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But can you be 99.99% confident that 1159 is a prime?

This doesn't affect the thrust of the post but 1159 is not prime. Prime factors are 19 and 61.

I agree that you can be 99.99% (or more) certain that 53 is prime but I don't think you can be that confident based only on the arguement you gave.

If a number is composite, it must have a prime factor no greater than its square root. Because 53 is less than 64, sqrt(53) is less than 8. So, to find out if 53 is prime or not, we only need to check if it can be divided by primes less than 8 (i.e. 2, 3, 5, and 7). 53's last digit is odd, so it's not divisible by 2. 53's last digit is neither 0 nor 5, so it's not divisible by 5. The nearest multiples of 3 are 51 (=17x3) and 54, so 53 is not divisible by 3. The nearest multiples of 7 are 49 (=7^2) and 56, so 53 is not divisible by 7. Therefore, 53 is prime.

There are just too many potential errors that could occur in this chain of reasoning. For example, how sure are you that you correctly listed the primes less than 8? Even a mere typo at this stage of the argument could result in an erroneous conclusion.

Anyway just to be clear I do think your high confidence that 53 is prime is justified, but that the argument you gave for it is insufficient in isolation.

Are the various people actually being presented with the same problem? It makes a difference if the predictor is described as a skilled human rather than as a near omniscient entity.

The method of making the prediction is important. It is unlikely that a mere human without computational assistance could simulate someone in sufficient detail to reliably make one boxing the best option. But since the human predictor knows that the people he is asking to choose also realize this he still might maintain high accuracy by always predicting two boxing.

edit: grammar

This is interesting. I suspect this is a selection effect, but if it is true that there is a heavy bias in favor of one boxing among a more representative sample in the actual Newcomb's problem, then a predictor that always predicts one boxing could be suprisingly accurate.

It is intended to illustrate that for a given level of certainty one boxing has greater expected utility with an infallible agent than it does with a fallible agent.

As for different behaviors, I suppose one might suspect the fallible agent of using statistical methods and lumping you into a reference class to make its prediction. One could be much more certain that the infallible agent’s prediction is based on what you specifically would choose.

You may have misunderstood what is meant by "smart predictor".

The wiki entry does not say how Omega makes the prediction. Omega may be intelligent enough to be a smart predictor but Omega is also intelligent enough to be a dumb predictor. What matters is the method that Omega uses to generate the prediction. And whether the method of prediction causally connects Omega’s prediction back to the initial conditions that causally determine your choice.

Furthermore a significant part of the essay explains in detail why many of the assumptions associated with Omega are problematic.

Edited to add that on rereading I can see how the bit where I say, "It doesn’t state whether Omega is sufficiently smart." is a bit misleading. It should be read as a statement about the method of making the prediction not about Omega's intelligence.


I have written a critique of the position that one boxing wins on Newcomb's problem but have had difficulty posting it here on Less Wrong. I have temporarily posted it here

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