Two envelopes problem

Am I really the one who is getting things conflated? Go back to the problem and look at step ten. The problem allows infinite swapping. You are doing an EV calculation, but your analysis is fundamentally flawed. Assign the policy of always switching. Your analysis is claiming that EVs can be calculated, but they can't. The EV is either zero or undefined for switching, because the recurrence relationship is an infinite sequence. Since 60 != 0 and 60 != undefined, but you claim that they are, something is very wrong with your calculations. You're using the wrong formalism. The policy is supposed to be able to vary, but you're treating EV as a concept that isn't dependent on policy.

Lets take a step back and learn the important lesson for more complex situations. Your policy influences your expected value. Not keeping that in mind is going to destroy the ability to correctly calculate expected value. You aren't trying to search for the best thing to do on the basis of expected value. You are searching for the right policy to provoke a high expected value. The difference is subtle, but essential.

How do we correct it? Well, the right formalisms that allow you to search for the correct policy comes from several fields, but one of them is game theory. In game theory when dealing with imperfect information, it is considered incorrect to do state-based reasoning under imperfect information. This is because you aren't in a state - you are in an information set. When you are playing the game you have to consider every game you could be in, because you don't know which you are in.

This is a second problem with the analysis, but I think you corrected this one.

They ask to be able to translate this into more complex situation. So the general lesson here is about considering counterfactuals in your analysis. An example of this in practice is Jeff Bezo's talking about his decision to found Amazon on the basis of regret minimization on account of his theory about how he would feel in various counterfactual futures. He didn't consider one EV, the founding of Amazon, but also other EVs like founding Amazon and failing and also not founding Amazon and doing other things.

I think I get why you think I'm conflating A, but I'm actually trying to point out that the wikipedia article is conflating A and so its hard to have a productive discussion due to our inheritance of their misuse of terms. I don't want to conflate A, but the Wikipedia article defined A in their expected value calculation and in that equation it ends up taking on a different meaning to what it means when it is defined to be 60. And their meaning ends up claiming things like 1=2 in practice, because of the properties of the hidden counterfactual part of their equations - just because they neglect to show them, doesn't mean they don't exist in the correct mathematics. So the logical contradiction is there - which is exactly the thing the problem asks us to identify.

Disregard the above; you basically said the same thing I did, but used a phrasing that made it hard for me to realize you had done so.

> 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.

> 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.

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.