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Wikipedia says

> While it has been claimed that Born's law can be derived from the many-worlds interpretation, the existing proofs have been criticized as circular.

So it seems you aren't alone in feeling this way.

Edit: Forgot the link - https://en.wikipedia.org/wiki/Born_rule

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Interesting. Following through the citations, this statement seems to be based on two quite old articles which criticise a different Born rule formulation based on frequency analysis:

https://link.springer.com/article/10.1007%2FBF02058273

https://arxiv.org/pdf/quant-ph/0409144.pdf

The latter article is more recent at 2005 but as far as I can tell Carroll's self-locating uncertainty ideas weren't introduced until around 2014.

Here's a blog post I wrote about this recently:

http://blog.rongarret.info/2019/07/the-trouble-with-many-wor...

The TL;DR is that the critics are correct. Deriving the Born rule begs the question because it makes an unjustified assumption (branching indifference) and also introduces an "invisible pink unicorn", i.e. a concept that, according to the theory, has physical significance but cannot be measured. In the case of MWI that concept is branch weights.

[UPDATE] The critique I wrote is based on Wallace (https://arxiv.org/abs/0906.2718). Carroll's argument appears to be somewhat different. I'm just now working my way through his paper (https://arxiv.org/abs/1405.7577) but I'd be very surprised if it did not also have some untenable assumption hidden in there somewhere.

Adrian Kent has a lengthy and detailed overview of MWIs pre-2009 which may interest you [0]. He discusses Wallace's formulation, contrasts it with other formulations and touches on branching indifference. He also has a more recent (and much more brief) article which refers to the Self-locating Uncertainty arguments [1].

I have previously come across Deutsch's formulation in terms of information flow [2], along with what seemed like a very strong criticism from Wallace and Christopher Timpson [3] that his model was not gauge-invariant.

[0] https://arxiv.org/pdf/0905.0624.pdf

[1] https://arxiv.org/abs/1408.1944

[2] https://arxiv.org/ftp/quant-ph/papers/9906/9906007.pdf

[3] http://users.ox.ac.uk/~bras2317/dhshort2.pdf

Thanks for the pointers. (This is turning into quite the little rabbit hole.)

What do you think of the simplified argument here: https://algassert.com/post/1902 ? Basically: ground the definition of probability into a reversible classical circuit then use that circuit's quantum behavior to generalize the definition to quantum (while assuming as few things as possible). In this case the assumptions are "limiting to amplitude 1 must mean limiting to probability 1" and no signalling.

Thanks for bringing that to my attention. My initial reaction is that the argument appears sufficiently cogent to merit respectful consideration. I predict that if someone were to dig into it, they'd find a question-begging assumption hidden somewhere. (The alternative is that this is a major breakthrough in physics, and my Bayesian prior on that is low.) But where that assumption is hiding is not immediately obvious to me. Looking for it seems like a worthwhile exercise.

I don't think the argument is novel, it's just a cut down counting/frequency argument. So any objections to those would presumably port over.

I dunno, I think you might be selling it short. This argument seems to be based on the mathematical continuity of probabilities, which is not something I can recall ever seeing before.

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