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Copenhagen-

  1. Suppose I ask you to bet on the outcome of 100 spin measurements. And you believe in the Copenhagen interpretation for now. The odds given by the Born rule, for each experiment, are 50:50 for spin_up : spin_down. You are only allowed to bet in favor of spin_up. It it comes spin_up, you win $100. If it comes spin_down, you lose $1000. Clearly, this bet shouldn't be taken.

  2. Now, I change the initial wavefunction of the experiment, such that the Born rule gives you 10000:1 in favor of spin_up. The monetary gains are the same. Now, the bet can be taken.

Many worlds:

  1. Now you believe in many worlds. The experiment is the same as in (1). If you take the bet, one world will get created where you win $100 and one world will get created where you lose $1000. This bet seems identical to (1) and hence shouldn't be taken.

  2. Same experiment as in (2). If you take the bet, I promise to create one world where you win $100 and one world where you lose $1000. But I will now clone the first world 10000 times. So there will be 10000 identical cloned worlds where you gain $100.

My question is, does the bet (4) really seem favorable over bet (3)? What does it matter if I clone the success world 10000 times?

The expectation value does turn out to be the same in (4) and (2), but the interpretation of the expectation value in (4) is something like: "If God were to randomly pick one of my futures, they are more likely to pick a successful future (because it's been cloned a lot)".

When you believe in the Copenhagen interpretation, you can interpet bets in terms of the likelihood that your current self has a good future. This is because your future is unique.

A "duplication of worlds" philosophy does not reproduce the same interpreration of probabilities as in Copenhagen. Statements like "My current self is more likely to end up in world A than world B" become meaningless, because your current self is the history of all the futures that are created. It does not end up in one world over another.

So again, Is a world duplication philosophy really equivalent to how we usually interpret probabilities?. I think a philosophy with a unique future is necessary to have the usual interpretation of probabilities. The probabilities then refer to the ratio of success:losses in repeated trials in that unique future. In Many Worlds, we are supposed to find a 10000:1 bet favorable, not because we think winning is more likely, but because we are supposed to care about the number of clones of our success world.

Ryder Rude
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  • If there are c clone worlds and a alternative worlds and I'm randomly thrown into a world then the probability I'll end up in a clone world is c/(c + a). I have a 50% chance of being in a clone world when c = a. As c > a, my odds improve. What does c > a mean? – Agent Smith May 14 '23 at 06:06
  • This question is about possible vs actual worlds. Nothing would change if you made them classical instead of quantum mechanical. – benrg May 14 '23 at 06:08
  • We can interpet bets in terms of current self's future not because said future is unique, but because the present self is continuous with the future self. And nothing prevents the current self from being continuous with multiple future selves. So those are not "clones". Of course, one may not care about their future self or selves anyway, after all, they are not the same as the current one, but in that case uniqueness/multiplicity makes no difference either. In other words, we are in the exact same position in Copenhagen and MWI as far as attractiveness of these bets. – Conifold May 14 '23 at 06:11
  • @AgentSmith I can intepret your question in two ways : 1. You are looking at the whole universe from a meta God-like perspective and choosing one branch randomly. In this case, yes the usual interpretation of probabilities is restored. I mentioned this in my post when I talked about expectation values. 2. You are considering yourself to be inside the universe, performing a probabilistic experiment. In this case, the statement that "you are randomly being thrown in one of the worlds" is meaningless. Your current self is the history of all the worlds. I've made the post about the second case. – Ryder Rude May 14 '23 at 06:16
  • @RyderRude, who would be doing the cloning? – Agent Smith May 14 '23 at 06:21
  • @AgentSmith If you believe in many worlds, the cloning happens automatically. – Ryder Rude May 14 '23 at 06:21
  • @RyderRude, I see, but you were very specific about what exactly is being cloned. – Agent Smith May 14 '23 at 06:25
  • @benrg Yes, I mostly just want to discuss the (non-) equivalence of probabilities and world duplication, with respect to the incentive to take bets. – Ryder Rude May 14 '23 at 06:30
  • @Conifold That is the intuitive way to think about it: since we are just discarding what we can't observe, it doesn't matter. But, tell me why do you care that your success world gets cloned 10000 times rather than just once? Please also see [this](https://physics.stackexchange.com/questions/762350/does-the-many-worlds-interpretation-trivialize-the-born-rule) post for one case where it does matter. It's not as simple as merely discarding what you can't observe. That post is about Many Worlds without world duplication – Ryder Rude May 14 '23 at 06:33
  • We do not care whether a world branches 10000 times, once or doesn't branch at all. This is why there is no difference between Copenhagen and MWI here. The magic of "unique" in your "because your future is unique" eludes me. All we care about (or rather all that a rational better should care about) is the probability of our future self winning $100 given the current self's position. *That* is the "interpretation", *in both cases*. One can imagine God picking futures to actualize in Copenhagen just as well, if they want to. Or not. Only odds matter, not mechanisms of their realization. – Conifold May 14 '23 at 07:14
  • @Conifold But the "world duplication" is *not* a mechanism to realise the usual probability odds, like I showed in my post. The usual probability interpretation talks about "The likelihood that my current self ends up in a particular world". This statement is meaningless in many worlds, because you current self is in the history all of the worlds that get created upon measurement. So the only reason you can find bets beneficial is that you care about the number of clones. – Ryder Rude May 14 '23 at 07:29
  • @Conifold A many worlder, if they truly believe in their philosophy, cannot claim prior to measurement that their current self is more likely to end up in a particular world. Now, you may say that some of the post- repeated-measurement-futures of the Many Worlder will happen to agree with their claim. But this follows trivially from combinatorics. And most of the futures will not agree with that claim. Please see [this post](https://physics.stackexchange.com/questions/762350/does-the-many-worlds-interpretation-trivialize-the-born-rule). – Ryder Rude May 14 '23 at 07:44
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    What you call "usual" is not usual and, strictly speaking, incoherent. The "current self" goes away along with the current moment. If "my current self ends up in a particular world" is to have a meaning "my current self" better not be used literally. You can use temporal contiguity, continuity of awareness or some such to connect current to future selves for this purpose, but in none of that is uniqueness salient. And that is what your argument lacks: a plausible account of self-identity through time where uniqueness matters, to distinguish Copenhagen and MWI. I am skeptical that one exists. – Conifold May 14 '23 at 07:54
  • @Conifold Yes, the "current self" itself cannot end up in the future. This is why I made precise what I meant by this. By "your current self ending up in a future", I mean that that future has your current self associated to it in its history. I've always used this "history" term in making my argument. So, a Many worlder cannot claim that their current self is more likely to end up in any future, because the current self ends up in all futures. – Ryder Rude May 14 '23 at 08:03
  • So it is not the "current self" after all. But if the current self cares about a different (future) self under Copenhagen anyway, why should she care any more or less about multiple future selves under MWI? Should a many worlder rephrase to "more of the current self will end up in good future" to comply with your linguistic convention? Actually, they should be given the same poetic license you give yourself and keep saying "current self is more likely to end up in good future". After all, when ending up in all futures current self is no longer a single self. Is there anything beyond verbiage? – Conifold May 14 '23 at 08:58
  • @Conifold Yes, this is what I'm trying to say. The Many Worlder can only claim: "My current self will end up in a higher number of clones of the good world than the bad world". They care about the number of clones. You're saying that this is equivalent to saying "My current self is more likely to end up in the good world". I don't think so. Interpreting in terms of likelihood is necessary to practice quantum mechanics. Any actual practitioner of many worlds interprets probabilities in terms of likelihood, even if their philosophy technically doesn't allow that – Ryder Rude May 14 '23 at 09:08
  • Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/146004/discussion-between-ryder-rude-and-conifold). – Ryder Rude May 14 '23 at 09:31
  • I am not saying it is equivalent, for that both phrases need to have independent meanings. I am saying that this is no less legitimate than your own creative rephrasing of "current self". Your use of "clones" seems to me misleading, even to yourself, and neither explains why uniqueness makes any substantive difference. You may be right that people have some intuitions attached to object permanence in a single timeline, but those are incoherent upon inspection. Once we make them coherent the supposed "usualness" vanishes, or any particular relation to use of likelihoods. – Conifold May 14 '23 at 09:41

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