If Omega’s coin flip comes up heads, then you make one decision, to pay or not pay, as a simulation. If it comes up tails, then you make two decisions, to pay or not to pay, as a real person. These each have probability 0.5, so you expect to make 0.5(1+2)=1.5 decisions total.
The expected total value is the sum of the outcome for each circumstance times the expected number of times that circumstance is encountered. You can figure out the expected number of times each circumstance is encountered directly from the problem statement (0.5 for simulation, 0.5 for reality day 1, 0.5 for reality day 2). Alternatively, you can compute the expected number of times a circumstance C is encountered as P(individual decision is in C) E(decisions made), which is (1/3)(3/2), or 0.5, for simulation, reality day 1, and reality day 2. The mistake that Stuart_Armstrong made is in confusing E(times circumstance C is encountered) for P(individual decision is in C); these are not the same.
(Also, you double-counted the 100 you lose in reality day 2, messing up your expected value computation again.)
Apparently I had gestalt switched out of considering the coin. Thanks.
(Also, you double-counted the 100 you lose in reality day 2, messing up your expected value computation again.)
the double counting was intentional. My intuition was that if your on reality day 1, you expect to lose 100 today and 100 again tomorrow since you know you will give Omega the cash when he asks you. However, you don’t really know that in this thought experiment. He may give you amnesia, but he doesn’t get your brain in precisely the same physical state when he asks you the second time. So the problem seems resolved to me. This does suggest another thought experiment though.
If Omega’s coin flip comes up heads, then you make one decision, to pay or not pay, as a simulation. If it comes up tails, then you make two decisions, to pay or not to pay, as a real person. These each have probability 0.5, so you expect to make 0.5(1+2)=1.5 decisions total.
The expected total value is the sum of the outcome for each circumstance times the expected number of times that circumstance is encountered. You can figure out the expected number of times each circumstance is encountered directly from the problem statement (0.5 for simulation, 0.5 for reality day 1, 0.5 for reality day 2). Alternatively, you can compute the expected number of times a circumstance C is encountered as P(individual decision is in C) E(decisions made), which is (1/3)(3/2), or 0.5, for simulation, reality day 1, and reality day 2. The mistake that Stuart_Armstrong made is in confusing E(times circumstance C is encountered) for P(individual decision is in C); these are not the same.
(Also, you double-counted the 100 you lose in reality day 2, messing up your expected value computation again.)
Apparently I had gestalt switched out of considering the coin. Thanks.
the double counting was intentional. My intuition was that if your on reality day 1, you expect to lose 100 today and 100 again tomorrow since you know you will give Omega the cash when he asks you. However, you don’t really know that in this thought experiment. He may give you amnesia, but he doesn’t get your brain in precisely the same physical state when he asks you the second time. So the problem seems resolved to me. This does suggest another thought experiment though.