That is what I just did. The envelopes cannot be in both cases at once, therefore it makes no sense to hedge your expections that both cases are possible. — Jeremiah

The argument that purports to show that the expectation from switching is 1.25X doesn't rely on both possible cases (possible consistently with the information that is available to you, that is) being actual at once. It only relies on them being equiprobable; or both equally likely to be true, consistently with everything that you know.

And I am saying that doesn't really matter because it will always be amount A and amount B. — Jeremiah

It doesn't really matter for what? It does matter for invalidating the fallacious argument that purports to show that your expected gain from switching, conditionally on having initially opened an envelope with the determinate amount X in it, is 1.25X.

See that was easy. — Jeremiah

Sure, but that's not what the equiprobability assumption is. What I have been referring to as the equiprobability assumption is the assumption that your credence in having picked the smallest envelope, which is 1/2 before you open it, remains 1/2 conditionally on there being the determinate amount X in it for any X. This is an assumption that can only be reasonably held (if at all) if the distribution of possible envelope contents is assumed to be unbounded and uniform.

I have two envelopes, one with amount A and one with amount B. I flip a fair coin to choose one. What is my chance of getting B? — Jeremiah

1/2

I think we are safe, I doubt anything will blow up. — Jeremiah

Things have blown up long ago. It is precisely the endemic oversight of the logical dependency at issue that is the source of the apparent paradox being presented in the OP. There is an illicit move from an assumption of equiprobability regarding the conditional probability of one having picked the smallest envelope (conditional on X, whatever X one might pick) to the assumption that there might be a bounded distribution of possible envelope pairs that is merely unknown. Those two assumptions are logically inconsistent. Either the unknown distribution isn't (as it indeed can't be, in realistic cases) uniform and unbounded or the equiprobability assumption is true. But if the equiprobability assumption is true, then the initial distribution for possible contents of the smallest envelope in each possible pair must be unbounded and uniform.

It absolutely can be ignored. — Jeremiah

To ignore logical dependencies between claims in rational arguments is a recipe for disaster.

The filling of the envelopes and the selecting of the envelopes are two separate events. — Jeremiah

I know that. But assumptions regarding the method for filling up the possible envelope pairs (and hence their distribution) entail logical consequences for the conditional(*) expectations of one having picked either the smallest or the largest one within one given pair. This dependency relation between the two successive events can't be ignored.

(*) Conditional on the observed value of the first envelope, that is.

Never said it was. — Jeremiah

OK. Once one is reminded that such an 'uninformed' prior, regarding the initial possible contents of the envelopes, isn't reasonable, then, it follows that an uninformed prior regarding the conditional probability that one has picked the smallest envelope, conditional on its value being X, is likewise unreasonable for at least some value of X. And that's because an unconditional uninformed prior of 50% (valid for any observed X value) entails a uniform and unbounded distribution for the possible envelope contents.

If a loaded coin flips H 9 out 10 times, without that knowledge, an uninformative of 50/50 prior is completely justified. — Jeremiah

Yes, for sure, but if someone hands me, as a gift, an envelope containing some unknown amount of dollars for me to keep, an 'uninformative' prior that is a uniform distribution (or continuous probability density function) from zero to infinity isn't reasonable for me to use for almost any purpose.

In this case, it's the fact that the Hotel has countably infinitely many rooms that enables the assumption of equiprobability to hold. — Pierre-Normand

I was rather careless in this post where I had made use of Hilbert's Grand Hotel for illustrative purpose. In my thought experiment the equiprobability assumption didn't actually hold because, after the guests have moved to their new rooms, they can often deduce from the room number they end up in where it is that they came from. Initially, the guests are equally distributed in the rooms numbered 2,4,6, ... Then, after flipping a coin and moving, accordingly, from room X to either room Y = X/2 or 2*X, there are three cases to consider. They may end up (case A) in a room within the range (1,3,5,...); (case B) within the range (2,6,10,...); or, (case C) within the range (4,8,12,...).

In case A, the players can deduce from the number Y of the room where they moved being odd that they moved there from room 2*Y and hence will want to move back if given the opportunity.

In case B also, they can deduce the same thing since, although Y/2 is an integer, it is an odd number and hence not in the initial range. They will also want to move back if given the opportunity.

Only in case C will they not be able to deduce where it is that they came from. About half of them, on average, will have moved to room Y from room Y/2 and the other half from room 2*Y. Their expectation for moving back to the room where they came from, conditional on them being in room Y, is 1.25Y (whereas, in cases A and B, the guaranteed outcome of moving back was 2Y)

So, even though the expectation that any guest who starts up in room X, from the initial range (2,4,8,...), is 1.25X after 'switching', the conditional expectations for moving back from Y to the initial room is either 2Y or 1.25Y. Rational (and greedy) guests should always accept the option to move, and then, wherever they end up, provided only that they don't know where they came from, they also should always accept the option to move back. This only makes sense in Hilbert's Grand Hotel, though, and not in any hotel that has a merely finite number of rooms.

I have a feeling though that Michael will still think that absent knowledge of the distribution, he can turn back to 50% as an assumption. — Srap Tasmaner

The best way to counter that assumption, it seems to me, just is to remind oneself that even though one may not know what the distribution is, assuming only that it is not uniform and unbounded, then, whatever this unknown distribution might be, the raised expectation from switching always is zero. So, it doesn't matter that the actual distribution is unknown. It's still known that it must be such as to make the raised expectation from switching zero, for the reason illustrated by @Andrew M in one specific case.

On edit: "the raised expectation from switching" is meant to refer to the average expectation of the always-switching strategy. The specific expectation of switching, conditionally on having found some determinate amount X in the first envelope, can only be guessed and can't be assumed to be zero.

I recall they did. Then there was a very strong rumour that Snr. helped draft a statement about the meeting after the news of it broke, whilst on Air Force One. I think that is one of the subjects of the 'obstruction of justice' part of the investigation. — Wayfarer

If you are talking about Don Jr's statement about the meeting (being about adoption, etc.), then it's more than a strong rumor that Donald J. Trump helped draft it. Trump's legal team acknowledged that he dictated it to Don Jr. This admission came after repeated denials that Trump Sr had anything to do about it.

I think I'm just reluctant to see the simple situation of choosing between two envelopes of different values in terms of the strange behavior of infinity. — Srap Tasmaner

Yes, it is indeed the strange behavior of infinity that generates the paradox.

(Edit: Opps, there are some mistakes below that I'll correct later today)
((Edit again: I'll correct them in a separate reply to this post))

Maybe I can get some more mileage from my analogy with Cantor's Hotel(*). Suppose you are hosted in this Hotel in some random even numbered room X. (The rooms are labeled 1,2,3,...). You are then offered to toss a coin and move to room X/2 if you flip tails, and move to room 2X if you flip heads. You can also choose not to toss the coin and stay in your room. Whatever you do, when you leave the hotel you are being awarded your room's number worth of dollars. Suppose everyone who occupies an even numbered room gets offered the same deal and everyone chooses to flip the coin. After the move, some rooms will be empty and some rooms will have one or (at most) two occupants. But that's immaterial. The point is that guests can expect to increase their rewards by 1.25 on average. It is also true that if they had chosen to flip the coin without looking at their room number, and only know the number Y of the room where they relocated as dictated by the result of the coin toss, they would still expect to increase their reward expectation to 1.25Y if offered to move back to whatever initial room they came from (assuming they had forgotten what it is that they flipped and hence couldn't deduce where it is that they came from before choosing whether to move back).

In this case, it's the fact that the Hotel has countably infinitely many rooms that enables the assumption of equiprobability to hold. Before the coin flip, each occupant is equally likely to double her reward than she is to halve it and hence is warranted to flip. But conversely, after she moved, she is equally likely to have moved there from a rooms that was twice as valuable than from a room that was half as valuable and hence is warranted to move back if given the opportunity, assuming only that she has forgotten where she came from!

(*) On edit: What I referred to as Cantor's Hotel is better known as Hilbert's Grand Hotel, or Hilbert's Infinite Hotel.

I'm still confused. This makes it sound like the switching argument isn't fallacious -- it just makes an unwarranted assumption. So if every value of X were equally probable, then it would be true that you can expect a .25 gain from switching? I see how the math works, but if that's true, isn't it true whether you know the value of your envelope or not? And if that's true, isn't it also true for the other envelope as well? — Srap Tasmaner

Yes, yes and yes, but the assumption isn't merely unwarranted, it is impossible that it be true in any real world instantiation of the problem where the available amounts that can figure in the distribution all are smaller than some finite upper bound. It follows from the equiprobability assumption not only that every possible amount in the distribution are equiprobable but also that the distribution is unbounded(*). In that case, your prior expectation, before you open the envelope, is infinite. The probability that you would be dealt an amount X that is lower than some finite upper bound M is vanishingly small however big M might be. But if you do wind up with some finite amount X in your envelope, then your conditional expectation upon switching is 1.25X. That doesn't make the unconditional expectation from the always-switching strategy any larger than the unconditional expectation of the 'always-sticking' strategy since 1.25 times aleph-zero is aleph-zero. The situation is rather akin to Cantor's Hotel where all the (countably infinitely many) rooms are filled up and there nevertheless still is 'room' for accommodating twice as many guests.

(*) Some distributions, such as the Gauss or Poisson distributions, are unbounded and well behaved (i.e. yielding finite expectations) but they are not uniform and so don't satisfy the equiprobability assumption.

Yes. But I think the OP is asking for a general solution for one run with no special assumptions about the context (such as whole dollar amounts or million dollar limits). — Andrew M

For sure, I agree, since any non-uniform prior would violate the equiprobability condition that alone grounds the derivation of the unconditional 1.25X expectation from switching.

(...) To remove that option, I recast the problem with the envelopes containing IOUs rather than cash, for an amount that is a real number of cents, with an arbitrary but large number of decimal places shown. The amount is only rounded to the nearest cent (or dollar) when the IOU is cashed in. — andrewk

That's clever.

Yes. You learn something about the distribution when you open an envelope (namely, that it had an envelope with that seen amount). But not enough to calculate anything useful. It's like getting a bicycle with one wheel. You might wonder whether you could get somewhere with it, but you probably can't. — Andrew M

I disagree. Suppose you were to engage in one million iterations of that game and find that the seen envelope contents converge on a specific and roughly uniform distribution of amounts all belonging in the discrete range ($1,$2,$4,$8,$16), with deviations from 1/5 frequencies that aren't very statistically significant. (I am assuming that the game only allows for amounts in whole dollars, for simplicity). Wouldn't that information be useful? I would argue that the useful knowledge that you thus gain is being accrued progressively, one little bit at a time, and could be represented by the successive updating from an allegedly very tentative long tailed initial prior distribution that represents your initial expectation that the dealer likely doesn't have access to an amount of money in excess of one million dollars, say, with the peak of the distribution somewhere around $50, say. This initial prior could be very wrong, but through (Bayesian) updating it after each iteration of the game, will lead to a prior that converges towards the 'real' distribution. Hence, each envelope that you look at provides some useful information since it is more likely than not to lead you, at each step, to updating your initial prior in the direction of a more reliable one.

I did wonder -- maybe a week ago? it's somewhere in the thread -- if there isn't an inherent bias in the problem toward switching because of the space being bounded to the left, where the potential losses are also getting smaller and smaller, but unbounded to the right, where the potential gains keep getting bigger and bigger. — Srap Tasmaner

Yes, you can have a uniform discrete distribution that is bounded between 0 and M such that the possible values are M, M/2, M/4, ... In that limit case also, if the player takes M as the cutoff for not-switching (that is, she only switches when she sees a value lower than M) her expectation is 1.25X whenever she switches and her expectation is X=M when she is dealt M (and therefore doesn't switch). Her overall expectation if she were rather to always switch would only be X. The limit case where M tends towards infinity also yields an always switching strategy (with M=infinity being the cutoff) with an expectation that is both X (=infinity) and 1.25X (=infinity).

The argument is sound, so I probably won't spend any more time trying to figure out how to simulate knowing nothing about the sample space and its PDF. — Srap Tasmaner

One way to represent "knowing nothing" might be to take the limit of an initial distribution that is uniform between 0 and M, where M tends towards infinity. In that case, the cutoff value that you can rely on to maximise your gain also tends towards infinity. In practice, that means always switching; and the expected added value from switching tends towards 0.25X, where X is the value of the envelope that you were initially dealt. This still appears to give rise to the same paradox whereby switching increases your expectation by 1.25 even though switching doesn't change anything to the probabilistic distribution of the value of the envelope that you end up holding. But the paradox only is apparent since your expectation from the non-switching strategy tends towards infinity and 1.25 times infinity (or aleph-zero) still is infinity. It is thus both true that your expectation from switching remains the same and is increased by a 1.25 factor. In this limiting case, though, it is infinitely unlikely that you will be dealt an envelope containing an amount smaller than M, however large M might be. This also explains why, in this limiting case, you should always switch.

But it is still false that you have an expected gain of 1/4 the value of your envelope. You really don't. All these justifications for assigning 50% to more possibilities than two envelopes can hold are mistaken. You picked from one pair of envelopes. This is the only pair that matters. You either have the bigger or the smaller. Trading the bigger is a loss, trading the smaller is a gain, and it's the same amount each way. — Srap Tasmaner

I agree that in the context of any real world instantiation of this problem, what you say is true (because there is no real world instantiation of this problem, with finite stocks of money, that satisfies the condition of equiprobability tacitly assumed in the original formulation). The challenge that @Michael would have to answer is this: Why is it, on his view, that it isn't rationally mandatory for him to switch his choice even before he has looked into his envelope? And if it is rationally mandatory for him to switch without even looking -- because whatever the content X of his envelope might be, the expected gain from switching is 0.25X -- then why isn't it also rationally mandatory for him to switch yet again, if given the opportunity, without even looking?

Sorry, I'm a bit confused by your response. Did you read me as saying "this isn't the same as"? — Michael

Indeed! I misread you precisely in this way. We agree then.

... Isn't this the same as: ... — Michael

Why are they not the same? In the case where the unseen envelope M is the smaller one, its content is indeed S = a/2.

If you right-click on a TeX formula and select 'Show Math As...' then 'TeX commands', then you can copy and paste the code for that in between ... — andrewk

Thanks so much. I'll do this from now on.

So far so good. But we cannot do this:

$\small p=\cfrac12\to E(N\mid M=a)=\cfrac{5a}{4}$ — Srap Tasmaner

(How should I proceed in order to quote your formulas correctly?)

Why couldn't you do that? If the initial set up calls for randomly assigning values for the two envelopes in the finite range ((1,2),(2,4),(4,8)) for instance, then, in that case, assuming the player knows this to be the initial set up (and hence uses it as his prior) then the posterior probability conditionally on observing any value of M that isn't either 1 or 8 (that is, conditionally on values 2 or 4 being observed) p will indeed be 1/2.

I was arguing that it can't be 1/2 regardless of the value being observed in the first envelope unless the prior being assumed is an infinite and uniform (and hence non-normalisable) distribution.

Random selection, which means equal probability, mitigates observational bias by treating each n in a population the same. — Jeremiah

The equiprobability that I am talking about is the posterior equiprobability between the two possible contents of the second envelope: either X/2 or 2*X. This posterior equiprobability only is guaranteed by an unbounded prior distribution. If one rather assumes a prior distribution that assigns the same probability to a finite population of possible envelope contents, then such a prior isn't uniform since it assigns a zero probability (or zero probability density) to all the conceivable envelope contents that fall outside of this finite discrete (or continuous albeit bounded) range. (One might also consider the case of unbounded albeit convergent, and hence normalisable, probability density functions. And those aren't uniform either).

1. If the player does not know the amount in the chosen envelope then the expected gain from switching is zero. — Andrew M

You might not find that everyone agrees on this first claim since, under conditions of equiprobability, the paradox arises whereby (1) acquiring knowledge of the content X of the first envelope yields and expected value of 1.25*X for switching and (2) merely acquiring knowledge of the content of the first envelope ought not to change anything to the already determined expectation of switching. This is what makes the assumption of equiprobability so problematical (since (2) can be inferred from it).

On edit: I looked at earlier posts of yours in this thread, such as this one, and I see that we are on the same page.

By this do you just mean that if we know that the value of X is to be chosen from a distribution of 1 - 100 then if we open our envelope to find 150 then we know not to switch? — Michael

That's one particular case of a prior probability distribution (bounded, in this case) such that the posterior probability distribution (after one envelope was opened) doesn't satisfy the (posterior) equiprobability condition on the basis of which you had derived the positive expected value of the switching strategy. But I would conjecture that any non-uniform or bounded (prior) probability distribution whatsoever would likewise yield the violation of this equiprobability condition.

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

You may call one interpretation the correct one in the sense that it provides a rational guide to behavior given a sufficient set of initial assumptions. But it this case, as is the case with most mathematical or logical paradoxes, the initial set of assumptions is incomplete, inconsistent, or some assumptions (and/or goals) are ambiguously stated.

Indeed, and that's where utility curves come in. If a parent has a child who will die unless she can get medicine costing M, and the parent can only access amount F, the parent should switch if the observed amount is less than M-F and not switch otherwise. — andrewk

Agreed. Alternatively, some player's goal might merely be to maximise the expected value of her monetary reward. In that case, her choice to stick with the initial choice, or to switch, will depend on the updated probabilities of the two possible contents of the second envelope conditional on both the observed content of the first envelope and on some reasonable guess regarding the prior probability distribution of the possible contents of the first envelope (as assessed prior to opening it). My main argument rests on the conjecture (possibly easily proven, if correct) that the only way to characterize the problem such that the (posterior) equiprobability of the two possibilities (e.g. ($10, $20) and ($10, $5)) is guaranteed regardless of the value being observed in the first envelope ($10 only in this case) is to assume something like a (prior) uniform probability distribution for an infinite set of possible envelope contents such as (... , $0.5, $1, $2, ...).

If my £10 envelope is Envelope X then switching to Envelope 2X gains me £10 and if my £10 envelope is Envelope 2X then switching to Envelope X loses me £5. — Michael

(This was Michael's response to a post by Baden on p.3 of this thread)

I just wanted to note that, as Michael may have realized by now, the higher expected value of the choice of switching, as compared with the choice of not switching, only is larger than zero if the act of opening the first envelope is assumed not to yield any additional (probabilistic) knowledge regarding the content of the second envelope. But if we assume any knowledge of the prior joint probability distribution of the contents of the two envelopes whatsoever, then, in that case, applying Bayes' Theorem in order to calculate the posterior probability distribution of the content of the second envelope yields an expected value of zero for the act of switching (as compared with not-switching).

On edit: The assumption that no additional knowledge regarding the content of the second envelope would be gained by opening the first only would valid in the case where the two distributions (prior probabilities) were independent, which they aren't.

I'm having trouble imagining what the source of this knowledge might be. — Srap Tasmaner

Since it's incomplete knowledge, or probabilistic knowledge, that is at issue, all that is needed is the lack of total ignorance. Total ignorance might (per impossibile) be represented by a constant probability density for all finite values, and hence a zero probability for all finite value intervals. The prior probability that the content of the first envelope (which represents your knowledge before opening it) is smaller than ten billion times the whole UK GDP would be zero, for instance. Any other (reasonable) expectation that you might have in a real world instantiation of this game would yield some probabilistic knowledge and then, therefore, lend itself to a Bayesian analysis whereby the paradox doesn't arise.

I just came upon this thread and didn't read though all of it. I did read the first few and the last few pages. It seems to me that @andrewk and @JeffJo have a correct understanding of the problem, broadly consistent with mine.

The paradox seems to me to stem a vacillation between assuming that the player possesses (and can thereafter make use of) some knowledge of the bounded probability distribution of the possible contents of the two envelopes, which can be represented by a joint prior probability distribution, and the alternative assumption (inconsistent with the first) that after opening up one envelope the posterior probability distribution of the content of the unopened envelope necessarily remains equal to 0.5 for the two remaining possible values of its content. This can only occur if the player disregards his prior knowledge (or happens by cheer luck upon a value such that the posterior probabilities for the two remaining possible values or the content of the unopened envelope are 0.5).

The value of relational QM, I think, is that it gives us a language for talking about the familiar world that we observe from our individual point-of-view rather than an idealized view-from-nowhere. — Andrew M

Let me just note that Rovelli and Bitbol both endorse relational approaches that share some features with Everett's interpretation. But they don't reify the multiverse anymore than they do its branches.

The value of relational QM, I think, is that it gives us a language for talking about the familiar world that we observe from our individual point-of-view rather than an idealized view-from-nowhere. Which is to say, we are each participants in a localized part of a much larger quantum universe that evolves unitarily. — Andrew M

Thanks for the reference to Wallace on Everett's interpretation. I just looked up his book The Emergent Multiverse: Quantum Theory According to the Everett Interpretation. The second part of the book, entitled Probability in a Branching Universe is of much interest to me. There is a short discussion on pp. 135-137 on the (pseudo-)problem of free will in the context of Everett's interpretation that I'm heartened to see appears consistent with my prior take on it in this thread. There is some discussion of the preferred basis problem that I will also look into. I'll postpone the task of making more explicit the grounds for my dissatisfaction with the metaphysical underpinnings of the multiverse approaches to quantum theory. I still have much ongoing readings to finish.

... Does that make freewill now a quasi-classical phenomenon? Well no. As I argued earlier, freewill is a much higher level socially constructed deal. It is about the construction of a "thinking self" that negotiates between a set of established cultural norms around behaviour, and some set of needs and feelings that represent "our selves" as a biological and psychological individual within that wider framework. — apokrisis

Yes, I quite agree; and so do I with most of the rest of your excellent post, with only minor reservations...

Which is also why a quantum interpretation that focuses on the observer rather than the observables, the complex epistemic relation rather than the simple ontic facts or events, would be the way forward.

Indeed!

Quantum Mechanics Unscrambled.
Just read it. Overly complicated - one suspects that at some points he is just showing off that he's technically proficient with QM formalism and complex analysis - and it is almost entirely devoid of any metaphysics, so I wouldn't bother wasting your time, unless you are interested in the technical idea that QM theory can be reshaped as a new kind of probability theory. At a couple of points he touches on the idea that what these "novel" approaches are proposing is just some kind of instrumentalism/operationalism, but he does nothing to actually argue that they should not be taken in precisely that kind of way. — jkg20

(We are veering a bit off-topic...)

I've now read about two thirds of it and let me demur. It seems to be an excellent paper. Rather than it being devoid of metaphysics I would rather say that it targets with great accuracy the metaphysical prejudices that sustain some of the most popular interpretations of QM. From what I see, he also is rather careful to distinguish his own pragmatist account from the cruder forms of positivism that it now has become fashionable to ascribe to Bohr and to Heisenberg. It is not entirely unfair to charge Bohr himself with operationalism but Delhôtel (just like Bitbol before him) also is careful to disclaim the idea of reducing quantum phenomena to classical 'observables'. He rather deflates the metaphysical implication of the quantum formalism through displaying how the generality and empirical adequacy of this formalism derives (and, indeed, can be mathematically derived) from principles that apply to classes of experimental contexts that obey some very general pragmatic requirements (such as the necessity to account for phenomena that are partially constituted and/or produced by the very circumstances of their observation) and simple norms of logical consistency.

I am getting to the point where Delhôtel seemingly is going to distinguish his approach from Bitbol's own approach (developed in Quantum Mechanics as Generalised Theory of Probabilities). This is quite interesting. I'll comment later.

I'm over my head here. But I've seen MWI described as superdeterministic. https://en.wikipedia.org/wiki/Superdeterminism
The future of all measurements is already known it's just that you happen to be in the one where the train of thought has completed. — JupiterJess

My argument, which was relying on the partial acceptance of the compatibilist notion of free will, addresses the sort of challenge posed by superdeterminism. What it is that evolves superdeterministically, on such a conception of the 'multiverse', never is, as you note, the trajectory of a conscious observer but rather the state vector which represents the 'state' of the whole multiverse. The effect from the manifold 'spitting' of observers over time ought not to be anymore troublesome to a compatibilist conception of free will than is the fact that, within any determinate history of a single observer, there are unlikely events that occasionally occur as the outcome of the uncontrolled amplification of quantum fluctuations. This is not sufficient to remove the agent's control where it matters, except in very restricted and artificial situations, since not all features of the emergent classical domain are subject to such uncontrollable fluctuations.

Presumably the worlds where your train of thought gets completed are the ones that are relatively normal with the illusion of the higher level regularities (breathable atmosphere ect). In that sense notions of identity and control are eliminable or instrumental.
I understand some say that it is not true and the state of things are that there are more (normal) worlds which is why when you think: "I raise my arm" the arm does go up rather than say your leg because there are more of the former than the latter. But I'm not sure why that is (more worlds of a certain kind than others).

To be honest, I am not sure either in what way, exactly, many-worlders account for the empirical verification of the Born rule in the individual 'worlds' (or individual splitting world-lines) of the agents/obervers (and it is a problem that used to trouble me greatly when I was myself a fan of David Deutsch and of many-world interpretation) but that is a problem that is quite distinct from the problem of superdeterminism (as it allegedly relates to the free will debate).

The constraint based physics being posted by Apokrisis here makes more intuitive sense to me but what seems intuitive might not be true

I am quite sympathetic also with the main drift of Apokrisis's constraint-based approach. But I think is it quite congenial to the pragmatist (or relational) interpretation of QM that I also favor over the alternative metaphysically 'realist' interpretations. It is indeed thanks to thermodynamical constraints that the structured and controllable 'classical world' emerges at all from the chaos of the homogeneous gas of the early expanding universe.

↪apokrisis
This would be a good starting point - https://www.nature.com/news/physics-quantum-quest-1.13711 — jkg20

@apokrisis By the way, there is a paper, which I haven't yet read, by Jean-Michel Delhôtel, discussing both Hardy's and Bitbol's approaches to the intepretation quantum mechanics: Quantum Mechanics Unscrambled.

Under GR gravity is not a force at all, it is the manifestation of the structure of spacetime, and is thus not something that can be transmitted from one body to another via particles like gravitons. — jkg20

Under GR the gravitational effects still are attributed to a field and to disturbances of this field by matter. This field just happens to be the structure of space-time in this case. The fundamental equations of GR are Einstein's field equations. They relate the metric tensor (characterizing the geometrical structure of space-time) to the source of the field (characterized by the stress-energy tensor, which registers the distribution of energy and momentum throughout space-time). So, the idea of a quantum theory of gravity is to quantize the field of gravity (that is, the perturbations in the structure of space-time) just as all the other fields are being quatized in quantum field theory. This is easier said than done, of course.

You can thus think of the graviton as a quantum of excitation of the metric of space-time roughly in the same way as you can think of the photon as a quantum of excitation of the electromagnetic field. Of course, your intuition is correct that the graviton can't be correctly conceived as a point particle that is traveling through space-time and somehow interacting with it. But this naive picture wouldn't be correct as applied the the photon's relation to the electromagnetic field either!

Under MWI, there will be infinitely many worlds in which all the bands are composed of the least likely events. So the bands will be exactly where they shouldn't be for an infinity of observers. — apokrisis

The MWI is a metaphysical gloss on Everett's relative-state interpretation. Everett's own interpretation is somewhat anti-metaphysical inasmuch as its main philosophical import is negative. It consists in denying the metaphysical reality (local realism) with respect the alleged collapse of the wave function. It is still somewhat 'realistic' inasmuch as it achieves this denial though reifying the state vector associated with the observer and then accounts for the singularity of the measurement result though relativising (one-to-one) the projected states of the observed system to the corresponding projected states of the observer that it is interacting with. This yields the problem of the determination of the privileged basis for the projections of the combined 'oberver+system' super-system. Decoherence theories seek to solve this privileged basis problem by means of an appeal to the interactions with the environment but run into other problems while attempting to factor out the quantum mechanical descriptions of the composite 'system + observer + environment' in a principled way. (This problem is intractable and ill-conceived, it seems to me, mainly owing to its reliance on the possibility of an un-situated God-eye-view on the whole universe (or its state vector) as a theoretically 'pure' standpoint from which to effect the factoring out of this universe onto the three components: oberved-system, observer and environment.)

The problem that you are raising for the MWI also arises within the framework of decoherence theories, but it seems to be mostly technical and relatively minor. If we don't reify the many-worlds as metaphysically real entities then we could account for the probability densities of potential quantum measurement (relative to 'observers') by means of coarse-grained descriptions of them. The main trouble with such interpretations, on my view, isn't so much the difficulty in accounting for the empirical verification of the probabilities derived from the Born rule so much as the ad hoc character of the definition of 'observers' (or or the 'worlds' of the MWI) seemingly devised for the sole purpose of rescuing metaphysical realism from the challenges posed to is by the profoundly relational character of the observables associated with quantum mechanical micro-physical 'states'. The realist interpretations seek to make QM palatable to the philosophically prejudiced theorist, with her 'classical' intuitions, but, in the process, obscure its most radically pragmatist implications.