So S∈P(S) is an event (a subset of elementary states, S⊆S).
E.g., we could have S be all the possible worlds; S={S1,...,Sn} be the possible worlds in which featherless bipeds evolved; and S7∈S be our actual world.
No, the codomain of gamma is the set of (distributions over) consequences.
Hammond’s notation is inspired by the Savage framework in which states and consequences are distinct. Savage thinks of a consequence as the result of behaviour or action in some state, though this isn’t so intuitively applicable in the case of decision trees. I included it for completeness but I don’t use the gamma function explicitly anywhere.
What’s bold S here?
It’s the set of elementary states.
So S∈P(S) is an event (a subset of elementary states, S⊆S).
E.g., we could have S be all the possible worlds; S={S1,...,Sn} be the possible worlds in which featherless bipeds evolved; and S7∈S be our actual world.
Is the idea that the set of “states” is the codomain of gamma?
No, the codomain of gamma is the set of (distributions over) consequences.
Hammond’s notation is inspired by the Savage framework in which states and consequences are distinct. Savage thinks of a consequence as the result of behaviour or action in some state, though this isn’t so intuitively applicable in the case of decision trees. I included it for completeness but I don’t use the gamma function explicitly anywhere.