4 ms·

This comment explains the fundamental reason why the reasoning is incorrect. I offer the same perspective in a different comment (https://news.ycombinator.com/i

This comment explains the fundamental reason why the reasoning is incorrect. I offer the same perspective in a different comment (https://news.ycombinator.com/item?id=31569991 https://news.ycombinator.com/item?id=31569991). In my opinion, the Wikipedia article is making a disservice to readers by mentioning unnecessarily complex mathematical arguments involving bayesian reasoning and infinite distributions. I believe all of these are distractions from the more fundamental typing error that invalidates the switching argument.

mike_hock 4y ago
They also fail the article's own premise > 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
- JoshCole 4y ago
  Honestly, the more I've thought about this problem the more that statement bothers me. It is just such a nonsense goal. The reason the calculations are wrong are fundamentally related to why correcting the calculations leads to the correct probabilities. If you understand why the problem gives you the wrong result, you proceed to correcting it, and you get the right result - that doesn't mean you didn't understand. It means you did. You can tell you did, because you have the correct answer. But if I show the calculations produce the correct answers, well, that is a step too far. We can now dismiss the understanding on the basis that it got the correct answer, apparently? It's no wonder the author of the wiki writes such a bullshit claim - that no one can agree on a definitive conclusion. Their terms of engagement are self-defeating. Correctness is error, because correctness means you didn't /really/ understand. Its a no true scottsman fallacy - get the right answer, and you aren't engaging with the 'real' problem. But its a flawed no true scottsman, because it defines something measurable: it tells us the goal, that it is to avoid this type of mistake in our thinking. And generalized algorithms for solving decisions problems that include this as a case which is successfully solved are many - and they just so happen to be in the imperfect information setting. And that setting has studied the problem of infinite recursion and has solutions for them. Which I can apply. To get the right answer. And, in general, avoid the problems they claim we aim to avoid. More importantly and to the point of this thread - this isn't an intractable debate, because its been solved and used in production settings for literally decades. So what if some people are going to pretend it isn't? This isn't an unsolved problem. The actual game theory math is /well/ beyond this level of complexity. Its contending with things like environments where you have so much complexity you have to reduce to a blueprint abstraction, not stumbling at a decision problem that is quite literally simpler than rock paper scissors.
  - mike_hock 4y ago
    You seem to misunderstand the point of that constraint. Correction is not an error or a "step too far," it's just insufficient. You can arrive at the correct conclusion either by pinpointing exactly where the original argument is incorrect, or you can come up with a completely different argument that does not have an error. The puzzle challenges you to pinpoint the error because coming up with the correct solution is trivial (and the puzzle is deliberately set up this way). This is not a "nonsense goal." If this came up in real mathematical research - two papers coming to contradictory conclusions - and no one could find where either paper went wrong, we would have a real paradox on our hands.
    - JoshCole 4y ago
      I think this, because I disagree with most people here about what the actual paradox is. I think the paradox is that the algorithm equates the expected value of the contents of an envelope with the expected value of a policy choice for a player. When I correct what I feel is the root of the paradox, my solution drastically differs in fundamental ways such that the way the problem restricts to pointing out the wrong step feels disingenuous. The entire structure is wrong, because even if you do correct the error that leads to the wrong EV for the envelope, you still haven't resolved the paradox. The right probabilities don't resolve the paradox, because they still imply that always switching has the same EV as not switching. If they were really equal, I could always choose switch, but I can't - so the paradox is still there. My resolution ends up being so critical of their argument that the entire way they go about solving gets thrown out. I end up seeing, not just a specific wrong EV calculation, but a decision problem that is just fundamentally using an inappropriate algorithm to determine the policy function.
      - mike_hock 4y ago
        With all due respect ... this is basic probability theory. It's not really controversial what the solution is. The article's failures are mainly pedagogical. We can agree that "the entire structure is wrong" because the "entire structure" is giving a wrong formula for the EV and saying "this is the formula for the EV." Yes, switching and not switching have the same EV and you can always switch.

mike_hock 4y ago

They also fail the article's own premise > 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

JoshCole 4y ago
Honestly, the more I've thought about this problem the more that statement bothers me. It is just such a nonsense goal. The reason the calculations are wrong are fundamentally related to why correcting the calculations leads to the correct probabilities. If you understand why the problem gives you the wrong result, you proceed to correcting it, and you get the right result - that doesn't mean you didn't understand. It means you did. You can tell you did, because you have the correct answer. But if I show the calculations produce the correct answers, well, that is a step too far. We can now dismiss the understanding on the basis that it got the correct answer, apparently? It's no wonder the author of the wiki writes such a bullshit claim - that no one can agree on a definitive conclusion. Their terms of engagement are self-defeating. Correctness is error, because correctness means you didn't /really/ understand. Its a no true scottsman fallacy - get the right answer, and you aren't engaging with the 'real' problem. But its a flawed no true scottsman, because it defines something measurable: it tells us the goal, that it is to avoid this type of mistake in our thinking. And generalized algorithms for solving decisions problems that include this as a case which is successfully solved are many - and they just so happen to be in the imperfect information setting. And that setting has studied the problem of infinite recursion and has solutions for them. Which I can apply. To get the right answer. And, in general, avoid the problems they claim we aim to avoid. More importantly and to the point of this thread - this isn't an intractable debate, because its been solved and used in production settings for literally decades. So what if some people are going to pretend it isn't? This isn't an unsolved problem. The actual game theory math is /well/ beyond this level of complexity. Its contending with things like environments where you have so much complexity you have to reduce to a blueprint abstraction, not stumbling at a decision problem that is quite literally simpler than rock paper scissors.
- mike_hock 4y ago
  You seem to misunderstand the point of that constraint. Correction is not an error or a "step too far," it's just insufficient. You can arrive at the correct conclusion either by pinpointing exactly where the original argument is incorrect, or you can come up with a completely different argument that does not have an error. The puzzle challenges you to pinpoint the error because coming up with the correct solution is trivial (and the puzzle is deliberately set up this way). This is not a "nonsense goal." If this came up in real mathematical research - two papers coming to contradictory conclusions - and no one could find where either paper went wrong, we would have a real paradox on our hands.
  - JoshCole 4y ago
    I think this, because I disagree with most people here about what the actual paradox is. I think the paradox is that the algorithm equates the expected value of the contents of an envelope with the expected value of a policy choice for a player. When I correct what I feel is the root of the paradox, my solution drastically differs in fundamental ways such that the way the problem restricts to pointing out the wrong step feels disingenuous. The entire structure is wrong, because even if you do correct the error that leads to the wrong EV for the envelope, you still haven't resolved the paradox. The right probabilities don't resolve the paradox, because they still imply that always switching has the same EV as not switching. If they were really equal, I could always choose switch, but I can't - so the paradox is still there. My resolution ends up being so critical of their argument that the entire way they go about solving gets thrown out. I end up seeing, not just a specific wrong EV calculation, but a decision problem that is just fundamentally using an inappropriate algorithm to determine the policy function.
    - mike_hock 4y ago
      With all due respect ... this is basic probability theory. It's not really controversial what the solution is. The article's failures are mainly pedagogical. We can agree that "the entire structure is wrong" because the "entire structure" is giving a wrong formula for the EV and saying "this is the formula for the EV." Yes, switching and not switching have the same EV and you can always switch.