Talk:Two envelopes problem/Arguments/Archive 4

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What is the paradox?

I have just thought of something that might shed some light on the disagreement on this problem.

What exactly do we think is required for the problem to be considered paradoxical? In the article itself the last steps in the argument are:

9) After the switch, I can denote that content by B and reason in exactly the same manner as above.
10) I will conclude that the most rational thing to do is to swap back again.
11) To be rational, I will thus end up swapping envelopes indefinitely.
12) As it seems more rational to open just any envelope than to swap indefinitely, we have a contradiction.

I think we all agree that for the player to swap endlessly is an absurd situation and therefore we have a clear paradox. I also suspect that everyone agrees that this paradox has been resolved for all cases.

No, this paradox has not been resolved. iNic (talk) 13:07, 25 January 2012 (UTC)

INic can you give in a new section below, or on the arguments page, an example where you think there is still an argument endless exchange. Please give a full and clear description of the exact setup and of the line of reasoning that you believe leads to a paradoxical conclusion. State when and if any person looks in their envelope. Martin Hogbin (talk) 20:10, 25 January 2012 (UTC)

Please read the Wikipedia article, it should give you the information you are looking for. iNic (talk) 17:21, 26 January 2012 (UTC)

The WP article, quite obviously. does not tell me which version you have in mind. Martin Hogbin (talk) 09:25, 31 January 2012 (UTC)

Aha in my mind? Well, in my mind there are only one paradox, only one version. All the different "versions" in the literature are due to the fact that all proposed solutions are flawed. A bad solution doesn't solve the problem if the problem is tweaked just a little bit, or slightly restated. Hence the series of never ending new "variants" of the problem. But the correct solution solves them "all" in one blow. iNic (talk) 09:36, 31 January 2012 (UTC)

At the end of his paper, Nalebuff talks of what he seems to consider a paradoxical situation that is more like the two neckties paradox. There is no suggestion that the player would swap endlessly but he does seem to consider it paradoxical that both players should find it to their advantage to swap in some cases. This seems to me to be a weaker paradox and I (and Nalebuff it seems) wonder if this paradox has been fully resolved in all cases. Martin Hogbin (talk) 11:38, 25 January 2012 (UTC)

To swap back and forth is in itself not paradoxical. Many processes go back and forth endlessly without being a paradox. Think of a pendulum for example. The paradox here is that we have rational decision theory urging us, as it seems, to do a series of rational steps that when thinking of it as a whole doesn't seem to be rational at all. This is the paradox here. How can a theory of rational behavior immediately lead to irrational behavior? The "twin construct" that is used in the neckties version is a little different. There the paradox is that something must be wrong if both persons think that they gain value by swapping their neckties. It seems to create value out of nothing. But if both gentlemen happen to think that the other tie is prettier, then they do in fact both gain if they swap. So in this case one need to be more accurate when presenting the story so that a paradox is really created. I guess this is why the common formulation of the paradox today is in terms of envelopes and money and not neckties. However, when looking in one of the the envelopes the twin construct is still useful to derive the paradox. iNic (talk) 13:07, 25 January 2012 (UTC)

In the necktie problem the gentlemen are not interested at all in which necktie is more pretty but which one cost their wives more money to buy for them. They apparently both think for the same reasons that on average they will individually get richer, in expectation value, by switching. Since they can easily imagine themselves in the position of the other person, they both think that together they will get richer on switching, yet they know that together the total cash price of their neckties did not change. So there was a paradox.

To swap back and forth imagining that each time you swap you get richer (on average) is a paradox, since you know that on switching twice you definitely have not got richer.

I think that the paradoxical nature of the different versions is pretty clear and moreover I think they have all been adequately resolved more than 20 years ago. Since then there has been a lot of repetition, a lot of re-inventing wheels, and a lot of noise. But of course ... that is just my personal opinion. On the other hand I think it is a pretty objective fact that there have not been any new "solutions" to the version (studied by mathematicians and economists) in which we imagine the writer of the TEP argument being concerned with conditional expectations since 1989, and that there have not been any new "solutions" to the version as imagined by the philosophers (and to a lesser extent psycholgoists and educationalists) who, not knowing about conditional expectation, thought that the writer was calculating an unconditional expectation value, since about 1995. There has been one claim, by Schwitzgebel and Dever, that they are the first to explain what went wrong in the philosophers' version, but I don't see any novelty in it (note: nor has their proposal been enthusiastically adopted by authoritative writers from philosophy, it has not even been discussed. Just a few papers give the citation in order to cite all known published papers in the field.)

There have been two "new" paradoxes: Smullyan's word game without probability, and Cover's trick with a randomized way to increase your expectation values when you do look in your envelope (which was already known in 1992). Smullyan's paradox has been studied in a small number of very academic papers from philosophy. Writers who are using different theories of counterfactual reasoning naturally find different solutions. But they all do find solutions.

There has been a lot of "noise" in the economics literature about infinite expectation values, in which people are endlessly arguing whether infinite expectations makes sense in real world practise. And this discussion is a repeat of the Saint Petersburg Paradox endless discussion. On the other hand this discussion says nothing about the question of where is the *logical* mistake in the sequence of apparently *logical* steps leading to a clearly nonsense conclusion. So though in the wikipedia article one can discuss this question a little, it seems to me of tangential interest. The essential paradox is resolved by demonstrating the illogical steps of the reasoning. Those steps have been well understood for a long time now, even if they happen to be different according to different interpretations of what the writer is doing. Richard Gill (talk) 14:58, 25 January 2012 (UTC)

You may find the paradoxical nature of the different versions is pretty clear but it surely would be good practice, for the benefit of laymen and experts alike, to make absolutely clear in each case exactly what is considered paradoxical. It also would seem a good idea to decide on exactly what constitutes a 'resolution' of the paradox. In my opinion a proper resolution requires identification of the error in the proposed line of reasoning not merely showing that exchange is not beneficial. I am sure that some of the noise that you complain about comes from this confusion. Martin Hogbin (talk) 10:04, 26 January 2012 (UTC)

The word "paradox" means different things in different contexts. The Wikipedia article gives a good overview. iNic (talk) 17:21, 26 January 2012 (UTC)

Yes, I know what a paradox is and I can see the paradoxical line of reasoning in WP but there are other cases where the setup and proposed paradoxical line of reasoning might be different. It never hurts to make things crystal clear and thus avoid pointless arguments. To just show that you should not swap is pointless, we all knew that at the start by reason of symmetry. Martin Hogbin (talk) 18:09, 26 January 2012 (UTC)

Exactly! This was what I tried to tell you when you used this argument yourself. I'm glad you realize now how pointless it is. iNic (talk) 16:24, 27 January 2012 (UTC)

I would be happy to point out the error in your line of reasoning of only you would give it, with the exact setup. Martin Hogbin (talk) 19:20, 27 January 2012 (UTC)

Please start right there and replace your pointless argument with a valid one. That will be interesting! iNic (talk) 09:36, 31 January 2012 (UTC)

I think the article does do just that, now, Martin. It shows for each of three or more variants of the problem, or if you prefer, different ways to imagine what the problem is, where the logical error is in the reasoning that leads to a clearly nonsensical conclusion. Now the Broome version is one which, if you are a mathematician, is easily mathematicallly (=logically) resolved, but it still remains a weird phenomenon for ordinary people. I think that is the reason why many authors slap on top of the logical analysis the further, strictly speaking superfluous, claim that in any case, infinite expectation values are stupid / unreal / whatever. Nalebuff does this, Falk does this. Many others do. Like other weird constructions in mathematics (eg Banach-Tarski) it is weird. Banach-Tarski is called a paradox because it insults our imagination and intuition, but there is no actual logical contradiction. Just an impossibility for our little brains to visualise the thing.

So I guess we can say that the Two Envelope Problem provides, among its many variants, paradoxes of both kinds: things which are logically correct but hard or impossible to imagine hence hard to believe; and things which are logically incorrect and where it suffices to point out the logical error.

I think this is a new insight in the field of TEP Studies. Thanks! Richard Gill (talk) 19:12, 26 January 2012 (UTC)

Exactly what new insight did Martin give you now? I didn't get it. iNic (talk) 10:07, 27 January 2012 (UTC)

Richard, I do not disagree with anything that you have written but you seem to have missed my point. In the article the only claimed paradox that I can see is that, by the given argument, the player should swap endlessly. Nalebuff, however sees a different paradox for some versions, namely that both players (note that there is only one in the setup given) expect to gain by swapping. These are not the same thing. Martin Hogbin (talk) 19:53, 26 January 2012 (UTC)

Yes, in the article we are copying a particular, later, statement of the problem due to Ruma Falk. It's not the original. Earlier versions tended to have two people each in possession of an envelope (wallet, box, necktie ...) and both standing to gain, on average. We, outsiders, can follow both persons' arguments so apparently must support both of their conclusions, yet we know that the total does not increase - so how come can we advise both to switch? Richard Gill (talk) 14:01, 29 January 2012 (UTC)

The new insight I got is that a problem can be solved, since we can find the error in the reasoning, but there can remain a paradox, in the sense that intuition still does not agree with the result of logical deduction. Richard Gill (talk) 14:33, 28 January 2012 (UTC)

The Broome Two Envelope Problem can be resolved, we can see where the argument breaks down. But we retain a Broome Two Envelope Paradox, since it remains counter-intuitive that if we repeatedly create two envelopes containing two numbers according to the Broome recipe, and give them to Alice and Bob, then on all those occasions that Alice has a number a in her envelope, the average of the contents of Bob's envelopes is 11a/10 > a, while simultaneously, on all those occasions that Bob has an a number b in his enveloe, the average of the contents of Alice's envelopes is 11b/10 > b; for all possible values of a and b >1. (When a or b =1, replace 11/10 by 2).

Yet the distributions of the contents of both Alice's and Bob's envelopes is the same!. Each possible value 1, 2, 4, 8, ... occurs, relatively speaking, equally often. Richard Gill (talk) 14:49, 28 January 2012 (UTC)

It's funny because Albers (2003) has exactly the opposite opinion: "The two-envelope paradox is easily explained, but the corresponding problem is more difficult." (p 29). So the total disagreement among scholars abound. Not even a simple high level description of the situation is possible to agree upon. iNic (talk) 09:55, 29 January 2012 (UTC)

These authors are colleagues of mine in the Netherlands, I know them well. When they say two-envelope "paradox" they refer to the paradoxical (in the sense of leading to a counter-intuitive result) derivation of always switching. When they talk about two-envelope "problem" they mean the problem of figuring out what the owner of Envelope A should do when he or she actually looks in his envelope first, in a real world situation. To solve this problem one needs more input. For instance: a prior and a utility and the principle of maximizing expected utility. If you accept the principle and you take a prior and a utility function as given, the problem has become soluble, it's a well defined mathematical optimization problem. But how do you figure out a prior and a utility? Anyway, why should you accept the principle of maximizing expected utility? In this sense the TE problem is not solved. You would probably agree, iNic, that in this sense it is not solved. That's my opinion too. In fact I would say it cannot be solved in general, and it's a waste of time to look for a general solution. Albers et al also say that in some situations it can be better not to give an answer at all. I too agree with this. Often the most valuable result of a statistical consultancy is that the statistician can explain to the client that his problem is ill-posed, un-answerable, and that the client should re-consider what it is he wants to do. Richard Gill (talk) 13:55, 29 January 2012 (UTC)

PS Albers et al. are not referring to the Broome paradox when they make these remarks. Richard Gill (talk) 14:03, 29 January 2012 (UTC)

OK so you and your peers in the Netherlands agree that for the Broome version the paradox is not solved but the problem is solved, while for the rest of the variants it's the other way around? Very confusing indeed. iNic (talk) 10:50, 30 January 2012 (UTC)

No. The other versions are easier. The problem is solved, there is no paradox either.

As you said yourself, the word paradox has a number of different meanings. I would like to distinguish a problem to be solved (where does an apparently logical deduction go wrong), and a situation which it counter-intuitive. Nothing to be solved, though there is something to be learnt: intuition need not be correct.

But first of all, let's define the Broome problem and paradox precisely. Suppose two numbers are generated and written down according to Broome's recipe and put in two envelopes, which are labelled completely at random (independently of the contents) A and B. The recipe is: take a coin with probability 1/3 of heads. Toss it just as long till you see the first head. Call the number of tails before the first head t. It could be 0, or 1, or 2, .... The two numbers are 2 to the power t and 2 to the power t+1. Let A and B denote the random numbers inside envelopes A and B.

Now comes the problem: what is wrong with the following reasoning? The owner of Envelope A reasons (1) given what is in his envelope, say a, the expected contents of the other envelope are 2a if a=1 and otherwise 11a/10; in any case, it is strictly larger than a. He reasons that therefore, (2), without looking in his envelope, he will become better off by switching to the other envelope.

The conclusion is obviously false since the marginal probability distributions of A and B are identical. Exchanging one closed envelope to the other does not improve his situation at all - it leaves his situation identically the same to what it first was, since the bivariate probability distribution of the ordered pair A,B is the same as the bivariate probability distribution of the ordered pair B,A.

Our problem, as I said, is to explain where the apparently logical reasoning breaks down. The answer is that it breaks down at step (2). In step (2) we are silently using a theorem which requires a condition which is not satisfied in this case. The conditions of the theorem do not apply hence we cannot deduce that the conclusion of the theorem is true.

Still, we are left with the paradoxical (in the sense of counter intuitive) fact that in many, many repetitions, for any particular amount in Envelope A, the average of what's in Envelope B is larger, and vice versa.

Problem solved, paradox remains. The paradox that remains does not have a solution and does not need a solution. We have learnt that our intuition about averages can break down when infinities are involved. This has been known for several hundred years. This is not a paradox which can be solved, but it is a situation to which one can become accustomed ... in other words, one can develop a better intuition.

Von Neumann said: one does not come to understand new mathematics, one gets used to it. This is the sort of paradox which goes away by being aware of the possibility and getting used to it. It doesn't go away by logical deduction.

Unfortunately, many writers on TEP are not happy with this situation. They use overkill by trying to argue that the Broome situation could never turn up in the real world. In other words: sorry guys, we are stuck with the paradox, but don't worry your little heads about it, you won't ever have to deal with it in practice. Richard Gill (talk) 13:43, 31 January 2012 (UTC)

Richard I agree with what you say and I have a suggestion that might help in your mathematicians vs the rest argument. When talking about the finite, physically reasonable cases it is alright to use a real physical representation of the problem, in particular to use terminology like 'envelope, and 'money'. When referring to the unbounded versions I think it would be better to use strictly mathematical terminology, 'set', 'number' etc.

It is quite right to say that no one will never have to deal with the untruncated Broome game in practice. Infinite amounts of money do not exist, neither do envelopes large enough to contain them. On the other hand, we can deal with infinities mathematically.

The problem is caused by a desire to glamourise the more spectacular results of mathematics. In the formal section on the Banach-Tarski paradox [[1]] we see the problem expressed mathematically. The problem is that it does not look very exciting so we are tempted to talk of gold balls which can be doubled in size. Of course, real gold balls cannot be doubled but the mathematics is perfectly correct. The same problem exists in QM where writers attempt to explain QM results in terms of classical physics. This produces exciting paradoxes but is not particularly helpful in understanding the subject. Martin Hogbin (talk) 18:19, 31 January 2012 (UTC)

Richard, you say that "In step (2) we are silently using a theorem which requires a condition which is not satisfied in this case. The conditions of the theorem do not apply hence we cannot deduce that the conclusion of the theorem is true." What theorem is that? iNic (talk) 10:23, 1 February 2012 (UTC)

Theorem. Suppose E(X) is finite. If E(Y | X=x) > x for all x, then E(Y) > E(X). Richard Gill (talk) 20:58, 7 February 2012 (UTC)

The 'simple solutions' explained

The simple solutions, as exemplified by the solution given by Falk in her 2008 paper, are correct but only in particular circumstances which have not thus far been clearly identified. In her paper, Falk says:

The assertion in no. 6 (based on the notation of no. 1) can be paraphrased ‘whatever the amount in my envelope, the other envelope contains twice that amount with probability 1/2 and half that amount with probability 1/2’. The expected value of the money in the other envelope (no. 7) results from this statement and leads to the paradox. The fault is in the word whatever, which is equivalent to ‘for every A’.

So far this resolution is in accordance with Nalebuff's solution.

She continues:

This is wrong because the other envelope contains twice my amount only if mine is the smaller amount; conversely, it contains half my amount only if mine is the larger amount. Hence each of the two terms in the formula in no. 7 applies to another value, yet both are denoted by A. In Rawling’s (1994) words, in doing so one commits the ‘cardinal sin of algebraic equivocation’ (p. 100).

It is not quite clear to me exactly what this bit is saying. What exactly is, 'the smaller amount'?

Falk explains further by giving a revised version of the paradoxical line of reasoning:

1. I denote by A the amount in my selected envelope.

2′. Denote by S the smaller of the two amounts.

The probability that A = S is 1/2 and that A = 2S is also 1/2.

3′. The other envelope may contain either 2S or S.

4′. If A=S the other envelope contains 2S.

5′. If A=2S the other envelope contains S.

6′. Thus the other envelope contains 2S with probability 1/2 and S with probability 1/2.

7′. So the expected value of the money in the other envelope is 1/2 2S+1/2S=3/2S.

8′. This is equal to the expected value of A, which is 1/2 S+1/2 2S=3/2S, so I gain nothing on average by swapping, and I am indifferent between keeping and swapping.

My assertion is that this argument, and many similar ones, are perfectly correct iff 'the smaller amount', as represented by S above, is a constant. In other words it has a fixed value. This means essentially that the solver of the problem knows the possible sums in both envelopes. In other words, our envelope space contains only two elements.

Later Falk says:

Had step 2 been adjusted to say ‘the probability that A is the smaller amount, denoted As, is 1/2, and that it is the larger amount, denoted Al , is also 1/2’, then the same symbol A would not have been used for the two cases in steps 6 and 7, and the paradox would have been avoided.

What are As and Al here, random variables? If so the argument above does not really make sense. On the other hand if they are both constants everything is clear.

In the Broome version of the game, because of the way the sums in the envelopes are determined, it is not true that the sum in the unchosen envelope is equally likely to be half or twice the sum in the chosen one - unless, after the two sums in the envelopes have been determined, we are told what they are, reducing our envelope space to only two elements.

In my opinion this is the crucial fact missed by many of the authors of papers on the subject. You cannot ignore how the sums in the envelopes are determined in your calculations unless, after they have been determined, you are told both sums. What you can never, in my opinion do, is have variable constants.

Richard, I would be interested to hear your view on this observation. Prof Falk, if you are reading this, I am sorry for using your paper as an example of what I consider to be incomplete thinking, but I would be interested to hear from you also.Martin Hogbin (talk) 10:22, 8 February 2012 (UTC)

I agree. Except, that the analysis does not require the amounts to be known. We can do probability calculations conditional on the values of the two sums of money (the smaller and the larger), without knowing them. I am afraid I have had so many fruitless discussions with amateurs about the fact that one can usefully think about E(Y|X=x), the expectation value of one random variable given that another takes on a particular value, without that value actually being known or given. Some of those amateurs were professional physicists, who certainly could do mathematics better than me, in the sense of having a fantastic intuition for finding really smart ways to do extremely difficult calculations. Richard Gill (talk) 10:51, 9 February 2012 (UTC)

Some more thoughts on this. The word "variable" has a very precise meaning within rigorous (precise, formal, academic, professional, theoretical, abstract, ... ) mathematics. And then always accompanied by a quantifier "for all" or "there exists". Examples: for all x in some set, x satisfies some relation; or: there exists x in some set, such that .... Brackets indicate the precise scope of the quantifier. The word "constant" is not used, and a word like "fixed" or "known" also has no meaning. Careful mathematics is like careful computer programming. Mathematical variables are just like variables in computer code, they are merely place holders, and one is careful to show which places are supposed to be occupied by the same thing (scope; local versus global variables), and what this thing can be (type).

Within probability theory, but only within probability theory (and branches of mathematics built on probability theory, like statistics) we use the words "random variable" and "constant" with rigorous, precise, formal meanings. Within probability theory the notion of conditioning is useful. One can say things like "the probability of A, given B" but this does not mean that B is known to have occured. It simply means (if you like frequentist language) "imagine many, many repetitions of the probability experiment we are talking about. Restrict attention to all those occasions, and only those occasions, when B occurs (completely irrelevant whether anyone knows this to the case, or not). What is the relative frequency of A within this sub-family of repetitions?".

Unfortunately, ordinary folk think that variables must vary and constants must be fixed and possibly also known, and that we can only talk about the probability of A given B if B is known (known to someone, to have happened). But in probability theory, a constant is merely a random variable with a degenerate probability distribution. A random variable is a (deterministic) function from the set of possible atomic outcomes of some probability experiment to some set of possible values, typically real numbers. Conditioning on an event, does not mean that we force the event to happen. Working given X=x does not mean that we know or force X to take the value x. It merely means a conceptual restriction to all those events, and only those events, when the random variable X happens to take the value x. The symbol x denotes a mathematical variable. The context must show what it can be, and whether the name x is only being used locally or whether it is being used globally (or something in between).

Another word with precise technical meanings, various ones in fact, within mathematics, is "independent". Another recipe for confusion. Richard Gill (talk) 09:28, 10 February 2012 (UTC)