You are no longer talking about just probability anymore, since you can now sample the distribution you are now engaged in statistics, which is outside the scope of the OP — Jeremiah

It is rather difficult to divorce discussion of probabilities from discussion of statistics. You can't really build an insulating wall between those two disciplines. It doesn't make much sense to talk about probabilities of events that are singular, unique, occurrences not belonging to any sort of distribution. Probability distributions do have statistical properties. You just seem to want to outlaw, by fiat, arguments that make trouble for your case.

You don't know the distribution, you don't know the limits and you only get once chance to switch. — Jeremiah

It is rational to want to maximize your expectation even when you only get one single chance to play, and it is irrational to dismiss your expectation merely on the ground that just one of the possible outcomes will be realized.

Suppose you are forced to play Russian roulette once. There are two revolvers, one with five bullets in the cylinder (and one empty chamber) and the other one with one bullet (and five empty chambers). The revolvers are truthfully labelled accordingly. You are free to pick any one. Are you arguing that since you only are going to play once, it's irrelevant which revolver you choose?

Statistics is a data science and uses repeated random events to make inference about an unknown distribution. We don't have repeated random events, we have one event. Seems like a clear divide to me. You can't learn much of anything about an unknown distribution with just one event.

It is also rational to want ice-cream on a hot day. You still don't know anything about the distribution.

Statistics is a data science and uses repeated random events to make inference about an unknown distribution. We don't have repeated random events, we have one event. Seems like a clear divide to me. You can't learn much of anything about an unknown distribution with just one event. — Jeremiah

None of my arguments relied on being able to lean "much" about a distribution from one single observation. My arguments rather relied entirely on logical relations between definite claims about possible distributions and conditional probabilities.

You are guessing. Then trying to model your guesswork.

It is also rational to want ice-cream on a hot day. You still don't know anything about the distribution. You are speculating then trying to model your speculations. — Jeremiah

When I say that this can be inferred from that, or that it is fallacious to infer this from that, then I am either right or wrong about it; and in the case where you think I am wrong, an argument is forthcoming. None of my claims purport to be empirical or speculative (except when I explicitly hedged some as conjectures earlier in the thread).

I take the two-envelopes paradox to be a puzzle about probability theory and there is little point speculating rather than arguing logically about it. The use of models is perfectly fine for illustrative purposes, for conveying a concept across, or for supplying proofs of existence.

Hey, if you feel lucky then switch, if you think you are close to the cap don't, feel this one out, but you are not going to be able to justly quantify a positive gain based on the information we have.

Hey, if you feel lucky then switch, if you think you are close to the cap don't, feel this one out, but you are not going to be able to quantify a positive gain based on the information we have. — Jeremiah

This is something I have never disputed. I have never purported to offer an optimal strategy or suggested that there is any way to come up with one. The two-envelopes paradox is, precisely, a paradox because under some widespread interpretations of "don't know" (regarding the prior distribution, and the probability that the open envelope is the smallest one) there appears to be two equally valid arguments that purport to conclude that switching yields a positive expectation or that it yields a null expectation. Since those two conclusions are inconsistent, the resolution of the paradox calls into finding the flaw in (at least) one of the two arguments. Considerations of well defined strategies only are meant (by me) to illustrating flaws in the arguments that purport to lead to two inconsistent conclusions on the basis of a common set of assumptions regarding the possible initial distributions. A few other participants in this thread (such as JeffJo, fdrake and both Andrews) have offered diagnoses similar to mines of the most common mistakes that lead one to the erroneous and paradoxical conclusions.

As far as I am concerned I already found the flaw. Take it or leave it, that is your choice.

As far as I am concerned I already found the flaw. Take it or leave it, that is your choice. — Jeremiah

I have chosen a third option, which is to point out the logical flaw in your purported identification of "the flaw". As I suggested earlier, your own resolution of the paradox relies on an argument that proves too much, since it leads to wrong inferences about expectations in specific cases.

So you think you always have a 1.25 expected gain in every possible instance?

If you are using the principle of indifference then criticizing people for using the principle of indifference, that is hypocritical.

So you think you always have a 1.25 expected gain in every case? — Jeremiah

Not at all. I have rather argued that there is an 1.25X expected gain from switching in one specific case of a known distribution {{5,10},{10,20}} where, by your own argument, switching ought to be no better than sticking since we are ignorant of the case and, according to this argument, the cases therefore can only be treated separately and don't justify the 1.25*$10 expectation.

It is rational to want to maximize your expectation even when you only get one single chance to play, and it is irrational to dismiss your expectation merely on the ground that just one of the possible outcomes will be realized. — Pierre-Normand

So say you know that one envelope contains twice as much as the other and that there's £10 in your envelope (and that you don't know how the amounts are selected). What's the rational decision?

I would say that knowing that there's £10 in my envelope doesn't provide me with information that allows me to reassess the initial probability of 0.5 that I will pick the smaller envelope, and so I will say that there's a probability of 0.5 that I have picked the smaller envelope, and from that calculate an expected gain of £2.50 for switching.

You seem to be saying that after picking an envelope (whether opened or not?) I have to go from saying that there's a probability of 0.5 that I will pick the smaller envelope to saying that the probability is unknown that I have picked the smaller envelope.

But what action does your answer entail? Switching or sticking? If you say it doesn't matter, and so you're being indifferent, isn't that the same as treating it as equally likely that the other envelope contains the larger amount as the smaller amount? And if you're treating them as equally likely then isn't it rational to switch?

So I would say that if you have no reason to believe that the other envelope is more likely to contain the smaller amount then it's rational to switch (assuming, of course, that you can afford to lose).

If you are using the principle of indifference then criticizing people for using the principle of indifference, that is hypocritical. Either accept that as a standard starting point or don't — Jeremiah

What I criticized merely was a failure to draw a logical inference from one particular application of the principle of indifference. The logical inference that must be drawn from the assumption that the principle of indifference can be applied unconditionally on X (the value of the seen envelope) is that the prior distribution must therefore be assumed to be unbounded and uniform. If the player correctly draws this inference, then she can still apply the principle of indifference and expect to gain 0.25X (on average) from switching from her envelope (containing $X) to the other one and this expected gain doesn't yield a paradox since, in the case of such an unbounded distribution, however large X might be, it was infinitely unlikely that it be so small and, also, her average expectation from an always-switching strategy isn't any larger than her average expectation from an always-sticking strategy since both are infinite.

You seem to be saying that after picking an envelope I have to go from saying that there's a probability of 0.5 that I will pick the smaller envelope to saying that the probability is unknown that I have picked the smaller envelope. — Michael

If there exists some bounded and normalized (meaning that the probabilities add up to 1) prior probability distribution that represents your expectation for the possible distributions of envelope pairs then, in that case, your average raised expectation for an always-switching strategy is zero, for reasons that many have expounded in this thread. However, any such prior bounded probability distribution which might represent your expectation is inconsistent with your being able to apply the principle of indifference to whatever case of X that you might observe in the first envelope. Your knowledge (or assumption) of this prior distribution rather allows you to calculate exactly the exact expectation conditionally on any X, and this is generally different from 1.25X.

If, on the other hand, you take the uniform and unbounded distribution to represent your prior expectation, then, in that case, you can apply the principle of indifference whatever X it is that you might observe in the first envelope. Under this wild assumption of a uniform expectation, for any number M, however large, your prior expectation was that it was infinitely more likely that X would turn out to be larger than M rather than smaller or equal to M. So, it is no surprise that you are expecting a gain from switching. (You can refer back to my Hilbert Grand Hotel example, earlier in this thread, for another illustration of the consequences that follow from assuming such a wild unbounded and uniform 'uninformed' probability distribution.)

In the case where you are confident that there is some bounded, and therefore non-uniform, prior distribution of envelope pairs, but you don't have a clue how to go about estimating what it might look like, the mere assumption that there exists such an unknown prior distribution is enough to rule out the wild degenerate case described in the prior paragraph. When faced with some determinate amount X, you will possibly not have a clue what the expectation from switching might be. But that is no reason for assuming it to be zero. Hence, there is no reason either for assuming (or inferring) that the expectation from switching is exactly 1.25X. What you can reasonably assume, rather, is that the higher the value of X is, the more risky it is to make the switch, and this knowledge that the risk of a loss increases roughly in proportion with the value of X is inconsistent with the unconditional application of the principle of indifference.

But what action does your answer entail? Switching or sticking? If you say it doesn't matter, and so you're being indifferent, isn't that the same as treating it as equally likely that the other envelope contains the larger amount as the smaller amount? And if you're treating them as equally likely then isn't it rational to switch?

Yes, it is rational to switch if you are justified in treating them as equally likely. But if it is axiomatic that they are equally likely, as most statements of the two-envelopes paradox seem to make it, then you must also infer that the prior distribution is uniform and unbounded with all the weirdness that such an ill-defined probability distribution entails.

Yes, it is rational to switch if you are justified in treating them as equally likely. But if it is axiomatic that they are equally likely, as most statements of the two-envelopes paradox seem to make it, then you must also infer that the prior distribution is uniform and unbounded with all the weirdness that such an ill-defined probability distribution entails. — Pierre-Normand

So what's the rational decision if you know that the prior distribution isn't uniform and unbounded? There's £10 in your envelope. Should you stick or switch?

So what's the rational decision if you know that the prior distribution is isn't uniform and unbounded? There's £10 in your envelope. Should you stick or switch? — Michael

I am fine with acknowledging that there isn't any such thing as the rational decision to make in the vaguely specified case where you merely have a reasonable expectation that the amount of money can't be infinitely large but you don't have any precise idea how very high the distribution might be tailing off.

Suppose for instance that tomorrow morning you are being called to play this game with real money in the context of some scientific experiment conducted by the psychology department of your local university. You are to play only once and keep the money. Suppose you open your envelope and find $96 in it. Is it rational to expect 1.25*$96 from switching? I am not committed to saying this. What I am committed to say merely is that it is increasingly irrational to expect 1.25*X (or more) from switching as the value that you find in your envelope increases to ever lumpier sums.

I am fine with acknowledging that there isn't any such thing as the rational decision to make in the vaguely specified case where you merely have a reasonable expectation that the amount of money can't be infinitely large but you don't have any precise idea how very high the distribution might be tailing off. — Pierre-Normand

But this just seems to be saying that there's no reason to believe that it's more likely that the other envelope contains the smaller amount and no reason to believe that it's more likely that the other envelope contains the larger amount and so you're effectively treating each as equally likely, in which case it would be rational to switch.

So my takeaway is that if it isn't rational to stick then it's rational to switch. There's no middle ground where it's either/or.

But this just seems to be saying that there's no reason to believe that it's more likely that the other envelope contains the smaller amount and no reason to believe that it's more likely that the other envelope contains the larger amount and so you're effectively treating each case as equally likely, in which case it would be rational to switch. — Michael

Yes, that is roughly true for some mid-range values of X. See the second paragraph of my edited post for more discussion about real cases.

So my takeaway is that if it isn't rational to stick then it's rational to switch. — Michael

Sure, but it is one thing to say that it is rational (or isn't irrational) to switch when you find X in your envelope ($10,000 say) and it is another to say that it is rational to behave, whatever your personal utility curve might be, as if the expected value(*) from switching always is exactly $12,500. That is rationally unjustified.

(*) I mean expected value in the technical sense, as the long term expected average of the individual pay offs.

Suppose for instance that tomorrow morning you are being called to play this game with real money in the context of some scientific experiment conducted by the psychology department of your local university. You are to play only once and keep the money. Suppose you open your envelope and find $96 in it. Is it rational to expect 1.25*$96 from switching? I am not committed to saying this. What I am committed to say merely is that it is increasingly irrational to expect 1.25*X (or more) from switching as the value that you find in your envelope increases to ever lumpier sums. — Pierre-Normand

Sure, the practical limitations of real life play a role, but I wonder if such limitations go against the spirit of the problem. What if instead of money it's points, and the goal of the game is to earn the most points? There isn't really a limitation, except as to what can be written on paper, but with such things as Knuth's up-arrow notation, unfathomably large numbers like Graham's number aren't a problem.

Sure, the practical limitations of real life play a role, but I wonder if such limitations go against the spirit of the problem. What if instead of money it's points, and the goal of the game is to earn the most points? There isn't really a limit, except as to what can be written on paper, but with such things as Knuth's up-arrow notation, unfathomably large numbers like Graham's number aren't a problem. — Michael

The practical limitations indeed go against the spirit of the idealized two-envelopes problem. That's because if the prior distribution is bounded, albeit unknown, then the paradox doesn't arise. The average raised expectation for the unconditional always-switch strategy, always is zero. In order that the expectation be exactly 1.25X, whatever X, and therefore also, the always-switch strategy superior to the always-stick strategy, then there ought to be no bound to the prior distribution of possible envelope values, not just an unfathomably large albeit finite bound such a Graham's number. If the upper bound is Graham's number, then the average raised expectation from the always-switch strategy still is exactly zero.

The average raised expectation for the unconditional always-switch strategy, always is zero. In order that the expectation be exactly 1.25X, whatever X, and therefore also, the always-switch strategy superior to the always-stick strategy, then there ought to be no bound to the prior distribution of possible envelope values, not just an unfathomably large albeit finite bound such a Graham's number. If the upper bound is Graham's number, then the average raised expectation from the always-switch strategy still is exactly zero. — Pierre-Normand

I thought we were talking about the rational decision for a single play of the game, not the payout for always-switch and always-stick strategies for repeated games?

My argument is that given how arbitrarily large the numbers in the envelopes can be (using points rather than money), there isn't really a number at which one would consider it more likely that your envelope has the larger value. If my envelope is 10 then it's rational to switch. If it's 1,000 then it's rational too switch. If it's 10¹⁰⁰ then it's rational to switch.

My argument is that given how arbitrarily large the numbers in the envelopes can be (using points rather than money), there isn't really a point at which one would consider it more likely that your envelope has the larger value. If my envelope is 10 then it's rational to switch. If it's 1,000 then it's rational too switch. If it's 10100 then it's rational to switch. — Michael

If the distribution is somewhat uniform with an unfathomably large (albeit finite) upper bound, and you know this, then you can't generally expect to get such puny values. If you do, conditionally on that, then for sure, you ought to switch, and your expectation will be close to or exactly 1.25X. But what are you rationally to do when you get "average" values from the distribution, which are unfathomably large? Is it still rational to switch? On what ground? There will be no reason then to ground your decision of an (average) expectation of 1.25X. The average expectation from switching still will be zero.

Suppose the expectation is (close to) 1.25X for purpose of reductio. You are being dealt some unfathomably large amount X which is typical from the actual distribution. We suppose that you are generally warranted to switch on the basis of the principle of indifference, and thus the expectation that switching yields the expected value of (roughly) 1.25X. After you've switched, but before you are permitted to look at the content of the second envelope, you are being given to opportunity to switch back. Is it rational for you to switch back? By the very same argument that justified your initial switch, you should deduce that the expectation for switching back is roughly 1.25Y, where Y is the content of the second envelope. But that's an inconsistency.

But what are you rationally to do when you get "average" values from the distribution, which are unfathomably large? — Pierre-Normand

I have no way of knowing that my value is "average". Perhaps the 10¹⁰⁰ in my envelope is a puny value because the upper bound is Graham's number.

I have no way of knowing that my value is "average". Perhaps the 10^100 in my envelope is a puny value because the upper bound is Graham's number. — Michael

Sure, you will never know for sure that the value that you get is close to the top of the distribution. But the main point is that you will have no reason to apply the principle of indifference to justify your switching decision on the basis an expected gain of 1.25X. If you were so justified, then you would be equally justified to switch back, on the ground of the very same argument, before you even looked into the second envelope. That's a reductio of the claim that the expectation of switching is 1.25X.

Suppose the expectation is (close to) 1.25X for purpose of reductio. You are being dealt some unfathomably large amount X which is typical from the actual distribution. We suppose that you are generally warranted to switch on the basis of the principle of indifference, and thus the expectation that switching yields the expected value of (roughly) 1.25X. After you've switched, but before you are permitted to look at the content of the second envelope, you are being given to opportunity to switch back. Is it rational for you to switch back? By the very same argument that justified your initial switch, you should deduce that the expectation for switching back is roughly 1.25Y, where Y is the content of the second envelope. But that's an inconsistency. — Pierre-Normand

If I know there's £10 in my envelope then the expected value for switching is £12.50, and the expected value for switching back is £10.