And I think I can still show that if you run TDT, you will decide to self-modify into CDT before starting this game
Well that should never happen. Anything that would make a TDT want to self-modify into CDT should make it just want to play D, no need for self-modification. It should give the same answer at different times, that's what makes it a timeless decision theory. If you can break that without direct explicit dependence on the algorithm apart from its decisions, then I am in trouble! But it seems to me that I can substitute "play D" for "self-modify" in all cases above.
First, if Omega's AIs know that you run TDT at the beginning, then they can use that "play D if you self-modify" strategy to deter you from self-modifying.
E.g., "play D if you play D to deter you from playing D" seems like the same idea, the self-modification doesn't add anything.
So who wins this game? (If someone moves first logically, then he wins, but what if everyone moves simultaneously in the logical sense, which seems to be the case in this game?)
Well... it partially seems to me that, in assuming certain decisions are made without logical consequences - because you move logically first, or because the TDT agents have fixed wrong priors, etc. - you are trying to reduce the game to a Prisoner's Dilemma in which you have a certain chance of playing against a piece of cardboard with "D" written on it. Even a uniform population of TDTs may go on playing C in this case, of course, if the probability of facing cardboard is low enough. But by the same token, the fact that the cardboard sometimes "wins" does not make it smarter or more rational than the TDT agents.
Now, I want to be very careful about how I use this argument, because indeed a piece of cardboard with "only take box B" written on it, is smarter than CDT agents on Newcomb's Problem. But who writes that piece of cardboard, rather than a different one?
An authorless piece of cardboard genuinely does go logically first, but at the expense of being a piece of cardboard, which makes it unable to adapt to more complex situations. A true CDT agent goes logically first, but at the expense of losing on Newcomb's Problem. And your choice to put forth a piece of cardboard marked "D" relies on you expecting the TDT agents to make a certain response, which makes the claim that it's really just a piece of cardboard and therefore gets to go logically first, somewhat questionable.
Roughly, what I'm trying to reply is that you're reasoning about the response of the TDT agents to your choosing the CDT algorithm, which makes you TDT, but you're also trying to force your choice of the CDT algorithm to go logically first, but this is begging the question.
I would, perhaps, go so far as to agree that in an extension of TDT to cases in which certain agents magically get to go logically first, then if those agents are part of a small group uncorrelated with yet observationally indistinguishable from a large group, the small group might make a correlated decision to defect "no matter what" the large group does, knowing that the large group will decide to cooperate anyway given the payoff matrix. But the key assumption here is the ability to go logically first.
It seems to me that the incompleteness of my present theory when it comes to logical ordering is the real key issue here.
Well that should never happen. Anything that would make a TDT want to self-modify into CDT should make it just want to play D, no need for self-modification. It should give the same answer at different times, that's what makes it a timeless decision theory. If you can break that without direct explicit dependence on the algorithm apart from its decisions, then I am in trouble! But it seems to me that I can substitute "play D" for "self-modify" in all cases above.
The reason to self-modify is to make yourself indistinguishable from pla...
Followup to: Newcomb's Problem and Regret of Rationality, Towards a New Decision Theory
Wei Dai asked:
...
All right, fine, here's a fast summary of the most important ingredients that go into my "timeless decision theory". This isn't so much an explanation of TDT, as a list of starting ideas that you could use to recreate TDT given sufficient background knowledge. It seems to me that this sort of thing really takes a mini-book, but perhaps I shall be proven wrong.
The one-sentence version is: Choose as though controlling the logical output of the abstract computation you implement, including the output of all other instantiations and simulations of that computation.
The three-sentence version is: Factor your uncertainty over (impossible) possible worlds into a causal graph that includes nodes corresponding to the unknown outputs of known computations; condition on the known initial conditions of your decision computation to screen off factors influencing the decision-setup; compute the counterfactuals in your expected utility formula by surgery on the node representing the logical output of that computation.
To obtain the background knowledge if you don't already have it, the two main things you'd need to study are the classical debates over Newcomblike problems, and the Judea Pearl synthesis of causality. Canonical sources would be "Paradoxes of Rationality and Cooperation" for Newcomblike problems and "Causality" for causality.
For those of you who don't condescend to buy physical books, Marion Ledwig's thesis on Newcomb's Problem is a good summary of the existing attempts at decision theories, evidential decision theory and causal decision theory. You need to know that causal decision theories two-box on Newcomb's Problem (which loses) and that evidential decision theories refrain from smoking on the smoking lesion problem (which is even crazier). You need to know that the expected utility formula is actually over a counterfactual on our actions, rather than an ordinary probability update on our actions.
I'm not sure what you'd use for online reading on causality. Mainly you need to know:
It will be helpful to have the standard Less Wrong background of defining rationality in terms of processes that systematically discover truths or achieve preferred outcomes, rather than processes that sound reasonable; understanding that you are embedded within physics; understanding that your philosophical intutions are how some particular cognitive algorithm feels from inside; and so on.
The first lemma is that a factorized probability distribution which includes logical uncertainty - uncertainty about the unknown output of known computations - appears to need cause-like nodes corresponding to this uncertainty.
Suppose I have a calculator on Mars and a calculator on Venus. Both calculators are set to compute 123 * 456. Since you know their exact initial conditions - perhaps even their exact initial physical state - a standard reading of the causal graph would insist that any uncertainties we have about the output of the two calculators, should be uncorrelated. (By standard D-separation; if you have observed all the ancestors of two nodes, but have not observed any common descendants, the two nodes should be independent.) However, if I tell you that the calculator at Mars flashes "56,088" on its LED display screen, you will conclude that the Venus calculator's display is also flashing "56,088". (And you will conclude this before any ray of light could communicate between the two events, too.)
If I was giving a long exposition I would go on about how if you have two envelopes originating on Earth and one goes to Mars and one goes to Venus, your conclusion about the one on Venus from observing the one on Mars does not of course indicate a faster-than-light physical event, but standard ideas about D-separation indicate that completely observing the initial state of the calculators ought to screen off any remaining uncertainty we have about their causal descendants so that the descendant nodes are uncorrelated, and the fact that they're still correlated indicates that there is a common unobserved factor, and this is our logical uncertainty about the result of the abstract computation. I would also talk for a bit about how if there's a small random factor in the transistors, and we saw three calculators, and two showed 56,088 and one showed 56,086, we would probably treat these as likelihood messages going up from nodes descending from the "Platonic" node standing for the ideal result of the computation - in short, it looks like our uncertainty about the unknown logical results of known computations, really does behave like a standard causal node from which the physical results descend as child nodes.
But this is a short exposition, so you can fill in that sort of thing yourself, if you like.
Having realized that our causal graphs contain nodes corresponding to logical uncertainties / the ideal result of Platonic computations, we next construe the counterfactuals of our expected utility formula to be counterfactuals over the logical result of the abstract computation corresponding to the expected utility calculation, rather than counterfactuals over any particular physical node.
You treat your choice as determining the result of the logical computation, and hence all instantiations of that computation, and all instantiations of other computations dependent on that logical computation.
Formally you'd use a Godelian diagonal to write:
Argmax[A in Actions] in Sum[O in Outcomes](Utility(O)*P(this computation yields A []-> O|rest of universe))
(where P( X=x []-> Y | Z ) means computing the counterfactual on the factored causal graph P, that surgically setting node X to x, leads to Y, given Z)
Setting this up correctly (in accordance with standard constraints on causal graphs, like noncircularity) will solve (yield reflectively consistent, epistemically intuitive, systematically winning answers to) 95% of the Newcomblike problems in the literature I've seen, including Newcomb's Problem and other problems causing CDT to lose, the Smoking Lesion and other problems causing EDT to fail, Parfit's Hitchhiker which causes both CDT and EDT to lose, etc.
Note that this does not solve the remaining open problems in TDT (though Nesov and Dai may have solved one such problem with their updateless decision theory). Also, although this theory goes into much more detail about how to compute its counterfactuals than classical CDT, there are still some visible incompletenesses when it comes to generating causal graphs that include the uncertain results of computations, computations dependent on other computations, computations uncertainly correlated to other computations, computations that reason abstractly about other computations without simulating them exactly, and so on. On the other hand, CDT just has the entire counterfactual distribution rain down on the theory as mana from heaven (e.g. James Joyce, Foundations of Causal Decision Theory), so TDT is at least an improvement; and standard classical logic and standard causal graphs offer quite a lot of pre-existing structure here. (In general, understanding the causal structure of reality is an AI-complete problem, and so in philosophical dilemmas the causal structure of the problem is implicitly given in the story description.)
Among the many other things I am skipping over:
Those of you who've read the quantum mechanics sequence can extrapolate from past experience that I'm not bluffing. But it's not clear to me that writing this book would be my best possible expenditure of the required time.