Sleeping Beauty and other kinds of multiverses

After I wrote that post about Quantum Sleeping Beauty, Sean Carroll wrote about it too.  At the risk of too much Sleeping Beauty, I also want to discuss the implications not just on Everettian Quantum Mechanics (EQM), but on other multiverse scenarios.

Yes, there are multiple multiverse scenarios.  For our purposes, we can use Max Tegmark's four-fold classification of multiverses:

Level I: There are different worlds separated by extremely large distances.  For example, if you believe that the universe is uniform and infinite, then obviously the observable universe, which is limited by the speed of light, is much smaller than the universe as a whole.  Thus there will be many copies of the observable universe with arbitrary configurations of stuff in them.

Level II: In some theories of cosmology, physical constants are also ultimately made of stuff, albeit the kind of stuff that gets decided very early on in a universe, and which is very stable thereafter.  But some inflationary cosmology scenarios predict many pocket universes forming, each with possibly different physical constants.

Level III: This is the multiverse described by EQM.

Level IV: The particular laws which govern our universe are arbitrary, and the equations could have had many other arrangements.  A level IV multiverse theory says that universes described by different math are not merely possible, but real.

[blockquote for organization, not because I'm quoting Tegmark]

As I explained in 2012, multiverses are not scientific theories by themselves, since they can't really be experimentally verified.  Rather, they are predictions of larger scientific theories.  The level I multiverse is predicted by most theories of uniform cosmology.  Level II is predicted by inflationary cosmology.  Level III is predicted by quantum mechanics, depending on your interpretation.  Level IV... well I think that's just speculation.

Multiverse scenarios live or die based on the larger theory that predicts them.  Nonetheless, some people, even physicists, have suggested that multiverses themselves can be confirmed or disconfirmed.  For instance, physicists argue that one of the reasons to prefer inflationary cosmology to other theories is because it predicts a Level II multiverse.  A Level II multiverse would explain why physical constants seem to be fine-tuned for the existence of life.  Thus, the very fact that we exist is evidence for a Level II multiverse, and thus a confirmation of inflationary cosmology.

What does this all have to do with Sleeping Beauty?

Imagine that God is a philosopher.  God flips a coin, and if it's heads, then he creates a multiverse, where there are many copies of Sleeping Beauty.  If it's tails, he creates a universe where there are very few copies of Sleeping Beauty (or maybe none at all).  Sleeping Beauty knows all this, and has just woken up.  What probability should Sleeping Beauty assign to waking up in the multiverse?

If you take the thirder position on Sleeping Beauty, then Sleeping Beauty should conclude that she most likely woke up in a multiverse.*  Possibly vastly more likely.  Possibly infinitely more likely.  In fact, you might ask why we even bother considering any non-multiverse theories.  Seen this way, the unimaginable largeness of the multiverse is itself evidence for the multiverse.

*Here I'm talking about Level I, II, or IV multiverse.  The case of the level III multiverse is analogous to the Quantum Sleeping Beauty problem discussed in the previous post.  That post argued that you can consistently take a thirder position in the classical Sleeping Beauty problem, and a halfer position in the Quantum Sleeping Beauty problem.

But that doesn't seem right.  My physicist intuition goes against it.  Even if the multiverse explains fine-tuned constants, this is at best very weak evidence for the multiverse.  Even physicists who make that argument don't take it to the inevitable conclusion that we are infinitely more likely to live in a multiverse.

Nonetheless, the philosophical consensus is that the thirder position is correct.  Mind you, philosophy consensuses are rarely very strong, and maybe it's just wrong in this case. But still, the thirder arguments are pretty solid, and this is a conflict that needs to be resolved.

To summarize, we have three options:

1. Physicists are wrong, and multiverse theories are infinitely preferred over their alternatives.
2. Philosophers are wrong, and the thirder view is incorrect.
3. There is a problem with the analogy between the two cases.

Some light can be shed by adopting the phenomenalist view of the Sleeping Beauty problem, which is a sort of middle ground between the halfers and thirders that everyone can agree with.  The phenomenalist observes that the problem becomes trivial when we attach consequences to the probabilities.

Say that Sleeping Beauty (in the original problem) gets a reward every time she guesses the coin flip correctly.  Therefore, she should make bets as if she were a thirder, because if the coin is heads, she'll get double the usual payoff.  On the other hand, say that Sleeping Beauty only gets a single reward after the experiment if she guessed correctly.  In this case, she should make bets as if she were a halfer.

Now take the Sleeping Beauty multiverse problem.  If each copy of Sleeping Beauty gets a reward for guessing correctly, then it seems better to guess that she's in a multiverse.  That way, many copies of Sleeping Beauty get rewards.  On the other hand, what exactly do we care about?  Do we care about the average* reward given to Sleeping Beauty copies?  Or do we care about the sum total of the rewards to all copies?  If the former, then Sleeping Beauty should make bets as a halfer.  If the latter, then Sleeping Beauty should make bets as a thirder.

*Tangentially, there's another complication when we talk about the average Sleeping Beauty in a multiverse with infinitely many copies of her.  How do we average over infinite copies?  We can't even average over finite volumes of space, because volume isn't even constant in inflationary cosmology.  This is known as the measure problem.

But this doesn't seem to be a very satisfying solution.  "Is there a multiverse or not?"  is not satisfyingly answered by "It depends, do you think average utility or total utility is more important?"  Philosophers need to get on this and figure out what's going on.