There is a particularly interesting and surprising difference that I am aware of between the MWI (many-worlds interpretation of quantum mechanics) and more general multiverse models like Tegmark's and especially Schmidhuber's. Even though the MWI is much better known and better accepted, it has a major unsolved problem that is actually dealt with much more successfully in larger multiverse models. That is the question of probability or measure.

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The MWI is basically modern quantum theory minus the concept of wave function collapse. Instead, the universal wave function is considered to just evolve as a single mathematical whole under a deterministic physical rule that is the generalized Schrodinger equation. In fact Everett's original work can be thought of as an attempt to investigate the question of what the world would look like if we didn't have random collapse and just had the smooth, deterministic evolution. Of course what he found was that even though there is no collapse, the universe would in a sense *appear* to have random collapses. Measurement-like interactions lead to effective "splits" in the mathematical expression of the wave function, separate branches that have essentially no more causal interactions between them. For all practical purposes the wave function can be split into pieces which evolve separately. Each piece corresponds to one possible outcome of the measurement. Each piece has an observer who "thinks" he sees that particular outcome. To him, the transition was discontinuous and random, exactly like what is called wave function collapse in the conventional interpretation. This is the amazing feature of the MWI, that by removing wave function collapse and simplifying the assumptions of the theory, we recover the prediction of an illusion of wave function collapse. It is IMO one of the most amazing philosophical results of the century. However, one piece is missing. Although Everett showed that the universe would, in effect, split and create separate observers who would observe separate outcomes, the question remains of how that relates to probability. The traditional quantum wave function collapse postulate not only says that we will observe a random outcome, it also tells what the probability of each possibility will be, based on what is called Born's rule. Deriving Born's rule from Everett's analysis has been difficult. In fact, it has been "solved" many times over the years, but none of the proposed solutions has really been satisfactory, which is why it keeps getting solved all over again. It is easy to show that the universe will split and observers will appear to observe random outcomes; it is hard to show that the most likely outcomes are the ones most likely to be observed. It is this problem, ironically, which is solved much more easily in the multiverse models we discuss. For example, in Schmidhuber's picture we see the multiverse as an ensemble produced by all possible computer programs. Each program produces output which is considered to be a universe. In this model, there is an easy and attractive argument for why some universes should have higher measure than others. If we consider only finite-sized programs, then an n-bit program is associated with a fraction of 1/2^n of all infinite bit strings. This means that if we imagine all such bit strings as the programs which are creating universes, shorter programs will occupy a greater fraction of the total space of universes. This leads to the most important (and perhaps only!) prediction of multiverse models: the universe we live in should be described by a relatively short and simple program (or mathematical description, in Tegmark's formulation). In principle, if we knew the program for our universe, we could calculate its measure (at least approximately). And likewise for the programs for other universes. And we have this simple and powerful argument that explains why certain programs should have much higher measure, and hence higher probability, than others. Yet no such argument appears to exist for the MWI. There, we have to rely on a variety of different assumptions to prove the Born rule, none of which have been widely accepted. It's interesting that these two conceptions of a multiverse have such different properties. Of course the branches in the MWI have a much closer mathematical relationship than the separate universes of a Schmidhuber multiverse, so the problem is more complicated with the MWI. It would be nice if there were a simple, information-based argument similar to the one used for the multiverse, that would produce the Born rule in the MWI. Hal Finney