I recently pointed out a curious implication of the no-duplicate-worlds view: whether you break a perfect symmetry on one side or the other, this amounts to the same possibility. Suppose our two-sphere world comes with an arrow in the middle which points directly to one sphere, which we dub Bob1. Could it have pointed to the other (Bob2) instead? Scrap the 'instead'. It could have pointed to Bob2, but this is not a possibility grounded in any other possible world. Bob1 is a perfect counterpart of Bob2, after all. So the possibility claim can be made true by this very same world.
Jack has an interesting post on his new blog which aims to make this a tougher bullet to bite. His trick is to ask us to imagine a variation where it's random which way the arrow ends up pointing. Now, the "two" possible results may be accommodated exactly as above. So there is the one result, and then there is the "other" possibility which simply consists in a counterpart-theoretic reinterpretation of this very same possible world. So far so good. But now I'm struck by a new concern: how are we to distinguish this randomized case from the deterministic arrow world I described previously? What is it that makes this newly described world an indeterministic one, if not a plurality of possible futures?
Here's my solution: randomness consists in symmetry. There is nothing in the initial makeup of the world which tilts in one direction rather than the other. (Imagine the arrow points straight up until an indeterministic physical process causes it to fall to one side and break the perfect symmetry.) In a deterministic world, by contrast, the initial state of the world is physically 'tilted' towards one sphere, so it stands in different qualitative relations from the other sphere; they are no longer perfect (incl. extrinsic) duplicates. Thus we can distinguish the worlds where the asymmetry is introduced randomly vs. deterministically.