Rather than transforming the problem in the way you did, transform it so that you move first—Omega doesn’t put money in the boxes until you say which one(s) you want.
Given this reframing, f(Always One-Box) seems like a fairly trivial solution, and neither paradoxical nor terribly difficult to represent mathematically...
As a decision problem, Newcomb’s problem is rather pointless, IMHO. As a thought experiment helping us to understand the assumptions that are implicit in game theory, it could be rather useful. The thought experiment shows us that when a problem statement specifies a particular order of moves, what is really being specified is a state of knowledge at decision time. When a problem specifies that Omega moves first that is implicitly in contradiction to the claim that he knows what you will do when you move second. The implicit message is that Omega doesn’t know—the explicit message is that he does. If the explicit message is to be believed, then change the move order to make the implicit message match the explicit one.
However, here, many people seem to prefer to pretend that Newcomb problems constitute a decision theory problem which requires clever solution, rather than a bit of deliberate confusion constructed by violating the implicit rules of the problem genre.
Rather than transforming the problem in the way you did, transform it so that you move first—Omega doesn’t put money in the boxes until you say which one(s) you want.
As a decision problem, Newcomb’s problem is rather pointless, IMHO. As a thought experiment helping us to understand the assumptions that are implicit in game theory, it could be rather useful. The thought experiment shows us that when a problem statement specifies a particular order of moves, what is really being specified is a state of knowledge at decision time. When a problem specifies that Omega moves first that is implicitly in contradiction to the claim that he knows what you will do when you move second. The implicit message is that Omega doesn’t know—the explicit message is that he does. If the explicit message is to be believed, then change the move order to make the implicit message match the explicit one.
However, here, many people seem to prefer to pretend that Newcomb problems constitute a decision theory problem which requires clever solution, rather than a bit of deliberate confusion constructed by violating the implicit rules of the problem genre.