TobyBartels comments on Newcomb's Problem and Regret of Rationality - Less Wrong
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Assume Omega has a probability X of correctly predicting your decision:
If you choose to two-box:
- X chance of getting $1000
- (1-X) chance of getting $1,001,000
If you choose to take box B only:
- X chance of getting $1,000,000
- (1-X) chance of getting $0
Your expected utilities for two-boxing and one-boxing are (respectively):
E2 = 1000X + (1-X)1001000
E1 = 1000000X
For E2 > E1, we must have 1000X + 1,001,000 - 1,001,000X - 1,000,000X > 0, or 1,001,000 > 2,000,000X, or
X < 0.5005
So as long as Omega can maintain a greater than 50% accuracy, you should expect to earn more money by one-boxing. Since the solution seems so simple, and since I'm a total novice at decision theory, it's possible I'm missing something here, so please let me know.
Your caclulation is fine. What you're missing is that Omega has a record of 70% accuracy because Omega always predicts that a person will one-box and 70% of people are one-boxers. So Omega always puts the million dollars in Box B, and I will always get $1,001,000$ if I'm one of the 30% of people who two-box.
At least, that is a possibility, which your calculation doesn't take into account. I need evidence of a correlation between Omega's predictions and the participants' actual behaviour, not just evidence of correct predictions. My prior probability distribution for how often people one-box isn't even concentrated very tightly around 70% (which is just a number that I remember reading once as the result of one survey), so anything short of a long run of predictions with very high proportion of correct ones will make me suspect that Omega is pulling a trick like this.
So the problem is much cleaner as Eliezer states it, with a perfect record. (But if even that record is short, I won't buy it.)
Oops, I see that RobinZ already replied, and with calculations. This shows that I should still remove the word ‘drastically’ from the bit that nhamann quoted.