There seems to be a strong case for the Everett or 'many-worlds' interpretation (MWI) of quantum mechanics. The world as revealed by MWI is so grotesquely superfluous — so offensively alien — that it's natural to imagine similarly bizarre ethical implications. If MWI is true, I am constantly 'splitting' into an unfathomable number of people. Shouldn't that make some difference to what I ought to do, some of the time?

Some suggestions have already been floated. For instance — if I take the wise option in this world, does that increase the chance of another 'me' taking the unwise option? Does that mean prudential behaviour is really selfish behaviour? And shouldn't I happily take up the smallest incentive to play quantum Russian roulette, given that I know I'll only ever experience winning? Doesn't the virtually guaranteed safety of at least some 'branches' imply we should care less about mitigating risks with low probabilities and high stakes — particularly the risk of snuffing out the human flame, and our cosmic significance along with it? And so on. These are muddled thoughts, for the most part. Just remember Egan's Law, Eliezer says: it all adds up to normality.

But I can't help thinking there is a sense in which it doesn't. This is because MWI seems to imply that many more minds will exist tomorrow than exist today — or in the next ten seconds, for that matter. So the effects of our actions on even the marginally further future dwarf the effects on the marginally nearer future in importance. And that often makes a practical difference to what we ought to do. The thought isn't complicated, which makes me suspect I've missed something obvious.

To restate in a wordier way: if you're a (total, aggregative) consequentialist, what determines the value of an act is its effect on the sum total of good and bad consequences. Just as some event is no less morally significant because it happened far away, so it seems wrong to discount consequences over time. And whether you decide to apply a discount rate for future consequences makes a material difference to which acts are best today. It seems like that the branching nature of MWI has the opposite effect of a social discount rate: it seems to imply that some outcome increases in importance as it recedes in time from the present moment into the future.


Take a toy model of MWI, where each 'branch' of the multiverse splits into two discrete copies every 24 hours. On Day 0, there is just our world (along with many other neighboring but causally isolated branches). On Day 1, there will be ten copies of our world and every other neighboring world — ten times as many worlds as Day 0. On Day 10, there are  branches of branches... of the world in Day 0, all now causally isolated from one another. Suppose on Day 0 I decide I'm due for a trip to the spa. Going to the spa is pure indulgence for me: I accrue no harms or benefits after the trip; nothing which could compound over time, no particularly fond memories or giddy antipication. I can go today for a 5-hour session, or in 10 days time for a 2-hour session. On the day I choose not to go to the spa, I'll be working, which I don't find particularly enjoyable. I need to book now in any case, and both treatments cost the same. Which do I choose? Even without discounting future welfare, the clear common-sense answer is to book the treatment today: it costs no more and lasts an extra 3 hours. No-brainer. But on this toy version of MWI, booking the 2-hour treatment means  copies of me get to enjoy the spa treatment, and only one of me has to endure an extra day of work. Booking the 5-hour treatment means one of me gets 5 hours of spa time, and  of me endures an extra day of work. It's not even close — choosing the 2-hour treatment is better by many, many orders of magnitude.

The example can be run where we're weighing between harms too. Suppose my dentist is due to receive some flashy new equipment next week which will cut the time for my procedure in half. I can receive the one-hour procedure today, or the half-hour procedure ten days from now. I don't dread dentist trips but I'm no fan of them — I just want to minimise the amount of time in the chair, feeling vaguely scared and uncomfortable. Which procedure do I take? Just like before, common-sense says: the procedure in ten days time. But on MWI, taking the half-hour procedure entails  hours of discomfort. The hour-long procedure entails an hour of discomfort. The decision is again absolutely unequivocal — I should want to take the procedure today even if it had to last all day!

Actually, these examples are potentially confusing insofar as they make it sound as if the question is about what becomes self-interestedly rational on MWI. Maybe MWI has similar implications with respect to self-interested rationality, but probably a full answer would involve thinking seriously about personal identity. Some quick thoughts: there's a naïve view that says 'I' will follow one and only one 'groove' along the many paths traced out by the branches of the multiverse. There's zero reason to think this is the case — it would involve some extra metaphysical paraphernalia in addition to MWI. Maybe instead I should weight the interests of a future copy of myself in inverse proportion to how many copies there are at that time. Or maybe I die a philosophical death at the moment of branching, so 'self-interest' is no guide to what counts as rational. In any case, the question of what is in my self-interest is is a red herring, and probably more complicated than the question of what is morally best, which is what I'm interested in. Either imagine you're an impartial altruist in the above examples, or else imagine you're deciding on behalf of another person. For instance, do I surprise my partner with a shorter spa trip next week, or a longer one today? What matters is that, in this toy example, consequentialism recommends choosing choosing the shorter + more temporally distant option when the world does branch, and the longer + temporally closer option when it doesn't.

It should be clear how my terrible examples generalise: on MWI, the stakes get higher as time goes on. The upshot is that it might sometimes be morally best to make 'sacrifices' in the short-term to avoid exponentially multiplied amounts of harm, or do exponentially multiplied amounts of good, in the longer-term — sacrifices which seem bizarre and unjustifiable on a non-branching interpretation of QM.

Note that you do not need to be a card-carrying consequentialist to buy into this. By its nature, the exponential multiplication of good and bad consequences might just crowd out other considerations even if you initially place relatively little weight on total, aggregated consequences. It is true that an average (versus total) consequentialism would not carry these odd implications, because average consequentialism is not going to be sensitive to differences in the sum total of good or bad things over time insofar as those differences don't affect the balance of good over bad things. Eliezer says something along these lines:

Average utilitarianism suddenly looks a lot more attractive - you don't need to worry about creating as many people as possible, because there are already plenty of people exploring person-space.

This doesn't strike me as a good reason to prefer average utilitarianism, and in any case AU has bizarre implications of its own, and seems to be fairly unpopular in the literature for that reason.

Note also that the strange implications of the toy example don't hinge on whether you're a hedonistic consequentialist. In other words, it doesn't matter what you think makes an outcome good or bad, as long as 'branching' the world makes there be more of it. In saying "many more minds will exist tomorrow than exist today" I'm ignoring the possibility that non-psychological things might matter, but you're welcome to think they do.

Lastly, the conclusion seems strong enough to force its way through empirical uncertainty about whether MWI is true. Like Pascal's wager and mugging, the stakes are so much higher on MWI than on non-branching interpretations that the option favoured by MWI is almost always going to win out unless your credence in MWI is stupidly small.

Reality Testing

I think there are two questions here: first, does the toy example imply the things I think it implies? Second, (how) do these implications carry over onto a sensible, modern understanding of MWI? I'll say something about this second question now.

One major difference between the toy example and worked-out version of MWI is that the branching takes place far, far more often. To my lights, all this means is that the practical differences between common-sense and 'Everettian' ethics become all the greater — from one fraction of a second to the next, there are many, many orders of magnitude more branches of 'me'.

Another major difference is that 'branching' or 'splitting' never literally occurs — these are cartoonish metaphors at best. De Witt (1970) popularised the picture of definite events cleaving the universe into countable, fully discrete and non-interacting worlds, but modern approaches appeal instead to decoherence. To the best of my understanding: decoherence describes how interference terms are suppressed as quantum systems becom eembedded within sufficiently complex macroscopic systems, such that macroscopic systems can be treated classically to a very close approximation while their quantum constituents remain superposed as described by the wave function. This large-scale interference explains why we do not appear to observe or measure superposed states. Yet, no clean ‘cut’ between classical and quantum objects is required. Rather, measurement apparatuses, and the 'worlds' they occupy, are understood as emergent macroscopic entities on par with cells, people, tables, and planets: explanatorily indispensable but not precisely definable. The 'worlds' that preponderate from one time to the next are therefore not countably discrete at all, nor are they perfectly causally isolated — they're approximations, or abstractions, or patterns — like squinting at an apple and modeling it as a sphere.

Does this make a moral difference from the toy example? I wouldn't have thought so. After all, people are imprecisely defined pattern-like things even in the (familiar, non-Everettian) world. And the argument hinges on there being more minds, or experiences, or people, or whatever else from one time to the next; but not that those things are always and precisely countable. However, I reached out to a philosopher of physics about this, who told me that they thought the conclusion I'm suggesting probably relies essentially on the ability to 'count branches'. If instead you think there is a measure over branches, where the total measure of branches is constant over time, then probably things really do just add up to normality.

How else might the conclusion from the toy example fail to carry over to serious versions of MWI? I can think of some plausible and some less plausible-sounding reasons, although I don't know enough to respond to them in any decisive way. One response points to another way in which the idea of new universes being 'created' by a process of 'splitting' is inaccurate: the many worlds have always existed, and measurement (broadly construed) just screens them off from interacting. This strongly suggests, to speak very loosely, that the minds split apart on Day 1 are already there on Day 0, 'overlapping' with but distinct from one another before they diverge. As such, rather than somehow becoming two people, observing e.g. the result of a Schrödinger's cat experiment just reveals which world you were in all along. Picture this animation, or the image of the fraying end of a rope: all the threads being present in the un-frayed end. The upshot is that there isn't radically more of anything that matters from one time to the next.

Another, similar, response says that my mind at time  splits into many branches at  such that the amount of what matters is conserved by a kind of 'dilution'. Suppose my mind splits into 10 between  and . Well, the amount those 10 minds matter is reduced tenfold — perhaps each util or whatever is a 10 times fainter, or less 'real'. Probably this view can be drawn from an analogy to the fact that MWI does not violate energy (or mass) conservation. Preumably stuff that matters is made out of stuff, and if the amount of stuff is conserved across time, then (all other things being equal) the sum total of what matters must be conserved across time. I'm not sure anyone thinks this, but it's clearly not the case that e.g. the value of a mental state is tethered to mass or energy, holding everything else constant. Is a whole-brain emulation running lead components more valuable than a functionally equivalent emulation running on lighter materials?

More to the point, imagine constructing a conscious whole-brain emulation out of wires and relays; and then meticulously splitting each wire and component into two. Eventually, you have two identical constructions; each performing identical computations. If the original machine was conscious, then the resulting two machines are presumably conscious also. But were both 'consciousnesses' somehow residing in the original machine all along? Surely not. This leads to a reductio — supposing we can keep splitting each subsequent machine ten times over, the 'there all along' view says that the original machine was home to  distinct, non-interacting loci of consciousness all along. This sounds like nonsense to me. Better to think that consciousness has much more to do with (or just is) computation or function. This bears on the first objection I mentioned, that if you thinking 'counting branches' doesn't make sense, and instead think that there's a measure over branches which is constant in time, then you don't get massive explosions in the amount of conscious experience or number of minds over time. But if a mind is more like a pattern, then can't many 'overlapping worlds' add up to a single mind? The image I have in my head is of a dozen sheets of cellophane, all printed with the same image, being stacked on one another in alignment. Are there a dozen images, or a single image? If they make up a single image, and 'overlapping' worlds make a single mind, then the idea of an 'explosion of value' is back on the cards. In any case, the question is: before a measurement, do multiple distinct loci of conscious experience 'overlap' one another, or not? Or does this make as much sense as the notion that  distinct minds must all be running in parallel on a single machine, because its wires can be split in two ten times? I suspect the answer requires a technical understanding of MWI which I don't have.

A final way in which the toy example may not carry over into grown-up MWI is that MWI in fact involves infinities of various kinds. Specifically, if there are infinitely many worlds at time  and infinitely many worlds at , it might not be possible to claim that there are more worlds, or more of anything that matters, at  than , even if it seems intuitively obvious. I don't really have any comments about this. Amanda Askell's thesis might be relevant.


I'm suggesting it seems plausible that MWI implies a constant exponential explosion of value over time — viz. the amount or number of things that mater, and in particular the number of minds or amount of conscious experience. This in turn seems to imply what would look like an insanely steep positive 'markup' rate for future welfare (to someone who doesn't buy into MWI). In this way, Egan's law fails: it all adds up to much, much more than normality. It seems like the major objection is that this idea relies essentially on the ability to 'count branches', and made false if there's a measure over branches which is constant in time. One way this could be true is if distinct, non-interacting minds 'overlap' before they 'split'. But it's unclear to me whether having a measure over branches actually precludes this constant explosion of value, considering how a kind of functionalism or computationalism about mental states suggests that value derived from conscious experience needn't be conserved with 'branching' in the way that e.g. mass is.

I should finish by underlining just how unqualified I am to draw any trustworthy conclusions: my entire knowledge of MWI comes from getting halfway through a QM textbook, reading a couple of popular science books about MWI, and taking a class in the philosophy of QM. I expect I'm almost certainly wrong about this, but I don't know enough to know why. Very curious to hear from people who know what they're talking about! Thanks for reading.


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I tend to agree with 'dilution' response, which considers branches with less Hilbert-space measure to be 'less real'. Some justification: if you're going to just count all the ways minds can be embedded in the wave function, why stop at "normal-looking" embeddings? what's stopping you from finding an extremely convoluted mapping such that the coffee in your mug actually instantiates 10^23 different conscious experiences? But now it's impossible to make any ethical comparisons at all since every state of the universe contains infinitely many copies of all possible experiences. Using a continuous measure on experiences, a la UDASSA, lets you consider arbitrary computations to be conscious without giving all the ethical weight to Boltzmann brains.

Another reason for preferring the Hilbert measure: you could consider weighting using the Hilbert measure to be part of the definition of MWI, since it's only using such a weighting that it's possible to make correct predictions about the real world.

Thanks for the response. I'm bumping up against my lack of technical knowledge here, but a few thoughts about the idea of a 'measure of existence' — I like how UDASSA tries to explain how the Born probabilities drop out of a kind of sampling rule, and why, intuitively, I should give more 'weight' to minds instantiated by brains rather than a mug of coffee. But this idea of 'weight' is ambiguous to me. Why should sampling weight (you're more likely to find yourself as a real vs Boltzmann brain, or 'thick' vs 'arbitrary' computation) imply ethical weight (the experiences of Boltzmann brains matter far less than real brains)? Here's Lev Vaidman, suggesting it shouldn't: “there is a sense in which some worlds are larger than others", but "note that I do not directly experience the measure of my existence. I feel the same weight, see the same brightness, etc. irrespectively of how tiny my measure of existence might be.” So in order to think that minds matter in proportion to the mesaure of the world they're in, while recognising they 'feel' precisely the same, it looks like you end up having to say that something beyond what a conscious experience is subjectively like makes an enormous difference to how much it matters morally. There's no contradiction, but that seems strange to me — I would have thought that all there is to how much a conscious experience matters is just what it feels like — because that's all I mean by 'conscious experience'. After all, if I'm understanding this right, you're in a 'branch' right now that is many orders of magnitude less real than the larger, 'parent' branch you were in yesterday. Does that mean that your present welfare now matters orders of magnitude less than yesterday? Another approach might be to deny that arbitrary computations are conscious on independent grounds, and explain the observed Born probabilities without 'diluting' the weight of future experiences over time.

Also, presumably there's some technical way of actually cashing out the idea of something being 'less real'? Literally speaking, I'm guessing it's best not to treat reality as a predicate at all (let alone one that comes in degrees). But that seems like a surmountable issue.

I'm afraid I'm confused by what you mean about including the Hilbert measure as part of the definition of MWI. My understanding was that MWI is something like what you get when you don't add a collapse postulate, or any other definitional gubbins at all, to the bare formalism.

Still don't know what to think about all this!

the experiences of Boltzmann brains matter far less than real brains

I think the actual reason is more like: there is nothing you can do to improve the average experience of Boltzman brains.

Why should sampling weight (you're more likely to find yourself as a real vs Boltzmann brain, or 'thick' vs 'arbitrary' computation) imply ethical weight (the experiences of Boltzmann brains matter far less than real brains)?

I think the weights for prediction and moral value should be the same or at least related. Consider, if we're trying to act selfishly, then we should make choices that lead to the best futures according to the sampling weight(conditioned on our experience so far), since the sampling weight is basically defined as our prior on future sense experiences. But then it seems strange to weigh other peoples' experiences differently than our own.

So in order to think that minds matter in proportion to the measure of the world they're in, while recognizing they 'feel' precisely the same, it looks like you end up having to say that something beyond what a conscious experience is subjectively like makes an enormous difference to how much it matters morally

I think of the measure as being a generalization of what it means to 'count' experiences, not a property of the experiences themselves. So this is more like how, in utilitarianism, the value of an experience has to be multiplied by the number of people having it to get the total moral value. Here we're just multiplying by the measure instead.

My understanding was that MWI is something like what you get when you don't add a collapse postulate, or any other definitional gubbins at all, to the bare formalism.

People like to claim that, but fundamentally you need to add some sort of axiom that describes how the wave function cashes out in terms of observations. The best you can get is an argument like "any other way of weighting the branches would be silly/mathematically inelegant". Maybe, but you're still gonna have to put it in if you want to actually predict anything. If you want to think of it in terms of writing a computer program, it simply won't return predictions without adding the Born rule(what I'm calling the 'Hilbert measure' here)

Got it, thanks very much for explaining.

correct predictions

"Correct" only in the sense that the measure of branches where it's not correct approaches zero. So only matters if you already value such a measure.

I mean that it correctly predicts the results of experiments and our observations -- which, yes, would be different if we were sampled from a different measure. That's the point. I'm taking for granted that we have some pre-theoretical observations to explain here, and saying that the Hilbert measure is needed to explain them.

I'm saying that the classical notions of prediction, knowledge, observations and the need to explain them in classical sense should not be fundamental part of the theory with MWI. It is a plain consequence of QM equations that amplitudes of the branches, where frequency of repeated experiments contradicts Born rule, tends to zero. Theory just doesn't tell us why Born probabilities are right for specific observables in absolute sense, because there are no probabilities or sampling on physical level and wavefunction containing all worlds continues to evolve as it did before. We can label "amplitudes of the branches, where x is wrong, tend to zero" situation as "we observe x", but it would be arbitrary ethical decision. The Hilbert measure is correct only if you want to sum over branches, but there is nothing in the physics that forces you to want anything.

I think some notion of prediction/observation has to be included for a theory to qualify as physics. By your definition, studying the results of e.g. particle accelerator experiments wouldn't be part of quantum mechanics, since you need the Born rule to make predictions about them.

It has some notion - that notion is just not classical and not fundamental. What happens when you study the results of any experiments or make predictions is described by the theory. It just doesn't describe it in classical or probabilistic terms because they are not real. And doesn't tell you how to maximize knowledge, because it's ambiguous without specifying how to aggregate knowledge in different branches.

I think you're misusing the word 'real' here. We only think QM is 'real' in the first place because it predicts our experimental results, so it seems backwards to say that those (classical, probabilistic) results are actually not real, while QM is real. What happens if we experimentally discover a deeper layer of physics beneath QM, will you then say "I thought QM was real, but it was actually fake the whole time"? But then, why would you change your notion of what 'real' is in response to something you don't consider real?

The main reason is the double-slit experiment: if you start with a notion of reality that expects photon to travel through either one or the other slit, and then the nature is like ~_~, it is already a sufficient reason to rethink reality. Different parts of probability distribution don't influence each other.

What happens if we experimentally discover a deeper layer of physics beneath QM

I mean, there is no need for hypotheticals - it's not like we started with probabilistic reality - we started with gods. And then everyone already changed their notion of reality to the probabilistic one in response to QM. Point is, changing one's ontology may not be easy, but if you prohibit continuous change then the Spirit of the Forest welcomes you. So yes, if we discover new better physics and it doesn't include interference between worlds, then sure, we dodged this bullet. But until then I see no reason to not assume MWI without special status for any measure. We don't even lose any observations that way - we just now know what it meant to observer something.

If we're cosmopolitan, we might expect that the wavefunction of the universe at the current time contains more than just us. In fact, the most plausible state is that it has some amount (albeit usually tiny) of every possible state already.

And so there is no good sense in which time evolution of the universe produces "more" of me. It doesn't produce new states with me in them, because those states already exist, there's just probably not much quantum measure in them. And it doesn't produce new quantum measure out of thin air - it only distributes what I already have.

Not sure if this is the correct place to put this comment, but I think it seems to relate. Layman here with big ideas, and little ability to back it up. Which is why I'm here hoping for some clarity. So anyway, this comment struck a chord with me:

"Probably this view can be drawn from an analogy to the fact that MWI does not violate energy (or mass) conservation. Preumably stuff that matters is made out of stuff, and if the amount of stuff is conserved across time, then (all other things being equal) the sum total of what matters must be conserved across time."

Taking into account that superposition might create discrepancies of relative location locally when instances of, say myself doing slightly different actions, are superimposed over one another if all possible worlds occupy the same space only in different dimensions, but not at different times, is it possible that it is the elusive 'dark matter' that accounts for the mass of the other worlds?

If we can only measure the arrangement of matter in the universe one instance at a time, yet the entirety of the mass of the Universe exists all at the same time just distributed equally among the various 'branches' of all possible worlds superimposed on top of each other, then it seems as though all the other worlds mass would seem to be 'unmeasurable' in this instance even as we can measure the mass in this world. 

We're pretty sure dark matter is just stuff that doesn't interact much except through gravity (see ). And we think we know how gravity behaves with quantum mechanics when gravity is weak (though we have experiments set to test what we think we know, see ).

Thanks, that's a nice framing.

My naive reaction is that this feels entirely too galaxy brained by about . If you start getting extremely counterintuitive results by adding entities from outside your lightcone, stop adding entities outside your lightcone. They're unknowable. They're not morally relevant. 

I'm only 65% confident in this, and I haven't done a deep dive into infinite ethics, but I still haven't seen any argument for why things that can never be known are morally relevant. 

Yes, I'm almost certain it's too 'galaxy brained'! But does the case rely on entities outside our light cone? Aren't there many 'worlds' within our light cone? (I literally have no idea, you may be right, and someone who knows should intervene)

I'm more confident that this needn't relate to the literature on infinite ethics, since I don't think any of this relies on inifinities.

I use 'light cone' to point at 'something which cannot ever conceivably casually affect anything in our present or future'. I don't know if the light-cone concept generalizes to the Everett QM branches; if not, the substitute 'anything which is in principle unknowable'. 

Epistemic status: gross over-simplification, and based on what I remember from reading this 6 months ago.

This paper resolved many quesitons I had left with MWI. Relevantly here, I think it argues that the number of worlds doesn't grow because there was already an infinity of them through space.

Observing an experiment is then equivalent to locating yourself in space. Worlds splitting is the process where identical regions of the universe become different.

Thanks, this is useful.

I don't know how far this generalizes, but in your toy example, you seem to be optimizing over 3D world states ("number of people who are having a positive experience") rather than 4D world states ("number of people who have had or will have or are having positive experiences"). If my choice is between 1 person having a positive experience today, and 10 having an equivalent experience tomorrow, then either way 2 days from now 100 people will look back on having had such an experience. 

When you introduce the model you assume there are no relevant anticipations, which makes sense to me since it creates measurable additional value to delaying the experience, but also no fond memories, which doesn't make sense to me. Assuming no value for memories, to me, gives the experience the same moral value as an experience that everyone involved in forgets due to being blackout drunk: maybe not none, but very very low, since it essentially stops being part of the story of anyone's life.

What I mean is, both options in the toy model (with or without any kind of conserved measure) result in the same number of people having equivalent experiences in their past and future light cones. Why should the moral value of those experiences depend on when in their light cones those experiences happened? For whom does this change the total utility of their experiences?

(Side note: I'm still confused about the implications of the idea of quantum immortality/suicide/Russian roulette, in that my brain rebels against anything I think about it).

I argue that counting branches is not well-behaved with the Hilbert space structure and unitary time evolution, and instead assigning a measure to branches (the 'dilution' argument) is the proper way to handle this. (See Wallace's decision-theory 'proof' of the Born rule for more).

The quantum state is a vector in a Hilbert space. Hilbert spaces have an inner product structure. That inner product structure is important for a lot of derivations/proofs of the Born rule, but in particular the inner product induces a norm. Norms let us do a lot of things. One of the more important things is we can define continuous functions. The short version is, for a continuous function, arbitrarily small changes to the input should produce arbitrarily small changes to the output. Another thing commonly used for vector spaces is linear operators, which are a kind of function that maps vectors to other vectors in a way that respects scalar multiplication and vector addition. We can combine the notion of continuous functions with linear operators and we get bounded linear operators.

While quantum mechanics contains a lot of unbounded operators representing observables (position, momentum, energy, etc.), bounded operators are still important. In particular, projection operators are bounded, and every self-adjoint operator, whether bounded or unbounded, has projection-valued measures. Projection-valued measures go hand-in-hand with the Born rule, and they are used to give the probability of a measurement falling on some set of values. There's an analogy with probability distributions. Sampling from an arbitrary distribution can in principle give an arbitrarily large number, and many distributions even lack a finite average. However, the probability of a sample from an arbitrary distribution falling in the interval [a,b] will always be a number between 0 and 1.

If we are careful to ask only about probabilities instead of averages, or even just to only ask about averages when the quantity is bounded, we can do practically everything in quantum mechanics with bounded linear operators. The expectation values of bounded linear operators are continuous functions of the quantum state. And so now we get to the core issue: arbitrarily small changes to the quantum state produce arbitrarily small changes to the expectation value of any bounded operator, and in particular to any Born rule probability.

So what about branch counting? Let's assume for sake of discussion that we have a preferred basis for counting in, which is its own can of worms. For a toy model, if we have a vector like (1, 0, 0, 0, 0, 0, ....) that we count as having 1 branch and a vector like (1, x, x, x, 0, 0, ....) that we're going to count as 4 branches if x is an arbitrarily small but nonzero number, this branch counting is not a continuous function of the state. If you don't know the state with infinite precision, you can't distinguish whether a coefficient is actually zero or just some really small positive number. Thus, you can't actually practically count the branches: there might be 1, there might be 4, there might be an infinite number of branches. On the other hand, the Born rule measure changes continuously with any small change to the state, so knowing the state with finite precision also gives finite precision on any Born rule measure.

In short, arbitrarily small changes to the quantum state can result in arbitrarily large changes to branch counting.

More Notes

Something very like the view I'm suggesting can be found in Albert & Loewer (1988) and their so-called 'many minds' interpretation. This is interesting to read about, but the whole idea strikes me as extremely hand-wavey and silly. Here's David Wallace with a dunk: “If it is just a fundamental law that consciousness is associated with some given basis, clearly there is no hope of a functional explanation of how consciousness emerges from basic physics.”

I should also mention that I tried explaining this idea to another philosopher of physics, who took it as a reductio of MWI! I suppose you might also take it as a reductio of any kind of total consequentialism also. One man's modus ponens...

David Lewis briefly discusses the ethical implications of his modal realism (warning: massive pdf), concluding that there aren't any. This may be of interest, but not sufficiently similar to the case at hand to be directly relevant, I think.

Another potential ethical implication: Hal Finney makes the point that MWI should steer you towards maximising good outcomes in expectation if you weren't already doing so (e.g. if you were previously risk-averse, risk-seeking, or just somehow insensitive to very small probabilities of extreme outcomes). The whole thread is a nice slice of LW history and worth reading.