A complexity of multiverse theories

One might think that a multiverse theory is in some sense simpler than a single universe theory. For where the single universe theory has to posit a particular value of a physical constant, the multiverse theory can allow that value to be random.

But unless there is a privileged probability distribution for that constant, randomness makes for a much greater number of degrees of arbitrariness—a much greater complexity. For instance, suppose we have some constant a on which the only constraint is that it is a real number. The single universe theory has one basic constant, a, and in that regard has one degree of arbitrariness. But the multiverse theory needs to specify the probability distribution for a. And that is an infinite number of degrees of arbitrariness. For instance, if the probability distribution is Gaussian, we need to specify the mean and the standard deviation—two basic constants. But it need not be Gaussian; there are infinitely many possible probability distributions, with no upper bound on the number of parameters of each.

Now it may be that in the end there will be a privileged probability distribution for these constants. But unless we currently know that there is such a "naturally privileged" probability distribution (and if we know that, then my argument does not apply), this is just as conjectural as thinking that what seems like an arbitrary value of a will in fact turn out to be in some way natural.

What if instead of randomness, which requires a probability distribution, we simply have a brute fact that there is an infinity of universes with different values of constants, and no natural distributions on the space of universes. In that case we have more arbitrariness than in a single universe scenario: where the single universe theory left one constant unexplained, the multiverse has one per universe unexplained.

A multiverse theorist who thinks all possible combinations of constants are realized has an answer to the problem, however. For there, there is neither randomness nor bruteness in the constants, at least as long as each combination is realized only once (or if some are realized more than once, there is some natural explanation why, perhaps due to symmetries of some sort).