Talk:Exchange paradox

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Merge with "Two envelopes problem"?[edit]

Should be deleted/merged with Two envelopes problem. —The preceding unsigned comment was added by (talkcontribs) 00:50, 1 March 2007 (UTC).

If this develops into an article about the decision problem per se it would be justified, I think. iNic 12:25, 11 April 2007 (UTC)
Where is the edit history for this page? Who added all the case 2 info and where is it from?--Dacium 00:37, 18 April 2007 (UTC)

Did a redirect to Two envelopes problem. iNic 09:59, 15 May 2007 (UTC)

Bad idea. The articles are very different. --X-Bert 22:42, 21 May 2007 (UTC)
Yes the articles are quite different, I agree there. But are the subjects different? Can you please explain in more detail why you think the ideas in this article aren't covered in the Two envelopes problem article? Does your article describe another subject than does the Two envelopes problem article? Or do you have an issue with what this problem/paradox should be called in Wikipedia? Would you prefer to rename the main article to Exchange paradox? iNic 13:13, 22 May 2007 (UTC)
The article "Two envelopes problem" is about a puzzle and finding the flaw, the erroneous step, in the presented switching argument. This article is about the question "Should the player swap, that is, exchange the envelopes?" and a paradox resulting in a special case. Hence I think that the title "Exchange paradox" fits well. Imho the article "Two envelopes problem" would better be titled "Two envelopes puzzle". --X-Bert 21:04, 22 May 2007 (UTC)
I think this article should be merged into Two envelopes problem, or possibly just made into a redirect page. X-Bert's analysis (distinguishing between a "puzzle" and a "paradox") is incorrect. Both articles cover exactly the same thing. And the other article does a better job of it. DavidCBryant 18:16, 23 May 2007 (UTC)
XBert, if you want a separate article about the decision problem per se I support that, as I've already stated. However, your article here isn't even close to an article about the decision problem. All you have done here is to repeat in a more complicated way (more formulas) what is already stated in the main article (Two envelopes problem). And if you want to start an article about the decision problem I think the title Exchange paradox is a bad choice of name. I will soon delete this article again as it adds nothing new or different to wikipedia. iNic 21:25, 8 June 2007 (UTC)
Dear iNic, obviously the topics of both articles are related, but the stated problems and the approaches to solve them are very different: In the article "Two envelopes problem" it is explicitly stated that it is about a puzzle and finding the flaw, the erroneous step, in the presented switching argument, while this article is about the question if and under what conditions a player should swap. So I think it would not make sense to merge one article into the other or delete one of them. If you don't like this article, please just ignore it, and if you would like to have an article about the decision problem, try to write one. --X-Bert 14:27, 10 June 2007 (UTC)
The two articles should be merged, per the arguments iNic and DavidCBryant give above. X-Bert, saying "please ignore this article" is not a good argument. —Lowellian (reply) 07:28, 13 August 2007 (UTC)
Actually, this wasn't my argument. My argument is given in the first two sentences. iNic's approach was to delete my article and replace it with a redirect to his article, which is definitely not appropriate. --X-Bert 09:37, 22 August 2007 (UTC)

There should be a single article mentioning the two variants. Petersburg 21:18, 13 August 2007 (UTC)

Yes. In fact I have already had a lengthy discussion once before when two different articles were covering this issue. Let me quote myself: "It's not the case that there are two different problems, one called the "two envelopes problem" and another called the "envelope paradox." It's just two different names of the same thing. Actually, the problem/paradox have gotten many different names over the years. Accordingly, there should be only one page at wikipedia devoted to this problem/paradox. All the different names should be redirected to one and the same page." And one of the other common names in the literature is "Exchange paradox," so the same holds for this name. iNic 12:06, 14 August 2007 (UTC)

I would also suggest merging the Necktie Paradox into the common article. Petersburg 21:24, 13 August 2007 (UTC)

I would suggest to keep that article separated, extend it and explain the relation to the exchange paradox. --X-Bert 09:37, 22 August 2007 (UTC)

iNic explicitly emphasized that his article describes a puzzle where a flaw in the presented reasoning is to be found [1]. This article is about the more general question if and under what conditions a player should swap. There isn't much overlapping between the two articles. Therefore, merging this articles would rather be a long concatenation which becomes difficult to read. In my opinion it's better to keep them separated, but connected by links, so the reader can easily get from one to the other. --X-Bert 09:37, 22 August 2007 (UTC)

Xbert, please understand that while this article can justifiably be described as your article (due to the fact that you are the only contributor), the Two envelopes problem article isn't "mine" in any way. I have contributed to it but so have many others too.
When it comes to the definition what this problem is all about, this is taken from the literature in the case of the Two envelopes problem article. Your definition what the problem is all about in this article I haven't seen anywhere in the literature. Advises of this kind pops up as spin-off effects for some authors, but are never the main theme. iNic 19:57, 15 November 2007 (UTC)

More merge discussion[edit]

  • Merge There are apperently several articles conserning the same topic. I started at Necktie Paradox (which is a stub), which referes to Two envelope paradox which it rightfully suggested merged with Exchange paradox. I would suggest that all are merged into one article. --webdahl (talk) 12:45, 21 February 2010 (UTC)
I redated the merge template to Feb 2010 as it appeared to be absent for some time. There is also the question of whether the articles have changed much since the original merge proposal. Given this, I think all the above could be irrelevant, so starting a new section for recent discussion seems appropriate. I will also take the opportunity to place a corresponding template on the other article, which it may have lacked. Melcombe (talk) 10:38, 16 February 2010 (UTC)
  • Merge necessary! I've been trying for some time to clean up the total mess at the "Two envelope paradox" page (despite an abusive editor). Apparantly, that article is under the control of an editor who insists that this is an "open problem in Bayesian decision theory". The same editor pointed out that there was an article on the same topic, but with another conclusion (!). Having two articles on the same topic (where one is VERY CLEARLY in the wrong) is unacceptable. This article gives the correct, properly referenced solution. I suggest we delete the "Two envelope case" article and add a section to this article about the controversy itself (ie. why people still conceive this as an open problem). Comments welcome. Tomixdf (talk) 12:22, 17 October 2009 (UTC)
I agree this merge should be done; the "Two envelope paradox" page is still somewhat of a mess and the topics are identical. Brainfsck (talk) 08:40, 3 January 2010 (UTC)
  • Delete this! This article is extremely misleading and uninformative. It seems to suggest you should switch envelopes which is clearly rubbish. The reasoning in part 2 is hard to follow and comes out with a false answer, and no explanation is given as to why it comes out with a false answer.
    The two envelopes article is much better, and at least attempts to tackle the issues at hand. —Preceding unsigned comment added by (talk) 10:34, 18 January 2010 (UTC)
The two envelopes article also suggest to swap the envelopes. If you find the math hard to follow, it doesn’t mean this article is necessarily a rubbish… // stpasha »
  • Merge. Two articles are clearly on the same topic: the two envelopes problem starts with some “common-sense” reasoning and tries to argue why common sense fails here. The present article, exchange paradox goes one step further and asks what should be the proper way of reasoning in this setup. The topic however is still the same, so the articles' content should be merged. // stpasha » 08:32, 16 February 2010 (UTC)
  • Merge or Delete. All three articles (necktie, two envelopes, exchange) should be deleted in their present form. In particular because none of them contains the correct analysis. Also this is not an open problem. It is impossible either in mathematical analysis or the real world to have a probability distribution that is constant over an infinite domain. If the articles are going to remain wrong, they should be deleted. THe preferred option is to merge them and have the correct explanation of the fallacy. As it stands, I currently use the two envelopes paradox as an example of where, in the words of Larry Sanger, Wickipedia is broken beyond repair. Zeitnot (talk) 19:17, 11 March 2010 (UTC)
Very well put. I agree completely. Tomixdf (talk) 08:34, 13 March 2010 (UTC)
Sorry. It is possible both in the real world and in mathematical analysis to have a probability distribution that is *almost* constant over an infinite domain! And one reading of the exchange paradox (and I mean, a reading which is common in the literature) is that that is actually what the "author" of the argument had in mind. I am not writing this to promote Own Research, but I'ld like you to take a look at my preprint [2]. Comments are welcome. I'm writing this paper as a service to the wikipedia community, not just pushing my own personal POV. I hope that the mathematical facts which are exposed there will be useful for editors since it is certainly clear that a lot of basic and elementary mathematical facts about these problems are not easy to find in reliable sources, precisely because they are so elementary that any professional easily finds them for themselves on the back of an envelope. However it is also amusing that many professionals made big mistakes. In the two envelopes context, it is the logarithm of the amount of money which should be taken as uniformly distributed on the whole real line (except, of course, that that is not possible), not the amount of money which should be taken as uniformly distributed on the positive half-line. Richard Gill (talk) 11:21, 10 August 2011 (UTC)
  • Delete all. OK, fair enough. Let's delete all three and start a completely new one. What should it be called? iNic (talk) 03:10, 14 March 2010 (UTC)
  • Merge Very similar articles. Jason Quinn (talk) 01:46, 22 March 2010 (UTC)
  • Merge The necktie paradox slowly mutated via two wallets into two envelopes and on the way got the alternative name of exchange paradox. Presently, the article on Two Envelopes problem is being most vigorously developed, taking account of the huge literature on all of the paradoxes, and (I think) slowly approaching a reasonable form: a decent overview and a decent synthesis is being attained. So I'd suggest that every who is active on this page go and join in the fun on the TEP page. Richard Gill (talk) 11:21, 10 August 2011 (UTC)
  • Merge This article adds nothing that the two envelopes article lacks except an incorrect solution. The present article suggests that the "correct" fallacy in reasoning is that each man's reasoning is false because he would need to revise his estimate of his own necktie's price downwards in the event that it is the cheaper of the two and he does not take this into account in his reasoning. This does not address the core of the paradox at all. The first man might know exactly the price of his necktie (say $20) but not know the value of the other man's. He would then reason that, should he lose, he will lose $20, but should he win, he would win more than $20. He would be correct in that. The second man could also know exactly the value of his own necktie without knowing the value of the first man's necktie. He would reason that if he lost, he would lose the value of his necktie (which he knows) and if he won, he would win more than the known value of his necktie. Hence the paradox remains even when each man knows exactly how much his necktie costs. The "solution" given on the page at present is just plain wrong. Somebody please delete this page. — Preceding unsigned comment added by (talk) 00:33, 18 November 2011 (UTC)
  • Do not merge Okay, I'm no genius, but it seems to me that these are two entirely different paradoxes. In one, one solution is more likely than the other is more likely than the first. This is totally different: a wager with no advantage. Not the same thing; so keep both pages. (talk) 00:42, 27 November 2013 (UTC)
  • Do not merge These two problems are indeed different. Actually there are two versions of the two envelope problem: the one where the generation of the envelopes follows a distribution with expected value of the second envelope always greater and the one where the envelopes are given and the switch after opening strategy creates a paradox. — Preceding unsigned comment added by (talk) 00:15, 30 April 2014 (UTC)
  • Merge The two are essentially identical. Oreo Priest talk 14:43, 18 June 2014 (UTC)


Mergers with Two envelopes problem, a.k.a. Two-envelopes paradox, were proposed four years ago and one year ago.

I do not see the case for covering the Exchange paradox separately. I suspect that "exchange paradox" was coined by someone who wished to focus on one aspect of the amorphous two-envelope paradox. Our two envelopes article now covers the amorphous quality explicitly and that can be elaborated if necessary to cover something distinct that might be here. --P64 (talk) 17:04, 9 May 2011 (UTC)

Sources. The first four of six sources identified here (footnotes 1-4) are NOT identified in the dozens of Notes and References, Further reading, and External links at Two envelope problem.
I have improved those four citations in my new sandbox.--P64 (talk) 16:21, 10 May 2011 (UTC)
It seems to me that it's time to do the merger (ie, this article becomes a redirect to the two envelopes problem). Richard Gill (talk) 19:11, 6 July 2011 (UTC)
Concerning the name, some of the main articles cited on two envelopes problem actually use the name exchange paradox in their titles. These really are synonyms, both names have been around for a long time. Richard Gill (talk) 11:19, 7 July 2011 (UTC)
Yes this is correct. It has many names but these two have become the most common. The name usually contains some of the words exchange/two/envelope/envelopes/paradox/problem/puzzle. iNic (talk) 22:57, 7 July 2011 (UTC)


I noticed that this article contained quite a big mix-up concerning continuous and discrete probability distributions. The notation was "discrete" throughout, yet a large part was specifically about continuous variables. The results in the two cases are actually different so it is important to be clear on this. I have corrected all that.

Compared to the two envelopes problem the present article contains quite distinct material so in principle if a merge is carried out, the material from here could be cut and pasted over to the other article. What is here belongs with the second variant of the two envelopes problem. It gives a simple necessary and sufficient condition on the distribution of the amount in the smaller envelope, such that whatever is in your envelope, you would want to switch if you base your decision on the expected amount in the other, given what is in yours.

Personally I would not suggest to copy the complete derivation of these criteria (one for continuous, the other for discrete distributions) but the results themselves seem to me to be noteworthy. Richard Gill (talk) 11:16, 7 July 2011 (UTC)

I think we can skip the contents in this article entirely. I can't see how all the greek symbols here would enhance the main article at all. The results can be stated as representing one opinion, sure, but the math itself is uninteresting. iNic (talk) 23:04, 7 July 2011 (UTC)
I don't see any Greek symbols! But I agree that the mathematical derivations are routine for those who would understand them, and meaningless for anyone else. However the conclusions are notable:
For a continuously distributed X (the smaller of the two amounts of money) with probability density the player should swap an envelope containing the amount a if
For a discrete X he should swap an envelope containing a if
For a finite discrete distribution of X the criterion cannot be satisfied at the maximal value of a (the amount in the first envelope), the other envelope must contain a/2. (The largest value which the smaller of the two amounts can take on, X, is then a/2, so while . ) However, there are distributions of X allowing arbitrarily large values for which the criterion is always fulfilled, e.g.
There are many examples of distributions of this kind in the literature. I think the first one by Broome is noteworthy enough to put in the main article. I don't see the point of adding more examples of the same kind. Also, the distinction in notation between the discrete case and continuous case is perhaps interesting to a mathematician but not to a philosopher or a general reader. We should only add more examples of this kind in case it illustrates a new original point of view or twist of the philosophical problem at hand. Remember, this is a problem within philosophy, not mathematics. iNic (talk) 10:32, 8 July 2011 (UTC)
I am amazed again, iNic, that you continue to insist that this is a problem in philosophy, not in mathematics. It actually is discussed in a great many mathematics books including undergraduate textbooks in probability and statistics, so it clearly is of interest to mathematicians, where-ever it actually belongs. Moreover, a good case can be made for the fact that these paradoxes continue to be thought of as *unresolved* paradoxes in philosophy, precisely because many philosophers don't know the tools which mathematicians developed precisely in order to resolve such silly paradoxes and to be able to move on and do some real science. Richard Gill (talk) 11:27, 10 August 2011 (UTC)
The references on which the article is based are also noteworthy - they are referred to many times in the literature by others. Richard Gill (talk) 07:40, 8 July 2011 (UTC)
These references were originally in the main article but deleted by someone that thought the list was too long. iNic (talk) 10:32, 8 July 2011 (UTC)
It's a good principle only to have references for sources which are actually mentioned in the text. If we add the just mentioned results we can also add the references. By the way, I must mention that this couple of articles (by philosophers) do fall into the category of those which I find woolly. If only the writers had learnt a bit of mathematics and especially, a bit of probability theory! They make such heavy going of it, especially through not distinguishing between a random variable and a possible value it might take. Well, that is of course a rather recent innovation in mathematics (Kolmogorov, 1933). You can't expect the philosophers of today to know modern probability theory. One of the great advantages of which was introducing a language and formalism capable of abolishing those old mixups. Richard Gill (talk) 19:30, 8 July 2011 (UTC)
I don't agree that we should add any superfluous examples to the article just to get an excuse to add some more references in the list of references. That is backwards thinking. (Side note: In general, philosophers are far better mathematicians than mathematicians are philosophers...) iNic (talk) 01:52, 10 July 2011 (UTC)
I strongly disagree with your side-note! But maybe we have different random samples of mathematicians and philosophers. Or are you able to support your claim through reliable sources? Richard Gill (talk) 11:27, 10 August 2011 (UTC)
Some random samples of philosophers could be Bertrand Russell, Kurt Gödel and Gottfried Wilhelm von Leibniz. I think they were quite skilled in mathematics as well, don't you? Now you give me your random picks of mathematicians that are also great philosophers. iNic (talk) 19:50, 10 August 2011 (UTC)
My random sample of great mathematicians who are also great philosophers might include Bertrand Russell, Kurt Gödel, Gottfried Wilhelm von Leibniz, as well as Pythagoras, Descartes and Luitzen Egbertus Jan Brouwer. But the question was about "in general". The kind of philosopher who writes about the two envelopes problem is manifestly not a very good mathematician. See also [3]. Richard Gill (talk) 10:45, 11 August 2011 (UTC)
Ha ha very funny! Who I call philosophers you call mathematicians. Great! This proves that you agree that the set of human beings can't be divided into disjoint subsets labelled "mathematicians", "philosophers", "physicists" and so on. It follows that your argument for why TEP should be called a mathematical problem breaks down, even by your own standards. Thank you. iNic (talk) 17:59, 11 August 2011 (UTC)


I don't quite catch the meaning of this sentence:

For a finite discrete distribution of X the criterion cannot be satisfied at the maximal value of a, the other envelope must contain a/2. (The largest value which the smaller of the two amounts can take on, X, is a/2, and P(X = a) = 0.)

Someone to explain this? Nijdam (talk) 16:05, 27 July 2011 (UTC)

Whoever wrote this means that for a distribution with bounded support, the criterion cannot be satisfied at the upper limit of the support. If the largest possible (larger) amount is a, then when you see a in your envelope, you know for sure the other envelope contains a/2. Richard Gill (talk) 11:30, 10 August 2011 (UTC)


I have redirected this page to the Two envelopes problem after a long discussion and a consensus to do so. That article is being expanded and improved. Martin Hogbin (talk) 14:48, 7 August 2011 (UTC)