Talk:Two envelopes problem/Archive 6

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Falk (2008), Falk and Nickerson (2009), Nickerson and Falk (2005)

I have also been in email contact with Ruma Falk and have read the three papers on TEP of which she is author or co-author. They are all excellent. Note that they are written for different audiences. Nickerson and Falk (2005) is perhaps the first paper on TEP which seriously surveys all the solutions known to the authors at that time. It explicitly points out that the different solutions make different assumptions as to context and intention of the writer. It pays attention to the Bayesian context with and without improper priors. Explains that improper priors give infinite expectation values and hence unreliable advice as to decision or action. It's all there. All three are now in my dropbox, if anyone else is interested to read them. I think that Nickerson and Falk (2005) is obligatory reading for any wikipedia editor of TEP. It's the only paper with a complete and neutral survey of as many solutions as possible. Richard Gill (talk) 13:42, 4 November 2011 (UTC)

The error in the switching argument

Please help to check which sources say the following, and help to improve the article accordingly:

The basic setup does not accent in any way the fact that there is a huge difference  between having chosen the envelope "filled at first with any determining amount of money", i.e. containing the "original amount" of X  (let's call that amount "Xoriginal", and let us denote that amount with the value of "1") with probability of 2/4, or having chosen the envelope that was filled thereafter with the dependent amount, say "the second envelope", containing dependently either 2X ("Xdouble", with the denoted value of "2") with probability of 1/4, or with equal probability of 1/4, containing only X/2 ("Xhalf", with the denoted value of "0,5").

For only in the case that (with probability of 1/2) you should have chosen the envelope with the determining amount of "Xoriginal", then with probability of 1/2 the other envelope will contain dependently "Xdouble" (2X) or will contain with equal probability of 1/2 only  "Xhalf" (X/2). And note that only in that case that, by chance, in 1/2 of cases you should have chosen the envelope containing the determining "Xoriginal" amount, all the further conclusions of the TEP, including "5A/4") do apply, otherwise not.

But fully neglecting that draconic precondition, "the basic setup" is based on this last variant only. Without saying so. The basic setup and all further conclusions do fully neglect the scenarios of the two other possible constellations with quite other assigned prospects, having equal probability of 1/4 each: That you could have chosen the dependent "second envelope" of 2X ("Xdouble"), having quite other prospects assigned, or that you could have chosen the dependent "second envelope" of X/2, again having quite other assigned prospects, contradicting all further conclusions of the TEP.

The value of 5A/4 is only correct if the candidate, by chance, should have chosen the determining envelope "filled at first with any amount of money". Then the dependent envelope B (its "slave") can be expected to contain 5A/4.

But if the candidate should have chosen the dependent envelope filled thereafter only, remunerated either with considerable 2X or with X/2 only, then in that case the expected determining value of the other envelope can never be 5A/4 but will forever be 4A/5 only. That's a given fact:  E(B) = (2/2+1/2*2)/2,5 = 4/5 A.

And – without knowledge which one of those both envelopes had been chosen, the determining first one or the dependent second one – the expected value of the other envelope is exactly 1A:

E(B) = ((2*1+1/2)+(2/2)+(0,5*2))/4,5 = 1A.

Which sources do show that coerciveness? It could help to better articulate the inevitable facts. Who can / will help? Gerhardvalentin (talk) 18:56, 23 November 2011 (UTC)

The standard TEP story has two envelopes filled with two positive amounts of money, one twice the other. Then the player picks one of the two envelopes completely at random. How the envelopes are filled is not part of the story. Some authors, by way of example, discuss particular ways in which the host could have chosen the two amounts of money. They show that for their particular story, one or more of the steps of the argument is actually wrong. But such examples do not solve the paradox. For one thing, for many authors the important thing is not how the host filled the envelopes, but what the player believes about the amounts of money finally in the two envelopes. For another thing, showing that there is an inconsistency in the TEP argument in one particular case does not prove that the argument is incorrect in all cases. It might help you to discover what is the mistake in general, but it might not.

Anyway, the general mathematical principles behind all knowns solutions is given in Samet, Samet, and Schmeidler and later improved by Gill. Richard Gill (talk) 00:05, 24 November 2011 (UTC)

Thank you.  Am no prob. champ, nor do I know all the sources, but you helped me to see that the 5A/4  result can never be based on any symmetric distribution, because in that case of symmetric distribution the only valid result E(B) = ((2*1+1/2)+(2/2)+(0,5*2))/4,5 = 1A.

And you helped me to see that one thing is quite obvious:
The TEP story and any game is based on two quite different types of envelopes. Type "α" had been filled first with any certain determining amount "X" (let us say that this amount is "1" e.g., and only "thereafter" another envelope of type "β" has been filled either with dependent "2X" or with "X/2".

And the 5A/4 result is only valid in case you know for sure that the determining envelope A is of type "α", otherwise never: 1/2 * 2A + 1/2 * A/2 =5A/4
(but the "basic setup" of the TEP ignores that fact and doesn't even mention this "a priori given draconic restriction".)

Whereas in case you know for sure that envelope A is of dependent type "β" then E(B) = (2/2 + 1/2 * 2) / 2,5 = 4A/5 only.

And as long as you do not know for sure whether you hold a determining "α-envelope" or a dependent "β-envelope", as this is not "part of the TEP story", then only E(B) = ((2*1 + 1/2)+(2/2) + (0,5*2))/4,5 = 1A can be correct. That's it.

	Envelope A	amount  A	Envelope B	amount  B	result of switching  to envelope B
A of type α (determining envelope):	Xoriginal	1	Xdouble	2	2Aα of Xoriginal
A of type α (determining envelope):	Xoriginal	1	Xhalf	1/2	1/2Aα of Xoriginal	5/4 Aα of Xoriginal
A of type β (dependent envelope):	Xdouble	2	Xoriginal	1	1/2Aβ of Xdouble
A of type β (dependent envelope):	Xhalf	1/2	Xoriginal	1	2Aβ of Xhalf	4/5 Aβ
total  (type unknown resp. mixed)		4,5		4,5	1A (if type unknown / mixed)

Result of switching to envelope B:  5/4 A, 4/5 A and 1A respectively (millionfold verified):

Only if envelope A is of type "α" with determining amount, then E(B) = 1/2 * 2A + 1/2 * A/2 = 5/4 A (as the article – without destinguishing - incorrectly claims for all types of envelopes)
Only if envelope A is of type "β" with dependent amount, then E(B) = (2/2 + 1/2 * 2) / 2,5 = 4/5 A
If type of envelope A ("α" or "β") is unknown:    E(B) = ((2*1+1/2)+(2/2)+(0,5*2))/4,5 = 1A

Is that correct distinction mentioned anywhere in the sources?  Regards, Gerhardvalentin (talk) 01:52, 24 November 2011 (UTC)

This distinction is mentioned in many papers. You should read the sources if you want to work on this article. If you like you are welcome to join my dropbox folder which contains all the important papers and many less important.

There is a problem with your notation. You relate E(B) to A. But in many formulations A is also random. If you want to *do* probability calculus, please *learn* probability calculus. There are excellent free eBooks easily available. Many of the sources point out that the error in the TEP argument is the error of not distinguishing between random variables and possible values thereof. If you do not make this distinction yourself you cannot contribute to writing an article which correctly summarizes what is in the sources!

Secondly, your conclusions are not correct, because in the case that we use probability also to represent prior ignorance of, say, the smaller amount of money x in the two envelopes, then it is the case that if we a priori are so collosally ignorant concerning x that we use the improper prior distribution which makes log(x) uniformly distributed on the whole real line, then it is true that E(B|A=a)=5a/4 for all a. If that interpretation was the intention of the writer, then we must resolve the paradox in another way. Richard Gill (talk) 18:47, 24 November 2011 (UTC)

Yes, thank you Richard. I meant to ask whether – for a better understanding of the very  "nature of this paradox"  for grandma and grandson – any sources do clearly accent that "E(B)=5A/4" is fully valid,  but fully valid only if you know for sure that envelope A had FIRST already been filled with any arbitrary determining amount and only afterwards it was decided whether the dependent envelope B was filled with dependent half or dependent duplicate of the already fixed determining amount of envelope A. Because only then, although the already "prefixed" determinng amount in envelope A  will equally likely be double or half of the "dependent" amount in envelope B,  "5A/4"  for envelope B is perfectly correct then, but never otherwise. Because "E(B)=5A/4" does not tolerate any prefixed amount in envelope B with some "dependent" content in envelope A. This given restriction should be made obvious and may never be kept secret, otherwise creating an apparent paradox. This given axiomatic precondition not explicitly being "part of the story",  E=(B)=5A/4 should never unrestrained and nonrestrictive be laconically quoted. Do any sources clearly accent this basic fact, just to help grandma and grandson for a better understanding of the very nature of this just assumed "mystic paradox"? Gerhardvalentin (talk) 11:58, 25 November 2011 (UTC)

The present text of the article says explicitly: there are two envelopes, you pick one completely at random, and that is Envelope A, the other is B. Now, before you even look in A, you consider whether or not you'ld like to switch for B. There is no point at all thinking about the wrong scenario, except to underline the point that it is wrong.

Sure, there are plenty of sources which compare the true TEP scenario to the wrong scenario.

But another point: why do write E(B)=5A/4? Is A supposed to be random or fixed? Please think carefully about your notation! You have got to get this straight!

Richard, but it's the article that misleadingly states:

7. So the expected value of the money in the other envelope is
${1 \over 2}(2A)+{1 \over 2}\left({A \over 2}\right)={5 \over 4}A$

8. This is greater than A, so I gain on average by swapping.

It's the article that misleadingly says that you can expect B to be 5A/4 or 5/4 A. – It's the article that incorrectly says so. Gerhardvalentin (talk) 12:55, 27 November 2011 (UTC)

Yes, the reasoning would be easier to follow and easier to see where it went wrong if a good notation was being used. I think that the writer was trying to compute E(B|A=a). He gets the answer 5a/4 by supposing that, given A=a, the two possible values of B are 2a and a/2, and the two conditional probabilities thereof are 1/2, 1/2. And that's where he is wrong. Richard Gill (talk) 14:11, 29 November 2011 (UTC)

Next point: it is not only in the special (wrong) scenario you just described that E(B | A=a )=5a/4 can be true for some values a. It can be true in the proper TEP scenario for almost all values a. The statement E(B | A=a )=5a/4 is true if and only if, given that A=a, envelope B is equally likely to contain 2a or a/2.

For instance, the chess-board example: the host chooses one of the 64 squares of the chess board completely at random. The squares were pre-numbered 0 to 63. Given that he chose square r, he puts amounts of money 2 to the power r and 2 to the power r+1 into the two envelopes, shuffles them; you pick one and call it Envelope A. Unless there is 2 to the zero, or 2 to the 64, in your envelope, it is equally likely that Envelope B contains half or double the amount. E(B | A=a )=5a/4 for almost all values of a. With probability 63/64 therefore, E(B|A)=5A/4.

Now consider bigger and bigger chess-boards. As they get bigger and bigger, the probability that E(B|A) is not equal to 5A/4 goes to zero.

The precursors of two envelopes (Schrödinger and Littlewood's two-sided cards problems, Kraitchik two neckties) were thinking of this limiting situation.

Another example. The host tosses a biased coin with probability of heads equals 1/3 till the first time he gets heads. Call the number of tails he saw before he got heads r. He puts two to the r in one envelope and 2 to the r+1 in the other, shuffles them, and you choose one and call it Envelope A. Whatever amount a is in your envelope, it's the case that E(B|A=a)>a. Do the calculations yourself, please! You need to learn some elementary probability and the good way to learn it is by carefully working through some simple examples. Build up your probabilitistic intuition, practice using a good notation which distinguishes random variables and possible values of random variables. Install the statistical language R to your computer (www.R-project.org) and do some simple simulation experiments. Learn! Only after learning, can you teach. Wikipedia editors have to be good teachers, they need to be on top of their material or they'll only confuse their pupils. That would be like the blind leading the blind.

I already wrote some R scripts for playing with various scenarios. Let me know if anyone wants them. Richard Gill (talk) 11:57, 27 November 2011 (UTC)

Thank you, but my point is what the article says:

I showed above that it is fully right that you can expect the dependent envelope B on average to contain 5/4 of the determining amount in envelope A.

But regard this draconic restriction, that the TEP-article hides and reprobately keeps secret: This is only true if you know for sure that the "determining" envelope A, independently from the empty envelope B, has been filled with any random amount first, and only afterwards it was decided to fill the "dependent" envelope B with the dependent double amount of "determining amount in A", or with equal probability with dependent half that "determining amount of A".

(added): This will apply only in one HALF of cases, never in the second half. And to get efficient for any analysis you definitely have to KNOW FOR SURE that it applies in the special case, otherwise you may never use it in your theorem. If it is unknown it cannot be considered in any way.

As this restriction is nowhere mentioned in the TEP article, and you never can nor will know which envelope is the substantially "determining" one and which is the "dependent" one in the TEP,  A or B, this theorem leading to 5/4 A may never be laconically considered for the TEP. If you have chosen an envelope with "dependent amount", you forever have to expect to get only 4/5 A by swapping to B (twice if A is the dependent half amount of envelope B, half if A is the dependent double amount of envelope B). And as I said above - in lack of knowledge - you only can expect B to contain ((2*1+1/2)+(2/2)+(0,5*2))/4,5 = 1A.

5/4 A,  4/5 A  and  1A.  All of this is millionfold verified. If you put the cards on the table, then there is no paradox at all. No more need for infinity. The article should be accessible for grandma and grandson also. With open cards on the table, as per the sources (Ruma Falk, e.g.). Regards, Gerhardvalentin (talk) 13:22, 27 November 2011 (UTC)

What the article says

Gerhard, read the article! It says

There are two indistinguishable envelopes,

each of which contains a positive sum of money,

one envelope contains twice as much as the other.

You pick one envelope at random.

You denote by A the amount in your selected envelope.

In short: Envelope A is a randomly chosen envelope from an indistinguishable pair, Envelope B is the other.

The introduction to the article, where standard TEP is described (the problem which most of the sources, and in particular most of the popular sources, write about), doesn't say anything about how the envelopes are filled. It doesn't have to. You don't know how.

That's right, Richard. Not knowing how, and not having to know it. But alone your experience teaches you that there must have been a certain history, and "if" the window glass has been crushed from outside, makes other conditions for the insurance company as if it had been crushed from within the flat. There are quite other conditions for both variants.

And as  "5/4 A"  will forever be valid in only one special HALF of cases, and never in the second half, and only if you already know for sure, just from the outset, that your envelope "A" is the determining envelope resp. the determining amount, but never never ever otherwise, you have to admit that - not knowing this significant condition indeed to be given for sure - makes  "5/4 A FORFEVER and EVER !!! ? ! ? !"  a really ridiculous farce, makes "5/4 A"  an unforgivable error and a gross fallacy, please ask the insurance company to get their expertise. Gerhardvalentin (talk) 17:52, 29 November 2011 (UTC)

You don't know the two amounts, you don't even get to look in your own envelope. That's the whole point. Yes you convince yourself to switch. And then you could convince yourself to switch back. It's a paradox because the conclusion is obviously wrong. The problem is to show where the apparently logical reasoning goes astray. There are alternative solutions depending on what you imagine the writer was trying to do and what he was assuming. We don't know either. Over the historic development of the paradox, and as it spread to different cultures (from mathematical recreations to economists, statisticians, philosophers), people tended to have different ideas what the writer was up to.

That is why the article is hard to write: because there is not one solution, but many, and they are different. The problem is how to organise the different solutions. My own original research is an attempt to find synthesis. But wikipedia editors are not allowed to make new syntheses so I am not allowed to edit the article. But I offer my findings for discussion and I am likely getting them published in a good peer-reviewed journal soon. The results are still getting better and better, I think, thanks to discussions here and elsewhere.

Sure, some sources are confused, and very many sources are bloody confusing. One source of confusion seems to be with Nalebuff (1988, 1989) who introduced a new version of TEP in which we know that envelope A is filled with x>0, amount unknown, by the host, then the host tosses a coin to put x/2 or 2x in envelope B. The player is given Envelope A and asked if he would like to switch. Nalebuff then gives a correct reasoning with the host and owner, Ali, of Envelope A would like to switch, followed by the paradoxical (wrong) TEP-like reasoning why the owner, Baba, of Envelope B would also like to switch.

At the end of his problem description he also mentions the original or standard version - the version described in the wikipedia article, and from which he invented his new problem.

At the time Nalebuff wrote his article the story was being discussed by statisticians and mathematical economists (he heard it from colleagues). Gardner also heard about it at the same time, and put it into his 1989 book "Penrose tiles to trapdoor ciphers", p.147-148. Gardner had previously (1982) discussed the Kraitchik 1942, 1953 two envelopes version, converting it to two envelopes, in his 1982 book "Aha, Gotcha". Gardner (1989) discusses our basic, "proper" TEP, and most of the people who wrote in to Nalebuff, whose solutions Nalebuff discusses in his second paper, also discussed our basic, proper TEP. The TEP described in Wikipedia, the version described by Falk, who even quotes from Wikipedia, and the version most discussed in the literature. Please let's focus on the basic TEP. Not on a variant.

Of course the difference with the variant is interesting. Nalebuff's new Ali and Baba TEP problem is different. Your grandmother and grandchild will understand that Ali should want to switch and Baba should want not to switch. In the original TEP problem they understand that there is no point in anyone switching. But the problem is not to realise what the right answer is, it is to understand where the reasoning which leads to an absurd answer goes wrong. Richard Gill (talk) 07:32, 29 November 2011 (UTC)

Thank you Richard for your incredible patience. Yes, the real "shame" is that "the given actual content of envelope A" and "All possible contents of envelope A" are confusingly mixed together. Let me repeat again:

Only items 1–3 of the 12 claims of the article (1-12) do apply in every case.

But in one full halve of cases, forever when "A" inevitably is to be the dependent amount, all steps 4–12 are completely inconsistent with the given facts, no matter whether you know the actual content of envelope A, or not. Yes, everything seems to be consistent, but this consistency does not exist: step 4 does never apply in 1/4 of cases, and step 5 does never apply in the remaining 1/4 of cases.

Once more: Step 4 is completely incorrect in 1/4 of cases, whether you know the given actual content of envelope A, or not. And step 5 is incorrect in another 1/4 of cases, whether you know the given actual content of envelope A, or not. So steps 4–12 are false allegations like "The Emperor's new clothes", if you just have a look there. Why is it so hard to name those false allegations, that might apply in exactly 1/2 of cases only, but never in the remaining 50 % of cases, where either item 4 is completely incorrect in 1/4 of cases, or item 5 is completely incorrect in the other 1/4 of cases.

So you are required to add to step 4: "Incorrect in 1/4 of cases", and to add to step 5: "Incorrect in another 1/4 of cases". And, before step 6, to add "As a consequence, the rest is nonsense in 1/2 of cases".

Just have a look to the four equal likely scenarios. It should be possible to object items 4 and 5, and as consequence the rest of 6 to 12. Regards, Gerhardvalentin (talk) 14:39, 29 November 2011 (UTC)

You say "four equally likely cases". You are still assuming things we don't know. You are assuming first the host picks an amount, then he halves or double it, then randomly one of the two envelopes becomes envelope A. So you are doing probability with just two independent coin tosses in the whole story. But other people do the probability differently. Very many people suppose the host first picks a number x, then puts x and 2x in two envelopes. One of the envelopes is picked at random and called Envelope A. Some people "do the probability" only using the final random pick of an envelope. Many other people also use probability to describe their uncertainty as to what x might be.

I'm not saying your analysis is wrong. I'm just saying it's relative to particular assumptions about what is fixed and what is random, and what the randomness is. Your choice is not common on the literature! From a mathematical point of view, my unified solution is a good solution because it applies to every single interpretation I know. Unfortunately however it is mathematical so your grandmother or little grandson won't understand it. This is the problem with writing the article. It must start with easy solutions which are however restricted to a particular interpretation. And the particular interpretation must be made absolutely clear each time. Unfortunately many of the sources are not very explicit about which interpretation they are taking, so the editors of wikipedia have to do some "reconstruction".

The problem is, in fact, that there hardly exists an authoritative source which discusses all interpretations. Possibly the only one so far is the Nickerson and Falk paper. And my up-coming paper. Richard Gill (talk) 08:16, 30 November 2011 (UTC)

Yes Richard, you are correct again, thank you – I steadily just am trying ... and I didn't "see" those aspects before. But what you said shows clearly the direction, where to search. Thank you once more. Kind regards, Gerhardvalentin (talk) 12:40, 30 November 2011 (UTC)

Falk and Konold (1992)

Just found a really great reliable source: Ruma Falk and Clifford Konold (1992). With the most succinct solution I have yet seen. I added the reference to the sources page and put a pdf in my Dropbox.(Ruma Falk sent me a photocopy - that's how I know about this one). Richard Gill (talk) 10:05, 30 November 2011 (UTC)

Thanks for the paper. Never read that before. iNic (talk) 13:53, 6 December 2011 (UTC)

Do you have the two papers by Zabell? Could you put them in the dropbox? Richard Gill (talk) 22:23, 12 December 2011 (UTC)

I just got the Zabell book Symmetry and its discontents which has the 1988 paper with the same name as its first chapter. Somewhat disappointingly this paper only states the TEP puzzle as a final illustration, a nice little challenge for the reader(!), without any real hint on his own solution to the problem. However, he ends with the comment "I will resist the temptation to explain what I take to be the paradox, other than noting that all hinges on [the player] A's apparently harmless symmetry assumption that it is equally likely that B holds the envelope with the greater or the lesser amount." iNic (talk) 00:20, 30 December 2011 (UTC)

PS my own "unified solution", latest version, is now on the TEP Talk/Arguments page, and on my own Talk page. I have been showing it to experts in the field. Richard Gill (talk) 09:31, 5 December 2011 (UTC)

Wow, so there are professional TEP experts now? How cool! iNic (talk) 13:53, 6 December 2011 (UTC)

There are probabilists who specialize in exchangeability (symmetry, under exchange). TEP is also called the exchange paradox and all of its variants (except Nalebuff's Ali-Baba problem) are about a joint probability distribution of two random variables which is unaltered under exchange of the two variables. Symmetry. Richard Gill (talk) 22:19, 12 December 2011 (UTC)

Chase

In the section called "Non-probabilistic variant" on Smullyan's presentation is the following:

"(Actually Smullyan only mentioned arguments 1 and 2; argument 3 was added later, by James Chase, who was the first to publish a solution of the paradox, see below)."

The claim that Chase's article contains a solution is not a matter of consensus. Furthermore, if the article does contain a solution, whether it is the first published solution is not a matter of consensus. I have therefore removed the phrase ", who was the first to publish a solution of the paradox".

It is also worth noting that "argument 3" is identical with "argument 1", except that the roles of the two envelopes are reversed. "argument 3" is therefore superfluous. ---Dagme (talk) 20:32, 26 December 2011 (UTC)

Good points! I deleted the third redundant argument and also the uninteresting fact that Chase was the one who added this argument. I also changed the headings from 'resolution' to 'proposed resolution' to remove the false impression this article otherwise conveys that the presented proposed resolutions are uncontroversial. iNic (talk) 01:01, 27 December 2011 (UTC)

Indeed argument 3 is identical to argument 1 with the roles of the two envelopes reversed. That does not make it superfluous, since the conclusion of argument 3 is opposite to the conclusion of argument 1. This observation supports Chase's reasoning that argument 1 must be wrong. Arguments 1 and 3 both make it favourable to go for one of the two envelopes, but a different one each time. Argument 2 makes it neutral.

It is a fact that Chase was the first to publish an attempted resolution of Smulyan's paradox. As far as I know, no-one disagrees strongly with his analysis; some have tried to improve or refine it. The only author who really disagrees is Byeong-Uk Yi. The fact that Li has a different proposed resolution doesn't change the fact that Chase was first to publish a proposed resolution.

One reason there is not complete concensus about the resolution is because there is no concensus among philosophers on the proper logic of counterfactual reasoning. To each different formal logic of counterfactual reasoning, there would have to be a different formal resolution to the paradox.

Another reason for apparent lack of concensus is that there is disagreement about the precise statement of the problem. For Chase, it is given as part of the problem that Envelopes A and B are defined by choosing one envelope at random. Because this ingredient is assumed, we know that we should be indifferent as to which envelope is chosen. Thus we know that arguments 1 and 3 must both be false. Argument 2 is argued to be correct because it is based on the most recent fork in history: which of the two envelopes is taken to be Envelope A. At that point in history, the contents of the pair of envelopes is fixed. Chase uses a particular approach to counterfactual reasoning which is certainly popular among philosophers and moreover easy to explain to non-specialists

As far as I know there is only one publication disagreeing completely with Chase's analysis: that is Byeong-Uk Yi (whose paper is not actually finished yet, it is "work in progress"). There are however two big differences between Yi's and Chase's background assumptions. One: Yi does not take as part of the problem statement that Envelope A was defined by picking one of the two envelopes completely at random. Two: Yi works with his own, different, logic of counterfactual reasoning. According to his framework, all three arguments are wrong since the statements they aim to derive are all completely meaningless. His solution, you could say, is that there is no solution.

In conclusion, I think the idea that Smullyan's paradox is still unresolved or that there is controversy about how to solve it, nonsense. Richard Gill (talk) 11:44, 1 January 2012 (UTC)

3 Contexts, 2 Aims gives 6 problems, 6 solutions

Our task is not to come up with new solutions of the TEP paradox but to survey the existing proposed solutions, of which there are many. In my humble opinion the multiplicity of solutions comes from the multiplicity of ways to imagine a context within which the argument is placed, and the actual aim of the writer of the argument. I can see 6=3*2 main possibilities. Most of them can be found in the literature.

There are 3 different implicitly understood contexts, or probability models. Context 1: the amounts of money in the two envelopes are fixed, say 2 and 4 Euros, and the only randomness is in the random allocation of one of these two amounts to Envelope A. Context 2: the amounts of money in the two envelopes are initially unknown, and our uncertainty about them is described by a (proper) prior probability distribution of, say, possible values of the smaller amount of money. For instance, it could be 1, 2, 3, ... up to 100 Euros, each with equal probablity 1/100. Context 3: we have no information at all about the possible amounts of money in the smaller envelope, so we use the standard non-informative Bayesian prior according to which the logarithm of the amount is uniformly distributed between -infinity and +infinity.

There are two different implicit aims of the writer. Aim 1: to compute the unconditional expectation of the amount of money in Envelope B. Aim 2: to compute the conditional expectation of the amount of money in Envelope B, given any particular amount, say a, which might be imagined to be in Envelope A. Note that this second aim does not depend on actually looking in Envelope A. In any of the three contexts we can imagine being informed the amount in Envelope A and thereupon doing the calculation. If the result of the calculation is that we would want to switch envelopes whatever the imagined a might be, then the calculation is useful even when we can't look in the envelope: it tells us to switch, anyway.

From studying the origins of TEP (two-necktie problem, two-sided cards problem) and knowing the authors of the papers which introduced all these problems, I think we can deduce that the writer is a sophisticated mathematician who is deliberately trying to trick the unwary not-mathematicially-sophisticated reader. This implies, for me, that the intended context of the problem is Context 2 (proper Bayesian) or perhaps even Context 3 (improper Bayesian); and that the aim of the writer was to compute the conditional expected amount in Envelope B given any particular imagined amount a in Envelope A. He uses the ambiguity of ordinary language to seduce us to take the conditional probabilities that the second envelope contains twice or half what's in the first, given any particular amount in the first as 50/50, whatever that amount (in the first envelope) might be.

It's a mathematical theorem that this last fact is impossible, at least, with a proper prior distribution over the amounts possible. We are easily seduced, since these conditional probabilities are in fact entirely correct, had we done formal calculations using the traditional improper prior which is conventionally often used to describe total ignorance.

However the philosophical literature is written by people who are often complete amateurs in probability theory, and they cannot even imagine such a sophisticated context as Context 2 or Context 3, nor such a sophisticated aim as Aim 2. This literature focusses on Context 1 and Aim 1. Personally I find this unsatisfactory since the more sophisticated context makes the paradox much more interesting; the "mistake" in the argument is much more subtle. On the other hand, if we assume Context 1 and Aim 1, then the writer of the TEP argument is not just making one stupid mistake but several. He is not only confusing random variables and possible values they could take, but he is also confusing expectation values and random variables.

The economic and decision theory literature tends to take the more sophisticated context and aim. Richard Gill (talk) 18:13, 1 January 2012 (UTC)

Sure I agree with most of what you say. (But you still don't even mention the non probabilistic versions of TEP when listing all the versions of TEP you think there are. Why you always omit them is still a mystery to me. It might be some kind of psychological blind spot that I don't understand.) I think the article now clearly states the fact that the problem has different interpretations and exist in different versions with different wordings. Your tree main versions are there for sure. Your Aim 1 and Aim 2 is relevant in the first version of TEP (when we don't look in an envelope) but I can't see the relevance of Aim 1 in version 2 and 3 where Aim 2 is stated explicitly. Which authors made such interpretations? Please remind me. I can't remember I read any accounts of that sort. If these interpretations really do exist in the literature we should represent them the article as well. However, the number of different proposed solutions are not only caused by the number of different versions and interpretations. There are genuine disagreements on how to solve all the versions. As well as fundamental disagreements on how many different versions there really are. This fact is still not explained with enough clarity in the article I think. iNic (talk) 13:02, 2 January 2012 (UTC)

I am only talking about the standard TEP problem as introduced in the article: the one with the 5/4 calculation, and where we do not look in Envelope A. As far as I can see the philosophical literature tends to choose as context either Context 1 or Context 2, and as aim always Aim 1. In my opinion the philosophical literature is wordy and fuzzy but there is broad agreement how "their" paradox is resolved. The statistical and the mathematical economics literature tend to go for Context 2 or Context 3, and as aim Aim 2. I don't see any disagreement concerning how it can be resolved. Regarding the educationalist and psychology literature, Ruma Falk seems to have moved from Aim 2 to Aim 1 over the years in her published papers, but in recent correspondence with me she seems to go back to her earliest position. Her first paper (Falk and Konold, 1992) also has explicitly the right improper prior (uniform prior on the logarithm of the amount), whereas later papers make the common mistake to think the prior being used is the wrong one (uniform prior on the amount itself).

I don't mention Smulyan's TEP without probability as I consider it has been adequately resolved and in any case much less interesting.

When we allow the player to look in Envelope A, the problem changes completely. There is no paradox any more. Depending on your prior, you will switch for some values of a, and not for others. Or if you don't want to make your decision by optimizing the expected gain relative to a given prior, you can use the trick of comparing the observed value of a with a random probe. These are interesting problems to look at, but there is nothing paradoxical any more.

I plan to go through all the papers I know and classify them according to my scheme. Do you have the papers by Zabell? If so could you put them in the dropbox?

My present impression is that authors who agree on the same context and aim, tend to agree as to resolution. Please tell us if you know any striking disagreements. Richard Gill (talk) 16:08, 2 January 2012 (UTC)

I have one of the Zabell papers, but only in print and not as a pdf file. I could scan it and put it in the dropbox if you want. I will go through all the papers as well and classify all different views. In some cases it was years since I read the papers so I need to refresh my memory. iNic (talk) 01:25, 4 January 2012 (UTC)

Yes please! I also need to sit down and rewrite my own paper, starting completely fresh, and reread the important papers. Richard Gill (talk) 09:51, 5 January 2012 (UTC)