Interesting Mathematical Paradox?

[dukemagic]

This is my first time posting in this subforum, so i apologize if I'm repeating an earlier problem, or if this belongs in probability.

My friend described this scenario to me the other day and I found it fascinating:

I have two envelopes, both of which have money inside. I tell you that one envelope has twice as much money as the other. I RANDOMLY choose one of the two envelopes and give it to you.

I then offer you the choice of changing envelopes before you open it. Is it in your advantage (EV wise) to do so? My initial though is no, because I randomly chose the envelope, and so you gain nothing by switching.

However, once you open the envelope and see how much money is inside (let's say its $20) I then offer you the chance to keep that $20 or change envelopes and take what is in the unopened one.

Do you then want to switch? The answer I arrived at is, yes, you do want to switch envelopes every time! I currently have $20. The other envelope muse have either twice as much or half as much- in other words, it must have either $40 or $10, so your EV by switching is $25 ($50 / 2).

So it seems to me that this is a paradox. There is no reason to change envelopes before you open them, but it is in your best interest to switch envelopes, after you open the first one, EVERY single time.

Anyone want to expand on this? There must be something I'm missing.

[MidGe]

You are making a mistake in your evaluation. Hint: in all cases you are guaranteed the minimum amount.

[dukemagic]

[ QUOTE ]
You are making a mistake in your evaluation. Hint: in all cases you are guaranteed the minimum amount.

[/ QUOTE ]
Do you mean that in all cases I expect the other envelope to have more money, no matter what I do? Is that all there is to it? So what is the correct plan of action in this case? I'm guessing it's irrelevant whether or not you switch pre-opening, but you always want to switch post-opening. Is that correct?

[MidGe]

[ QUOTE ]
Is that correct?

[/ QUOTE ] No. The switching does not change the EV.

[dukemagic]

The one thing I wasn't sure about was whether or not you want to switch before you open up the envelope. You could use the same logic that you apply to the open envelope:

Let's say the envelope I offer you first is Envelope 1, containing A amount. Envelope 2 would then contain either A/2 or 2A. So the expected value of switching would be (A/2 + 2A) / 2, or 5/4 A (B = 5/4 A). So it seems you do want to accept the offer to switch.

But if you do accept this offer to switch and open envelope 2, you're still going to change back to Envelope 1 EVERY single time since now you expect Envelope 1 to have more than Envelope 2, on average. In fact (if my math is correct) you expected Envelope 1 to have (5/4) B, or 25/16 A which is greater than the 5/4 A you expected Envelope 2 to have. Is this a paradox? What am I missing?

[repulse]

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

[dukemagic]

[ QUOTE ]
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

[/ QUOTE ]
Ah this is perfect. I'll read it now. Thanks!

[jason1990]

[ QUOTE ]
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

[/ QUOTE ]
Here are some quotes from the Wikipedia article.

[ QUOTE ]
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.

[/ QUOTE ]
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.

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

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

[ QUOTE ]
...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.

[/ QUOTE ]
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:


[ QUOTE ]

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.


[/ QUOTE ]

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:

[ QUOTE ]
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,...

[/ QUOTE ]
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.