How is it not a simulation? [...] How do you define simulation in a way that [...]
I'm not sure how to answer either question, and I'm not sure anyone has a perfectly satisfactory definition of "simulation". I'm pretty sure I've never seen one. But consider the following scenario.
Omega looks at the structure of your brain, and deduces a theorem of the following sort: For a broad class of individuals with these, and those, and these other, features, they will almost all make the one-box-or-two decision the same way. (The theorem doesn't say which way. In fact, the actual form of the theorem is: "Given features A, B and C, any two individuals for which parameters X, Y and Z are within delta will go the same way with probability at least 1-epsilon", and features A,B,C are found about as often in one-boxers as in two-boxers. Finding and proving such a theorem doesn't in itself give Omega much information about whether you're likely to one-box or two-box.) Omega then picks one of those individuals which isn't you and simulates it; then Omega knows with high confidence which way you will choose.
Let's suppose -- since so far as I can see it makes no difference to the best arguments I know of either for one-boxing or for two-boxing -- that Omega tells you all of the above. (But not, of course, which way the prediction ended up.)
In this scenario, what grounds are there for thinking that you could as easily be in Omega's simulation as in the outside world? You know that the actual literal simulation was of someone else. You know that the theorem-proving was broad enough to cover a wide class of individuals besides yourself, including both one-boxers and two-boxers. So what's going on here that's a simulation you could be "in"?
(Technical note: If you take the form I gave for the theorem too literally, you'll find that things couldn't actually be quite as I described. Seeing why and figuring out how to patch it are left as exercises for the reader.)
How would such a device work?
There's a wormhole between our present and our future. Omega looks through it and sees your lips move.
Omega then picks one of those individuals which isn't you and simulates it; then Omega knows with high confidence which way you will choose.
Seems like Omega is simulating me for the purposes which matter. The "isn't you" statement is not as trivial as it sounds, it requires a widely agreed upon definition of identity, something that gets blurred easily once you allow for human simulation. For example, how do you know you are not a part of the proof? How do you know that the statement that Omega tells you that it simulated "someone else&qu...
Finding a good decision theory is hard. Previous attempts, such as Timeless Decision Theory, work, it seems, in providing a stable, effective decision theory, but are mathematically complicated. Simpler theories, like CDT or EDT, are much more intuitive, but have deep flaws. They fail at certain problems, and thus violate the maxim that rational agents should win. This makes them imperfect.
But it seems to me that there is a relatively simple fix one could make to them, in the style of TDT, to extend their power considerably. Here I will show an implementation of such an extension of CDT, that wins on the problems that classic CDT fails on. It quite possibly could turn out that this is not as powerful as TDT, but it is a significant step in that direction, starting only from the naivest of decision theories. It also could turn out that this is nothing more than a reformulation of TDT or a lesser version thereof. In that case, this still has some value as a simpler formulation, easier to understand. Because as it stands, TDT seems like a far cry from a trivial extension of the basic, intuitive decision theories, as this hopes to be.
We will start by remarking that when CDT (or EDT) tries to figure out the expected value or a action or outcome, the naive way which it does so drops crucial information, which is what TDT manages to preserve. As such, I will try to calculate a CDT with this information not dropped. This information is, for CDT, the fact that Omega has simulated you and figured out what you are going to do. Why does a CDT agent automatically assume that it is the "real" one, so to speak? This trivial tweak seems powerful. I will, for the purpose of this post, call this tweaked version of CDT "Simulationist Causal Decision Theory", or SCDT for short.
Let's run this tweaked version though Newcomb's problem. Let Alice be a SCDT agent. Before the problem begins, as is standard in Newcomb's problem, Omega looks at Alice and calculates what choice Alice will make in the game. Without to much loss of generality, we can assume that Omega directly simulates Alice, and runs the simulation through the a simulation of the game, in order make the determination of what choice Alice will make. In other formulations of Newcomb's problem, Omega figures in out some other way what Alice will do, say by doing a formal analysis of her source code, but that seems intuitively equivalent. This is a possible flaw, but if the different versions of Newcomb's problem are equivalent (as they seem to be) this point evaporates, and so we will put it aside for now, and continue.
We will call the simulated agent SimAlice. SimAlice does not know, of course, that she is being simulated, and is an exact copy of Alice in all respects. In particular, she also uses the same SCDT thought processes as Alice, and she has the same utility function as Alice.
So, Alice (or SimAlice, she doesn't know which one she is) is presented with the game. She reasons thusly: