Ok, we agree that in the formula (2X+.5X)/2 there exists a term that will ultimately not be present.
I fully understand.
My point is that when you use the formula (2X+.5X)/2, you are being told something different about the situation than what you think you are being told.
No.
My answer is: don’t use the formula (2X+.5X)/2 to choose envelopes because it is telling you something different than what you think it is telling you.
You can apply that to both envelopes and they both have the same result.
Whether you use (M+2M)/2 for both envelopes
or (2X+.5X)/2 and (2Y+.5Y)/2 for both envelopes, in both cases both envelopes have a formula that does not indicate either one has an advantage.
(2X+.5X)/2 is not telling you the other envelope has more money on average.
Only if it is impossible for you to see k.
Apply what, the reasoning that E[Y] = 1.25 * E? Yes, that’s the paradox!
And there’s nothing wrong with that reasoning. It is mathematically a fact that E[Y] = E[E[Y | X]]; this is the law of total expectation. And E[Y | X] = E[Y | X, Y = 2X] * P(Y = 2X | X) + E[Y | X, Y = X/2] * P(Y = X/2 | X) = 2X * 1/2 + X/2 * 1/2 = 1.25 X; there’s nothing wrong with that calculation. So E[Y] = E[1.25 X] = 1.25 E, by linearity of expectations. And all of these calculations are correct.
And you get that E[Y] = 1.25 * E, while at the same time, symmetrically, E = 1.25 * E[Y], which seems paradoxical until you realize that E = E[Y] = ∞.
There’s nothing wrong with the calculation E[Y] = E[E[Y | X]] = E[1.25 X] = 1.25 E. You keep claiming there’s something wrong with it, but there isn’t. It’s not the only way you can break E[Y] down, but it is one perfectly legitimate, mathematically sound way of calculating E[Y]. It happens to lead to this counterintuitive (aka, paradoxical) result, but you can’t explain that away by saying “Do another calculation instead”. You have to point out what happened in this calculation. And nothing erroneous happened. It just happens to be the case that one way of writing E[Y] is as the infinite quantity 1.25 * 1.25 * 1.25 * 1.25 * 1. 25 * 1.25 …, and one way of writing E is as the (same), infinite quantity 1.25 * 1.25 * 1.25 * 1.25 * 1.25 * …, so each is equal to 1.25 * the other, with no contradiction.
Little Nemo, my apologies for not yet finishing the discussion I started in my last post about infinite series. I keep getting distracted, but I will return sometime today and wrap up the correct explanation of what’s going on in your paradox.
E[Y]=1.25 * E and E=1.25 * E[Y]
I don’t think you read my post.
What is wrong is thinking (2X+.5X)/2 indicates you should switch envelopes.
It’s wrong because it is not telling you that envelope 2 has more value than envelope 1 and envelope 1 has less value than envelope 2, on average.
Do you think (2X+.5X)/2 is telling you that, on average, you should choose envelope 2 because it will have more value than envelope 1?
How did you get that P(Y = 2X | X) = 0.5 and P(Y = X/2 | X) = 0.5?
That’s right, that’s the paradox!
In a situation which didn’t involve ill-behaved infinite series, the fact that E[Y] = 1.25 * E would tell you that E[Y - X] = 0.25 * E > 0. Do you disagree that this would tell you that, on average, the profit from switching from the X envelope to the Y envelope is positive?
It is true that one way of calculating the average profit from switching from the X envelope to the Y envelope indicates that profit to be positive:
The average profit from switching from the X envelope to the Y envelope, over those particular cases where the X envelope contains $1, is 25 cents.
The average profit from switching from the X envelope to the Y envelope, over those particular cases where the X envelope contains $40, is $10.
The average profit from switching from the X envelope to the Y envelope, over those particular cases where the X envelope contains $72, is $18.
Etc., etc.
The average profit from switching from the X envelope the Y envelope overall would appear therefore, by the law of total expectation (which just says you can take the average of a rectangle of values by first averaging each column, then averaging those results), to be the average of {25 cents, $10, $18, …}, which is a positive value, since there are no negatives in there. This would appear to indicate that the average profit from switching from the X envelope to the Y envelope is positive.
That is the paradox.
What step went wrong?
Implicit stipulation. It’s not an interesting problem without that assumption.
(And if you say “But that’s incompatible with the Kolmogorov axioms”, you’re not engaging with the problem. People can and did talk about probability in systems other than the Kolmogorov axioms; e.g., abandoning countably infinite additivity so as to allow a notion of a uniform distribution on the integers is no conceptually different from standardly abandoning uncountable additivity so as allow a notion of a uniform distribution on the real interval [0, 1])
So, you’re right that if we let “Which of X and Y is larger?” be non-independent from “What is X?”, then the calculation is incorrect and the resolution is in the fact that the weights to be used in E[Y | X] = something * 2X + something * X/2 aren’t 0.5 and 0.5 But that’s fairly clearly not the problem the OP is fundamentally interested in. The OP is interested in the problem in which learning the value of envelope 1 provides no information as to whether envelope 2 is larger or smaller.
Amy?
250 posts and we’ve arrived at the point where we started. Now that’s progress.
I mean, I can tell you what went “wrong”. I’m just trying to outline the paradox for the people who are attacking it for irrelevant reasons.
What goes wrong is that the average profit from switching from the X envelope to the Y envelope is like the average integer; it’s defined by one of those series that you can make add out to different values (even changing whether it’s positive, zero, or negative) depending on how you arrange it.
Normally, we expect that re-arranging the order in which we calculate a sum doesn’t change the outcome. But that’s only true if there isn’t an infinite large amount of positive and negative terms in the series. If there is an infinitely large amount of positive and negative terms in the series, they can be re-arranged to either cancel out exactly, or mismatch to come out to any other value you like, making the total sum sensitive to order-arranging (in other words, not well-defined).
That’s what happens with E[Y - X], just like it happens with E[Z] for a random integer Z. I’d like to outline this in simpler detail, but the above is the gloss of what happens.
(2X+5.X)/2 is telling you something about the relationship between X and Y.
If instead of M and 2M we had M and M+5, then you wouldn’t have (2X+.5X)/2 you would have ((X+5)+(X-5))/2 which is just X.
The formula is telling you more about the relationship between the values that can appear in the 2 envelopes than it is about what can be found in the other envelope.
I understand. I was just making a joke that after 250 posts in search of an answer, we’re still debating what exactly the question is.
I’m worried that this is going to be the point where you will lose me. I have to admit that your summation above did not resolve the problem for me. I’m not saying you’re wrong; I’m just saying I don’t see it.
Right, that is why I would like to outline the same point in simpler detail, when I have time to craft a really clear and easy to follow explanatory post that hopefully is a beacon of understandability. Later today, maybe.
But you do somewhat recognize the difficulty in figuring the average earnings of the “Earn Z dollars, where Z is a random integer” game, right? The integers are symmetric around 0, so you might think the average earnings are $0. But the integers are symmetric around 17, so you might think the average earnings are $17. Etc. Turns out, there isn’t a well-defined center of the integers. Which means there isn’t a well-defined average earnings for that game. What I want to eventually make clear is that it’s the same way with the expected profit from switching envelopes, but I’ll have to wait till later to give a better explanation.
(Or another way of putting it is that if I give you an infinite line and ask you to find the center, any point’s as good as any other; there’s no well-defined middle point of an infinite line. Anyway, more on this later)
Yes, I see your point on this. And I have heard of a variation of the two envelope paradox which involves the idea you’re explaining. But I don’t yet see how it applies to the two envelope paradox.
I’m loving this puzzle – although the thread has blown to 6 pages before I have had a chance to get into it. I still haven’t read it all. Apologies if some of my thoughts have been covered well already.
Here’s where I got to.
If M is the minimum envelope and 2M the maximum, then the expected value of the envelope is 3M/2. Happy and comfortable with that. In a real situation this would be my normal approach to analysing the situation. This suggests that the situation is symetrical and that there is nothing to be gained on average by switching envelopes. Seems common sense. Again I’m happy.
If X is the unknown value of the envelope in my hand then the possible values of the other envelope are X/2 and 2X with equal probability. This suggests that the expected value of the contents of the second envelope are 1.25X. I don’t see any error with this analysis but it is obviously the source of the paradox.
If the first envelope is opened, I don’t see any significant change to the above scenario. Suppose the opened envelope reveals $100. Then the other one is $50 or $200 with equal probability leading to an expected value for the game being $125.
Now I’m starting to get uncomfortable and unsure of my calculations. Should I switch? Should I be prepared to pay $125 for this game?
I love Indistinguishible’s typical clarity in this thread.
Firstly his stating that the situation is equivalent to weights placed on the lines Y=2X and Y=X/2 and determining the centre of mass of the system – every vertical line has the centre of mass above the line YX. Every horizontal line ha the centre of mass below the Y=X. I get that. Makes sense. (post 106 and 220)
Secondly his resolution of the paradox by pointing out that the expected value of either envelope is infinite or at least undefined. Therefore there is no difference between calculating the expected value of Y as 3M/2 or calculating it as 4M/3 (post 238) Again, I am happy with that when no information is given about the values in the envelopes.
In the situation when an envelope has been opened, I can’t see that this reasoning holds. However, the paradox still remains.
Thirdly the indeterminate nature of summation of the integers (post 202). I follow perfectly, but I’m not sure I fully grasp its relevance to this situation. (I probably missed reading something.)
So, getting back to practicalities – marrying all of this to some decision-making
Given one envelope that contains either $50 or $200 with 50% probability – I am happy to pay $125 for that game.
Given an open envelope containing $100 and the knowledge that the other contains either $50 or $200 with 50% probability I would make the switch. (Am I right to do so?)
Given two unopened envelopes one marked with an X, I can mathematically demonstrate that the expected value of either of them is 1.25 times the other. However, I cannot justify switching much less repeated switching due to the indeterminite nature of the problem.
(How am I doing so far?)
This causes me to doubt my decision when the $100 was revealed. Running a quick simulation lets me know that it is the correct decision – however weird that seems.
Mathematically, the grass is always greener on the other side of the fence! I’m not sure that I could actually convince someone of that though.
Now. Have I summed up adequately? What am I missing?
You appear to have summed the problem all up. Now I just need somebody to give me an answer I can follow.
The paradox seems to be that a situation which apparently should be completely symmetric somehow spontaneously appears to develop asymmetry. So what happened? Was the initial symmetry false? Was the subsequent asymmetry false? Or was there some point in the process where it switched from symmetry to asymmetry?
Up to #258 and OP is still unserved.
We agree to call this envelope’s contents X, the other’s Y.
Suppose we know that two envelope-pair settings are possible, (50 & 100) and (100 & 200). If we open and reveal X = 100, we know
E[Y] = $125 = 1.25 X
But, and this is a key point, if we see X = 50 or X = 200, we then have E[Y] = 100 = 2 X and E[Y] = 100 = 0.5 X respectively.
(Now, if X = 50 and X = 200 are equally likely, then E[Y/X] = 1.25, but this is just a red herring. I mention this in parentheses because AFAIK Mr. Nemo doesn’t fall for this trap.)
Now Nemo may be saying "Yes, Yes, but
E[Y] = .25 E[Y | X = 50] + .50 E[Y | X = 100] + .25 E[Y | X = 200]
Substituting from above this is
E[Y] = 100/4 + 125/2 + 100/4 = 112.25
But here this is equal to
E = 50/4 + 100/2 + 200/4 = 112.25
Above we wrote, for a specific probability example, E[Y | X=50] = 100 = 2 X
but please note that 100 = 2X is a “local” equality. Without the care taken above, substitution into the aggregating equation fails. (Is this your query, finally, why is the substitution leading to E[Y | any X] = E improper?)
Finally, perhaps Nemo is saying “Yes Yes, I see that asymmetries can arise, but shouldn’t we be in a particularly symmetric case?”
To this I reply that (X, X+10) is “more” symmetric than (X, 2 X) in the way Nemo hopes for (though improper probability arithmetic is still possible). Since it is seldom the case that both arithmetic and geometric means solve a problem, the fact that (X, X+10) has a desired symmetry suggests that (X, 2 X) will not !
Hope this helps.
At what point did we agree to all this? There was never any time when you were told how much money was being put in the envelopes, so why come up with a solution that’s based on information you don’t have? You might as well say you’re going to solve the liar’s paradox by assuming that Epimenides was actually from Rhodes.