I initially thought two-box, but on thinking about it more, I’m going for one-box.
For simple numbers, let’s suppose that the lottery has a 50% chance of choosing a prime number, and that if Omega could select the same number as the lottery, he’ll do so with 10% probability.
Three simple strategies:
1) Always one-box: Gets Omega’s payout every time, wins the lottery 50% of the time. Average total payout $2M. (numbers are the same 10% of the time when the lottery is ‘prime’)
2) Always two-box: Omega never pays out, wins the lottery 50% of the time. Average total payout $1.001M. (numbers are the same 10% of the time when the lottery is ‘composite’)
3) Normally one-box, two-box when numbers are the same. Omega pays out 95% of the time. Lottery pays out 50% of the time. Average total payout $1.95M. (Numbers are the same 10% of the time when the lottery is ‘composite’)
The trick is that the question tries to lead you to the wrong counterfactual by drawing your attention to the situation where the numbers are the same. Whether you see the numbers being the same depends on your decision. In the counterfactual world where you decide something else, the lottery number doesn’t change to match Omega’s prediction. Instead, in the counterfactual world, the lottery number and Omega’s number are different.
I’m not sure where you got that 95% number from for your strategy #3; it sounds like the “both numbers are the same” situation only happens once ever several thousand of runs.
Anyway, if you’re using strategy 1, then if the two numbers are the same, that means that the number is prime, and your payout for this scenario is only 1 million dollars (you lost the lottery). If you’re using strategy 3, then that means the number is not prime, and the payout is $2.001 million dollars (the number is not prime, because you’re going to double box.)
There is no difference between strategy 1 and strategy 3 except in the one scenario where both numbers are the same, and the one scenario where both numbers are the same, strategy 3 is better. Therefore, strategy 3 is always better.
I initially thought two-box, but on thinking about it more, I’m going for one-box.
For simple numbers, let’s suppose that the lottery has a 50% chance of choosing a prime number, and that if Omega could select the same number as the lottery, he’ll do so with 10% probability.
Three simple strategies:
1) Always one-box: Gets Omega’s payout every time, wins the lottery 50% of the time. Average total payout $2M. (numbers are the same 10% of the time when the lottery is ‘prime’)
2) Always two-box: Omega never pays out, wins the lottery 50% of the time. Average total payout $1.001M. (numbers are the same 10% of the time when the lottery is ‘composite’)
3) Normally one-box, two-box when numbers are the same. Omega pays out 95% of the time. Lottery pays out 50% of the time. Average total payout $1.95M. (Numbers are the same 10% of the time when the lottery is ‘composite’)
The trick is that the question tries to lead you to the wrong counterfactual by drawing your attention to the situation where the numbers are the same. Whether you see the numbers being the same depends on your decision. In the counterfactual world where you decide something else, the lottery number doesn’t change to match Omega’s prediction. Instead, in the counterfactual world, the lottery number and Omega’s number are different.
The sixth virtue is empiricism. Nice job.
I don’t see the relevance. The commenter contemplated a hypothetical scenario through abstract thinking, there’s no empiricism here.
Actually doing the math, rather than just relying on intuition about what sounds right.
I’m not sure where you got that 95% number from for your strategy #3; it sounds like the “both numbers are the same” situation only happens once ever several thousand of runs.
Anyway, if you’re using strategy 1, then if the two numbers are the same, that means that the number is prime, and your payout for this scenario is only 1 million dollars (you lost the lottery). If you’re using strategy 3, then that means the number is not prime, and the payout is $2.001 million dollars (the number is not prime, because you’re going to double box.)
There is no difference between strategy 1 and strategy 3 except in the one scenario where both numbers are the same, and the one scenario where both numbers are the same, strategy 3 is better. Therefore, strategy 3 is always better.