The envelope problem, and a possible solution

[BBB]

[ QUOTE ]
Hi, BBB. You said: "All we are told in the original problem is that one of the envelopes contains twice the other. It tells us nothing about the chances that the other envelope contains twice as much, and to simply assume that these chances are 50% is not correct."

Do you then imply that we choose an envelope in a non random way? Or do you think that those probabilities do not depend on the probability with which you choose the initial envelope? I would think that I have 50% chance op picking up N or 2N.

[/ QUOTE ]

NarobisDad,

You do have a 50% chance of initially picking up on N or 2N. And clearly, once you choose an envelope, there is a 50% chance that the other envelope contains N more than the one you chose, and 50% that it contains N less. If you do not gain or infer any usable information upon viewing the contents of your envelope, then there is still a 50% chance that the other envelope contains N more than yours, and a 50% chance that it contains N less. If you see $100, you now know that N is either $50 or $100. So argument 1 concludes: Since there's a 50% chance that the other envelope contains N more than mine (which is true whether N is $100 or $50), then there's a 50% chance that N contains $200, and a 50% chance that it contains $50. The first part of the preceding statement is correct, BUT THE CONCLUSION IS NOT VALID. The conclusion is only correct if N is equally likely to be $100 or $50 (which may or may not be the case).

If we have some basis of information on which to determine the probability that N is $100 versus the probability that it is $50, for example using what we know in general about how people such as our benefactor might be willing to put in envelopes and give away, it turns out that we should clearly switch if we determine that the probability that N is $100 is more than half the probability that N is $50, and we should clearly not switch if N is more that twice as likely to be $50 as it is to be $100. But to simply guess that N is just as likely to be $100 as it is to be $50 and going from there is totally baseless and meaningless. That would be like if I told you I had a coin in my pocket that was not necessarily fair and I asked you what were the chances that it would come up heads if I flipped it. Unless you had some kind of information on which to determine what the coin might be like, it would be totally meaningless for you to gess 1/2 just because that would be the answer if the coin were fair.

[AaronBrown]

No, I'm saying something different.

The actual envelope problem doesn't come up a lot. The point of this paradox is not to figure out what to do if it happens, it's to show you that two common reasoning devices can conflict, so you have to be careful using them.

For example, one common argument for "resolving" the envelope paradox starts by assuming you have a consistent prior distribution on amounts that might be in the envelope, then demonstrating that if you always switch, that distribution must have infinite expectation. If you expect infinity, any finite amount will be a disappointment, so you switch.

That's fine, but in reality we don't have consistent prior beliefs about everything that might happen. Assuming we do in this case is hiding from the paradox, not resolving it.

We often have problems of the sort, do I stop now or try for more at the risk of losing what I have? We rarely have consistent prior beliefs that would give us a full Bayesian solution. But we still have to decide.

Consider Maurice Kratchik's original formulation that started all of this. He knows nothing about fashion or style. He's wearing a necktie and another fashion-impaired nerd offers him a bet. They'll show their neckties to a style expert, and the with the worse necktie will win both ties. He reasons, "I will either lose my necktie or win a better tie, my chances of winning are equal, so I should take the bet." Of course, the other person can reason the same way.

Now where is the consistent set of Bayesian beliefs? Is there some "tie quality" that would make you take the bet or not? Is it entirely obvious that this bet cannot make both people better off? Does it imply infinite expectations to take the bet?

If we didn't have the paradox to teach us, most of us would unhesitatingly agree that (a) when you pick one of two envelopes with different amounts at random, there is a 50% chance that you'll get the one with the higher amount, and (b) if you have a 50/50 chance to win $2 or lose $1 you have positive expected EV. The fact that these "obvious" statements can conflict is important. Deciding which one is more reliable in the envelope example is not important, unless you get offered a lot of bets about amounts in envelopes.

[PairTheBoard]

I'm not sure why your "No" was in reply to my post Aaron.

[ QUOTE ]
For example, one common argument for "resolving" the envelope paradox starts by assuming you have a consistent prior distribution on amounts that might be in the envelope, then demonstrating that if you always switch, that distribution must have infinite expectation.

[/ QUOTE ]

I think you misspoke here. I don't think this makes sense. I do believe it has been shown that if the assumed prior distribution has finite expectation then always switching is EV neutral. Also, if the prior distribution does not have finite expectation then it's possible for Always Switching to produce infinite expectations. But then so does Never Switching. Even then, One infinity does not improve over the other.

[ QUOTE ]
That's fine, but in reality we don't have consistent prior beliefs about everything that might happen. Assuming we do in this case is hiding from the paradox, not resolving it.


[/ QUOTE ]

I have no idea what you mean by "consistent prior beliefs".
However, assuming a prior distribution for the envelopes to have finite expectation is probably pretty safe since there's only a finite amount of money in the world.



[ QUOTE ]
"I will either lose my necktie or win a better tie, my chances of winning are equal, so I should take the bet." Of course, the other person can reason the same way.


[/ QUOTE ]

Or they might reason, "I will either win the better tie which I don't have or lose the better tie that I do have."

You also give an example in another post of betting $1000 against 500 pounds that the dollar will - I believe - drop against the pound. You claim the EV is positive for both parties. The key here is that you are making computations based on a variable unit measure.

[ QUOTE ]
If we didn't have the paradox to teach us, most of us would unhesitatingly agree that (a) when you pick one of two envelopes with different amounts at random, there is a 50% chance that you'll get the one with the higher amount, and (b) if you have a 50/50 chance to win $2 or lose $1 you have positive expected EV. The fact that these "obvious" statements can conflict is important.

[/ QUOTE ]

I don't think these statements are in "conflict". If you pick an envelope (a) is true. (b) is true on it's own merits. But if you open an envelope and see $2 what's not true is that you have a 50-50 chance to win $2 or lose $1. That statement is not implied by the previous one. It's not a matter of conflicting modes of reason or chosing which is more reliable. It's a matter of being careful about what implies what.


The point I would drive home here is that when you talk of probablities and expectations, think of what experiment or game you could be repeating under identical conditions whereby you could see the probabilties/expectations prove out over numerous trials. This is why I gave the most simple example where the prior distribution for the envelopes is simply the delta distribution. The conditions that brought about the envelope amounts were predetermined and Fixed. If we want to talk probablities and expectations on that basis we can do so by using the same envelope amounts and letting numerous people play the game, none of whom know the contents. In that scenario it's easy to see how the paradox evaporates.

PairTheBoard

[AaronBrown]

For your first point, these are two different ways of saying the same thing, but with an essential difference of order. I say the common argument is "if you always switch, that distribution must have infinite expectation." You say "if the assumed prior distribution has finite expectation, then always switching is EV neutral."

The difference is I start my summary of this argument with the assumption of a consistent prior distribution, which I object to. You start with "it has been shown" and bury the word "assumed" as an adjective in the middle.

This gets to your second point. We agree (I think) that if you have consistent prior beliefs about everything, as required for Bayesian statistical calculations, that either switching is EV neutral or you have infinite prior expectation. It's not clear that infinite prior expectation is irrational, despite there being only a finite amount of money in the world. For one thing, maybe there's an infinite amount in the universe. For another, you don't an infnite realization to support an infinite expectation.

But that's theory stuff, my concern is that people don't have consistent prior beliefs. Bayesian statistics achieves consistency by ignoring things that are important to real decision-making. What is a young person's prior about his lifetime income or amount of marital happiness? Isn't the choice of going to college or dumping a girlfriend somewhat like the envelope choice? Do you think understanding the Bayesian "resolution" of the paradox will help him decide?

Sometimes our best information about possible outcomes comes from the first choice we are offered. We have no idea what things cost until someone offers us something. We instinctively turn it down, because we realize the chance of the first offer being the best is pretty low. We get a few more offers, until someone offers us something that appears to be a low price relative to what we've been seeing from the others. Isn't this a pretty good description of how you make some decisions? Would you be surprised if I proved that it shows you have an infinite prior expectation?

Yes, the USD/GBP example involves two people measuring expectation in different units. I think this is very common in real decisions. Even if we are both betting money, if our credit ratings are different, then we are not computing in the same units. And since the money is presumably a means to some end, if there is price uncertainty about what we plan to spend it on, there's also a difference.

I don't say (a) and (b) imply each other, I say that if you assume both, you get the envelope paradox. The Bayesian resolution rejects (a). It says that after seeing the amount in the envelope you pick you have to change your estimate of the probability it is the higher amount. Bayesians define probability as subjective belief, and subjective belief can change even when objective facts don't (looking at the amount in the envelope can't change whether it is or isn't the higher amount). That's fine for feeling intellectually consistent, but rejecting (a) puts you at a big disadvantage when faced with practical choices. It's a very useful principle and refusing to use it on the grounds of theory is foolish. But the paradox demonstrates that it's not universally reliable, which is why the paradox is valuable.

The other common "resolution" of the paradox rejects (b). It insists on zero EV for a zero sum exchange. Before you look at your envelope the EV of exchange is clearly zero, objectivists don't believe looking at the amount can change the EV. What they give up is the idea that $1 is always $1. $1 before you look at the envelope is something different from $1 afterwards. Again, they've jettisoned some important common sense in order to get to the result they know is correct.

The Bayesians get consistency, at the cost of making some stupid decisions, the objectivists get the right decisions, at the cost of consistency. But both of them have to torture reality in the process. Both ways of thinking are useful sometimes, but we have the envelope process to remind us how shaky their philosophic foundations are.

[PairTheBoard]

[ QUOTE ]
For your first point, these are two different ways of saying the same thing, but with an essential difference of order. I say the common argument is "if you always switch, that distribution must have infinite expectation." You say "if the assumed prior distribution has finite expectation, then always switching is EV neutral."


[/ QUOTE ]

And I insist that my statement makes sense while yours does not. You say, "if you always switch, that distribution must have infinite expectation". That's simply not true. The prior distribution can have finite expectation AND you can always switch. You have mispoken. Your statement is incomplete. I suppose I should guess that what you mean to say is, "If a prior distribution is assumed and it's assumed that always switching is not EV neutral then the prior distribution cannot have finite expectation". That statement is implied by mine. I agree the form is different. I don't think the difference is essential.

I agree this is one approach to resolving the paradox - whether the person taking it adopts your statement of the fact or mine. It speaks to one of the hidden assumptions people mistakenly make when they think that envelope amounts can be chosen from a uniform distribution on an infinite scale of money. It's really an elaboration on that point and thus succeeds in breaking down the paradox at that false premise.

However, I don't think it gets to the heart of the paradox. One of the best explanations I've seen for this was made by BBB when he said,

[ QUOTE ]
BBB -
If we have some basis of information on which to determine the probability that N is $100 versus the probability that it is $50, for example using what we know in general about how people such as our benefactor might be willing to put in envelopes and give away, it turns out that we should clearly switch if we determine that the probability that N is $100 is more than half the probability that N is $50, and we should clearly not switch if N is more that twice as likely to be $50 as it is to be $100. But to simply guess that N is just as likely to be $100 as it is to be $50 and going from there is totally baseless and meaningless. That would be like if I told you I had a coin in my pocket that was not necessarily fair and I asked you what were the chances that it would come up heads if I flipped it. Unless you had some kind of information on which to determine what the coin might be like, it would be totally meaningless for you to gess 1/2 just because that would be the answer if the coin were fair.

[/ QUOTE ]


You said,
[ QUOTE ]
This gets to your second point. We agree (I think) that if you have consistent prior beliefs about everything, as required for Bayesian statistical calculations, that either switching is EV neutral or you have infinite prior expectation. It's not clear that infinite prior expectation is irrational, despite there being only a finite amount of money in the world. For one thing, maybe there's an infinite amount in the universe. For another, you don't an infnite realization to support an infinite expectation.


[/ QUOTE ]

I still don't know what you mean by "consistent prior beliefs about everything". You have yet to explain this. It may be that you are not dealing in the realm of mathematics. If that's the case you should be more clear about where you depart from mathematics and explain more precisely just what you are saying in whatever realm you are dealing. I believe there is some controversy about certain so called "Baysian" approaches which lack sound mathematical foundations. I suppose if this is where you're at there's not much basis for argument with you. We will have to take your pronouncements on authority.


[ QUOTE ]
But that's theory stuff, my concern is that people don't have consistent prior beliefs. Bayesian statistics achieves consistency by ignoring things that are important to real decision-making. What is a young person's prior about his lifetime income or amount of marital happiness? Isn't the choice of going to college or dumping a girlfriend somewhat like the envelope choice? Do you think understanding the Bayesian "resolution" of the paradox will help him decide?


[/ QUOTE ]

"Isn't the choice of going to college or dumping a girlfriend somewhat like the envelope choice?"

I see no relationship between the two.

[ QUOTE ]
Sometimes our best information about possible outcomes comes from the first choice we are offered. We have no idea what things cost until someone offers us something. We instinctively turn it down, because we realize the chance of the first offer being the best is pretty low. We get a few more offers, until someone offers us something that appears to be a low price relative to what we've been seeing from the others. Isn't this a pretty good description of how you make some decisions? Would you be surprised if I proved that it shows you have an infinite prior expectation?


[/ QUOTE ]

You would have to explain your model more precisely for us to see its implications. I don't see how it relates to the envelope paradox.


[ QUOTE ]
Yes, the USD/GBP example involves two people measuring expectation in different units. I think this is very common in real decisions. Even if we are both betting money, if our credit ratings are different, then we are not computing in the same units. And since the money is presumably a means to some end, if there is price uncertainty about what we plan to spend it on, there's also a difference.


[/ QUOTE ]

Maybe people do this. But I don't see how the observation that people often do bad math relates to the envelope paradox.

[ QUOTE ]
I don't say (a) and (b) imply each other, I say that if you assume both, you get the envelope paradox.

[/ QUOTE ]

Let's be clear here. Your statement of (a) and (b) are

(a) when you pick one of two envelopes with different amounts at random, there is a 50% chance that you'll get the one with the higher amount

(b) if you have a 50/50 chance to win $2 or lose $1 you have positive expected EV.

Now you say, "if you assume both, you get the envelope paradox"

There's no need to assume (b). (b) is simply a true statement. Also, if by (a) you mean that if you chose one of the two closed envelopes there's a 50-50 chance it will have the larger amount then that is simply a true statement as well. It's you who are missing the heart of the paradox. It's right here. It's when someone opens an envelope, sees $2 in it and says that (a) implies the premise of (b) that they are making a mistake.

Instead, you remain unclear as to what you mean by (a) so that you can conclude,

[ QUOTE ]
The Bayesian resolution rejects (a). It says that after seeing the amount in the envelope you pick you have to change your estimate of the probability it is the higher amount.

[/ QUOTE ]

There's no need to reject (a) unless you were unclear as to what it meant. If you are clear with (a) and mean that if you chose one of the two closed envelopes there's a 50-50 chance it will have the larger amount then there is a repeatable experiment you have in mind that can prove out this statement of probabilty. The experiment is for numerous people to pick one of the two envelopes at random. The more people who pick, the closer you will see half picking the higher amount and half the lower amount. There's a 50-50 chance that any one of them will pick the higher amount.

Consider this 2 envelope game. One envelope contains a dollar and the other envelope contains nothing. Again you have an
(a) when you pick one of two closed envelopes at random, there is a 50% chance that you'll get the one with the dollar

and a
(b) The expectation for a person picking an envelope at random is 50 cents.

Now a person picks an envelope, opens it sees it contains nothing and is given the option of switching. (a) remains true as clearly stated. (b) remains true. There's no deep philisophical conflict here between subjective Baysians and Objectivists. There's simply elementary conditional probabilty saying that once seeing the content of the first envelope the probablity for the contents of the second changes from the apriori probabilty. The exact same thing happens in the Original Two Envelope Problem. It's just not as obvious because people tend to percieve all envelope amounts as being equally likely - which we know they are not under any apriori conditions.


PairTheBoard

[AaronBrown]

I think we each have stated our positions as clearly as we can, and still think the other misunderstands us. Not much point in going forward, but I will pick out a few sentences that demonstrate our disagreement.

You seem to treat this as a paradox designed by people who have envelopes and don't know whether or not to switch. It is in fact a paradox designed to demonstrate that common statistical reasoning can lead to contradictory conclusions.

[ QUOTE ]
I still don't know what you mean by "consistent prior beliefs about everything".

[/ QUOTE ]
Bayesian statistics defines probability as subjective belief. It assumes you can always determine a prior belief before the experiment (in this case, before the envelope is opened). The information in the experiment (in this case, the amount in the envelope) adjusts that belief to the posterior distribution. To a Bayesian, showing that no consistent prior distribution justifies switching regardless of the amount you observe, shows that always switching is irrational. My objection is that in real decisions, people often don't have consistent prior distributions.

[ QUOTE ]
I don't see how the observation that people often do bad math relates to the envelope paradox.

[/ QUOTE ]
It's not bad math not to have a consistent prior distribution. It's impossible to have consistent beliefs about everything. It's not bad math to compute expected value in different units from someone else. Everyone has different units if you look carefully enough.

[ QUOTE ]
There's no need to assume (b). (b) is simply a true statement. Also, if by (a) you mean that if you chose one of the two closed envelopes there's a 50-50 chance it will have the larger amount then that is simply a true statement as well.

[/ QUOTE ]
You insist both statements are true and regard them as not only beyond argument, they are so obvious, they are beyond the need to state as assumptions. The paradox was designed to teach you they cannot be both true all the time.

There are senses in which each of them are true, but they are different senses.

[ QUOTE ]
if you chose one of the two closed envelopes there's a 50-50 chance it will have the larger amount then there is a repeatable experiment you have in mind that can prove out this statement of probabilty. The experiment is for numerous people to pick one of the two envelopes at random. The more people who pick, the closer you will see half picking the higher amount and half the lower amount. There's a 50-50 chance that any one of them will pick the higher amount.

[/ QUOTE ]
This is the frequentist argument. If that's your definition of probability, then (b) is not always true. This is the criticism that led to Bayesians to reject that definition of probability. Bayesians achieve consistency by rejecting (a) instead.

I understand the desire to make both (a) and (b) always true, but no one has discovered a way to do it.

When you have nothing in one of the envelopes, (b) no longer applies. It only covers cases of $1 and $2, not $1 and $0. You can't reconstruct the paradox in this case.

[PairTheBoard]

[ QUOTE ]
You seem to treat this as a paradox designed by people who have envelopes and don't know whether or not to switch.

[/ QUOTE ]

Not at all.

[ QUOTE ]
It is in fact a paradox designed to demonstrate that common statistical reasoning can lead to contradictory conclusions.


[/ QUOTE ]

I would say it's designed to show the need to be careful when applying mathematical tools like expected value and probabilties. When you speak in those terms you need to have a clear mathematical model in mind to which you are applying them.

[ QUOTE ]
Bayesian statistics defines probability as subjective belief.

[/ QUOTE ]

This is why Mathematical Probabilists and Statisticians don't attend conferences of such so called Bayesians. I would define probability as a branch of mathematics and expect the full rigor of mathematics in its application. There's nothing subjective in such rigor.

[ QUOTE ]
To a Bayesian, showing that no consistent prior distribution justifies switching regardless of the amount you observe, shows that always switching is irrational. My objection is that in real decisions, people often don't have consistent prior distributions.


[/ QUOTE ]

I really don't know much about the Bayesians. My impression is that they sort of create a prior distribution out of thin air, do sampling, then apply actual mathematical Bayesian techniques to adjust the original distribution depending on the sample. They then claim to have evidence for the real original distribution when what they really have are conclusions based on their original assumptions.

If that's the case I don't blame you for having a bone to pick with them. However I don't think it's central to the two Envelope Paradox. In the 2E paradox people are simply not being careful in how they apply the mathematical tools of probabilty and expectation.

However, if you'd like to explain more fully and clearly what Bayesians do and give examples of real life situations where you would disagree with their techniques due to "no consistent prior distribution" - whatever that means - I'd be interested in reading it.

[ QUOTE ]
It's not bad math to compute expected value in different units from someone else. Everyone has different units if you look carefully enough.


[/ QUOTE ]

You misquoted me here. It was "variable" units I objected to. Computing expected value with a unit of measure that changes over time and treating it as if it's fixed. That's not only bad math but presenting conclusions from such a calculation to make your case is downright misleading.

[ QUOTE ]
"PTB -
There's no need to assume (b). (b) is simply a true statement. Also, if by (a) you mean that if you chose one of the two closed envelopes there's a 50-50 chance it will have the larger amount then that is simply a true statement as well."

You insist both statements are true and regard them as not only beyond argument, they are so obvious, they are beyond the need to state as assumptions. The paradox was designed to teach you they cannot be both true all the time.


[/ QUOTE ]

Not at all. The paradox was designed to force you to make the statements in a clear way and to understand what you mean by them. When you do so they are both true as I showed in my last post. The paradox forces you to see where you mistakenly use them to jump to a false conclusion as I also showed in that post. If you are going to quote me, quote this rather than the phrase above out of context.

[ QUOTE ]
PTB -
" if you chose one of the two closed envelopes there's a 50-50 chance it will have the larger amount then there is a repeatable experiment you have in mind that can prove out this statement of probabilty. The experiment is for numerous people to pick one of the two envelopes at random. The more people who pick, the closer you will see half picking the higher amount and half the lower amount. There's a 50-50 chance that any one of them will pick the higher amount."


This is the frequentist argument. If that's your definition of probability, then (b) is not always true. This is the criticism that led to Bayesians to reject that definition of probability. Bayesians achieve consistency by rejecting (a) instead.


[/ QUOTE ]

I am not defining probabilty by way of frequencies. Probabilty has rigorous mathematical definitions. However, out of those definitions comes the law of large numbers which produces such frequencies. This is mathematical probabilty which is what we are usually talking about in this forum. If the Bayesians work with something else they call subjective probability then you need to clearly explain what they do and why it matters.

You say, "This is the frequentist argument. If that's your definition of probability, then (b) is not always true."

Yet look at (b) as you stated it,

(b) if you have a 50/50 chance to win $2 or lose $1 you have positive expected EV.

I challenge you to show this as not true under a standard mathematical definition of probabilty - where the law of large numbers - aka frequency - holds.

You say, "This is the criticism that led to Bayesians to reject that definition of probability. Bayesians achieve consistency by rejecting (a) instead."

Then what is a Bayesian definition of a probabilty space? I can give you a standard mathematical definition of a probabilty space for which the law of large numbers - aka frequencies - will hold. Having rejected this, What is the Bayesian definition of a probabilty space? Do they actually have something they can do math on?


[ QUOTE ]
I understand the desire to make both (a) and (b) always true, but no one has discovered a way to do it.


[/ QUOTE ]

I think I have done it simply by stating them clearly and understanding what they mean. I agree though that there's no way that all misinterpretations of them can be made to be true.

[ QUOTE ]
When you have nothing in one of the envelopes, (b) no longer applies. It only covers cases of $1 and $2, not $1 and $0. You can't reconstruct the paradox in this case.

[/ QUOTE ]

You missed my point here. If the Bayesians reject (a) on principle then they should reject it in this case as well. Yet it's clear there's no good reason to reject a clearly stated and well understood (a) that says all participants choosing one of the two envelopes at random have a 50-50 chance of choosing the $1 envelope. Just because Mr Lucky opens an envelope, sees it's empty, and now has a better than 50% chance of getting the $1 envelope if he switches there's no good reason to reject the statement (a) that future participants have a 50-50 chance of choosing a $1 envelope. Yet the Basesians should do so according to you on the same principle they rejected it in the 2E case, because (a) can't always be true? Because (a) is no longer true for Mr Lucky? Right. Because (a) can no longer be true when misconstrued.


PairTheBoard

[punter11235]

AAron, I would love to see you explaining more about "bayesians" point of view here. For now I agree with PTB post and my impressions are that your position on that matter is inconsistent in several places.

Best wishes

[NaobisDad]

Dear, PTB. You state that the place where argument 1 breaks down is at the part where it assumes 50% probablility. However, unless the probabilities now take on specific values, switching will still be either +EV or -EV.

If this is true, then that leaves a bit of a strange situation. I could switch, and then after I did, switch again on the basis of the exact same reasoning, and then again, and again, and in the end I'd have to be rich.

This is basically stating what has been stated before, argument 1 would create +EV situations from both ends of the switch.

How do you reason your perspective deals with this problem?

@BBB, thank you as well for your elaborate explanation. As far as I can see my reply to PTB is also valid as a response to your post and I ask you the same thing.
If I missed something in your post that would show that the above does not relate well to your story, please let me know.

[jason1990]

Practicing Bayesian statisticians use the same rigorous mathematics (in particular, the same definition of a probability space) as we probabilists do. Here's a toy example inspired by poker.

Let m be my winrate in BB/100 and s my standard deviation. I could model the outcome of a single 100 hand session by

X = m + sN,

where N is a Normal(0,1). Suppose I know that s=15, but I don't know m. However, I have "beliefs" about m. I believe that m=2. Of course, I'm not certain of it, it could be higher or lower. But I am fairly certain that I am at least a winning player. I might decide to model these beliefs by saying that

m = 2 + M,

where M is a Normal(0,1), independent of N. In "reality", m is a fixed, deterministic number, but I am representing it as a random variable in order to model my beliefs. Similarly, in the envelope paradox, the amounts in the envelopes are fixed, deterministic values, but we may choose to represent them as random variables in order to model our beliefs about the contents of those envelopes.

Now suppose I play a session of poker and lose 100 BB. I now need to adjust my "beliefs". This, of course, is analogous to the act of opening one of those envelopes. The adjustment is done by computing the conditional law of m, given X, and plugging in the observed value X=-100. We get the conditional CDF by computing

P(m <= x | X).

Upon computing this expression and differentiating, we find that the conditional law of m is again normal. According to my hasty calculations, its mean is

(225/226)*2 + (1/226)*X

and its variance is simply 225/226 (regardless of the value of X). Plugging in X=-100, our updated "belief" about our winrate is that it is 350/226=1.55 and our "belief" has a standard deviation of about 0.998.

Obviously, the process gets more complicated when we try to apply it to a sequence of sessions, and we might find it useful to apply the mathematical tools of filtering theory. One aspect of this that I find interesting, as it relates to poker, is that you can get analogs of confidence intervals which appear much more realistic over a smaller sample size. Of course, those estimates always contain some residue of our original "beliefs". For what it's worth, that residue goes away in the limit as the sample size increases.