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cousin_it comments on Problematic Problems for TDT - Less Wrong
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You're right about problem 2 being a fully general counterargument, but your philosophical conclusion seems to be stopping too early. For example, can we define a class of "fair" problems that excludes problem 2?
One possible place to look is that we're allowing Omega access not just to a particular simulated decision of TDT, but to the probabilities with which it makes these decisions. If we force it to simulate TDT many times and sample to learn what the probabilities are, it can't detect the exact balance for which it does deterministic symmetry breaking, and the problem goes away.
This solution occurred to me because this forces Omega to have something like a continuous behaviour response to changes in the probabilities of different TDT outputs, and it seems possible given that to imagine a proof that a fixed point must exist.
Fair point - how does Omega tell when the sim's choosing probabilities are exactly equal? Well I was thinking that Omega could prove they are equal (by analysing the simulation's behaviour, and checking where it calls on random bits). Or if it can't do that, then it can just check that the choice frequencies are "statistically equal" (i.e. no significant differences after a billion runs, say) and treat them as equal for the tie-breaker rule. The "statistically equal" approach might give the TDT agent a very slightly higher than 10% chance of winning the money, though I haven't analysed this in any detail.
If the subject can know the exact code of TDT, Omega can know the exact code of TDT, and analyse it however it likes. That means it can know exactly where randomness is invoked - why would it have to sample?
This was my first thought: Omega can just prove the choosing probabilities are equal. However, it's not totally straightforward, because the sim could sample more random bits depending on the results of its first random bits, and so on, leading to an exponentially growing outcome tree of possibilities, with no upper size bound to the length of the tree. There might not be an easy proof of equality in that case. Sampling and statistical equality is the next best approach...
It looks like the issue here is that while Omega is ostensibly not taking into account your decision theory, it implicitly is by simulating an XDT agent. So a first patch would be to define simulations of a specific decision theory (as opposed to simulations of a given agent) as "unfair".
On the other hand, we can't necessarily know if a given computation is effectively equivalent to simulating a given decision theory. Even if the string "TDT" is never encoded anywhere in Omega's super-neurons, it might still be simulating a TDT agent, for example.
On the first hand again, it might be easy for most problems to figure out whether anyone is implicitly favouring one DT over another, and thus whether they're "fair".