Talk:Two envelopes problem/Arguments

Variant with no constrains

https://docs.google.com/file/d/0B7Z3B3DjCNt6VjJLT284TDVSZGs/edit?usp=sharing

Suppose you are allowed to open the envelope before you make the decision to swap.

1. Suppose the amount in the envelope you opened is $10
2. The other envelope may contain either $5 (meaning the one you chose is the higher value of the two with P=.5) or $20 (meaning the one you chose is the lower value of the two with P=.5).
3. So the expected value of the money in the other envelope is

$E(X)=\$5*1/2+\$20*1/2=\$12.5>\$10$

This is greater than the one you opened. So you gain on average by swapping. This applied to any amount of money you discover in the envelope and suggesting you should always swap no matter what you find. This implies that opening the envelope and finding the amount is not needed in deciding the action of swapping.

This is contradictory to the fact that the probability of choosing the envelope with more money is .5, and swapping the envelop will not increase that probability (1-.5=.5).

Solution:

The probability, P=.5 of you choosing the envelope with the higher (or lower) amount does not imply that the other envelope have 50% chance containing 1/2 of amount you see and 50% chance containing 2X of the amount. There’s only 2 possible cases in this game.

Case #1: The game organizer put a $5 and $10 in each envelope. You have 50% chance of getting the $10 and once you chose the envelope of $10. The amount that the other contains is $5 with 100% probability.

Case #2: The game organizer put a $20 and $10 in each envelope. You have 50% chance of getting the $20 and once you chose the envelope of $10. The amount that the other contains is $20 with 100% probability.

The probability of the game organizer doing Case #1 is P and Case #2 is Q. The expected value of the other envelope once you open the $10 envelope is:

$E(X|X_{0}=\$10)=\$5*P+\$20*Q$

However, from your perspective, P and Q is unknown. So the expected value of the other envelope cannot be determined.

The probability of you choosing the higher value in either case is 0.5. So the probability of winning this game (chooing the envelope with the higher amount) is

$.5*P+.5*Q=.5*1=.5$

So no swapping is needed.

Modification to the setup:

To satisfy the original conclusion of "always swapping", the following is needed:

You are given the following information before the game:

The game organizer flip a fair coin.

1. if heads, they will put $5 and $10 in each envelope
2. if tails, they will put $10 and $20 in each envelope

Once you open the envelope that contains $10, the expected value of the other envelope is indeed higher and hence you should swap.

Question

Why is it better to switch in nalebuff? — Preceding unsigned comment added by 2601:58B:4280:1180:2914:7D36:9319:2337 (talk) 04:24, 13 October 2021 (UTC)

Error in formulation?

"The switching argument" (and the rest of the article) keeps listing a formula that doesn't match the description of the paradox.

Imagine you are given two identical envelopes, each containing money. One contains twice as much as the other.

• Envelope A contains 1$.
• Envelope B contains 2$. (Suppose $ is some scalar constant).

You may pick one envelope and keep the money it contains.

• The expected value of the envelope you have = 1.5$

Having chosen an envelope at will, but before inspecting it, you are given the chance to switch envelopes. Should you switch?

• Expected value of the other envelope = $1.5
• If you picked envelope A, the expected value of switching to envelope B = +1$
• If you picked envelope B, the expected value of switching to envelope A = -1$
• Expected value of switching to the other envelope = (1$-1$)/2 = 0

Why does the article keep saying this is 5/4? This is clearly wrong.

Awwright (talk) 03:43, 31 August 2023 (UTC)

It's supposed to be wrong. Firefangledfeathers (talk / contribs) 03:47, 31 August 2023 (UTC)

The formulation is presented as a serious problem in mathematics with no solution. If we know it's wrong then the article needs to say that. Awwright (talk) 03:57, 31 August 2023 (UTC)

It's a semi-serious puzzle in mathematics with multiple solutions. Yours above is essentially the same as the one presented at the beginning of §Example resolution. Firefangledfeathers (talk / contribs) 04:01, 31 August 2023 (UTC)

So then based on what you're saying, there's a problem in the article, because the text says:

No proposed solution is widely accepted as definitive

If there really is a mathematical solution I would expect the article to be forthright about that; and multiple solutions, multiply so.

But the problem I'm raising here is there's an actual error in the formulation which is being presented as correct to our readers. The particular problem is in this line:

The other envelope may contain either 2A or A/2.

This is an outright error. A is being re-used with different values. If you picked the smaller envelope, then value of A will be less, and thus your "percentage gain" will be larger; and if you picked the larger envelope, your "percentage loss" will merely be 50%. Percentage changes do not average like this! This is the same arithmetic mistake as averaging together 2A and 1/2 A: (2 + 1/2)/2 = 5/4

The formula described in the article describes the following game instead:

Imagine you are given two identical envelopes, each containing money. One contains twice as much as the other. You may pick one envelope. If you pick the larger envelope, you will be taxed half the money when you leave. You're then given the chance to switch envelopes. Should you switch?

Yielding:

• Envelope A = 1
• Envelope B = 2
• Expected value of staying with A = 1
• Expected value of staying with B = 2/2
• Expected value of switching from A to B: 2
• Expected value of switching form B to A: 1/2
• Expected value in your envelope: 1
• Expected value in other envelope: 5/4

And yet No proposed solution is widely accepted as definitive? Really? Awwright (talk) 05:32, 31 August 2023 (UTC)

• This article is kind of a low-budget version of Monty Hall problem. EEng 06:18, 31 August 2023 (UTC)

  User talk:EEng#s, this thread specifically discusses a problem with specific passages of the article. Please do not remove it. Awwright (talk) 04:30, 7 September 2023 (UTC)

  It wasn't removed; it was moved to Talk:Two_envelopes_problem/Arguments, which is where stuff like this belongs. As already observed, this article attracts a lot of ill-considered commentary. We can leave it here if you wish for a time, and you can argue with whoever wants to engage what you're saying -- which I can't understand -- but in the meantime your change to the article cannot stand. EEng 06:06, 7 September 2023 (UTC)

  I don't care all that much about where this discussion lives, but I agree with EEng that Awwright needs to stop edit warring their change in. Among other issues, it's not helpful to duplicate the point of §Example resolution earlier and with less clarity. Firefangledfeathers (talk / contribs) 12:16, 7 September 2023 (UTC)

  It doesn't matter if it was "moved" or not; you removed it from this page, which is where discussion about editing the article belongs. Do not remove it.

  You know the rules, if you want to revert an edit that's not obviously vandalism then you must engage with the contents of what you're reverting. You can't simply say you don't like it. I will revert such unjustified reverts in accordance with the rules. Awwright (talk) 17:07, 7 September 2023 (UTC)

  Hi EEng, I would like to again invite you to explain what your concern with my edit is. Here, I describe a specific way in which readers are being mislead, so I made an edit that partially rectifies that. It was not intended to be a complete fix and if you have additional improvements to suggest then I would like to hear those, or see additional improvements on top of my edit. (But do not simply ignore the issue then silently revert any changes based on what I've identified.) Awwright (talk) 17:29, 7 September 2023 (UTC)

  User:Awwright Let's take your formulation above, and imagine that you open the envelope you've chosen, finding one hundred dollars. This establishes that your "$" ("some scalar constant") is either 100 dollars or 200 dollars; we don't know which. This 100 dollars is the value of what you are calling A: the amount in the other envelope is either 50 dollars (half of A) or 200 dollars (double A). How do you get from this to "A is being re-used with different values."? MartinPoulter (talk) 18:18, 7 September 2023 (UTC)

  either 100 dollars or 200 dollars

  No, when I say "constant", I mean constant; it must have a specific, unchanging value before any envelope is chosen, such as $100, or 50 utils. It doesn't have to be "known" but it must at least be independent of the envelope selection.

  To permit A to be a dependent variable here is a fallacy, and one of the causes of the paradox; it's the same class of error as saying "I double then halve a stack of pennies, which is 200% and 50% = (200% + 50%)/2 = 125% which is more than I started with". (The correct formulation would of course be 200% * 50% = 100%.)

  If instead I had two equal stacks of pennies and I doubled one and halved the other, then I indeed would have 5/4ths the pennies that I started with—but this is not the game that's described at the start of the article. The source cited by the introduction does in fact explain this, but this is wrongly omitted. If an article is going to describe a fallacy, known to the cited sources to be a fallacy, then it should point out the fallacy, instead of leaving it as a "puzzle," or presenting the error as correct. Awwright (talk) 20:02, 7 September 2023 (UTC)

  @Awwright You're writing a lot of text that has no connection to the problem in the article or the arguments you are responding to. Of course the scalar constant is a constant. Of course it has a specific, unchanging value before any envelope is chosen. The person opening the envelope does not know what that unchanging value is, but it is still a specific, unchanging value. Of course it is independent of the envelope selection. None of that changes the fact that if you open one envelope and find 100 dollars, that means the scalar constant is either 100 dollars or 200 dollars. You don't know which it is, but it must have one of those two values. That you think this is the same class of error as doubling then halving just shows you're not understanding the basics of the problem. You've failed to justify your statement "A is being re-used with different values." MartinPoulter (talk) 12:16, 8 September 2023 (UTC)

"Double then halve" isn't the situation described. It's "I have a .5 chance of doubling something and .5 chance of halving that thing", and for that (200% + 50%)/2 = 125% does indeed apply; the question (or one of them, anyway) is whether that thing is well defined. EEng 23:03, 7 September 2023 (UTC)

It's nonetheless the same class of error. The first source talks about this in detail. I don't even know what content is getting pulled from that source, the passages with the citation completely mismatches the source.

I have a .5 chance of doubling something and .5 chance of halving that thing

...is also not the game described in the introduction. If that were the game being played, you would be assigned an envelope, then told you can switch to one that has a 50% chance of being double, and 50% of being half. In that case then yes, you would benefit from switching envelopes—but only once; if given the chance to switch again, you may be giving up an envelope with as much as 4x variation in possible value, resulting in an expected value of 4/5ths if you switched.

(Further: if you let Monty then take both envelopes from you, randomize them, then give them back to you, and he lets you switch, there's no expected value to switch. But if he then says "the other envelope has a 50% chance of being half, 50% chance of double"—that means he gave you the same envelope you started with, and it's in your interest to switch—but again, only once.)

Again, the sources talk about all of this; and as I look through the article history and people keep editing these explanations out and saying "it's an unsolved problem" and this is wrong. Awwright (talk) 00:42, 8 September 2023 (UTC)

Who are you quoting there? Who is saying "it's an unsolved problem"? That's certainly not what the article says. MrOllie (talk) 00:48, 8 September 2023 (UTC)

Would a reasonable reader be able to read through the introduction and first heading, and understand that there's an arithmetic and probability error being made in one of the steps?

Would a reader see "The puzzle is to find the flaw in the line of reasoning in the switching argument..." and understand that the sources being cited, in fact, identify a specific cause of the paradox? (I feel like there's a WP:NOTPUZZLE page, I'm pretty sure articles are not supposed to pose "left as a problem for the reader" questions.)

If you read "There have been many solutions proposed," is that not worded as if none of them are definitive or correct?

Would you be able to read the article and explain what causes the paradox? Awwright (talk) 01:03, 8 September 2023 (UTC)

It's worded as if it is an ambiguously stated problem that has different resolutions depending on one's unstated assumptions, which is what it is. Given your responding to my question with questions of your own, I'll assume you weren't quoting anyone. Please don't do that. MrOllie (talk) 01:10, 8 September 2023 (UTC)

No, I can perfectly well do that; you asked who I was quoting when I said "it's an unsolved problem" and so I quoted for you all the passages in the article that I was referring to. If you want to disagree with the summary fine, but please don't throw the "straw man" argument at me, you know what parts of the article I'm talking about.

Also, the questions I posed are not rhetorical; I would like to know if you can read the article and succinctly explain what fallacy produces the paradox. (Hint: it's with bullet point 3.) This is so if you're also under the impression that there's not a solution, I can quote from the article sources and persuade you. Awwright (talk) 01:23, 8 September 2023 (UTC)

None of the passages in the article contain the text you initially quoted. That you seem to think the article claims the problem is unsolved is illuminating as to what's going on in this talk page discussion, but that is not what the article actually says. We can perhaps try to figure out how to word the article more clearly, because it does not say what you seem to think it says (I also think that is probably true of your conception of other people's responses in this talk page discussion). MrOllie (talk) 01:29, 8 September 2023 (UTC)

Well then apologies for the confusion with paraphrasing, which is why it's fair for you to ask "who said that" and for me to elaborate about what I meant. But I don't think that amounts to a "straw man"; my argument is purely that the article is inconsistent, misleading to readers, and how poorly it summarizes the sources (the sources simply don't seem to be used any time until "Example resolution").

I'm also not trying to claim anyone's opinion here is "this paradox is completely unsolved" but instead, that a reader would draw this conclusion. Though I do find it vexing that nobody so far is willing to point out the fallacy. That strikes me as essential knowledge to do any substantive work on this article.

Look, Necktie paradox gets to the heart of fallacy that causes the paradox in less than 200 words, and hints that the paradox is caused by an error in about 100.

In contrast, reading this article, I'm strongly led to believe it's still an active problem in math, and it's just not. It's an incorrect use of a dependent variable where only an independent variable can be used, and bullet (3) should read "2A or A" not "2A or A/2"—and most people do not know enough probability to realize this, so this easily-made mistake should be called out. Awwright (talk) 01:55, 8 September 2023 (UTC)

I've got a degree in statistics so I assure you I know enough probability for present purposes. Item (3) at Two_envelopes_problem#The_switching_argument reads correctly as is: "2A or A/2". And the Necktie Problem article does not, in fact, explain the fault in the originally stated reasoning; it simply gives another line of reasoning which leads to a different conclusion. EEng 03:38, 8 September 2023 (UTC)

No, as identified in literally the first source cited by the article, (3) is indeed incorrect; if "2A or A/2" were correct then that would imply the highest possible value of the envelope would be 4x the lowest possible value of the envelope. But it's not, the difference in possible values is merely 2x.

The Necktie Problem article could perhaps use some elaboration, sure, but my point is that it's not as misleading as this article is. Awwright (talk) 22:38, 8 September 2023 (UTC)

I give up. EEng 01:42, 9 September 2023 (UTC)

@Awwright You're treating your failure to understand the problem discussed in the article as though it overrides the description of the article in reliable sources. Wikipedia is not for publishing your original research and isn't meant as an outlet for people who feel really sure the experts are wrong. There are plenty more forums where you're free to express your opinions, including Talk:Two_envelopes_problem/Arguments; stop trying to edit them into the Wikipedia article about the topic. MartinPoulter (talk) 12:25, 8 September 2023 (UTC)

You misunderstand my point, with the exception of me inventing games that are described by the given formulation (to help illustrate), I'm merely summarizing the sources used in this very article. If you want to disagree with my summarization, that you think I'm interpreting them wrong, please make that point, I would like to hear that.

Also please note, original research is not actually prohibited on talk pages, only on article content:

This policy does not apply to talk pages and other pages which evaluate article content and sources, such as deletion discussions or policy noticeboards

Awwright (talk) 22:43, 8 September 2023 (UTC)

@Awwright To the extent you are saying that you think this article needs to be shorter and more to the point for the general reader I agree with you one hundred percent. Not many people have the time and energy to read the whole article anymore. I stopped reading the whole article long ago. It does not have to be this way. I think we can delete a lot of fringe ideas and just explain the more common ideas in a succinct way. And delete a lot of unnecessary math. iNic (talk) 20:34, 8 September 2023 (UTC)

@Awwright Another point you highlight is that the introduction lacks sources. This is correct and the reason for this is that there is still not a single secondary source on which to base an introduction. There is just a lot of primary sources. Since 1987 on average five research papers per year have been published, and not a single secondary source. Wikipedia is not even allowed to base an article on only primary sources, which creates a real dilemma for this article. If we would interpret this Wikipedia rule strictly this article has to be deleted immediately. But as no one so far has called for its deletion the editors of the article has tried their best to explain some of the simplest ideas found in the primary sources, and write an introduction that is balanced and fair. iNic (talk) 20:34, 8 September 2023 (UTC)

AI generated solutions

Extended content

At last we can put an end to this long discussion. The correct answer is really easy. Question: What is the correct solution to the two envelope problem? ChatGPT: The two envelope problem is a classic problem in probability theory. The basic premise is that you are given two envelopes, each containing a certain amount of money. One envelope contains twice as much money as the other. The question is, if you are allowed to open one envelope and see its contents, should you switch to the other envelope to maximize your chances of getting the larger amount of money? The correct solution to this problem is to always switch envelopes. This may seem counterintuitive, but the reason it works is because of the way the problem is set up. Since one envelope contains twice as much money as the other, if you choose an envelope at random, you have a 50% chance of choosing the envelope with the smaller amount of money and a 50% chance of choosing the envelope with the larger amount of money. If you choose the envelope with the smaller amount of money, and you then switch to the other envelope, you are guaranteed to get the larger amount of money. This is because the other envelope must contain the larger amount of money, since the one you chose contains the smaller amount. Therefore, by switching envelopes, you will always get the larger amount of money. On the other hand, if you choose the envelope with the larger amount of money, and you then switch to the other envelope, you will only get the smaller amount of money. This is because the other envelope must contain the smaller amount of money, since the one you chose contains the larger amount. Therefore, by switching envelopes, you will only get the smaller amount of money in this case. Therefore, the correct strategy is to always switch envelopes. This will maximize your chances of getting the larger amount of money.iNic (talk) 11:59, 9 December 2022 (UTC) Hard to believe no one thought of that before! EEng 12:22, 9 December 2022 (UTC) That's no way to make money. Invest in crypto. Tell all your friends! Thincat (talk) 10:37, 10 December 2022 (UTC) Personally I don't think we should have the Talk:Two envelopes problem/Arguments sub-talk page at all—but if any topic belongs there, shouldn't it be this one? Awwright (talk) 18:05, 7 September 2023 (UTC) Yes, a joke post passing on a ridiculous AI-blather "solution" to the problem is what we need here for sure. EEng 19:17, 7 September 2023 (UTC) Q:   What is the correct answer to the Two Envelopes Problem? A:   The "Two Envelopes Problem" is a classic paradox in probability and decision theory that challenges our intuition about expected value calculations. Here's a breakdown: The Problem Setup: You are presented with two envelopes. One contains twice as much money as the other, but you don’t know which is which. You pick one envelope (say Envelope A), but before opening it, you’re given the option to switch to the other envelope (Envelope B). The question is: Should you switch envelopes? The Paradox Suppose Envelope A contains an amount x. If you switch, Envelope B might contain: 2x (if Envelope A is the smaller amount). x / 2​ (if Envelope A is the larger amount). From this reasoning, the expected value of switching looks like this:         E(switch) = 0.5⋅2x + 0.5⋅x​/2 ​= 1.25x​. Since 1.25x ​> x, it seems you should always switch! But this reasoning applies no matter which envelope you initially choose, creating a paradox because it suggests an infinite loop of switching. The Resolution The paradox arises from a misunderstanding of the problem setup and the incorrect use of probabilities. Here’s the key issue: Flawed Assumption of Symmetry: The calculation assumes you can freely assign probabilities to the envelopes containing 2x or x​/2 without accounting for the actual distribution of the amounts. However, x is not fixed—it depends on which envelope you initially choose. The "expected value" calculation mixes up these scenarios, leading to inconsistent reasoning. Correct Framework: The paradox dissolves when you consider the prior distribution of the amounts in the envelopes. For example: If the amounts are drawn from a bounded or finite distribution, you cannot apply the same logic to all x. If the amounts are unbounded or drawn from an infinite distribution, there is no well-defined expectation to calculate. Practical Answer: Without additional information about how the amounts in the envelopes are determined, there is no rational basis to favor switching or sticking. Both options are equally good because the reasoning is symmetric. Conclusion The correct answer is that switching or staying is equivalent unless you have more information about the prior distribution of the amounts in the envelopes. The paradox is a product of misapplying probability concepts and ignoring the dependencies in the problem setup. iNic (talk) iNic (talk) 17:25, 28 January 2025 (UTC)