Interesting Mathematical Paradox?
 

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.

Re: Interesting Mathematical Paradox?
 

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

Re: Interesting Mathematical Paradox?
 

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?

Re: Interesting Mathematical Paradox?
 

[ 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?

Re: Interesting Mathematical Paradox?
 

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

Re: Interesting Mathematical Paradox?
 

[ QUOTE ]
Is that correct?

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

Re: Interesting Mathematical Paradox?
 

[ 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!

Re: Interesting Mathematical Paradox?
 

[ QUOTE ]
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, ...


[/ QUOTE ]

Yes. However you Don't know that the probability is 50% for the $40 and 50% for the $10. Therefore you cannot draw the conclusion,

[ QUOTE ]
... so your EV by switching is $25 ($50 / 2).


[/ QUOTE ]

Just because there are two possibilties doesn't make them equally likely.

PairTheBoard

Re: Interesting Mathematical Paradox?
 

tldr. Reminded me of Monty Hall Problem. Here's a wikipedia article on a two envelope problem. Not sure if the formulation is the same.

Re: Interesting Mathematical Paradox?
 

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!

Re: Interesting Mathematical Paradox?
 

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).

Re: Interesting Mathematical Paradox?
 

[ QUOTE ]
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!

[/ QUOTE ]

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

Re: Interesting Mathematical Paradox?
 

[ QUOTE ]
[ QUOTE ]
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!

[/ QUOTE ]

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

[/ QUOTE ]

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]

Re: Interesting Mathematical Paradox?
 

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
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!

[/ QUOTE ]

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

[/ QUOTE ]

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]

[/ QUOTE ]

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!

Re: Interesting Mathematical Paradox?
 

make that number 100,000 times bigger.

Would you still switch?

Re: Interesting Mathematical Paradox?
 

[ QUOTE ]
make that number 100,000 times bigger.

Would you still switch?

[/ QUOTE ]


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.

Re: Interesting Mathematical Paradox?
 

[ 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.

Re: Interesting Mathematical Paradox?
 

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.

Re: Interesting Mathematical Paradox?
 

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.

Re: Interesting Mathematical Paradox?
 

[ 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.


[/ QUOTE ]

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

Re: Interesting Mathematical Paradox?
 

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

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

[/ QUOTE ]
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.

Re: Interesting Mathematical Paradox?
 

[ 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.

[/ QUOTE ]
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.

Re: Interesting Mathematical Paradox?
 

[ QUOTE ]
[ QUOTE ]
make that number 100,000 times bigger.

Would you still switch?

[/ QUOTE ]


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.

[/ QUOTE ]

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.

Re: Interesting Mathematical Paradox?
 

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?

Re: Interesting Mathematical Paradox?
 

[ QUOTE ]
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?


[/ QUOTE ]

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

Re: Interesting Mathematical Paradox?
 

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.

Re: Interesting Mathematical Paradox?
 

[ QUOTE ]
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.

[/ QUOTE ]

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]

Re: Interesting Mathematical Paradox?
 

[ QUOTE ]
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.


[/ QUOTE ]

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.

[ QUOTE ]

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

[/ QUOTE ]

This is a really great line.

Re: Interesting Mathematical Paradox?
 

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.

Re: Interesting Mathematical Paradox?
 

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


[/ QUOTE ]

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

Re: Interesting Mathematical Paradox?
 

[ QUOTE ]
[ QUOTE ]
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.


[/ QUOTE ]

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.


[/ QUOTE ]

No. That's not where the mistake is being made. If you accept the information that one envelope contains twice the other, seeing $20 does tell you something you didn't know before about the other envelope. You certainly now know that it doesn't contain $100. You didn't know this before seeing the $20. Logically, the conclusion is forced that your new, incomplete, state of knowledge about the other envelope is that it must have either $10 or $40. Another way to put this would be, seeing the $20 now eliminates all other possible amounts for the other envelope except for $10 and $40.

The mistake comes from taking these two remaining possiblities for the other envelope and assigning them both probabilities of 50%. As you pointed out before, that makes no sense. That would mean that when you open the other envelope, half the time you would see $10 and half the time you would see $40. That just doesn't happen. Every time these envelopes are offered and someone sees $20 there is only one possibility for the other envelope. That's where you're correct. As far as we know now, it could be $10 or $40. That's where TimWillTell's statement is correct. But whichever it is, that's what it will be every time. That's where the 50% probabililty assumption is wrong for purposes of computing EV with respect to switching the $20 envelope.

PairTheBoard

Re: Interesting Mathematical Paradox?
 

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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.

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The indifference principle is interesting. Let me say more about it here, since it relates to what I will say later in this post.

Imagine I take a "normal" (fair) coin and flip it. What is the probability it will land heads? We say that it is 1/2, and we base this on the symmetry of the coin. A die lands on 4 with probability 1/6 because of the symmetry of the die. A roulette ball lands in the 00 slot with probability 1/38 because of the symmetry of the roulette wheel. We understand that these symmetries are not perfect and, therefore, our probability statements about equal likelihood are not perfect either. But they are very close in practice and we are satisfied with them. One thing which is particularly satisfying about these claims is that they are based on the symmetries of the physical system we are considering. In other words, the claim that the coin will land heads with probability 1/2 is a claim about the physical nature of the coin itself. In some sense, then, it is an objective scientific claim.

Now imagine I mangle the coin. What is the probability it will land heads? Now it could be anything between 0 and 1. We cannot know. (Suppose for now that we cannot effectively analyze the way in which it is mangled and how that affects the probability.) The indifference principle says this: there are two possible outcomes -- heads or tails -- and we have no information about the likelihoods of these outcomes; therefore, we should take the probability to be 1/2. According to the indifference principle, the answer is the same as the answer for the unmangled coin. But this time, it is not a statement about the physical symmetry of the coin. Instead, it is a statement about the symmetry in our lack of knowledge. It is no longer an objective (though perhaps imperfect) statement about a physical system, but is now an epistemological statement about our mental state. In short, it is no longer science, but is now philosophy.

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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.

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Imagine my brother is the one stuffing the envelopes. Imagine also that one envelope contains 10 times the other. I select an envelope and do not open it. I am perfectly happy with the claim that I have the smaller amount with probability 1/2. Now, I open it. It contains $50. There is no way in hell I would switch. I know my brother very well and there is definitely not $500 in that other envelope. In this particular example, seeing amount A definitely changes the probability that I have the smaller envelope.

You seem to be imagining some kind of idealized situation in which the observation of A tells us absolutely nothing, and then applying some version of the indifference principle which says that A is therefore independent of everything. I think the psychological dissatisfaction you are experiencing comes from the fact that the indifference principle "feels" true and we ought to be able to apply it.

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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.

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Agreed.

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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.

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I do not agree here. If, in the situation above with my brother, I saw $50 in my envelope, I would definitely want to call off my bet.

There is a fundamental difference between the two situations. When the envelopes are closed, then there is a genuine physical symmetry in the system. The envelopes are (presumably) identical. The method of shuffling and picking was presumably symmetric. (Perhaps I flipped a fair coin to make my choice.) So there is an objective reason for saying that my envelope contains the smaller amount with probability 1/2.

But once I open my chosen envelope, that symmetry disappears. I see A. The other envelope contains either 0.5A or 2A. There is no natural symmetry about money or about the person stuffing the envelopes or about anything else for that matter which would give an objective reason for saying that these are equally likely. Moreover, the indifference principle cannot even be applied here, since it leads to a mathematical contradiction. The envelope puzzle, in fact, seems to do a very good job of refuting the indifference principle as a valid principle which can be applied universally.

So once I open the envelope and see A, what is the probability I have the smaller amount? I must answer this in order to decide whether or not to call off my bet. I would say that the only objective answer one can give is "I don't know," and this does not translate into 1/2.

By the way, I like your prop-bet description. I think it isolates a key misunderstanding that a lot of people have. But I think you may be sweeping under the rug this issue I have brought up here.

Re: Interesting Mathematical Paradox?
 

Right. I'm sweeping it under the rug with the word "probably". If the situation were like with your brother then seeing the amount in the envelope does give you more information. I'm not exactly sure what information it might give in general.

However, here's the sense in which I think my statement has validity. Suppose many different people are offered such envelopes. Each of them wagers $100 that they have the smaller envelope. Each of them ignores the amount they see in their envelope and continues their $100 bet. At 2-1 odds they will on average make money on their $100 bets. But they will not on average make money from the Envelope Switch. This I think is at the heart of the psychological conundrum.

PairTheBoard

Re: Interesting Mathematical Paradox?
 

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Right. I'm sweeping it under the rug with the word "probably". If the situation were like with your brother then seeing the amount in the envelope does give you more information. I'm not exactly sure what information it might give in general.

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Seeing the amount in the envelope gives you very specific information: the amount in the envelope. In the situation with my brother, I knew how to deal with that information. In general, you may not know how to deal with it. You may not know how it affects the probability of having the smaller envelope. But that does not imply that the information is independent of having the smaller envelope. To assume so is to apply a variation of the indifference principle.

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However, here's the sense in which I think my statement has validity. Suppose many different people are offered such envelopes. Each of them wagers $100 that they have the smaller envelope. Each of them ignores the amount they see in their envelope and continues their $100 bet. At 2-1 odds they will on average make money on their $100 bets. But they will not on average make money from the Envelope Switch. This I think is at the heart of the psychological conundrum.

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Of course, if they ignore the amount, then it is as if they never opened the envelope at all. The $100-wagerers will make money without opening, and the Envelope Switchers will not. But this does not mean that the information they ignored did not affect their conditional probabilities.

I think your observations here are correct and I think they demonstrate nicely the mistake people make when they fail to acknowledge that the amount they are wagering (half their chosen envelope) is a random variable tied to the result of their bet. But I do not see how that is at the heart of the psychological conundrum. You said,

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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.

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It seems that this is the heart of the psychological conundrum. It should not seem unreal that seeing A causes some sort of change. It definitely changes the experiment. At the very least, it removes the symmetry from the experiment, as I mentioned in my previous post. You may not know how this change affects the probability of having the smaller envelope, but that does not mean you can assume that it does not change it at all.

Re: Interesting Mathematical Paradox?
 

This problem is troubling me as well. It seems like it must be due to some sloppy definitional issue but I can't put my finger on what it is.

To try to get at the root of it consider a similar situation, but where there is no hint of paradox. I have two envelopes and tell you that envelope A has $10, and envelope B has a 50/50 chance of being either $5 or $20 dollars. In this case you would obviously just pick envelope B because it has higher expected value. In a sense this is exactly what you're doing, albeit more abstractly, to get to the paradox in the case outlined in the OP. You're saying "let my envelope have $x, so the other one has a 50/50 chance of being either .5x or 2x." It seems like you're invalidly breaking some type of symmetry here. The situation is that the envelopes are distributed either as [(x, 2x) or (2x, x)]. By using the reasoning above you're changing the distribution to [x, (0.5x) or (2x)], which I'm pretty sure is not equivalent to the original situation. Given that you've recieved no additional useful information by picking an envelope, or looking even looking inside of one, it seems like you can't do this and still have to speak only in terms of expected value and using the (x, 2x) or (2x, x) formulation. And, as somebody has mentioned, there is no paradox in this case as long as you calculate the EV correctly (by recognizing the fact that you always get at least the low value). Because now the EV of switching is

EV(switch) = x + 0.5(x) + 0.5(0) = 1.5x

which is the same expected value you had when initially picking the envelope (and still have currently).

Thinking about it this way makes it a little more clear to me, but I'm not anywhere near to an answer I find rigorous/satisfactory. I do follow and agree with the explanations that mention the issue of not being able to come up with an equal-probability distribution over an infinite set and hence in some since the conditional probability questions you're asking aren't meaningful... but my gut tells me there must be a simpler explanation. Of course, my gut's definitely been wrong many times before.

Re: Interesting Mathematical Paradox?
 

Here is a puzzle which I feel is, at least tangentially, related to the envelope puzzle.

I have two coins: one fair and one mangled. I flip them so that they both land on a table behind a curtain where you cannot see them.
<ul type="square">
[1] What is the probability that they have both landed on the same face?
[2] How would your answer change if I partially pulled the curtain back to reveal the mangled coin?
[3] How would your answer change if I revealed the fair coin?
[4] Are your answers to the previous questions "facts" or "opinions"?[/list]

Re: Interesting Mathematical Paradox?
 

I'll have to give some thought to what you're saying about the psychology. Since the math is well understood it seems to me that the psychology is the most interesting thing left to consider. It would be interesting to look at how our 2+2 treatment of this problem has evolved through the many threads on it over the years.

Something that bothers me. You said,

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jason1990 -
It should not seem unreal that seeing A causes some sort of change. It definitely changes the experiment. At the very least, it removes the symmetry from the experiment, as I mentioned in my previous post. You may not know how this change affects the probability of having the smaller envelope, but that does not mean you can assume that it does not change it at all.

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and

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jason1990 -
In general, you may not know how to deal with it [the information]. You may not know how it affects the probability of having the smaller envelope. But that does not imply that the information is independent of having the smaller envelope. To assume so is to apply a variation of the indifference principle.


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Which I of course agree with. But in my scenario:

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PairTheBoard -
However, here's the sense in which I think my statement has validity. Suppose many different people are offered such envelopes. Each of them wagers $100 that they have the smaller envelope. Each of them ignores the amount they see in their envelope and continues their $100 bet. At 2-1 odds they will on average make money on their $100 bets. But they will not on average make money from the Envelope Switch. This I think is at the heart of the psychological conundrum.


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Suppose there are two such groups. The First deals with the information by ignoring it (applying the indifference principle?). The Second deals with the information by saying upon opening the envelope that the amount changes the experiment but they don't know how so they call off the $100 wager. The First Group on average makes money while the Second doesn't.

PairTheBoard

Re: Interesting Mathematical Paradox?
 

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Suppose there are two such groups. The First deals with the information by ignoring it (applying the indifference principle?).

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I would not call this the indifference principle. There are three Stages:
<ul type="square">
[1] The Information. I see amount A.
[2] The Effect. ???
[3] The Decision. Switch or not. Call off bet or not.[/list]The indifference principle assumes that there is no effect. This produces a mathematical contradiction. Hence, there must be some effect, though it is unknown.

The people in your First group may very well realize this. They simply decide to not call off their bet, regardless of the information.

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The Second deals with the information by saying upon opening the envelope that the amount changes the experiment but they don't know how so they call off the $100 wager.

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Why should ignorance in Stage 2 lead one to think that calling off the bet in Stage 3 is the best decision?

What about the hypothetical group of Random Switchers who use their pocket calculator to generate an exponentially distributed random variable, and switch envelopes when A is less than the number they generated? They do better on average than the Always Switchers and the Never Switchers. This, by itself, should be convincing enough evidence that The Effect exists.

Re: Interesting Mathematical Paradox?
 

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Suppose there are two such groups. The First deals with the information by ignoring it (applying the indifference principle?).

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I would not call this the indifference principle. There are three Stages:
<ul type="square">
[1] The Information. I see amount A.
[2] The Effect. ???
[3] The Decision. Switch or not. Call off bet or not.[/list]The indifference principle assumes that there is no effect.

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Is that correct? This Indifference Principle is new to me. But from your description of the Bent Coin I don't see that it assumes there is no effect from Bending the Coin. I see it as saying that since as far as we know, the Bend can bias Heads just as well as Tails, for the purposes of making a Bet on the First Coin Flip we may as well figure the chances are still 50-50 even though we know they are probably not.

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What about the hypothetical group of Random Switchers who use their pocket calculator to generate an exponentially distributed random variable, and switch envelopes when A is less than the number they generated? They do better on average than the Always Switchers and the Never Switchers. This, by itself, should be convincing enough evidence that The Effect exists.

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I'm not arguing that there is no effect. Certainly this shows that there is. This raises an interesting question though. The Decision by Calculator improves the results of the Switchers. But if the $100 bettors use the same method to decide whether to continue their bets or call them off, does it improve their results? Will they also always get better results than by just ignoring the amount A, and always continuing the bet?

PairTheBoard

Re: Interesting Mathematical Paradox?
 

I was a little dissatisfied with my Random Switchers, since I was supposed to be replying to your idea about calling off the fixed $100 bet. So here is an idea about Random Call-Offers.

There is a sequence of Groups. Group n decides whether or not to call off the bet in this way. They generate a random number U, uniform on (0,1), and call off their bet if

A &gt; sqrt{n|ln(U)|}.

Their EV is

50(2e^{-y^2/n} - e^{-4y^2/n}),

where the values in the envelopes are y and 2y. For small n, Group n will not do very well, since they will be calling off their bet too often. But the EV will increase monotonically with n. At about n = 1.64y^2, the EV of Group n will be about the same as the Never Call-Offers, $50. But the EV will continue to increase, reaching a maximum at about n = (3y^2)/ln(2). This Group will have an EV of about $59. After that, the EV will decrease monotonically, with the limit being $50. In other words, all Groups with n sufficiently large will outperform the Never Call-Offers.