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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.

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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 at 6:06
  • This question is about possible vs actual worlds. Nothing would change if you made them classical instead of quantum mechanical.
    – benrg
    May 14 at 6:08
  • 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 at 6:11
  • @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 at 6:16
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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 at 7:54

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