If you don't use a random sampling method your event could become skewed by observational bias. I am sure there are cases to use weighted selection methods, but you need to justify their use, not just throw them out there just because. The goal is to mitigate observational bias; however, it can never be fully scrubbed. Still we need to try and let the distribution reveal itself instead of injecting it full of our opinions. We live in a house of subjectivity, and I just want to get as close to the objective as I can, even if that is an impossible task. Maybe you didn't pick up on it, but I use to have a significant interest in philosophy.

Why is equiprobability simple but other priors aren't? — fdrake

Equiprobability is unbiased.

The simplest explanation is that our gain/loss is either x or-x.

If you introduce a known distribution then you are making assumptions about the distribution. Classical or Bayesian, it does not matter, that is true for both. If you insist on defining it on these terms then why are we not adhering to the Law of Parsimony?

The whole distribution thing is faulty. I agree in the strictness sense of the definition there is an unknown distribution, but as far was we know it was whatever was in his pocket when he filled the envelopes. We can't select a distribution to use, as we have no way to check it. We don't know what the limits are and have no way to check that. So it is not that I disagree with making assumptions, it is making unnecessary assumptions; aka Occam's razor.

philosophy in general - which you regularly express contempt for. — fdrake

I picked up my contempt for philosophy from these forums many years ago. I was a member of the old forums.

As soon as you apply an expectation operator to the random variable in the envelope, you're already required to apply the context of long term behaviour and the independent repeatability of the trial. — fdrake

Which is why you need to stop and ask yourself, "Is this correct?" You need to make sure what you are doing is applicable. It is math, not magic, you need to make sure your approach fits.

if his assumptions are — fdrake

Talk about "charitable reading" . . . I think I have fully expressed that my main problem is with the assumptions.

As I said many, many times algreba is the correct tool to use here. Not Baysian or Classical statistics, but simple algreba. Simply because I talk about the stats doesn't mean I have changed my mind on that.

I really do not care about this Classical vs Baysian nonsense you all have going on in these forums. That mind set didn't even exist to me before you all started in about it. Which makes me constantly question what other bad habits I could pick up here.

We went from being totally uninformed at the start of the thread to the knowledge gained after thousands of iterations with predefined conditions never included in the OP. Did you really map the OP, or did you map your assumptions?

This is one of the reasons we need to constantly question, "Is this correct?"

It is an interesting idea in terms of exploring unknown distributions; however, no one has actually justified this approach with respects to the OP. People are making far too many assumptions that they have no way of confirmimg.

A "switching strategy" is essentially machine learning. It is also out of scope of the OP, there is no guarantee such methods are applicable.

The best way to sovle this is still the simple algreba, it introduces no new assumptions. However, if you are going to simulate this it should be done in a way were the distribution and bounds remain as unknowns, instead of being set beforehand, if that could be done than it could become a useful tool for mapping unknowns distributions.

It needs a simulation with a distribution but gives no new information of the distribution. Perhaps by scrambling the possible outcome than assigning random arbitrary labels. If the distribution is not known then you can't approximate it and get accurate results. You'd still have the problem of bounds though. Maybe scramble those as well.

A "switch strategy" is just a way to approximate the distribution through iterations. Or the transformed distribution.

I choose 10 billion.

I already ran my own simulations, Michael, and the idea that if this was a game you'd get many turns to learn the range is very unrealistic. This is what I am not interested in. I didn't get into statistics to map people subjective assumptions. I can do that without math.

Even if this was a game show, they are not going to give you 20,000 goes to figure out their range. That makes no sense. This is what I mean by being subjective. I rather remain objective and get a model that approaches reality.

Don't get me wrong, Michael, I found your linked article here an interesting read, but it moves beyond the scope of the problem in the OP, and all these assumptions are subjectively based.

Basing it on "previous games" is an assumption. That information was never in the OP.

No, I used the same scale in my code.

It is this:


You code is based off your assumptions.

If you are doing it based on the your X, X/2 sample space, you'll get your expected gain. I have no doubt about that. The math already proves that.

To simulate this we need agreed upon assumptions.

If you use a distribution you are making assumptions not included in the OP. I pointed this out before.

One thing we can be sure of from the OP is that one has X and one has 2X.

I like using simulations for empirical investigation; however, you and I, deeply disagree on the underlying assumptions and the code will reflect that if we are not careful.

Thank you, sorry for getting riled, I just hate going over someone else's code without notes.

You still need to notate your code. Especially on forums like these. I don't know why you don't do that. It almost makes it seem like you are trying to hide your assumptions in the code.


You are still using your incorrect sample space.

I am not going to look at any more code that is not notated. I mean that should just be common sense if you are going to post it here.

I only read R.

As I pointed out the thread is just going in circles.

I already commented on the assumptions that follow the use of distributions earlier in this thread. So basically we are back to once more modeling a pile of assumptions.

They are making a lot of assumptions. Conditions and information not included in the OP.

You should be willing to empirically justify your theory.

No amount of "math" will ever change the actual contents of the envelopes. Some of you are just pushing numbers around on the page, but math is not magic, it can't change the actual dollar values.

And I know I am just speaking in the wind at this point, but we could physically set this game up, using various dollar amounts and you'd never get the expected gain some here are predicting. You can't just think about the numbers, you have to think about the actual outcome.

Like making a bunch of assumptions based on Y.

This is why I became disillusioned with philosophy. I wanted real answers, not made up ones

In any science you should repeatedly stop and ask yourself, "Is this correct?"

Each probability space serves a different purpose. We can ask whether a probability space appears to help towards achieving its aim, but it makes no sense to ask whether a probability space is correct. — andrewk

It absolutely makes sense to ask if it is correct, and that should be the first question you ask yourself whenever you model something.

I will not further debate such specifics.

I read that part. In fact I read it several times.

I am already convinced my approach is correct, Michael. I have no doubt about it, and I no longer care about arguing or proving that point. You can interpret this article however you wish. If you learn something from it great; if you decide to think of it as personal validation that is your choice. I have been reading a number of papers on this problem with a number of different approaches and outcomes and I found this guy's approach interesting. And I think it is wise if I keep my thoughts about Andrewk’s approach to myself.

You might find this interesting:

https://www.researchgate.net/publication/266706577_The_fallacy_of_the_two_envelopes_problem

, so this is pointless. — JeffJo

Funny how I already said something very much like this.