3 ms·

> The first part of that quote is incorrect: the written formula at bullet point 7 does not claim that one envelope contains 2A and the other envelope contains

> The first part of that quote is incorrect: the written formula at bullet point 7 does not claim that one envelope contains 2A and the other envelope contains A/2. It claims that the first envelope contains A and the second envelope has a 50% probability of containing 2A and a 50% probability of containing A/2. But it's wrong! Try with literally any A. > If you read bullet point 6, it is very clear: "the other envelope contains 2A with probability 1/2 and A/2 with probability 1/2". Notice that bullet point 6 does not claim "One envelope contains 2A and the other envelope contains A/2". Sure but this bullet point is also dead wrong and possibly the start of the scam. If you say A is lowest value, Either it contain A with probability 1/2, or 2A. Otherwise you are changing the reality of A midway through the sentence. To meet you halfway, what you may actually want, or see in your head is the possibility that A (in the lowest value sense) may vary in the experiment, and try to compute the expectation of that new problem. But that's a different problem. And if you want to compute the expectation of this problem with A varying in a range you have to write an integral over the p(A) around the (corrected) equation 7. But the equation 7, even in this integral, need to use the same consistency for A: either you have A being the lowest envelope value and you only write 2A ever, or the highest value and then you only write 0.5A

baobabKoodaa 4y ago
> But it's wrong! Try with literally any A. As I said, I agree that the formula is wrong. I disagree with you regarding where the mistake is and how to fix it. > Sure but this bullet point is also dead wrong and possibly the start of the scam. If you say A is lowest value, Either it contain A with probability 1/2, or 2A. I think maybe you have some typos here, because I don't understand what you're trying to say here. In any case our disagreement concerns whether the formula in Wikipedia refers to the same envelope in both parts of the formula, or different envelopes. You kept stating that it refers to different envelopes, and I wrote a long post to refute that. Now you came back saying that the bullet / formula / Wikipedia text is "wrong" and "possibly the start of the scam". Ok, sure, but the EV formula still clearly refers to the same envelope in both parts of the formula, despite the fact that you claimed otherwise with literally "absolute certainty". You were wrong about something that you claimed to know with "absolute certainty", so perhaps you should in the future adjust those estimates downwards with a bit of uncertainty added in? > Otherwise you are changing the reality of A midway through the sentence. Nope, A is defined as the "value of the firstly-chosen envelope" in the beginning of the sentence, and A is defined as the "value of the firstly-chosen envelope" in the end of the sentence. The definition for A does not change midway through the sentence. > But that's a different problem. And if you want to compute the expectation of this problem with A varying in a range you have to write an integral over the p(A) around the (corrected) equation 7. But the equation 7, even in this integral, need to use the same consistency for A: either you have A being the lowest envelope value and you only write 2A ever, or the highest value and then you only write 0.5A That's not the only way to compute the expectation for this problem. I actually ran some small simulations for this today, representing wagers on this problem. The simulations demonstrate the following claims: - Steps 1 through 7 in the Wikipedia page are correct, in terms of calculating the "expected value in the other envelope, relative to the value in the firstly-chosen envelope". The expected value for the switch, relative to the firstly-chosen envelope's value, is in fact positive (+25%). - At the same time, the "expected value for the switch in absolute terms" is zero. - These claims do not contradict each other. - The first error in Wikipedia's line of reasoning is step 8 that states "they stand to gain by swapping". This indicates that the player should try to maximize goal "expected value relative to firstly-chosen envelope", which is incorrect. The player should instead maximize goal "expected value in absolute terms". If you disagree with any of these, let's formulate our disagreement in the form of a wager that can be simulated in code.
- deleted 4y ago
  [deleted]
- JoshCole 4y ago
  I don't think you're wrong exactly, but I just want to try and give you a different way of framing your disagreement. A man walks to a house with a ladder. He explains that ladders are climbing devices. He tries to set it up against the house, killing plants that he sets it on. The ladder falls over. One person watches this happen and correctly explains that the ladder is a climbing device and says many very true things about the intentions and proves the genuine thoughtfulness of the ladder users approach. He isn't wrong. Another person looks at everything from a different perspective. They are a time traveler and they experience the world in reverse chronological order. They say the ladder isn't a climbing device - it is a thing which falls on ground. They say it is a flower killing device. They claim that when the person explains that the ladder was for climbing, they were wrong, because it isn't. They fundamentally disagree about what the ladder is, because they viewed the ladder from a different perspective. He isn't wrong. They start arguing about what the ladder is. The first person is trying to argue that the ladder isn't a falling device, because it isn't defined to be so. The other person is arguing - this is very critical by the way - not that it is a falling device, but that the way it got used meant it was one and that this disagrees with the statements about what the ladders purpose was. Now, because they aren't noticing this distinction, they argue and they try to prove that the ladder isn't what the other person is claiming it is. Perhaps they think they are winning the argument by pointing out the contradiction, but pointing out the contradiction is agreement that the contradiction exists. You are using a programmatic understanding of what is happening. You are having to proceed forward from statements in order to get meaning. But there are different paradigms in programming type resolution which might help you understand why you get such sharp disagreement. In programmatic terms you can actually have this backward flow too. b [unknown]; foo(int); foo(b) -> b [int] is implied. You can propagate the type backwards from the function call. Critically, this is extremely common in math. Mathematics has identities. The relationships between operations flow along those identities. When you let the identities flow backwards, the ladder you two are discussing stops being a ladder. When they make the choice to use a known algorithm, getting the expected value for a subgame under imperfect information, we know the relationships which flow backward from a correct usage. We know what was 'supposed to' be there. They had an incorrect usage. Their ladder wasn't intended to be for falling. But we're experiencing the world in a backwards direction. So we don't know that yet, like you know it. We're treating it as it was used, not as it ought to have been used - flowing what it is backwards from what they mean in the equations, not forward from how they are defined. So he started trying to explain his thinking to you by moving backward and you noticed that he was saying the ladder was a falling over device, when the ladder maker told you it was something else. And you are right. He isn't wrong so much as his understanding of what the terms are was decided by a different process. And he agrees that you are right that the terms are defined differently! You aren't really disagreeing about as much as you think you are, because you and him both already agree that the ladder maker said the ladder was for climbing and he said the ladder was for falling over. When you find the contradictions, well, I'm not sure you are actually proving him wrong so much as you are agreeing with each other that a contradiction exists.

baobabKoodaa 4y ago

> But it's wrong! Try with literally any A. As I said, I agree that the formula is wrong. I disagree with you regarding where the mistake is and how to fix it. > Sure but this bullet point is also dead wrong and possibly the start of the scam. If you say A is lowest value, Either it contain A with probability 1/2, or 2A. I think maybe you have some typos here, because I don't understand what you're trying to say here. In any case our disagreement concerns whether the formula in Wikipedia refers to the same envelope in both parts of the formula, or different envelopes. You kept stating that it refers to different envelopes, and I wrote a long post to refute that. Now you came back saying that the bullet / formula / Wikipedia text is "wrong" and "possibly the start of the scam". Ok, sure, but the EV formula still clearly refers to the same envelope in both parts of the formula, despite the fact that you claimed otherwise with literally "absolute certainty". You were wrong about something that you claimed to know with "absolute certainty", so perhaps you should in the future adjust those estimates downwards with a bit of uncertainty added in? > Otherwise you are changing the reality of A midway through the sentence. Nope, A is defined as the "value of the firstly-chosen envelope" in the beginning of the sentence, and A is defined as the "value of the firstly-chosen envelope" in the end of the sentence. The definition for A does not change midway through the sentence. > But that's a different problem. And if you want to compute the expectation of this problem with A varying in a range you have to write an integral over the p(A) around the (corrected) equation 7. But the equation 7, even in this integral, need to use the same consistency for A: either you have A being the lowest envelope value and you only write 2A ever, or the highest value and then you only write 0.5A That's not the only way to compute the expectation for this problem. I actually ran some small simulations for this today, representing wagers on this problem. The simulations demonstrate the following claims: - Steps 1 through 7 in the Wikipedia page are correct, in terms of calculating the "expected value in the other envelope, relative to the value in the firstly-chosen envelope". The expected value for the switch, relative to the firstly-chosen envelope's value, is in fact positive (+25%). - At the same time, the "expected value for the switch in absolute terms" is zero. - These claims do not contradict each other. - The first error in Wikipedia's line of reasoning is step 8 that states "they stand to gain by swapping". This indicates that the player should try to maximize goal "expected value relative to firstly-chosen envelope", which is incorrect. The player should instead maximize goal "expected value in absolute terms". If you disagree with any of these, let's formulate our disagreement in the form of a wager that can be simulated in code.

deleted 4y ago
[deleted]
JoshCole 4y ago
I don't think you're wrong exactly, but I just want to try and give you a different way of framing your disagreement. A man walks to a house with a ladder. He explains that ladders are climbing devices. He tries to set it up against the house, killing plants that he sets it on. The ladder falls over. One person watches this happen and correctly explains that the ladder is a climbing device and says many very true things about the intentions and proves the genuine thoughtfulness of the ladder users approach. He isn't wrong. Another person looks at everything from a different perspective. They are a time traveler and they experience the world in reverse chronological order. They say the ladder isn't a climbing device - it is a thing which falls on ground. They say it is a flower killing device. They claim that when the person explains that the ladder was for climbing, they were wrong, because it isn't. They fundamentally disagree about what the ladder is, because they viewed the ladder from a different perspective. He isn't wrong. They start arguing about what the ladder is. The first person is trying to argue that the ladder isn't a falling device, because it isn't defined to be so. The other person is arguing - this is very critical by the way - not that it is a falling device, but that the way it got used meant it was one and that this disagrees with the statements about what the ladders purpose was. Now, because they aren't noticing this distinction, they argue and they try to prove that the ladder isn't what the other person is claiming it is. Perhaps they think they are winning the argument by pointing out the contradiction, but pointing out the contradiction is agreement that the contradiction exists. You are using a programmatic understanding of what is happening. You are having to proceed forward from statements in order to get meaning. But there are different paradigms in programming type resolution which might help you understand why you get such sharp disagreement. In programmatic terms you can actually have this backward flow too. b [unknown]; foo(int); foo(b) -> b [int] is implied. You can propagate the type backwards from the function call. Critically, this is extremely common in math. Mathematics has identities. The relationships between operations flow along those identities. When you let the identities flow backwards, the ladder you two are discussing stops being a ladder. When they make the choice to use a known algorithm, getting the expected value for a subgame under imperfect information, we know the relationships which flow backward from a correct usage. We know what was 'supposed to' be there. They had an incorrect usage. Their ladder wasn't intended to be for falling. But we're experiencing the world in a backwards direction. So we don't know that yet, like you know it. We're treating it as it was used, not as it ought to have been used - flowing what it is backwards from what they mean in the equations, not forward from how they are defined. So he started trying to explain his thinking to you by moving backward and you noticed that he was saying the ladder was a falling over device, when the ladder maker told you it was something else. And you are right. He isn't wrong so much as his understanding of what the terms are was decided by a different process. And he agrees that you are right that the terms are defined differently! You aren't really disagreeing about as much as you think you are, because you and him both already agree that the ladder maker said the ladder was for climbing and he said the ladder was for falling over. When you find the contradictions, well, I'm not sure you are actually proving him wrong so much as you are agreeing with each other that a contradiction exists.