# The Ethics of Copying Conscious States and the Many-Worlds Interpretation of Quantum Mechanics

Suppose that a digital computer is having a conscious experience. We could then make an *exact* copy of this conscious experience and run it on a second computer. Does this act have any moral significance? If the experience is a happy one, have we increased the amount of happiness in the world? If the experience is a painful one, have we increased the amount of pain?

It is plausible that we should answer "no" to each of the above questions. The argument for this view claims that the question "How many physical copies of a conscious experience are there?" is not well defined, and therefore cannot matter. I will refer to this as the *Ignore-Copies* view.

The purpose of this post is to explore what I think is a tension between the Ignore-Copies view, and the Many-Worlds Interpretation of Quantum Mechanics (MWI).

Intuition behind the tension
============================

I think the fastest way to convey the intuition behind why there might be a tension here is to consider something which is *not* the Many Worlds Interpretation of Quantum Mechanics:

> **Not** the Many-Worlds Interpretation: When quantum mechanics predicts that outcome A is twice as likely as outcome B, this is because twice as many (otherwise identical) copies of you experience outcome A as experience outcome B.

Importantly, this is *not* how MWI actually works, but suppose for the moment that it was. Hopefully the tension between *this* statement and the Ignore-Copies view is clear. On the one hand, we have an argument that the question "How many physical copies of a conscious experience are there?" is not well defined. On the other hand, we have an argument that questions of this kind are ultimately responsible for some of the most well tested empirical predictions in all of science.

In the rest of this post, I will review what MWI actually involves, and I will argue that there really is a tension between MWI and the Ignore-Copies view. It is not simply an artefact of my butchered re-statement of MWI above. I argue that some of the most appealing ways to derive the Born rule for quantum mechanical probabilities in MWI are seriously undermined by the Ignore-Copies view.

This is something that has bothered me for a while, although I've not seen this tension explicitly referred to anywhere else. I've only recently decided to spend a little bit of time digging into some of the literature on these questions to see if I could resolve my confusion. I have come away feeling that the tension is real, and thought some people might be interested enough to justify sharing this write up.

I imagine a lot of people will not find this particularly interesting, but if like me you put non-negligible credence in each of the following propositions:

*   It is possible for conscious experiences to be perfectly copied (this follows, for example, if it is possible for a digital computer to be conscious).
*   Exact copies of conscious experiences carry no extra moral weight.
*   MWI is correct.

then this tension seems worth understanding. Is it resolvable?

Outline of the argument
=======================

This post is fairly long and involved, so I will start by giving the argument in outline, with reference to the different sections of the post.

*   **Why the Ignore-Copies view is plausible**: I begin by briefly giving a defence of the Ignore-Copies view. I defend the claim that the question "How many physical copies of a conscious experience are there?" is not well-defined.
*   **Defeating Dr Evil with self-locating belief**: I review Elga's paper "Defeating Dr Evil with Self-Locating belief". Elga's argument has nothing to do with quantum mechanics, but considering it carefully will be very helpful in what follows. This paper considers situations involving so-called "Self-Locating Uncertainty". It makes a compelling argument that if you know there are N identical physical copies of your conscious state, then you should assign probability 1/N to being each one of them. If we consider a scenario in which 2 out of 3 identical copies of you each receive an identical reward, Elga's argument implies that you should prefer this to a scenario in which only 1 out of 3 copies receives the reward. This is because in the first scenario the probability that you are a copy who will receive the reward is 2/3, whereas in the second scenario it is only 1/3. A defender of the Ignore-Copies view must reject Elga's argument. They believe that N, the number of copies of a conscious experience, is not well-defined, and they believe that the two scenarios just described should be treated identically (assuming that in either scenario we only end up with *two* distinguishable conscious entities after the reward has been given). I argue they can consistently do so by denying that the concept of self-locating uncertainty among identical conscious states is well defined, and by denying that the concept of a self-interested rational agent makes sense in situations which involve branching personal identity.
*   **The Many-Worlds Interpretation**: I briefly summarise MWI (can be skipped if already familiar). MWI provides a compelling resolution to the Measurement Problem in quantum mechanics, but it renders quantum mechanics fully deterministic, so that the origin of quantum mechanical probabilities (the 'Born rule') requires explanation. In MWI, the Born rule must be derived, not assumed.
*   **Derivations of the Born rule from Self-Locating uncertainty**: I review Sebens and Carroll's derivation of the Born rule based on self-locating uncertainty. Here, the tension with the Ignore-Copies view is most apparent. Sebens and Carroll argue that Born rule probabilities arise from an observer's self-locating uncertainty in the time between making a measurement and becoming aware of the result, while the conscious states of their parallel-world copies are identical. A defender of the Ignore-Copies view believes that the concept of self-locating uncertainty among identical conscious states is incoherent, and so cannot be moved by Sebens and Carroll's argument. I speculate on whether Sebens and Carroll's argument could be adapted into a form that a believer in Ignore-Copies would be happy with, and conclude that it is not easy to do so.
*   **Decision-theoretic derivations of the Born rule**: I give two closely related arguments that decision-theoretic derivations of the Born rule for MWI are in tension with the Ignore-Copies view. First, like Elga's classical argument on Self-Locating Uncertainty, they appeal to the concept of a self-interested rational agent in situations which involve branching personal identity. As previously argued, a defender of the Ignore-Copies view should deny that this concept is well-defined. Second, I explain that each of the steps in Deutsch's original decision-theoretic derivation of the Born rule can be straightforwardly lifted and applied to classical situations involving self-locating uncertainty, where they provide an alternative derivation of Elga's result. A classical version of Deutsch's argument can be applied to a scenario where 2 out of 3 identical copies of you receive an identical reward. The argument "proves" that you should prefer this situation to an alternative where only 1 out of 3 copies receives the reward. A defender of the Ignore-Copies view must deny the validity of this argument in the classical setting, so it would seem strange for them to accept the validity of the quantum version.
*   **Other derivations of the Born rule**: I explain that as far as I am aware, the many other proposed derivations of the Born rule are likely not in direct conflict with the Ignore-Copies view, but that they are also less appealing. They proceed by showing that the only probability measure consistent with some stated assumptions is the Born rule, but the *reason* why probabilities should enter into the theory at all is left unexplained. If these are the only derivations that a believer in the Ignore-Copies view is left with, then it would appear that the plausibility of MWI has been at least somewhat reduced.
*   **Conclusion**: Despite all of the above, I explain why I still think there is a significant likelihood that both MWI and the Ignore-Copies view are true. I would like to see someone provide a satisfying derivation of the Born rule within MWI which does not clash with the Ignore-Copies view. I speculate on what such a derivation might look like.

Why the Ignore-Copies view is plausible
=======================================

I first encountered the thought experiment I describe here in a [Facebook post](https://www.facebook.com/olliesayeed/posts/pfbid027cpeTkY84QZfLorQTLWhA1aDBGBLxZYm2D44q5uZkiCmp9YT1kUfV4Y73GMg8R7El) by Ollie Sayeed, not himself a proponent of the Ignore-Copies view, summarising an argument by Mihnea Maftei, who is (or at least was at the time).

Suppose we accept that a digital computer is capable of having conscious experiences. A digital computer is usually made out of tiny electrical circuits built on pieces of silicon, but it does not need to be. You could also build a computer out of [dominos](https://www.youtube.com/watch?v=OpLU__bhu2w) if you wanted to, and in principle any program that runs on an electronic computer could also be run on a domino computer. In principle, you could even 'run' any computer program manually, working through each of the steps by hand, using nothing more than a pencil and paper in place of the computer's memory.

What happens if we take an array of dominoes that is instantiating some conscious experience, and then we cut each domino in half? Now we have two copies of the same system: one instantiated by the left halves of all the dominos, and one instantiated by the right halves. Are there now two minds, or one?

Alternatively, suppose that we are running the computer program by hand using a pencil and paper. What happens if we now make two pencil strokes in every place where we previously made one? What if we just use a thicker pencil?

You might be inclined to answer that these acts create two minds where previously there was one, but as Ollie puts it:

> But then we have to face the question of what exactly it was about the domino-splitting process that “created” all those extra minds. What if we cut them almost in half, or cut them in half and glued them back together – is each domino one or two? We don’t necessarily even have to cut them at all: even if a domino is still intact, its left half still exists, so we can think about the mind instantiated by each intact domino’s left half. So even if we think there can be multiple identical minds instantiated in the world, we have this awkward mereological problem of deciding exactly how many minds are instantiated. The way out is to say that all of these systems are still only instantiating one mind.

I will describe someone who takes this "way out" as subscribing to the Ignore-Copies view. To give a more explicit definition: a believer in the Ignore-Copies view is someone who believes that the question "How many physical copies of a conscious experience are there?" is not well defined.

There are many more things that could be said here. Ollie gives his argument against the Ignore-Copies view in the same linked post. Nick Bostrom argues against it in a paper [here](https://nickbostrom.com/papers/experience.pdf) (although his argument involves conceding that the question "How many physical copies of a conscious experience are there?" can be answered with fractions!) But the purpose of this post is not to review these arguments in detail. I simply want to explain why the Ignore-Copies view is at least plausible, which I have done, and then to explain why it appears to be in tension with MWI.

Defeating Dr Evil with Self-Locating Belief
===========================================

In the paper [Defeating Dr Evil with Self-Locating Belief](https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1933-1592.2004.tb00400.x), Elga makes a compelling argument that if we know there are N physical copies of our conscious state, then we should assign probability 1/N to being any one of them.

In particular, suppose that 3 identical copies of a conscious state are running on 3 distinct digital computers. Suppose that if event A occurs, then 2 of these copies will receive some reward. If event B occurs, then only 1 will receive the reward. Elga would argue that each copy of the consciousness should assign a 2/3 probability to receiving the reward in event A, and a 1/3 probability of receiving the reward in event B. If they have a choice, they should therefore prefer event A to occur instead of event B.

Elga's arguments have nothing to do with quantum mechanics, but it will be useful for what follows to understand how a defender of the Ignore-Copies view would respond to Elga. A defender of the Ignore-Copies view must reject Elga's claim. They believe that event B should be valued exactly the same as event A (as long as the two copies who receive the reward in event A, and the two copies who do not receive the reward in event B, remain identical to each other). In this section we review Elga's argument and explain how a defender of the Ignore-Copies view should respond.

Elga considers the following thought experiment:

> Safe in an impregnable battlestation on the moon, Dr. Evil had planned to launch a bomb that would destroy the Earth. In response, the Philosophy Defense Force (PDF) sent Dr. Evil the following message:
> 
> Dear Sir,  
> (Forgive the impersonal nature of this communication—our purpose prevents us from addressing you by name.) We have just created a duplicate of Dr. Evil. The duplicate—call him “Dup”—is inhabiting a replica of Dr. Evil’s battlestation that we have installed in our skepticism lab. At each moment Dup has experiences indistinguishable from those of Dr. Evil. For example, at this moment both Dr. Evil and Dup are reading this message. We are in control of Dup’s environment. If in the next ten minutes Dup performs actions that correspond to deactivating the battlestation and surrendering, we will treat him well. Otherwise we will torture him.  
> Best regards,  
> The PDF.
> 
> Dr. Evil knows that the PDF never issues false or misleading messages.  
> Should he surrender?

Elga claims that after reading the message, Dr Evil must assign probability 1/2 to being either copy, and therefore should surrender as long as he prefers unnecessary surrender to being tortured. The paper defends this claim by making an ingenious argument involving conditional probabilities, and an irrelevant additional coin toss. It then generalises this to cases involving more copies, to derive the 1/N result. I won't reproduce the details here, because I do not believe the details are necessary in order to understand how a defender of the Ignore-Copies view could reply to Elga. I imagine a hypothetical dialogue here between a defender of Ignore-Copies (IC) and a defender of Elga's argument on Self-Locating Uncertainty (SLC):

> **IC:** The question "How many physical copies of my conscious state are there?" does not have a well defined answer. Similarly, the question "Which physical copy of my consciousness am I?" is incoherent. It is like a consciousness inhabiting a domino computer asking what the probability is that they live in the left vs the right halves of each domino. Attempting to answer this question, or attempting to assign probabilities to the different possible answers, does not make sense. Self-locating uncertainty does not make sense. There are not two copies of Dr Evil before their conscious experiences diverge. There is only one Dr Evil, who happens to be instantiated in two places.
> 
> **SLC:** The two copies have identical conscious experiences, but they are also different in important ways. For example, the causal consequences of the decisions made by each copy are different. The Dr Evil on the moon has control of the battlestation, and his actions can cause it to be deactivated. The duplicate Dr Evil does not. Should this fact not be relevant? Can we not use it to distinguish the copies? Can each copy not reasonably ask himself, "Am I the copy of Dr Evil who has control of the battlestation?"
> 
> **IC:** If the conscious states of the two copies are truly identical, then no. By assumption, they will make identical decisions, so it makes no sense to consider scenarios in which they make different decisions. They will, as one mind, make one decision, and they know what the consequences of that decision will be. The question "Am I the copy of Dr Evil who has control of the battlestation?" is also incoherent.
> 
> **SLC:** Ok, you might think that the question is incoherent, but Dr Evil must still make a decision here about how he is going to act. Once the conscious states of Dr Evil and his clone(s) have diverged, you cannot deny that there are multiple versions of Dr Evil *then*, so we are led to wonder whether the eventual fate of his clone(s) should have any bearing on his decisions now. Crucially, he is evil. He does not care about anyone else, even one of his own clones. But perhaps we can use an argument like Elga's to show that the only rational way for Dr Evil to make decisions when his consciousness has been duplicated N times is to act *as if* he has a 1/N chance of being each physical copy.
> 
> **IC:** I agree with Elga's argument that all probability assignments other than 1/N are irrational. I just take this one step further. The probability assignment of 1/N is irrational as well, for the simple reason that N itself is not well defined. There is no rational way for Dr Evil to act according to his stated values in this scenario, because his values are incoherent. The concept of a self-interested rational agent does not make sense in a situation involving branching personal identity, and under Ignore-Copies, this is exactly what this situation is. For example, Parfit explores such branching scenarios in [Reasons and Persons](https://academic.oup.com/book/12484) and uses them to reject the Self-Interest theory of rationality.

To conclude, I do not believe that Elga's argument poses any fresh challenges for the Ignore-Copies view. A defender of the Ignore-Copies view will simply assert that self-locating uncertainty among identical conscious states is an incoherent concept. A consequence of this appears to be that the concept of a self-interested rational agent no longer makes sense in situations which involve the divergence of multiple physical copies of the same conscious experience.

The Many-Worlds Interpretation of Quantum Mechanics
===================================================

We are now ready to discuss Quantum Mechanics. This section gives a rapid introduction to Quantum Mechanics and the Many-Worlds Interpretation (can be skipped if already familiar).

Quantum mechanics (QM), like all good theories of physics, tells you how to predict the state of a system at some future time, given the state of the system now. In QM, states can evolve in two different ways. When a system is undisturbed, its state evolves continuously and deterministically, much like the state of a system in classical physics. We can predict exactly what the state of a quantum system will be in 5 seconds' time if we are told what its state is right now. On the other hand, when a system is *measured*, then in general its state will change randomly. We cannot generically predict which state it will end up in as a result of the measurement, we can only assign probabilities to the different possible outcomes. The possible final states and their probabilities depend not only on the state of the system prior to the measurement, but also on what measurement we choose to make.

For any possible observable that we can measure, X, there are some special states, called eigenstates, associated with X which will be unchanged by the measurement, and for which the measurement outcome can be predicted with certainty. It is common to denote a state which yields a measurement outcome $x$ with certainty as $\left|x\right>$, using these strange brackets.

Furthermore, if we measure X in a generic state, the state will change randomly, but it will change randomly *into one of the eigenstates of X*. This means that if we were to immediately repeat the measurement again, then we would get the same result as before with certainty.

For any possible quantum state, there will be some measurements which do not change the state and which yield a definite outcome (so with respect to which the state is an 'eigenstate' as described above) and there will be some other measurements which do change the state and yield an unpredictable outcome. No state gives a definite result to every possible measurement. The state $\left|x\right>$ is unchanged by measuring X, but would be changed in a non-deterministic way if we chose to measure some other observable instead. For example, a state with a definite position would change randomly and yield an unpredictable result if we were to try to measure its momentum, and vice-versa. This is quantified in the famous Heisenberg uncertainty principle.

Suppose we have a quantum system and a measurement on the system that can yield one of two outcomes, 0 or 1 (such a system is usually called a *qubit*). Here is an example of a state which would yield either outcome with probability 1/2:

$\frac{1}{\sqrt{2}} \left| 0 \right> + \frac{1}{\sqrt{2}}\left|1\right>$

The square roots here are just a convention because physicists like their state 'vectors' to have length 1 (the length has no physical significance in the theory). Importantly, this is not the *only* quantum state that would give these measurement probabilities. We can actually use any complex numbers of magnitude 1/2 in front of the eigenstates $\left|0\right>$ and $\left|1\right>$ and if the ratio of the coefficients changes, then we get a distinct quantum state (it cannot be distinguished by a measurement of the observable 0/1 since it predicts identical probabilities for that measurement, but it could be distinguished by the measurement of some other observable). This is a crucial feature of the theory which allows different possible measurement outcomes to 'interfere' with each other prior to the measurement being made. This is ultimately the reason why it is challenging to explain away quantum mechanical uncertainty as merely stemming from unseen hidden variables. But for now, we can ignore this. I will not use complex numbers in what follows.

We could also have a state containing different mixtures of  $\left|0\right>$ and $\left|1\right>$, corresponding to different probabilities of each outcome. For example, the state:

$\sqrt{\frac{2}{3}}\left|0\right> + \sqrt{\frac{1}{3}}\left|1\right>$

would have a 2/3 probability of yielding the outcome 0, and a 1/3 probability of yielding the outcome 1. The rule is that after normalizing the state vector to have length 1, and expressing it as a combination of the eigenstates of some observable, the probability of seeing a particular outcome when measuring that observable is given by the square of the magnitude of the coefficient of that outcome's eigenstate. This rule is known as the Born rule.

So that's (some of) quantum mechanics, as currently understood. This picture works. It leads to amazingly accurate experimental predictions. But it also begs the question: what counts as a measurement? We know intuitively what 'measurement' means. It is what happens when an experimenter points a measuring device at some quantum system. And in practice, this intuition is usually enough for us to figure out how to apply the theory to any given situation. But ultimately, the measuring device and even the experimenter themselves are also made up of tiny particles which move according to the rules of quantum mechanics. What is to stop us from treating the system being studied, the measuring device, and the experimenter herself, as a single self-contained quantum system? And if we were to do that, why shouldn't this system then evolve deterministically, as undisturbed quantum systems are supposed to? How are we supposed to know when a 'measurement' happens? When does randomness kick in?

This is known as the *measurement problem*, and the competing "interpretations" of quantum mechanics attempt to resolve it in different ways. Arguably, MWI is the simplest resolution to the measurement problem. The big claim of MWI is that we should just remove the measurement part of QM entirely, because the non-measurement part of the theory can already by itself explain all of our observations.

How does this work? Lets consider a qubit in state $\frac{1}{\sqrt{2}} \left| 0 \right> + \frac{1}{\sqrt{2}}\left|1\right>$, where the 0,1 observable is measured by an experimenter, Alice. What happens if we try to analyse the situation without treating measurement as something special? Instead, we will just apply the standard deterministic rules of non-measurement QM to the qubit, the measuring device, and to Alice herself.

It turns out that as Alice's measuring device interacts with the qubit, ordinary non-measurement QM predicts that the state of the system, detector, and Alice, should change to:

$\frac{1}{\sqrt{2}}\left|0 \text{ and Alice sees outcome 0}\right> + \frac{1}{\sqrt{2}}\left|1\text{ and Alice sees outcome 1}\right>$

It also predicts that once the simple qubit has become 'entangled' with the complex macroscopic system of Alice and the measuring device in this way, then these two 'branches' of the state corresponding to the two measurement outcomes will no longer interfere with one another. From this point on, they will evolve independently. The name for this phenomenon is 'decoherence'.

We see that the quantum state now encodes two copies of Alice, and that each copy has observed a different measurement outcome. As far as each copy is concerned, it will now appear that the state of the qubit has spontaneously changed into an 'eigenstate' of the 0,1 observable. And we have arrived at this conclusion without needing to make any special assumptions about the role of measurement in quantum mechanics. It follows simply by taking seriously the predictions of the standard and well-tested non-measurement parts of the theory.

So it looks like the non-measurement part of the theory is already capable of explaining the mysterious state changes that occur when a measurement is made! At least as long as we are comfortable with accepting that there are many parallel worlds containing slightly different copies of ourselves. The only thing left unexplained is where probability fits in. How can probability play any role in what is now a fully deterministic theory? The Born rule says that Alice should assign probability 1/2 to either outcome when she measures the above state, but in MWI everything about the evolution of the quantum state is known in advance. What is she doing when she uses probability here?

Of course, Alice will only ever be consciously aware of seeing one outcome or the other, not both, and the idea that Alice should assign probability 1/2 to each of the two possible outcomes seems at least natural by symmetry. But the role of probability in MWI is made especially mysterious when we consider a state like:

$\sqrt{\frac{2}{3}}\left|0\right> + \sqrt{\frac{1}{3}}\left|1\right>$

which MWI tells us will evolve to;

$\sqrt{\frac{2}{3}}\left|0 \text{ and Alice sees outcome 0}\right> + \sqrt{\frac{1}{3}}\left|1\text{ and Alice sees outcome 1}\right>$

The Born rule now tells us that Alice should assign probability 2/3 to seeing outcome 0 and 1/3 to seeing outcome 1, when she makes this measurement. But where do *these* probabilities come from? There is still seemingly just one version of Alice encoded in the wave function who sees either outcome. Why are the odds not still 50/50?

A satisfying defence of MWI should include a 'derivation of the Born rule'. It should explain why it is natural for Alice to assign probability 2/3 to outcome 0 when she knows that the quantum state of her mind is about to evolve as above. And it should also give a satisfying account of the meaning of these probabilities. Why does it make sense for Alice to talk about probabilities at all when she is dealing with a fully deterministic theory? And why does she find that the many observations that she has already made in her past conform to the Born rule predictions so well?

Derivations of the Born rule from Self-Locating Uncertainty
===========================================================

The derivation of the Born rule which is in clearest tension with the Ignore-Copies view is that of [Sebens and Carroll](https://arxiv.org/pdf/1405.7577).

Sebens and Carroll attempt to explain the origin of probability in MWI through the concept of Self-Locating Uncertainty. They consider the time after Alice has made a measurement, but before she has become consciously aware of the result. Immediately after the measurement has been made, there are two physical copies of Alice, one on each branch of the quantum state, but the conscious states of each copy are supposedly still identical. If the copies of Alice asks themselves, "which branch am I on?", then it is claimed that the answer to this question is for each of them uncertain, due to self-locating uncertainty (the same concept introduced by Elga, who they cite). They claim it is this uncertainty which justifies the use of probability to describe the different possible answers to the question. They next defend a principle that they call the 'Epistemic Separability Principle' or ESP. They describe the "gist" of this principle as follows: "The credence one should assign to being any one of several observers having identical experiences is independent of the state of the environment." They finally give their argument that the Born rule is the only way for Alice to assign probabilities to the question of which branch she is on that is consistent with both the principles of quantum mechanics and with ESP.

Given that the whole argument only concerns a brief period of time between Alice making a measurement and learning the outcome, you might wonder why Alice is supposed to use Born rule probabilities when considering measurements that she is yet to make. Their answer is that although Alice is not uncertain about anything prior to making a measurement, she knows that she will briefly *become* uncertain about something after the measurement is made (namely which branch she is on) and she also knows which probabilities it will be rational for her to assign at that time. This is the sense in which Born rule probabilities make sense for Alice prior to the measurement as well. For example, if she is offered a bet whose outcome depends upon the result of some quantum measurement, she should use the Born rule probabilities to judge the value of the bet, because in doing so she knows that she is acting in a way which the uncertain future copies of her will be satisfied with.

The tension between Sebens and Carroll's argument and the Ignore-Copies view is immediately clear. As we saw in the discussion of Elga's paper, a defender of the Ignore-Copies view believes that the concept of self-locating uncertainty among identical conscious states is incoherent. When there are multiple identical copies of the same conscious state, it does not make sense for that conscious state to wonder "Which copy am I?" Self-locating uncertainty therefore cannot be invoked to solve the mystery of where probabilities come from in MWI.

It is interesting to ask whether Sebens and Carroll's argument can be modified into a form that a defender of the Ignore-Copies view would be happy with. For example, suppose we can find a time *after* the conscious states of the two copies of Alice have diverged, but *before* she has been able to figure out whether the qubit was a 0 or a 1 (perhaps the display on the measuring device is distinctive but difficult to interpret). If we can do this, then even a believer in Ignore-Copies would agree that there are now two distinct versions of Alice, and that each version of Alice could reasonably ask herself "Am I the Alice who will deduce that the qubit's value was 0, or the Alice who will deduce that the qubit's value was 1?". Could we apply a modified version of Sebens and Carroll's argument to this situation, together with a modified version of their ESP principle, to re-derive the Born rule probabilities?

I think the answer is: perhaps, but in doing so, it not only becomes more complicated, but also less convincing. Consider the plausibility of the ESP principle itself. On first look, ESP seems plausible. It seems plausible to assume that changes to the environment outside of Alice and the system being measured should not affect Alice's probability assignments regarding the measurement. But now suppose we were to consider changes to the environment *which also change the number of copies of Alice*? The idea that such a transformation should not affect Alice's probability assignments feels like it requires more justification. And this is not a minor quibble. Transformations of the environment which change the number of copies of Alice are a crucial part of Sebens and Carroll's quantitative derivation of the Born rule probabilities.

They consider this issue in the paper, and conclude that their principle is "well-motivated but not established beyond any doubt". One strong argument in favour of their ESP principle over what they describe as the simple "branch-counting" approach to probability assignment is that branch-counting is not something that can be well defined in quantum mechanics in general. In the absence of a simpler alternative principle, ESP becomes more plausible. But now recall that we are trying to modify the argument, and the ESP principle, to apply it to self-locating uncertainty among *distinct* rather than *identical* conscious states. In this context we now *do* have a simpler principle that we could consider using in place of ESP, that you might call "distinct conscious state counting" (or if you are a familiar with anthropic reasoning, the "Self Sampling Assumption"). I think that the question, "How many distinct conscious states are encoded in this quantum state?" ought to be well-defined, so that the need to appeal to a new principle such as ESP is reduced. It is also not at all obvious why applying the Self-Sampling Assumption to this situation should lead to the Born rule probabilities. I discuss this line of thought further in the post's conclusion.

Nevertheless, whatever you think of this updated form of the argument, it is clear that the Ignore-Copies view is inconsistent with Sebens and Carroll's argument for Born rule probabiltities as stated in their original paper.

Decision theoretic derivations of the Born rule
===============================================

A variety of derivations of the Born rule have been proposed which are based on decision theory. They consider rational agents who are trying to make decisions in situations where their reward depends on the outcome of some quantum measurement. They defend a set of principles which rational decision makers should abide by. These principles are then argued to imply that decision makers should act *as if* the measurement outcomes were governed by Born rule probabilities. They should therefore act according to these probabilities even though there is not in fact anything which they are uncertain about.

On the face of it, this seems to be a compelling way to justify the origin of probabilities in MWI. It is also not immediately clear why such a derivation should be in tension with the Ignore-Copies view. But it is, as I shall now explain.

The first thing that should make us concerned is the appeal to the concept of a self-interested rational agent in a situation which involves branching personal identiy. I argued above that a defender of the Ignore-Copies view should deny that this is coherent, in order to be able to fully refute the arguments of Elga's paper.

We can also reveal the tension between Ignore-Copies and decision theoretic derivations of the Born rule explicitly. Consider a fully classical situation in which we have 12 computers each having an identical conscious experience. In event A, we give 8 of these computers an identical reward ('identical' meaning that they continue to remain in the same conscious state as each other after receiving the reward, although they must diverge from the other 4). In event B, we only give this reward to 6 of the computers. According to a defender of the Ignore-Copies view, event A and event B should be valued in the same way. The number of copies is irrelevant. But now we can consider the [original decision theoretic derivation of the Born rule](https://arxiv.org/abs/quant-ph/9906015) due to Deutsch. I argue that each of the steps in Deutsch's paper can be straightforwardly lifted and applied to this classical situation, where they would appear to 'prove' that the computer consciousness should prefer event A to event B. A defender of the Ignore-Copies view must deny the validity of this argument in the classical setting, so it would be strange for them to accept the quantum version.

Here is a rapid overview of how the argument works (for further justifications and details, I refer you to Deutsch's paper, which I claim is the same in the relevant respects):

Consider a game in which 6 computers will receive reward $x_1$ and the other 6 receive reward $x_2$. Denote the value of the game by $V(x_1, x_2)$. If the computer consciousness is a rational decision maker then we claim they should adopt a value function with the following properties:

$V(x_1 + k, x_2 + k) = V(x_1, x_2) + k$  
$V(-x_1, -x_2) = -V(x_1, x_2)$  
$V(x_1, x_2) = V(x_2, x_1)$

Substituting $k = -x_1 -x_2$  and applying some algebra shows that

$V(x_1, x_2) = \frac{1}{2} (x_1 + x_2)$

Next, if we have a game where two computers receive $x_1$, two computers receive $x_2$, ..., two computers receive $x_4$, it should be valued as:

$V(x_1, x_2, x_3, x_4) = \frac{1}{4} (x_1 + x_2 + x_3 + x_4)$

This follows because we can break the game into two half/half games performed one after the other.

Now we can deduce $V(x_1, x_2, x_3)$ (where three groups of 4 computers receive different rewards). We claim that a rational decision maker should pick value functions so that:

$V(x_1, x_2, x_3, V(x_1, x_2, x_3)) = V(x_1, x_2, x_3)$

which using our result for the 4-reward game implies:

$V(x_1, x_2, x_3) = \frac{1}{3} (x_1 + x_2 + x_3)$

Finally, consider an unbalanced game where 8 computers receive reward $x_1$ and 4 receive reward $x_2$. Call the reward function $U(x_1, x_2)$ (U for 'unbalanced').

Consider combining this unbalanced game with a subsequent third/third/third game where the computers in the $x_2$ group receive nothing, half the $x_1$ group receive reward $y$, and the other half receive reward $-y$. The value of the combined game is then:

$V(x_1 + y, x_1 - y, x_2) = \frac{2}{3} x_1 + \frac{1}{3} x_2$

But we know from previous results that the two conscious states which result from the first game will be indifferent about playing the follow up game (it has value 0 for each of them), so they should also be indifferent before discovering the result of game, so the combined game should have the same value as the unbalanced game, and we have:

$U(x_1, x_2) = \frac{2}{3} x_1 + \frac{1}{3} x_2$

So in particular, if we are considering whether to give a reward $x$ to either 8 computers or 6 computers, this argument implies that the computer consciousness itself should prefer to have the reward given to the 8, because $U(x, 0) > V(x, 0)$.

A defender of the Ignore-Copies view must reject this argument. It seems they should therefore also reject the quantum version.

Other derivations of the Born rule
==================================

As far as I am aware, the many other proposed derivations of the Born rule are likely not in direct conflict with the Ignore-Copies view, but they are also less appealing. The nice feature of the Born rule derivations discussed so far is that they tackle head-on the philosophical problem of how probabilities can play any role in a fully deterministic theory. In contrast, the other derivations that I am aware of take for granted the fact that we are looking for some probability measure over quantum state branches, and then proceed by showing that the only probability measure consistent with some reasonable sounding assumptions is the Born rule. The reason why probabilities should enter into the theory at all is left unexplained. For me, the big problem with this is that it is then unclear against what standards we are supposed to judge the plausibility of their assumptions.

If these are the only derivations of the Born rule that a believer in the Ignore-Copies view is left with, then it would appear that the plausibility of MWI has been at least somewhat reduced. I imagine intuitions will vary here, but personally I think that if we lose the derivations based on self-locating uncertainty and decision theory, then this poses a significant problem for MWI.

Possible Conclusions
====================

If you accept the arguments that I have made here, then I think there are a number of ways that you could respond. I review some of these possibilities now.

Ignore-Copies and MWI are both false
------------------------------------

If you have little credence in either the Ignore-Copies view or MWI, or if you reject the premise that it is possible for a conscious experience to be perfectly copied, then the arguments here have little relevance to you. This post has been a purely academic exercise with no bearing on the real world. Or at best, perhaps it has further weakened the plausibility of each of these views.

Ignore-Copies implies MWI is false
----------------------------------

If you are convinced by the Ignore-Copies view, then you may choose to take the arguments here as a reason to believe that MWI false. My own confidence in it has been significantly reduced during the writing of this post.

MWI implies Ignore-Copies is false
----------------------------------

If you are convinced by MWI, then you may choose to take the arguments here as reason to believe that Ignore-Copies is false. I find this possibility remarkable! On the face of it, Ignore-Copies is an extremely abstract philosophical position relating to consciousness and ethics. But under this interpretation, we would essentially be saying that *we can test the Ignore-Copies view empirically* every time that we measure Born rule probabilities! And it fails the test!

MWI and Ignore-Copies are both true, but we need a better derivation of Born rule probabilities
-----------------------------------------------------------------------------------------------

I think this possibility is the most interesting. Despite all the arguments I have made, I still put significant credence on this possibility. None of the arguments I have made prove that MWI and Ignore-Copies are inconsistent. They only show that there are problems with some of the Born rule derivations that have been proposed *so far*. Perhaps there is a derivation of the Born rule still to be discovered (or which exists but I am unaware of) that a believer in Ignore-Copies would be satisfied with.

I have explained already why I think the Ignore-Copies view is plausible, but I did not spend much time defending MWI. I will try to explain now why I still find it so compelling.

Quantum mechanics has this ugly feature: there is a thing called 'measurement' which behaves differently to everything else in the theory, and the theory does not even tell us precisely what should count as a measurement and what shouldn't. On the other hand, it seems like the theory *wants* to solve this problem by itself. The non-measurement part of the theory contains this phenomenon called "decoherence" which already explains almost everything about what happens when a measurement is made. The only missing ingredient is the Born rule. But it seems that the Born rule is at least in some sense "natural", as evidenced by the many "derivations" of it that have been proposed in the literature. It is very tempting to conclude that we should take this part of the theory seriously, ditch the concept of measurement completely, and accept MWI.

So lets suppose for a moment that MWI and Ignore-Copies *are* both true. What would a convincing derivation of the Born rule look like? It would be nice if something like the following conjecture were true:

> **Anthropic Born Rule Conjecture**: The principles of quantum mechanics imply that the vast majority of the *distinct* conscious states that are encoded in the quantum state of the universe will have histories that are consistent with the Born rule probabilities.

If this conjecture were true, then I think we would be done. The Born rule could be defended on anthropic grounds. There would be no tension with the Ignore-Copies view, since the conjecture explicitly refers to distinct conscious states.

It might appear that this conjecture is obviously false. In our discussion of Alice measuring the state $\sqrt{\frac{2}{3}}\left|0\right> + \sqrt{\frac{1}{3}}\left|1\right>$, did we not find that there were only two distinct copies of Alice after the measurement? If we then repeat this measurement N times, it seems we should expect exactly $2^N$ distinguishable copies of Alice, one for each possible sequence of measurement outcomes. If N is large, then the vast majority of these copies would see a sequence of measurement outcomes consistent with probability 1/2 and 1/2, not 2/3 and 1/3.

But this simplistic analysis ignores that the measurement of our qubit is not the only way in which the universe will be branching. Countless quantum "measurements" are being made all of the time (indeed, the consideration of additional measurements is crucial to all derivations of the Born rule that I have seen, even if the assertion is usually made that it is not necessary for these additional measurements to actually happen). If these additional measurements can also have an effect on the conscious state of Alice, then the question of how many distinguishable copies of her there will be after some time is made much more complicated. Perhaps we should not be so quick to dismiss the above conjecture out of hand.

The trouble is, although the conjecture may not be as ridiculous as it first sounds, it is also hard to see why there is any reason to expect that it should be true. If we were to attempt to rigorously prove the conjecture within the mathematical framework of quantum mechanics, we would immediately run into some big questions that we have glossed over in this post so far: What is consciousness? How do we know when two conscious states are 'distinct'? What does it mean for a conscious state to be 'encoded' in a quantum state?

But even putting these questions to one side, there seem to be good reasons to doubt that a proof of the above conjecture can be made to work. We can understand the difficulty by considering a recent derivation of the Born rule due to [Short](https://arxiv.org/pdf/2106.16145). This derivation is of the third kind discussed above. It is taken for granted that we are looking for a probability measure over worlds, some axioms of the probability measure are stipulated, and it is then shown that the Born rule is the only measure consistent with those axioms. It is interesting then to consider these axioms while trying to hold in our mind the interpretation that this measure ought to be counting *distinct* conscious states. We hit a problem as soon as we encounter this axiom:

> Weak connection with transformations - If the  
> set of worlds can be partitioned in such a way  
> that a transformation T acts separately on each  
> part, then T will preserve the total probability  
> of each part. This captures the intuition that  
> probability cannot ‘flow’ between worlds that are  
> uncoupled by the dynamics.

Something like this axiom seems to be common to most of the Born rule derivations that I have seen. The problem is that this axiom is obviously not true if we want our probability measure to be counting distinguishable conscious states. To see this we just need to consider a transformation which creates (or destroys) additional observers in one of the 'parts' (for example, by causing additional branching). If we would like to prove our conjecture, it seems we cannot do so by simply adapting one of the existing derivations of the Born rule. Something new would be required.

But if our conjecture is false then I have a hard time seeing how a defender of Ignore-Copies could ever be satisfied with MWI. If a significant proportion of the distinguishable conscious states in the multiverse do not observe Born rule probabilities when they look into their past, then it would seem that the only way to recover the Born rule would be to claim that for some strange reason, some of these conscious states should carry more 'weight' than others. A believer in Ignore-Copies should find such a claim mysterious.

References
==========

Bostrom, N. Quantity of experience: brain-duplication and degrees of consciousness. *Minds & Machines* **16**, 185–200 (2006). [https://doi.org/10.1007/s11023-006-9036-0](https://doi.org/10.1007/s11023-006-9036-0)

ELGA, A. (2004), Defeating Dr. Evil with Self-Locating Belief. Philosophy and Phenomenological Research, 69: 383-396. [https://doi.org/10.1111/j.1933-1592.2004.tb00400.x](https://doi.org/10.1111/j.1933-1592.2004.tb00400.x)

Parfit, Derek (1984). Reasons and Persons. Oxford, GB: Oxford University Press.

Sebens, Charles T. and Carroll, Sean M. (2018), Self-locating Uncertainty and the Origin of Probability in Everettian Quantum Mechanics. The British Journal for the Philosophy of Science, 69: 25-74. [http://dx.doi.org/10.1093/bjps/axw004](http://dx.doi.org/10.1093/bjps/axw004) 

[Deutsch David](https://royalsocietypublishing.org/author/Deutsch%2C+David) 1999, Quantum theory of probability and decisions, *Proc. R. Soc. Lond. A.***455**3129–3137 [http://doi.org/10.1098/rspa.1999.0443](https://doi.org/10.1098/rspa.1999.0443)

Short, A. J. (2023). Probability in many-worlds theories. *Quantum*, *7*(971), 1-10. Article A2. [https://doi.org/10.22331/q-2023-04-06-971](https://doi.org/10.22331/q-2023-04-06-971) 

[Saunders Simon](https://royalsocietypublishing.org/author/Saunders%2C+Simon), 2021, Branch-counting in the Everett interpretation of quantum mechanics, *Proc. R. Soc. A.* **477**20210600 [http://doi.org/10.1098/rspa.2021.0600](https://doi.org/10.1098/rspa.2021.0600)

Mateus Araújo, Why I am unhappy about all derivations of the Born rule (including mine), Blogpost, [https://mateusaraujo.info/2021/03/12/why-i-am-unhappy-about-all-derivations-of-the-born-rule-including-mine/](https://mateusaraujo.info/2021/03/12/why-i-am-unhappy-about-all-derivations-of-the-born-rule-including-mine/)