6 ms·

Basically when the wiki says 2A because less than and A/2 because greater than things are okay, but we're in dangerous territory. If we ever introduce uncertain

Basically when the wiki says 2A because less than and A/2 because greater than things are okay, but we're in dangerous territory. If we ever introduce uncertainty the imperfect information scenario we are in is going to force us to acknowledge that we don't know which game we are in. Which means if that happens we don't have just 2A, but a information set {2A, A}. And we don't just have A/2, but an information set {A/2, A}. In game theory if you have two subtrees and they share an information set than the information sets are equal to each other. You can't tell which subtree you are in, so the information set contains both subgames. You can think of one as factual and the other as counterfactual and vice versa, but the elements in that set need to be equal. When we get to step seven we do a calculation that reintroduces our uncertainty. We multiply by the probability of being in each case. However, we don't actually know we are in one case and we don't know we are in the other case. So combining these two situations leads to the counterfactual part of subgames emerging. So we have {A, 2A} and {A/2, A}. The sets aren't equal. We can choose some parts of this, like grabbing A and comparing it to 2A. So now we have an identity rule that just let us say 2A=A. 2=1. Our contradiction is found. > 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. Eh, I mean, I guess I'm cheating by claiming an interpretation they aren't trying to use. This is why I detest the "no true scottsman" setup to the question. We can create an identity that maps their wrong step into my formalism where what they did is wrong in the way I claim it is. Even though they didn't "try to do it" doesn't mean I can't see that they did do it. But apparently - even though game theory notation neatly avoids these pitfalls - we need to stick to the footgun notation. It just seems stupid to me. If you want to avoid the problem, use the notation that trivializes avoiding the error. I really like the other person's analogy to type errors, because it is such a similar idea to what I'm saying, but they just use a different part of math to assert it. There are ways you can just make this decision fail to typecheck. If you don't want to have these type of errors? Be stricter about your typing so at to prevent them.

baobabKoodaa 4y ago
> 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. Nobody makes this claim, directly or indirectly. If you got this result by formalizing the plain English statement into a proof assistant, then the error was introduced during the step where you interpret the plain English into a formal statement. If the claim "Z = 2Z" was inherent to the plain English version of the problem, then it wouldn't be possible to run the wager as a simulation, but it is. The entire Two Envelopes problem - including step 7 - is possible to simulate in code: https://pastebin.com/jPyZVrkx https://pastebin.com/jPyZVrkx What the simulation demonstrates is: - It's perfectly possible to define "expected value of the final envelope in relation to the value of the firstly-chosen envelope", as the Wikipedia page describes. No contradiction exists that would prevent this computation. - It's also perfectly possible to define "expected value of the final envelope in absolute terms". - These are different goals to maximize. A rational actor should maximize goal 2 ("expected value of the final envelope in absolute terms") in this variant of the game. It's also possible to construct another variant where a rational actor should NOT maximize goal 2, but should instead maximize goal 1 ("expected value of the final envelope in relation to the value of the firstly-chosen envelope"). - The error in Wikipedia's "compelling line of reasoning" appears to be at step 8 where they conclude that "they stand to gain by swapping". This implies that a rational actor should maximize goal 1, when in fact a rational actor should maximize goal 2 instead. This is the error in Wikipedia's line of reasoning. > When we get to step seven we do a calculation that reintroduces our uncertainty. We multiply by the probability of being in each case. However, we don't actually know we are in one case and we don't know we are in the other case. So combining these two situations leads to the counterfactual part of subgames emerging. So we have {A, 2A} and {A/2, A}. The sets aren't equal. We can choose some parts of this, like grabbing A and comparing it to 2A. So now we have an identity rule that just let us say 2A=A. 2=1. Our contradiction is found. I'm not a mathematician, so I don't understand expressions like "counterfactual part of subgames emerging" or "there is an identity implied by being under imperfect information". I appreciate you writing back at length, but the majority of what you wrote went straight over my head. The impression I have is that you formalized the problem in a manner that lead to a contradiction. If this contradiction was inherent to the original English problem statement, it wouldn't have been possible for me to simulate the problem. But it is. So it seems to me that there is no inherent contradiction in the English problem statement, it seems that the contradiction was introduced during the formalization step. > I detest the "no true scottsman" setup to the question [...] If you want to avoid the problem, use the notation that trivializes avoiding the error. I wouldn't describe the setup as a "no true scottsman" scenario, because the setup describes what counts as a scottsman: determine which step in the line of reasoning is incorrect, why and under what conditions. I appreciate that you tried to do this when you identified step 7 and explained what was wrong with it in your opinion. This is also what I tried to do in my explanation, with regards to step 8 and the comparison between the two goals. Besides, it's very trivial to conclude that nothing can possibly be gained by switching (in absolute money terms, which should be the goal a rational actor chooses to maximize). 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. It would be akin to looking at an illusion that displays a man or a cliff depending on how you look at it, then shouting "this is not an illusion! it's just a cliff! there is no man in this picture!"
- JoshCole 4y ago
  > 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

baobabKoodaa 4y ago

> 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. Nobody makes this claim, directly or indirectly. If you got this result by formalizing the plain English statement into a proof assistant, then the error was introduced during the step where you interpret the plain English into a formal statement. If the claim "Z = 2Z" was inherent to the plain English version of the problem, then it wouldn't be possible to run the wager as a simulation, but it is. The entire Two Envelopes problem - including step 7 - is possible to simulate in code: https://pastebin.com/jPyZVrkx https://pastebin.com/jPyZVrkx What the simulation demonstrates is: - It's perfectly possible to define "expected value of the final envelope in relation to the value of the firstly-chosen envelope", as the Wikipedia page describes. No contradiction exists that would prevent this computation. - It's also perfectly possible to define "expected value of the final envelope in absolute terms". - These are different goals to maximize. A rational actor should maximize goal 2 ("expected value of the final envelope in absolute terms") in this variant of the game. It's also possible to construct another variant where a rational actor should NOT maximize goal 2, but should instead maximize goal 1 ("expected value of the final envelope in relation to the value of the firstly-chosen envelope"). - The error in Wikipedia's "compelling line of reasoning" appears to be at step 8 where they conclude that "they stand to gain by swapping". This implies that a rational actor should maximize goal 1, when in fact a rational actor should maximize goal 2 instead. This is the error in Wikipedia's line of reasoning. > When we get to step seven we do a calculation that reintroduces our uncertainty. We multiply by the probability of being in each case. However, we don't actually know we are in one case and we don't know we are in the other case. So combining these two situations leads to the counterfactual part of subgames emerging. So we have {A, 2A} and {A/2, A}. The sets aren't equal. We can choose some parts of this, like grabbing A and comparing it to 2A. So now we have an identity rule that just let us say 2A=A. 2=1. Our contradiction is found. I'm not a mathematician, so I don't understand expressions like "counterfactual part of subgames emerging" or "there is an identity implied by being under imperfect information". I appreciate you writing back at length, but the majority of what you wrote went straight over my head. The impression I have is that you formalized the problem in a manner that lead to a contradiction. If this contradiction was inherent to the original English problem statement, it wouldn't have been possible for me to simulate the problem. But it is. So it seems to me that there is no inherent contradiction in the English problem statement, it seems that the contradiction was introduced during the formalization step. > I detest the "no true scottsman" setup to the question [...] If you want to avoid the problem, use the notation that trivializes avoiding the error. I wouldn't describe the setup as a "no true scottsman" scenario, because the setup describes what counts as a scottsman: determine which step in the line of reasoning is incorrect, why and under what conditions. I appreciate that you tried to do this when you identified step 7 and explained what was wrong with it in your opinion. This is also what I tried to do in my explanation, with regards to step 8 and the comparison between the two goals. Besides, it's very trivial to conclude that nothing can possibly be gained by switching (in absolute money terms, which should be the goal a rational actor chooses to maximize). 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. It would be akin to looking at an illusion that displays a man or a cliff depending on how you look at it, then shouting "this is not an illusion! it's just a cliff! there is no man in this picture!"

JoshCole 4y ago
> 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