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I don't think Nomega has to simulate you interacting with Omega in order to know how to would react should you encounter it, in the same way that you can predict the output of many computer programs without simulating them.
By the time you get mugged, you could be 100% sure that you are in the Omega world, rather than the Nomega world, but the principle is that your decision in the Omega world affects the Nomega world, and so before knowing UDT commits to making the decision that maximizing EV across both worlds.
This logic operates in the same way for the coin coming up tails -- when you see the tails, you know your in the tails world, but your decision in the tails world affects the heads world, so you have to consider it. Likewise, your decision in the Omega world affects the Nomega world (independent of any sort of simulation argument).
Thus, in a situation where UDT has seen Omega, it has influence over the Omega world and Nomega/Omega world, but no influence over the normal world and Nomega world. Since the Omega world has so much more weight than the Omega/Nomega world, UDT will effectively act as if it's in the Omega world.
This argument would also suggest that by the time you see tails, you know you live in the tails world and thus should not pay up.
But UDT's decision on how to interact woth Omega does direct affect worlds in which Nomega exists instead of Omega.
Again overly simplistic prior:
50% chance: Omega exists, and we get counterfactually mugged, half of the times heads and half of the times tails.
50% chance: Nomega exists, guesses what we would do if Omega existed and the coin came up tails, and pays out accordingly.
There is only one decision -- do you pay if Omega exists and the coin comes up tails, and that decision affects both (or all three) possible worlds.
Even once you see that Omega exists, UDT already recognized that in order to maximize utility it should precommit (or just decide or whatever) to not pay.
If the answer is that you have a higher prior towards Omega before the mugging, then fine that solves the problem. But if you think Omega is more likley to exist only because you see Omega in front of you, then doesnt that violate UDTs principle of never updating?
Hmm perhaps I am still a little confused as to how UDT works. My understanding is that you don't make your decisions based on the information you have observed, but instead, when you "boot up" your UDT, you consider all of the possible world states you may find yourself in and their various mesures, and then for each decision, "precommit" to making the one that maximizes your expected utility across all of the possible world states that this decision affects.
If this understanding is correct, then unless we have some sort of prior telling us, when we "boot up" UDT and thus before we interact with Omega, that Omega is more likley to exist than Nomega, then I don't see how UDT could tell us to pay up.
I think it is somewhat likley that I am missing something here but I dont know what.
Whoops -- EV re-updated.
Perhaps I am misunderstanding the setup of the counterfactual mugging -- do we live in a world in which Omega is a known being (and just hasn't yet interacted with us), or do we live in a world in which we have roughly equal credence of the existence of Omega vs Nomega (vs any other arbitrary God-like figure). If it's the former, then sure UDT says precommit and pay.
But if its the latter, I still don't see why UDT tells us to pay -- not because not precommitting is some sort of default (which is I agree UDT says isn't relevant) but because when making decisions based on the possible existence of some sort of God while ignoring the possible existence of other God's isn't fair or an effective way to maximize you expected utility. Perhaps some sort of Occam's Razor / Solomon Induction argument could be made that Omega is simpler and thus more likely to exist, but this seems fairly difficult to rigorize.
Hi Vladimir, thanks for your response.
Upon further reflection, I think the crux of my argument is that by precommiting you are essentially pascals wagering yourself -- you are making a decision looking to maximize yoir reward should a certain type of God (Omega) exist. Unless (before you get mugged) you have some reason to believe that this type of God is more likley to exist then the opposite type (Nomega), then precommiting is getting wagered (as far as I can tell). You cant wait until you find out that Omega exists to preccomit because by then you have aready learned that the coin is tails -- you have to do so blind.
I dont think this proves to much becuase in other problems (Prisoners Dilema, Newcombs Paradox, etc.) considering what to do if a random God shows is wagering, so you just ignore it. Here, though, precommiting is wagering, so (it seems to me) that youshould just ignore it as well and not precommit.
Good point on EV numbers -- they are now updated although the actual numbers are not super important to the crux of the argument.