Well, the simple way to put is that "A" isn't fixed. The "expected value" argument in steps 6-7 is acting like "A" is a single value when "A" will be larger or smaller depending on what envelope you picked.
It doesn't actually say that. It says the other envelope will either contain 2A or A/2. The envelope you're holding is A. The other envelop is either twice that, or half that.
the "writing" doesn't say it but the written formula at bullet point 7 says it. That's the mistake, the formula is wrong and does not describe reality.
This is incorrect. Bullet point 7 in the written formula does not refer to a corresponding pair of envelopes, as you imply. Instead, it refers to "the other" envelope only: 1/2 chance that "the other" envelope contains 2A, and 1/2 chance that "the other" envelope contains A/2. Notice that both references reference the same object.
> The right formula is changing either the 0.5A by A or the 2A by A.
No it's not, that's not how you make EV calculations. As I already said in my previous post, the parts in the EV calculation do NOT refer to the pair of envelopes. Both parts refer to the same object, not to different objects. You are trying to change the EV calculation such that the different parts refer to different envelopes. This does not make any sense.
> Try with A being a real like 100 usd numbers to see it.
...and that's not a way to prove / demonstrate the correctness of math. Yes, if you change these numbers in this way, you get EV 0, which is the correct answer. Just because the answer is correct does not mean the calculation is correct. In this case it isn't.
My explanation is correct and I can't say much more to convince you that I've already said.. yours is wrong in precisely why people are confused with the problem and I'm kinda sorry you don't see it.
Since you seem to be making some authority arguments just know that I have a PhD in ML / stats and was top 30 in my country in olympiad level competitive maths so I DO know what I do and in this case with absolute certainty.
Look for the other guys answer if you want more details
> Since you seem to be making some authority arguments just know that I have a PhD in ML / stats and was top 30 in my country in olympiad level competitive maths so I DO know what I do and in this case with absolute certainty.
Let's review what you're claiming with absolute certainty. This is the first claim that I'm contesting:
> they say "one envelope contains 2A and the other A/2".
When pointed out that they actually don't say this, you clarified your point:
> the "writing" doesn't say it but the written formula at bullet point 7 says it. That's the mistake, the formula is wrong and does not describe reality.
The second part of that quote is correct: bullet point 7 has the mistake, the formula is wrong and does not describe reality.
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.
You correctly identify where the mistake is, but you misidentify what the mistake is.
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".
To illustrate my point, consider a simple coin toss game. If the coin comes up heads, you win $2. If it comes up tails, you win $0.50. How might we construct the EV calculation for this coin toss?
0.5 * $2 + 0.5 * $0.50
Notice how both parts of the formula refer to the same coin. They don't refer to different coins. You have one coin that you flip, and depending on how that one coin lands, there is some probability that you get $2, and some probability that you get $0.50. If you looked at that formula, you wouldn't think that the first part refers to different coin that the second part.
Similarly, here we have one envelope (similar to having one coin) and we have uncertainty about the what the envelope contains (similar to having uncertainty about how the coin will land). We have 50% chance that the envelope will contain 2A, and we have 50% chance that the envelope will contain A/2. Thus, we arrive at the (incorrect) formula:
0.5 * 2A + 0.5 * A/2
It's very obvious that both parts of the formula refer to the same envelope, just like in the coin toss formula both parts of the formula refer to the same coin. To be extremely clear, I am not claiming that the formula is correct. I am claiming that both parts of the formula refer to the same envelope, not to different envelopes. To be specific, I am refuting this claim that you made:
> they say "one envelope contains 2*A and the other A/2".
They don't say that.
Furthermore, you make an additional incorrect claim:
> The right formula is changing either the 0.5A by A or the 2A by A.
Here you are just randomly changing the formula in order to make it output 0EV (which is the correct answer). Just because you get a correct answer does not mean your computation was correct. In this case it is incorrect, because A was defined badly. In order to fix the formula you would have to fix the definition of A. If you keep the incorrect definition of A and just shuffle symbols around until you get the correct result, your computation is still incorrect.
> 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
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.
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.
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.
> 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.
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.
> 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
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!"
> 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.
> 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.
> 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.
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.
> 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.
Yes, I've done dynamic programming in competitions. I'm not familiar with bellman equations, and the explanation you provided about convergence, policy functions, etc. went over my head, sorry.
> If you're actually interested in this sort of thing, I suggest checking out the poker research papers by Noam Brown [...] 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.
This sounds really interesting! It was a huge deal in the poker scene when the poker AI developed by Brown and Sandholm defeated pro human players. I read the first few pages of the paper now, but the paper becomes very math-heavy after that. I don't have the necessary background to understand the notation they use. That said, the "mistake" in the way "subgames influence each other" that you referenced, I suspect it was of a far simpler kind - the kind that Brown explains in chapter 2 which he concludes with:
This shows that a player’s optimal strategy in a subgame can depend on the strategies and outcomes in other parts of the game. Thus, one cannot solve a subgame using information about that subgame alone. This is the central challenge of imperfect-information games as opposed to perfect-information games.
Although Brown says "imperfect-information games" here, he actually means a specific type of imperfect-information games: the type where the opponent's strategy is not fixed. We're talking about games where your opponent can change their strategy in order to exploit weaknesses in your strategy. This property was a key requirement of the "coin toss" example game that he provided in chapter 2. If you modified the coin toss game such that the opponent's strategy was fixed, then the situation would change completely. Crucially, Two Envelopes Game is not one of those games where the "opponent" can adapt their strategy according to your strategy. The strategy of the opponent is fixed in Two Envelopes Game. That's why you can solve a single subgame in Two Envelopes game independently of other subgames, even though it's an imperfect-information game. If you are suspectful of this claim, we can verify it by simulations.
> the explanation you provided about convergence, policy functions, etc. went over my head, sorry.
I was agreeing with you when I talked about convergence. You said that adding a cost would solve the problem and I agree that it does. I think you were probably thinking of a literal cost like "one cent". If you choose a cost like that then switching an infinite number of times has an infinite cost. You could instead think of the cost as a fraction of your expectation, the cost is 1% of whatever you end up getting back. Now if you switch an infinite number of times you end up having a cost of zero. That might seem counterintuitive, but recall that 1/3 is .3333 repeating. So when you sum 1/3 + 1/3 + 1/3 you get .9999 repeating. Yet 1/3 + 1/3 + 1/3 is equal to one. Infinitely close to something else is basically being the thing you are infinitely close to. Even though we never get the reward we know that the fraction is becoming infinitely close to zero. People call it "converging" when we have an infinite sequence we can sum to a real value. We know before ever seeing the value that we'll be multiplying it by zero. So we can refactor the equation to be 0*ev(switch) and then take advantage of the identity of 0x=0 to declare the result to be 0. Thus, the calculation converges to zero.
> Although Brown says "imperfect-information games" here, he actually means a specific type of imperfect-information games: the type where the opponent's strategy is not fixed. We're talking about games where your opponent can change their strategy in order to exploit weaknesses in your strategy.
This isn't really the central problem of imperfect information. Consider that in a perfect information game, your opponent will also adjust their strategy to exploit weakness in your strategy. So if it was the central challenge, why does it happen in both? You can rule it out as what he was referring to, because it doesn't discriminate between the two types of games. The central challenge in imperfect information games is you have to play with respect to your information set, not the subgame you are in. So the policies and outcomes of other subgames influences the expected value of the subgame you are in. In perfect information, the only game in your information set is the subgame you are in. So you only have to play with respect to the subgame. That is what makes perfect information different from imperfect information. It discriminates between the two game types.
> This isn't really the central problem of imperfect information. Consider that in a perfect information game, your opponent will also adjust their strategy to exploit weakness in your strategy. So if it was the central challenge, why does it happen in both? You can rule it out as what he was referring to, because it doesn't discriminate between the two types of games.
The type of issue I was referring does not occur in perfect information games.
As a practical example, consider the concept of "balancing your range" in poker. If you play poker without doing that - if you play with a purely exploitative strategy where you are only trying to maximize your EV for each hand - then your strategy will be very easily exploitable by an adaptive opponent. You will frequently end up in river situations where your opponent can deduce whether you have a strong or weak hand, so they can fold to your strong hands and bluff you out of weak hands. In contrast, if you attempt to "balance your range" - that is, consider all the subgames - then you won't end up in these situations as badly. For example, when you make a particular river bet, your opponent might deduce that you have a strong hand 70% of the time and a bluff 30% of the time (as opposed to 100% and 0%).
This issue does not exist in perfect information games. Yes, my definition of "adjusting your strategy to your opponent's strategy" was overly broad to define this issue. But if you think about the poker example, where a poker player will might make a decision like "I need to bluff with this part of my range, because I need to support my strong hands [other subgames] by having some bluffs in my range in this situation" - you won't find a corresponding example from perfect information games like chess. This class of problems is unique to imperfect-information games in which an opponent is allowed to adapt their strategy to yours. If you fix the opponent's strategy, the issue disappears. If you turn the game into a perfect-information game, the issue disappears. Both requirements must be present for this issue to exist.
I thought that Noam Brown was talking about this issue in chapter 2. He discussed a simple coin toss game where one player took a strategy, and then the other player adapted by taking the optimal (exploitative) strategy against them. Then the other player changed their strategy, and the other player again adapted their strategy. And then he described a balanced (GTO) strategy. Then he said this as a conclusion:
> This shows that a player’s optimal strategy in a subgame can depend on the strategies and outcomes in other parts of the game. Thus, one cannot solve a subgame using information about that subgame alone. This is the central challenge of imperfect-information games as opposed to perfect-information games.
I thought that this corresponds perfectly to my poker example. If it doesn't, and it means something completely different, ok, sure. I'm not a mathematician. I can't even read the notation that's used in subsequent part of the paper.
> I thought that this corresponds perfectly to my poker example. If it doesn't, and it means something completely different, ok, sure. I'm not a mathematician. I can't even read the notation that's used in subsequent part of the paper.
I think you understood his point very well. I just think you're making a mistake in trying to recast his claim from "imperfect information games" have this property to "this narrow subset of imperfect information games" has this property. Both imperfect games with an opponent and imperfect games without an opponent have the property of needing to play as if you are in multiple subgames, because the definition of the problem is that you don't know which subgame you are in. His claim was that the policy in one subgame could influence the EV of another subgame - and in this problem, it does. I believe you're thinking of the EV of the envelope when you think you are proving this game doesn't have that property via calculation.
To see his claim applies to this game consider the case where P(Switch)=1. Being able to calculate the EV of the envelope's contents is a bit different from solving the subgame. Here, we have two subgames E1 and E2. We can know a priori what the expected values of envelope's contents in E1 and E2 are. But if you change your policy in E1, it changes your expected value in E2. If you doubt this, remember the core of the paradox again - choose to always switch and your EV is no longer the EV of the envelopes. Ergo, the EV of the subgame E1 is dependent on the strategies and policies of another subgame, E2.
> I think you understood his point very well. I just think you're making a mistake in trying to recast his claim from "imperfect information games" have this property to "this narrow subset of imperfect information games" has this property. Both imperfect games with an opponent and imperfect games without an opponent have the property of needing to play as if you are in multiple subgames, because the definition of the problem is that you don't know which subgame you are in.
If we take the poker example and we modify it by "locking" our opponent's strategy, then this property is removed. Suddenly the optimal strategy for us no longer includes any GTO-like thinking such as "balancing our range", we should simply maximize our EV for each hand "in a vacuum" without any thought to other subgames. We no longer care how our range looks to our opponent, because they are no longer able to change their strategy.
> To see his claim applies to this game consider the case where P(Switch)=1. The policy you chose in one subgame just changed the EV of another subgame.
Sorry, but I don't understand this. This sounds to me like we fix the probability of switching to 1, which means that we end up in infinite loop and the outcome can not be computed. I don't understand this premise, nor its implications for the EV of the other subgame.
> This is a subtle distinction that I mentioned earlier - the EV of the envelope as in the wikipedia problem isn't the EV of the subgame. So you're not solving the subgame if you figure out the EV of the envelope.
The expression you use "EV of the envelope" is ambiguous in this context. I'm not sure if you mean EV relative to the value of the total amount of money in the game, or if you mean EV relative to the value of the firstly-chosen envelope.
When we're talking about "solving a game", we're talking about finding the optimal decisions within a game to maximize the expected value from the game as a whole. In the Two Envelope game we have just one decision: switch or not. So we need to find out if EV(switch) > EV(stay), where both EVs are relative to the total amount of money in the game (not relative to the value of firstly-chosen envelope). We have many ways to conclude that both of these actions have expected value zero. We don't have to be able to compute the "EV of the envelope". We only need to know the relative difference between the EV of these 2 actions. There's many ways of computing them, and all of those ways lead us to the conclusion that the EV of both actions is zero. I'm not aware of any "incorrect" way of computing those EVs such that we would get a nonzero result.
I guarantee you that your solution is going to have to incorporate a set somewhere that includes both subgames. You might forget it does, because you simplify to a scalar, but it is going to be there. In perfect information, it isn't there. In imperfect information it is.
> I guarantee you that your solution is going to have to incorporate a set somewhere that includes both subgames. You might forget it does, because you simplify to a scalar, but it is going to be there. In perfect information, it isn't there. In imperfect information it is.
I'm having a lot of trouble identifying what exactly it is that we disagree about. If you have identified what it is that we disagree on, can you please formulate the disagreement as a wager that can be simulated with code? That way we can easily resolve the disagreement (or conclude that we actually don't have a disagreement).
Your opponents strategy in poker is fixed; they will always play the nash equilibrium strategy, they will never play another strategy. Their strategy is fixed. They will never change it from this setting.
You've claimed that subgame perfect play can be calculated without respect to the subgame you aren't in, because you can make a choice on the basis of the EV of the subgame you are in without respect to the subgames you aren't in.
I disagree. I think you still need to account for every subgame you are in as if you are in all of them.
Let the subgame you are in be you having KK and your opponent having AA. However, obviously - you only know that you have KK.
Therefore, you should be able to compute the strategy which is the best response to 37 suited and according to your logic it should be equal to the best response to AA. After all, you have no means of determining which subgame you are in. So you have to have the same response in both subgames.
So compute the best response for KK to AA and prove that this is also the best response to 37 suited.
However, you've claimed you don't need to calculate this with respect to other subgames. So your computation of 37 suited and your computation for AA must not be equal to each other - if they are, then you share terms. You calculated them with respect to each other.
Let Br = Best response.
Write a program which shows Br(p1, p2, I[KK]) != Br(p1, p2, I[KK]) and Br(p1, p2, I[KK]) = Br(p1, p2, I[KK]) simultaneously. (That is to say, both your policy and your opponents policy are fixed)
My contention is that you can't do this. You claim you can. You are free to use a simpler variant of poker - Kuhn poker - so that the computation becomes more tractable.
> That's why you can solve a single subgame in Two Envelopes game independently of other subgames, even though it's an imperfect-information game.
Obviously the EV of envelope 1 is the contents of envelope 1. If you know the probability of reaching it you can calculate the expected value of that envelope by multiplying by the probability of reaching it. But why are you multiplying by 1/2? Probability is defined in terms of sets. What are the set that makes it 1/2? Does that set contain only the subgames that are part of the subgame you are in?
> 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.
# Defining Counterfactuals
Consider a fair coin flip. You have {Heads, Tails}. Lets assume you are going to get heads, take it as a given - that is what actually happens. It actually happens. It is factual. However, for the purposes of analysis, sometimes it doesn't really matter that we know what happened. We need to consider all the cases that didn't happen. Tails in our analysis would be the counterfactual. These two situations, the factual heads and the counterfactual tails, they're associated with each other. There is a set {HT} that contains both of them. E.g P(head) = |{H}|/|{H, T}|.
When I'm saying counterfactual I'm referring to the events that we didn't assume to be factual, but which we want to keep track of. Probability kind of drops these terms when it says "assume" because it says |{H}|/|{H, T}| becomes |{H}|/|{H}| which is equal to one. This is perfectly fine from a math perspective. The thing is that just like saying 5/x = y lets us move to 5=xy is valid, it has some assumptions built into it. Namely that x can't be zero. Our counterfactuals are a bit like the x, because they offer an easily hidden constraint on what it is valid to do. If you reintroduce uncertainty about whether or not you are in H, you have to do so in a way that takes you back to having the set {H, T}.
Let me show you a practical example of that to make the point very clear:
Let the value of heads be one and the value of tails be zero. P(Head) = 0.5 But assume heads on the same coin flip. P(Head|assumptions) = 1.0 But assume heads again on the same coin flip. P(Head|assumptions) = 1.0. Now since P(Head) is actually true with 1/2 probability: Ev(Head) = 1/2*P(Head|assumption) + 1/2P(Head|assumption)?
Well, it is certainty true that 1/2P(Head|assumption) is a correct term. So you can't say this is wrong on the basis of the assumptions alone. Being very precise, the problem is the neglected counterfactual wasn't handled. Every term leading up to the equation was technically true, but obviously we just did something really weird right? And to be very precise, we neglected the counterfactual associated with heads.
# Defining subgames
Most games can be written out as a game tree. I showed one in my previous post. When you move down the tree, you are in a subgame of that tree.
In a perfect information game like chess if you are in a subgame, a portion lower on the tree, the other parts of the tree don't matter anymore. Your results are independent of the rest of the game tree.
In imperfect information though, just because you are in a subtree doesn't mean you know which subtree you are in. Your actual view into the game is through the information you have. Just like in the {HT} case you have to consider the potential that you have both H or T, in a subgame you have to consider the potential you are in every subgame that is reachable given the information you've seen.
> This game tree only makes sense if you change the definition of A.
Switching away from using A to avoid ambiguity.
> 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.
Yes it does, actually. Well, to me. I'll go more in depth so it less fuzzy for you.
The issue is that when you handle imperfect information correctly you have counterfactuals implied by your state-oriented reasoning, because you can't actually escape into perfect information world - you're not in it. When you combine the state-derived probabilities the - I don't know the word for this, so forgive me, but the - superposition of the two counterfactuals implied by those state-derived probabilities end up occupying the same space. They exist, because they are implied by the imperfect information game. They just aren't writing them out in the equations on wikipedia.
Edit: My 'spatial' probabilistic reasoning in my head is explained here, but I show you the identity rule that proves this in the next section.
> I never claimed that A = 2A
You aren't, but 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. The two cases were under two different games. So their implied counterfactual components don't agree with each other and we should never have been allowed to combine them.
We agree on this!
You're just more focused on the they come from different games part and not noticing the other part of what I'm telling you: you can equate the counterfactual components that they don't show between the two games, because there is an identity implied by being under imperfect information. Look at the set relationship closely across the two subgames and you'll realize that the set is defined to be equal to itself across all subgames where the set is used.
That is quite literally one of the logical contradictions. This is where they claim that 2=1. The sets aren't equal to each other after their operations. So by the identity implied by the set, they've claimed 2=1.
And yet when I try to claim this is a core aspect you throw out this...
> 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.
This is a really dumb nitpick with very poor support. If you're going to be this fallacious, you might as well go all out. Why not say I'm wrong because first graders learning to add numbers aren't learning how to handle imperfect information subgames? After all, your argument for me being wrong is that the pedagogical value of problems is independent of the subproblems they contain.
> Nope! I was very clear about which very specific claims I was refuting.
You have a more subtle point than I thought you were making. I'll check out your other post.
> 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.
Well, he does have another error; EV isn't defined over all policy choices. It is a problem on many of the wikipedia solutions too. He does see it - its why he mentioned other people talking about infinite series - but his correction attempt was more like "halt here because of type error" which is something that I as a programmer can respect.
This may be progress, but I don't see how it eliminates the "paradox". The claim in the "paradox" answer is that when you switch envelopes, "A" is not one of the possible outcomes. If "A" isn't a possible outcome then it shouldn't show up in the expectation value.
I agree that's true once we know that the two envelopes have 100 and 200 dollars. For me the tricky part is, how do we know it's 100 and 200? Maybe we have two envelopes with 100 and 50 dollars instead.
I think that subtly misses the point. The problem is that you're implicitly using a distribution that... isn't a distribution. And with this particular not-a-distribution, whether you should switch or not does not depend on the value of A. But with any actual distribution (... I think?) it does, at which point... no paradox. It's true that it's not clear (at least to me, but perhaps more generally) what distribution we should assume, but as long as we avoid treating something that isn't a distribution as if it were one we avoid the worst of it.
No, joe_the_user is exactly correct. The expectation they compute in step 7 requires that A is a constant. The formula seems to just be the regular formula for expectation: Value that B can take times probability that it takes that value, summed over all possible values.
But the "possible values" they put in are A/2 and 2A, which makes no sense. A is a random variable.
Don't forget that random variables are really functions on a probability space with an implicit argument, so (1) when multiple random variables appear in a formula, they're implicitly evaluated at the same argument omega, and the formula must hold for all omega from the probability space, and more importantly, (2) there can't be free-standing omegas in a formula for expectation because expectation is an integral (or sum in this case where the probability space is discrete) over all possible omegas.
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.
> 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
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.
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.
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.
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.
> 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."
We aren't in agreement about this. I realize we have to fix this, but fixing it doesn't resolve the paradox. It is a red herring.
> Yes, switching and not switching have the same EV and you can always switch.
With all due respect, this isn't true and asserting this doesn't resolve the paradox. See my other reply for why it doesn't resolve the paradox.
Mike, even when you get the correct result using the right rules the expected value of an envelope is not the expected value of a policy. I'm not agreeing with you that the problem is the expected value calculation. I'm telling you the problem is deeper than that. Fix the expected value calculation and you still have a paradox, because you are making a decision not on your expected value but on the expected value of the envelope. These are two different things and a rational person shouldn't eliminate the dependency between their policy and the reward they get.
To try and stress to you how big a problem this is, pretend you were playing the game and you wanted to find the right move that was going to make your EV the highest. Now, if you go with the logic of the problem, you are allowed to select not on the basis of your EV, but on the basis of some other EV. So instead of choosing the envelope, why not choose something else that has no relationship with our EV? Say, the weather in Alaska. If it is sunny, we like sunny. So switch. If it isn't sunny, we don't like that. So keep. It is crazy to do this, because there is no relationship between the EV you are using as a selection criteria and the EV your policy gets. This is the same situation as using the EV of the envelope. It sounds really crazy when you use the Alaska example, because it is so obviously unrelated. It sound so reasonable when you use the envelope example, because it isn't as obvious that they are unrelated. Yet for the policy of P(switch)=1, the ev of the envelope and the ev of the policy with respect to the game are not the same thing.
Now imagine the wikipedia article for the Alaska problem variant of the two envelope problem. Do you really think everyone would be so focused on the EV of the envelope as the step that was wrong? How could they? We have the same paradox still, but there is no EV calculation for the envelope included in the problem. If we can remove the EV calculation, yet still have the same paradox, it seems to me the paradox is not the expected value calculation.
So what is my solution? Well, to actually find your correct policy function you need to get the argmax of the policy with respect to the game. There are multiple ways to do this:
- Reinforcement learning does it by finding argmax pi with respect to Q_pi(s, a) = R(s') + P(keep)Q_pi(s',keep) + P(switch) Q_pi(s', switch).
- Game theory sets it up a bit differently. You define a similar graph using a different formalism, but simplified to operate over information sets. You can use a thing called regret matching; basically it turns out that if you play in proportion your normalized counterfactual regret, the average of those policies is the optimal best response.
In both cases you need to do something about the fact you're actually on an infinite graph. So the actual solution in the general case looks very very different from their way of solving the problem. It isn't just simple probability; I mean, it is, but taking the limit of an infinite sequence and taking advantage of the properties of markov chains aren't usually what I think of when someone tells me that something is simple probability. That is one formalism. In the other, we do have simple probability, but it isn't necessarily obvious that the central limit theorem gives us the optimal policy when we play in proportion to not just our regret, but our counterfactual regret. So yes, simple probability, but also, most people who know simple probability don't necessarily even know what a counterfactual is. So maybe not that simple after all.
But lets say we stick to the problem. We are here to learn how to avoid this problem, right? Nope. If you don't do things like this, you'll just be wrong in more complicated situations. Because the EV of the envelope is not the EV of the game with respect to your policy. This gets increasingly true as your imperfect information games get more complicated; it is very true of complex real world situations. The value of a wallet with a hundred dollars in the real world is different depending on whether you got that wallet with a policy function of robbing people versus earning it at your work. I feel sticking to their formalism means you end up conceited with regard to your ability to protect yourself from this paradox, because you consider yourself a master of the expected value of the envelope, but you're still vulnerable to the paradox, because the expected value of the envelope isn't the expected value of the game with your policy. So sticking with the problem is the opposite of protecting yourself.
I'm so far from what they want the problem to focus on, but they are wrong to focus on that. They aren't protecting themselves from making the same mistake. They're dooming themselves to use the wrong tools for solving this problem.. So they will make this mistake and they'll even be more confident in themselves as they do it, because they were clever and did the wrong thing in a better way, calculating the EV correctly, but staying within the land of paradox despite that.
Well taking the mean of products doesn’t make much sense anyway… but if you take a log and average then you end up with the typical solution. If you take a geometric mean then you end up with 1, which implies there’s no difference either.
I think the assumption that "for all x, p(x) = p(2x)" implies a uniform distribution on the natural numbers.
Imagine we have a distribution that satisfies that assumption, and then someone tells you they've sampled from that distribution and found that the result is of the form (say) 7*2^k, for some k > 0. That conditional distribution for k would seem to have to be uniform, right?
> We're not dealing with the natural numbers, right?
Well, you can either assume that the envelope's numbers are specified only up to pennies. Or you can just round and say you don't care about sub-pennies.
Or you can go with your suggestion, and look at the lack of a uniform distribution over the reals / rationals.
