorithm changes enough so that it really annoys me, because I have this intuitive feeling that I'm changing the solution so much so that I'm no longer working within the spirit of the puzzle. That is why I feel like "no true scottsman"; to use a metaphor, they gave me a coloring book and told me to color something beautiful but that I had to stay in the lines, but the lines of are a skunk and I want to draw a sunrise. I don't want to correct just the one particular step. I hate the framework that forces me into the confusion by falsely implying that if I get the right envelope EV, I know my EV.

baobabKoodaa 4y ago
> If both are worth the same, well, why not always switch? I feel like this is a detour that we can avoid by adding a small cost to switching. > I hate the framework that forces me into the confusion by falsely implying that if I get the right envelope EV, I know my EV. I love it. I love it in the same way I love the trick of a magician who fools me. Also, the answer of "two EVs" is particularly satisfying for me, because it resembles a similar confusion in poker tournaments where you need to account for chip EV and dollar EV separately.
- JoshCole 4y ago
  > I feel like this is a detour that we can avoid by adding a small cost to switching. I don't think its a detour we actually want to avoid; the implications are really fascinating and help us to understand how to solve games more generally. I agree with adding cost as a way to do it. Actually, that is why I've been saying it is zero or undefined. You know about dynamic programming right? Well, one of the reasons it was invented is kind of related to what we're talking about right now. There are these things called the bellman equations. They can look like this when you think of things in terms of a markov decision process. https://wikimedia.org/api/rest_v1/media/math/render/svg/0e358db69fc04ad39781acdc8300059e7a93791a https://wikimedia.org/api/rest_v1/media/math/render/svg/0e35... Since they're recursively defined, you can compute them more quickly by caching the computation back to front then by computing them front to back. Anyways, putting aside that bit of trivia, do you see the symbol that looks a bit like a y? When we have an infinite sequence like we would if we kept switching you can set that term to be some constant, for example 0.9999999999999. That lets you take the limit, because the infinite sequence is obviously going to converge to zero. Check out the formula again and notice one of the things that it does which the wikipedia article doesn't. Do you see the symbol for pi? That is talking about the concept of a policy function. A strategy, that way you compute what the agent would get if they played the game, rather than the envelope contents. Under this formalism we've doing an argmax over the policy function for the equation in order to get the policy that has the highest expected value. > I love it. I love it in the same way I love the trick of a magician who fools me. If you're actually interested in this sort of thing, I suggest checking out the poker research papers by Noam Brown. He had a talk at NIPS 2017 where he won best paper award. In 2019 he was runner up for best scientific advancement of the year. His work is about applying game simplification to poker to create a bluneprint of the game which is simpler, solving the blueprint using a modified form of counterfactual regret minimization, and then refining the solution during actual play with reach subgame solving. You might have heard of his work, because he was involved in the entire AI now better than humans at poker thing even in no limit breakthrough. I have some belief, but not certainty, that his paper contains a correction of someone who made this style of error. He mentions in the research paper that there was a mistake in another paper where they didn't account for the way subgames influence each other. I see that as the problem here, but I haven't read the other paper, so I don't know if it was the same category of error. Find the paper he referenced, read through it, and see if you're tricked. It is potentially a kind of real world two envelope problem to see if your tools for reasoning about these things is actually helping you to avoid the error in more complicated situations - though, since you disagree with me that this is about handling of imperfect information (or more precisely, the counterfactuals - there is a variant on wikipedia where they removed the probabilites, but counterfactual reasoning still applied to resolve the paradox) maybe you won't see it as related errors.

baobabKoodaa 4y ago

> If both are worth the same, well, why not always switch? I feel like this is a detour that we can avoid by adding a small cost to switching. > I hate the framework that forces me into the confusion by falsely implying that if I get the right envelope EV, I know my EV. I love it. I love it in the same way I love the trick of a magician who fools me. Also, the answer of "two EVs" is particularly satisfying for me, because it resembles a similar confusion in poker tournaments where you need to account for chip EV and dollar EV separately.

JoshCole 4y ago
> I feel like this is a detour that we can avoid by adding a small cost to switching. I don't think its a detour we actually want to avoid; the implications are really fascinating and help us to understand how to solve games more generally. I agree with adding cost as a way to do it. Actually, that is why I've been saying it is zero or undefined. You know about dynamic programming right? Well, one of the reasons it was invented is kind of related to what we're talking about right now. There are these things called the bellman equations. They can look like this when you think of things in terms of a markov decision process. https://wikimedia.org/api/rest_v1/media/math/render/svg/0e358db69fc04ad39781acdc8300059e7a93791a https://wikimedia.org/api/rest_v1/media/math/render/svg/0e35... Since they're recursively defined, you can compute them more quickly by caching the computation back to front then by computing them front to back. Anyways, putting aside that bit of trivia, do you see the symbol that looks a bit like a y? When we have an infinite sequence like we would if we kept switching you can set that term to be some constant, for example 0.9999999999999. That lets you take the limit, because the infinite sequence is obviously going to converge to zero. Check out the formula again and notice one of the things that it does which the wikipedia article doesn't. Do you see the symbol for pi? That is talking about the concept of a policy function. A strategy, that way you compute what the agent would get if they played the game, rather than the envelope contents. Under this formalism we've doing an argmax over the policy function for the equation in order to get the policy that has the highest expected value. > I love it. I love it in the same way I love the trick of a magician who fools me. If you're actually interested in this sort of thing, I suggest checking out the poker research papers by Noam Brown. He had a talk at NIPS 2017 where he won best paper award. In 2019 he was runner up for best scientific advancement of the year. His work is about applying game simplification to poker to create a bluneprint of the game which is simpler, solving the blueprint using a modified form of counterfactual regret minimization, and then refining the solution during actual play with reach subgame solving. You might have heard of his work, because he was involved in the entire AI now better than humans at poker thing even in no limit breakthrough. I have some belief, but not certainty, that his paper contains a correction of someone who made this style of error. He mentions in the research paper that there was a mistake in another paper where they didn't account for the way subgames influence each other. I see that as the problem here, but I haven't read the other paper, so I don't know if it was the same category of error. Find the paper he referenced, read through it, and see if you're tricked. It is potentially a kind of real world two envelope problem to see if your tools for reasoning about these things is actually helping you to avoid the error in more complicated situations - though, since you disagree with me that this is about handling of imperfect information (or more precisely, the counterfactuals - there is a variant on wikipedia where they removed the probabilites, but counterfactual reasoning still applied to resolve the paradox) maybe you won't see it as related errors.