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DanielLC comments on Counterfactual Mugging - Less Wrong
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I really fail to see why you're all so fascinated by Newcomb-like problems. When you break causality, all logic based on causality doesn't function any more. If you try to model it mathematically, you will get inconsistent model always.
They don't require breaking causality. The argument works if Omega is barely predicting you above chance. I'm sure there are plenty of normal people who can do that just by talking to you.
There are also more important reasons. Take the doomsday argument. You can use the fact that you're alive now to predict that we'll die out "soon". Suppose you had a choice between saving a life in a third-world country that likely wouldn't amount to anything, or donating to SIAI to help in the distant future. You know it's very unlikely for there to be a distant future. It's like Omega did his coin toss, and if it comes up tails, we die out early and he asks you to waste the money by donating to SIAI. If it comes up heads, you're in the future, and it's better if you would have donated.
That's not some thing that might happen. That's a decision you have to make before you pick a charity to donate to. Lives are riding on this. That's if the coin lands on tails. If it lands on heads, there is more life riding on it than has so far existed in the known universe. Please choose carefully.
Arguments like these remind me of students' mistakes from Algorithms and Data Structures 101 - statements like that are very intuitive, absolutely wrong, and once you figure out why this reasoning doesn't work it's easy to forget that most people didn't go through this ever.
What is required is Omega predicting better than chance in the worst case. Predicting correctly with ridiculously tiny chance of error against "average" person is worthless.
To avoid Omega and causality silliness, and just demonstrate this intuition - let's take a slightly modified version of Boolean satisfiability - but instead of one formula we have three formulas of the same length. If all three are identical, return true or false depending on its satisfiability, if they're different return true if number of one bits in problem is odd (or some other trivial property).
It is obviously NP-complete, as any satisfiability problem reduces to it by concatenating it three times. If we use exponential brute force to solve the hard case, average running time is O(n) for scanning the string plus O(2^(n/3)) for brute forcing but only 2^-(2n/3) of the time, that is O(1). So we can solve NP-complete problems in average linear time.
What happened? We were led astray by intuition, and assumed that problems that are difficult in worst case cannot be trivial on average. But this equal weighting is an artifact - if you tried reducing any other NP problem into this, you'd be getting very difficult ones nearly all the time, as if by magic.
Back to Omega - even if Omega predicts normal people very well, as long as there are any thinking being who is cannot predict - Omega must break causality. And such being are not just hypothetical - people who decide based on a coin toss are exactly like that. Silly rules about disallowing chance merely make counterexamples more complicated, Omega and Newcomb are still as much based on sloppy thinking as ever.
I don't know any reason why a coin toss would be the best choice in Newcomb's paradox. If you decide based on reason, and don't decide to flip a coin, and Omega knows you well, he can predict your action above chance. The paradox stands.