Interesting Mathematical Paradox?

[PairTheBoard]

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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.


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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

[jason1990]

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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.

[PairTheBoard]

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

[jason1990]

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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.

[AWoodside]

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.

[jason1990]

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.
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[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]

[PairTheBoard]

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

[jason1990]

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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:
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[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.

[PairTheBoard]

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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

[jason1990]

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.