How superduper does a physicalist supervenience need to be?

Note: this is the first published presentation and defense of the 'proper subset strategy' for making sense of non-reductive physicalism or the associated notion of realization; this is sometimes, inaccurately, called "Shoemaker's subset strategy"; if people could either call it the 'subset strategy' or better yet, add my name to the mix I would appreciate it. Horgan claims that physicalism requires "superdupervenience" -- supervenience plus robust ontological explanation of the supervenient in terms of the base properties. I argue that Horgan's account fails to rule out physically unacceptable emergence. I rather suggest that this and other unacceptable possibilities may be ruled out by requiring that each individual causal power in the set associated with a given supervenient property be numerically identical with a causal power in the set associated with its base property. I go on to show that a wide variety of physicalist accounts, both reductive and non-reductive, are implicitly or explicitly designed to meet this condition, and so are more similar than they seem. In particular, non-reductive physicalism accounts typically appeal to a relation plausibly ensuring that the powers of a higher-level property are a proper subset of those of its physical base property.