“subagent [C] that could choose U”—do you mean U⊆Ctrl(C) or U⊆Ensure(C) or neither of these? Since Ctrl is not closed under unions, I don’t think the controllables version of “could choose” is closed under coarsening the partition. (I can prove that the ensurables version is closed; but it would have been nice if the controllables version worked.)
ETA: Actually controllables do work out if I ignore the degenerate case of a singleton partition of the world. This is because, when considering partitions of the world, ensurables and controllables are almost the same thing.
“subagent [C] that could choose U”—do you mean U⊆Ctrl(C) or U⊆Ensure(C) or neither of these? Since Ctrl is not closed under unions, I don’t think the controllables version of “could choose” is closed under coarsening the partition. (I can prove that the ensurables version is closed; but it would have been nice if the controllables version worked.)
ETA: Actually controllables do work out if I ignore the degenerate case of a singleton partition of the world. This is because, when considering partitions of the world, ensurables and controllables are almost the same thing.