Interesting Mathematical Paradox?

[jogger08152]

One envelope holds an amount X. The other envelope holds amount 2X. If you switch, you will gain X half the time (when you started with envelope X), and you'll lose X the other half of the time (when you started with envelope 2X).

[dukemagic]

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Not exactly the same formulation as the Monty Hall problem.

In the OP's case you are betting as to whether or not you got the small or the big amount, in the first envelope. In any case, you are guaranteed the small amount!

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I don't understand what you mean when you say we are guaranteed the minimum amount? Care to expand on what you are saying?

[MidGe]

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Not exactly the same formulation as the Monty Hall problem.

In the OP's case you are betting as to whether or not you got the small or the big amount, in the first envelope. In any case, you are guaranteed the small amount!

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I don't understand what you mean when you say we are guaranteed the minimum amount? Care to expand on what you are saying?

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To quote jogger: "One envelope holds an amount X. The other envelope holds amount 2X."

In all possible outcomes you are guaranteed X. That is the neutral expectation, if you could call it that. Since after the first one is opened , you don't know whether it is X or 2X, as you formulated your OP, the only possible outcomes are, now, to loose X (if the first one contained 2X or to win an extra X (If the first one contained X - and the second 2X, as per your OP). To me, that's 50/50 you win or loose the same amount.

PS Let me know which poker tables you are playing. [img]/images/graemlins/smile.gif[/img]

[TimWillTell]

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Not exactly the same formulation as the Monty Hall problem.

In the OP's case you are betting as to whether or not you got the small or the big amount, in the first envelope. In any case, you are guaranteed the small amount!

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I don't understand what you mean when you say we are guaranteed the minimum amount? Care to expand on what you are saying?

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To quote jogger: "One envelope holds an amount X. The other envelope holds amount 2X."

In all possible outcomes you are guaranteed X. That is the neutral expectation, if you could call it that. Since after the first one is opened , you don't know whether it is X or 2X, as you formulated your OP, the only possible outcomes are, now, to loose X (if the first one contained 2X or to win an extra X (If the first one contained X - and the second 2X, as per your OP). To me, that's 50/50 you win or loose the same amount.

PS Let me know which poker tables you are playing. [img]/images/graemlins/smile.gif[/img]

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So if I understand correctly, you don't switch because it doesn't make any difference?

So now were making a little change.
Two boxes exactly alike, in one a unknown amount in the other an amount 1000 time bigger.
Pick one!
No need to change after you made your pick; you can if you want to.
Now open the one you have picked.
There is a thousand dollars in it.
Now you are allowed to switch!
I don't care how right you are, but I know I am even willing to pay another 1000 dollars to be allowed to make that switch.! [img]/images/graemlins/smile.gif[/img]

This paradox is making me crazy!

Double cheers!

[Spence]

make that number 100,000 times bigger.

Would you still switch?

[TimWillTell]

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make that number 100,000 times bigger.

Would you still switch?

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Would totally depend on the amount of money in the box that I have opened.
If it was 1000 dollars I would take the money and not switch, because no way I could ever believe that there would be 100 million dollars for me waiting in that other box; I'd be sure there would be this ugly one cent coin!

But this does not solve the paradox.
Tomorrow, I'll ask my shrink for some stronger medication; I wish I never read the OP.

[jason1990]

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This problem has definitely come up on 2+2 before (Probability forum I think?), it's where I was first introduced to it. The wikipedia entry for it details a number of good explanations for it, I like it: link

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Here are some quotes from the Wikipedia article.

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The two envelopes problem is a puzzle or paradox within the subjectivistic interpretation of probability theory; more specifically within Bayesian decision theory. This is still an open problem among the subjectivists as no consensus has been reached yet.

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It may be true that philosophers still argue about this. I honestly do not know what philosophers argue about these days. But it is not an open math problem. Probability theory is a branch of mathematics and its practitioners are called probabilists. I am a probabilist. Probabilists are not distinguished according to their philosophical beliefs about probability (e.g. subjectivist/Bayesian, frequentist, etc.) Probabilists do not need philosophy to do their work. They are simply doing mathematics. Whatever (if any) philosophy they may adhere to in their hearts will have no bearing whatsoever on the mathematics they do. To a probabilist, this is not a genuine paradox, nor is it an open problem. It is simply a puzzle which confuses people and, therefore, offers us an opportunity to try to teach our craft.

Similarly, it is not an open problem for working statisticians. Statisticians work on mathematical models which are built upon the formalism of probability theory, and they apply the mathematics of probability theory. Their philosophical stance may influence their decisions about which tools to use, but they are still only using tools which come from the mathematics of probability. And within that mathematical framework, this is not an open problem.

But, as I said, maybe philosophers still argue about this. But even if that is the case, the above quote is still a bit misleading since it says this is an open problem in "probability theory," which typically refers to the branch of mathematics, not a branch of philosophy.

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Comment: Because the subjectivistic interpretation of probability is closer to the layman's conception of probability...

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This comment does not belong in an "encyclopedia." This is an unverifiable statement which, at best, represents the opinion of the author.

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...for a working statistician or probability theorist endorsing the more technical frequency interpretation of probability this puzzle isn't a problem, as the puzzle can't even be properly stated when imposing those more technical restrictions.

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This statement is wrong on multiple levels. Working statisticians and probability theorists (i.e. probabilists) do mathematics. This quote suggests that there is a set of mathematical concepts which are valid under one philosophy and invalid under another, and vice versa. This is not true. There is one set of mathematical ideas in which to do probability, and it is valid regardless of philosophy. Now, when we apply these concepts, we must build a model. And our choice of model may be inspired by philosophy. But it is not true that models inspired by the frequency interpretation are more technical than those inspired by subjectivism. In fact, in general, the opposite is true.

Moreover, it is definitely not true that this puzzle "can't even be properly stated" by a frequentist. Properly stating the puzzle, for a statistician or probabilist, means translating it into a mathematical model. This can be done without any reference to philosophy.

In summary, this envelope puzzle is not a paradox or an open problem in mathematics, even though it may be a problem for some philosophers. Mathematicians, even probabilists, are unaffected by philosophical debates over the foundations of probability. We work in a rigorously established mathematical framework which is independent of how our models are interpreted by philosophers. The wikipedia article at least implies that this is not the case. But in that regard, the article is wrong.

If you are interested in reading about a probabilist's perspective on the Bayesian vs. frequency philosophical debate, http://www.math.washington.edu/~burd...sophy/book.pdf has a lot to say on it.

[bigpooch]

Yes, wikipedia has many mistakes, even very simple ones:

from "Centroid" article:


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A similar result holds for a tetrahedron: its centroid is the intersection of all line segments that connect each vertex to the centroid of the opposite face. These line segments are divided by the centroid in the ratio 2:1.


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As mentioned before, the more interesting result is that
there are infinitely many switching strategies that have
+EV in the two-envelope paradox. Also, you only need to
have the contents of the envelopes to differ in amount for
there to be infinitely many +EV switching strategies.

[jason1990]

What is true is that there are infinitely many models you can build to try and represent this envelope situation in which p(x), the conditional probability that you hold the smaller envelope, given that the envelope contains x, is greater than 1/3 for all x. There are no models in which it equals 1/2 for all x.

However, in all of the models for which 1/3 < p(x) < 1/2, your expected gain from switching is undefined.

Consider the example in the Wikipedia article:

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Suppose that the envelopes contain the integer sums {2^n, 2^(n+1)} with probability 2^n/3^(n+1) where n = 0, 1, 2,...

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In this case, p(x) = 2/5 for all x > 1.

The unconditioned expected gain from switching (i.e. your EV if you switch without looking) is undefined. It is not zero. In infinite probability spaces, the expected value of a random variable X is undefined whenever the expected value of |X| is infinite. In this case, if X is your net profit from switching, then

P(X = 2^n) = 2^(n-1)/3^(n+1) for n = 0, 1, 2, ...
P(X = -2^n) = 2^(n-1)/3^(n+1) for n = 0, 1, 2, ...

A naive calculation would lead one to think that EX = 0. But really EX is undefined, since E|X| = infinity. The expectation of X is undefined (and not zero) for the same reason that the integral from -infinity to infinity of x^3 dx is not zero.

The conditional expectation of X, given that your envelope contains x, is also undefined. It is not 11x/10, as the Wikipedia article claims. The reason is simply because the conditional expectation of a random variable X, given another random variable Y, is undefined whenever the (unconditional) expectation of |X| is infinite.

All of this formal mathematical machinery may go against a student's intuition, but that is precisely why this example is a good teaching tool. Division by zero, infinity minus infinity, and all the other "technicalities" that we mathematicians rule out are ruled out for a reason. They do not make sense, they lead to apparent contradictions, and they cannot be supported with a rigorous logical foundation.

[Aleo]

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It may be true that philosophers still argue about this. I honestly do not know what philosophers argue about these days. But it is not an open math problem. Probability theory is a branch of mathematics and its practitioners are called probabilists. I am a probabilist. Probabilists are not distinguished according to their philosophical beliefs about probability (e.g. subjectivist/Bayesian, frequentist, etc.) Probabilists do not need philosophy to do their work. They are simply doing mathematics. Whatever (if any) philosophy they may adhere to in their hearts will have no bearing whatsoever on the mathematics they do. To a probabilist, this is not a genuine paradox, nor is it an open problem. It is simply a puzzle which confuses people and, therefore, offers us an opportunity to try to teach our craft.


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Out of curiosity, I'd be interested in hearing more about this. I can't tell whether you are stating that this is a problem for philosophers and not mathematicians, or whether you are stating that it isn't a problem at all, and philosophers are confused in finding it so.

I think it is often the case that mathematicians, physicists, and other working scientists look at the philosophical foundations of their discipline as a non issue, and in turn look to the philosophers interested as confused about what they are doing.

In fact, I think philosophical foundations are very important, and in my experience, most philosophers doing serious work in those areas have strong professional level knowledge of the disciplines they lend their philosophical work to. In many cases, those doing the best philosophical work are converts from within the discipline itself. That this is so should suggest it is not a confusion.

As for the two envelopes problem, I cannot comment on its importance. It may simply be a confusion. There are, however, real issues in the philosophical foundations of probability and I suspect they have more relevance to actual practice than you give credit for.

For an authoritative article, I'd recommend the Stanford Encyclopedia of Philosophy:

http://plato.stanford.edu/entries/pr...ity-interpret/

It is possible, perhaps likely, that you are familiar with all of this and I do not mean to suggest otherwise. If not, it is worth reading.

Regards
Brad S