This post was inspired by taw urging us to mathematize Newcomb's problem and Eliezer telling me to post stuff I like instead of complaining.
To make Newcomb's problem more concrete we need a workable model of Omega. Let me count the ways:
1) Omega reads your decision from the future using a time loop. In this case the contents of the boxes are directly causally determined by your actions via the loop, and it's logical to one-box.
2) Omega simulates your decision algorithm. In this case the decision algorithm has indexical uncertainty on whether it's being run inside Omega or in the real world, and it's logical to one-box thus making Omega give the "real you" the million.
3) Omega "scans your brain and predicts your decision" without simulating you: calculates the FFT of your brainwaves or whatever. In this case you can intend to build an identical scanner, use it on yourself to determine what Omega predicted, and then do what you please. Hilarity ensues.
(NB: if Omega prohibits agents from using mechanical aids for self-introspection, this is in effect a restriction on how rational you're allowed to be. If so, all bets are off - this wasn't the deal.)
(Another NB: this case is distinct from 2 because it requires Omega, and thus your own scanner too, to terminate without simulating everything. A simulator Omega would go into infinite recursion if treated like this.)
4) Same as 3, but the universe only has room for one Omega, e.g. the God Almighty. Then ipso facto it cannot ever be modelled mathematically, and let's talk no more.
I guess this one is settled, folks. Any questions?
Aren't these rather ducking the point? The situations all seem to be assuming that we ourselves have Omega-level information and resources, in which case why do we care about the money anyway? I'd say the relevant cases are:
3b) Omega uses a scanner, but we don't know how the scanner works (or we'd be Omega-level entities ourselves).
5) Omega is using one of the above methods, or one we haven't thought of, but we don't know which. For all we know he could be reading the answers we gave on this blog post, and is just really good at guessing who will stick by what they say, and who won't. Unless we actually know the method with sufficient confidence to risk losing the million, we should one-box. ([Edit]: Originally wrote two-box here - I meant to say one-box)
3b) Our ignorance doesn't change the fact that, if the scanner is in principle repeatable, reality contains a contradiction. Type 3 is just impossible.
5) If I were in this situation, I'd assume a prior over possible Omegas that gave large weight to types 1 and 2, which means I would one-box. My prior is justified because a workable Omega of type 3 or 4 is harder for me to imagine than 1 or 2. Disagree? What would you do as a good Bayesian?