Newcomb's problem was introduced in a paper by Nozick called "Newcomb's Problem and Two Principles of Choice". The two principles are dominance and expected value. The idea is that Newcomb's problem is a case where these principles give conflicting recommendations: expected value says one-box, but dominance says two-box. This seems impossible because dominance implies higher expected value.
Savage calls a mapping from a state of the world to a consequence an "act". I suppose in Newcomb's problem, the states of the world are money under one or both boxes, the acts are one-box or two-box, and the consequences are the amount of money you get.
Two states of the world? Nozick would say four. So let's back up: what is a state of the world?
Wittgenstein famously began his Tractatus by saying the world is everything that is the case. But Savage says the world is "the object about which the person is concerned". For example, if we are wondering whether an egg is rotten, the world is the egg. He does acknowledge that we may want to consider a more expansive notion of the world:
In application of the theory, the question will arise as to which world to use in a given context. Thus, if the person is interested in the only brown egg in a dozen, should that egg or the whole dozen be taken as the world?
He also considers as a possibility something more like what I think a philosopher thinks of when they see the phrase "the world": the "exact and entire past, present, and future history of the universe". But this, he says, is "vague", and:
It may also be added that the use of modest little worlds, tailored to particular contexts, is often a simplification, the advantage of which is justified by a considerable body of mathematical experience with related ideas.
I think Savage misses the main issue, though, which is the inclusion of the decision-maker. Nozick doesn't necessarily consider the whole universe, but to him, there are four states of the world, not two. In fact, it seems to me that what Nozick calls a state of the world is precisely what Savage calls a consequence. Or, at least, it determines it: the state of the world includes not just what money is under the boxes but which boxes you take, and therefore determines the consequence, that is, how much money you get. In fact, Savage writes:
Consequences might appropriately be called states of the person as opposed to stage of the world.
Doesn't he see that this requires some limitation in the scope of "the world", not just as a simplification, but as a requirement to exclude the person?
I would say one act dominates another if it's consequences preferred in every state of the world. Or, that's strict dominance, we have weak dominance if they're not dispreferred, and preferred in at least one state. Nozick's definition of dominance is:
Dominance Principle: If there is a partition of states of the world such that relative to it, action A weakly dominates action B, then A should be performed rather than B.
Action A weakly dominates action B for person P iff, for each state of the world, P either prefers the consequence of A to the consequence of B, or is indifferent between the two consequences, and for some state of the world, P prefers the consequence of A to the consequence of B.
But what's this about a partition? I think that what Savage calls states of the world, Nozick considers partitions of states. "There is money in both boxes" contains "There is money in both boxes and I take both" as well as "there is money in both boxes and I take one". Four states of Novick's world, partitioned into two states of Savage's world. He goes on to say:
The dominance principle as presented here speaks of dominance relative to a partition of the states of the world. This relativization is normally not made explicit, which perhaps accounts for the fact that writers did not mention that it may be that relative to one partition of the states of the world, one action A dominates another, whereas relative to another partition of the states of the world, it does not.
