Interesting Mathematical Paradox?

[jason1990]

There are two things one could mean when speaking of the "foundations of probability." On the one hand, probability theory is a branch of mathematics and, therefore, an axiomatic system. Almost all probabilists and working statisticians use the system that results from the axioms of measure theory. Questions about the foundations of this axiomatic system, as well as questions about the relevancy of alternative axiomatic systems, are certainly relevant and important, and form the foundation of my discipline. These questions, however, have nothing to do with the envelope puzzle as it is typically presented.

On the other hand, "foundations of probability" could refer to the philosophical issues that arise when we try to interpret these mathematical statements and connect them to the real world. Strictly speaking, these questions have nothing to do with formal mathematics. A working statistician might be somewhat concerned with these issues, since he or she wants to select a mathematical model that accurately represents something in the real world. But once that model is selected, and he or she begins doing mathematics, these kinds of philosophical issues are not relevant.

The envelope puzzle, however, is not typically presented as a paradox about the interpretation of probability statements in the real world. It is presented as a mathematical paradox, similar in spirit to the puzzles one often encounters that give a "proof" that 0 = 1. To claim that it is a legitimate mathematical contradiction is absurd.

Insofar as philosophers stick to philosophy, I cannot judge whether or not they got anything wrong. But if they make mathematical claims, I have a chance of judging those. The Wikipedia article got the math wrong in at least one place. In Section 3.3.1, they say

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But in every actual single instant when you open an envelope the conclusion is justified: you should switch!

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This is false. It is not justified. The conditional expectation of the contents of the other envelope, given that your envelope contains x, is not 11x/10. It is undefined, since the unconditional expectation of the contents of the other envelope is infinity.

It is natural for a non-probabilist to make this mistake. For example, a non-probabilist would likely assume it is a tautology that E[Z | Z = x] = x. (Even a probabilist might assume this for a moment.) But it is not. The statement is not true when E|Z| is infinite. In that case, the left-hand side is undefined. It should not be surprising that assuming something is true when it is actually undefined can prove a lot of absurd statements. This false assumption about conditional expectations is similar, in spirit, to assuming you can divide by zero, which is the trick by which people "prove" that 0 = 1.

[jason1990]

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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.

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Aha... I think you are referring to something which is not mentioned in the Wikipedia article and which has not been brought up yet in this thread. If so, then I agree. That aspect of it is very interesting.

[Magic_Man]

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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.

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Your change to the OP is easily accounted for by considering a non-linear utility function. If we exagerate your scenario a little bit and say:

There are 2 boxes. One contains X, the other contains 100,000*X. You open A, and it contains $100. Do you switch?

I'm sure almost everyone here would switch. It's perfectly reasonable to not mind paying $100 to try to win 10 million, and most economists could easily describe this choice. On the other hand, if the box you opened contained $100,000, switching boxes becomes a little more silly. It's all about risk tolerance.

[mosta]

I've chimed in on this topic in several previous threads, and I'm not going to try to debate or go on at length. But let me ask if I've gleaned correctly that you are saying the EV of switching is undefined?

while not a mathematician, I've been satisfied with the argument that when you define the problem as two envelops and two quantities P and Q, and then at the next step you get quantity A and start to analyze the possibilities of A, .5A and 2A--you have simply introduced a contradiction: 3 possibilities versus 2 possibilities. At that point you've gone off track into nonsense. And there is no need to introduce any talk of the distribution of the original quantities to address the error.

And when you keep it straight as 2 quantities only, the EV is zero.

at least one mathematician in a previous thread disagreed that that is the/an answer. but a friend of mine who is also a mathematician (phd from a top school) but not a probabilist agreed with me entirely. so if I'm wrong I guess I'm not inarguably wrong?

[PairTheBoard]

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I've chimed in on this topic in several previous threads, and I'm not going to try to debate or go on at length. But let me ask if I've gleaned correctly that you are saying the EV of switching is undefined?


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I think Jason was talking about a specific example in the case where the amount in the smaller envelope is modeled to have come from a particular probability distribution so that the expected value of the amount in the smaller envelope is undefined. If the expected value of the amount in the smaller envelope is undefined then so is the expected value of switching.

I believe your argument is basically correct if you consider the amounts in the envelope fixed. If someone insisted on a prior distribution for smaller envelope you could just assume it is a point mass one, which you just don't know.

Then if you look in the envelope and see the amount A, it's not the case that there is a 50% chance the other envelope has .5A and a 50% chance it has 2A. What's true is that there is either a 100% chance the other envelope has .5A or a 100% chance it has 2A, you just don't know which. This sounds funny to some people because they don't have it clear in their minds what they mean by "50% probabilty". In this case, with the Envelope Amounts Fixed, we would be looking at the repeatable experiment of shuffling and choosing from the same two envelopes, then looking at what happens whenever someone opens to see the amount A. We would certainly not see the other envelope containing .5A half the time and 2A half the time. That's what the 50% probabilty would mean, but it wouldn't happen. We would either see .5A all the time or 2A all the time. We just don't yet know which. I think that's basically the same thing that you are saying and it is correct.

Some people are not satisfied with this though. They say that the smaller envelope amount must have been chosen by the Envelope Stuffer according to some decision making process and that the repeatable mind experiment we should consider is the Stuffer choosing different envelope amounts each time he offers them. They then, maybe without thinking, invoke what I believe is called the indifference principle, which says something like if there are two possibilites with no knowledge about either you should consider them equally likely. Something like that. From this they argue that there is a 50% chance the other envelope has .5A and 2A. From there they get the EV paradox.

The problem with this model is that there is no prior probabilty distribution for the smaller envelope amount such that all amounts are equally likely. The indifference principle might be a good rule of thumb for certain reasoning but it does not make for a good probability model in this case. And that's what you need to make the probability statements of 50% and the resulting EV calculation.

If you insist on the perspective of a prior probabilty distribution for the envelope amounts then you have to have a valid one. Once you start looking at valid ones you can draw the conclusions Jason explained in his post.

I've felt that although the above explanations are correct and solve the "paradox" they don't seem fully psychologically satisfying. Someone says, before I open the envelope, surely I have a 50% chance of switching to the larger. It just seems unreal that seeing amount A in the envelope should change that when it doesn't really give me any new information about whether it's the smaller amount.

Here's my attempt at satisfying this psychological conundrum. First notice that switching is equivalent to betting half your envelope at 2-1 odds that it's the smaller envelope. If you could bet any amount you want that you have the smaller envelope that would surely be a good bet. You're getting 2-1 odds. Certainly before you open the envelope you would bet, say $100 that you have the smaller envelope at 2-1 odds. Even after you see amount A in the envelope I think you'd be happy to continue your $100 bet. You probably wouldn't call it off. It's +EV before you open. Seeing amount A probably won't convince you it's not still +EV. So why isn't switching +EV as well? After all, switching is equivalent to betting half your envelope that it's the smaller. Same bet right? Same 2-1 odds.

The difference between the bets is that in the second one you are being forced to bet an amount A that has been dictated by the already determined outcome of the bet. It's like saying you can bet $1 at 2-1 odds that tails will come up. Except you will also be forced to bet $2 when head comes up and you lose.

Consider this proposition bet. I have a standard deck of shuffled cards and I deal one to you face down. I'm going to let you bet that it's a black card. And I'm going to pay you off 2-1 when it's black and you win. Certainly you would be happy to place a $100 wager on that basis. Except it's not that simple. I've written numbers on the back of each card in invisible ink. I deal the card out on the table. At that point you'd still like to make your $100 bet. Now I put some lemon on the card and expose the number A. You'd probably still like to make the $100 bet. But then I tell you that you must bet the amount A written on the card. Do you still think it's a +EV bet?

PairTheBoard

[MusashiStyle]

this is just an exercise in conditional probability which often seems counterintuitive. The standard way to present this type of problem is:

1. You are a contestant on a game show.
2. There are three closed doors
3. behind 2 doors is a goat and behind one is a car.
4. After you choose one door (but before opening it) the gameshow host suddenly opens one of the other two doors and reveals a goat.
5. Is it in your best interest to switch to the other remaining (closed) door given this new information?


It turns out that the answer is not so simple and requires a fair amount of conditional probability. However, it is not a true paradox.

[TimWillTell]

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this is just an exercise in conditional probability which often seems counterintuitive. The standard way to present this type of problem is:

1. You are a contestant on a game show.
2. There are three closed doors
3. behind 2 doors is a goat and behind one is a car.
4. After you choose one door (but before opening it) the gameshow host suddenly opens one of the other two doors and reveals a goat.
5. Is it in your best interest to switch to the other remaining (closed) door given this new information?


It turns out that the answer is not so simple and requires a fair amount of conditional probability. However, it is not a true paradox.

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i like this one very much, but i know the answer and can understand it perfectly.
The two envelops, i can not understand.
Some of the posters here understand the two envelops and they would not switch...?

Suppose you open the envelop and there is 20 dollars in it.
So in the other envelop there is either 10 dollars or 40 dollars.
I think, before opening there is no need for switch and therefore there cannot be a need for switch after opening.
Jet, if I see the 20 dollars, I have to switch...

I feel like the monkey that has trapped himself by holding on to the banana. [img]/images/graemlins/confused.gif[/img]

[mosta]

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Suppose you open the envelop and there is 20 dollars in it.
So in the other envelop there is either 10 dollars or 40 dollars.


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I'd say: as soon as you say this, realize it's a mistake, a contradiction, and "erase" your statement. Seeing one envelop tells you nothing about the other. But some people who probably know what they're doing don't think that's enough to say.

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I feel like the monkey that has trapped himself by holding on to the banana. [img]/images/graemlins/confused.gif[/img]

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This is a really great line.

[mosta]

Thanks for clarifying his use of a distribution. I suspected I didn't read carefully enough. I still have to think about your prop bet some more.

[PairTheBoard]

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I still have to think about your prop bet some more.


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Knowing that I'm in the business of offering sucker prop bets, what kind of numbers do you think I wrote on the back of the Red and Black cards? How is being forced to wager the amount written on the card that it's black, different from chosing the amount you wager yourself?

PairTheBoard