The math is actually quite straight-forward, if anyone cares to see it. Consider a generalized Newcomb’s problem. Box A either contains $A or nothing, while box B contains $B (obviously A>B, or there is no actual problem). Let Pb the probability that you 1-box. Let Po be the probability that Omega fills box A (note that only quantum randomness counts, here. If you decide by a “random” but deterministic process, Omega knows how it turns out, even if you don’t, so Pb=0 or 1). Let F be your expected return.
Regardless of what Omega does, you collect the contents of box A, and have a (1-Pb) probability of collecting the contents of box B.
F(Po=1)= A + (1-Pb)B
F(Po=0)=(1-Pb)B
For the non-degenerate cases, these add together as expected.
F(Po, Pb) = Po(A + (1-Pb)B) + (1-Po)[(1-Pb)B]
Suppose Po = Pb := P
F(P) = P(A + (1-P)B) + [(1-P)^2] B
=P(A + B—PB) + (1-2P+P^2) B
=PA + PB - (P^2)B + B − 2PB + (P^2)B
=PA + PB + B − 2PB
=B + P(A-B)
If A > B, F(P) is monotonically increasing, so P = 1 is the gives maximum return. If A<B, P=0 is the maximum (I hope it’s obvious to everyone that if box B has MORE money than a full box A, 2-boxing is ideal).
I’m not sure why you take Po = Pb. If Omega is trying to maximize his chance of predicting correctly then he’ll take Po = 1 if Pb > 1⁄2 and Pb = 0 if Pb < 1⁄2. Then, assuming A > B / 2, the optimal choice is Po = 1⁄2.
Actually, if Omega behaves this way there is a jump discontinuity in expected value at Po=1/2. We can move the optimum away from the discontinuity by postulating there is some degree of imprecision in our ability to choose a quantum coin with the desired characteristic. Maybe when we try to pick a coin with bias Po we end up with a coin with bias Po+e, where e is an error chosen from a uniform distribution over [Po-E, Po+E]. The optimal choice of Po is now 1⁄2 + 2E, assuming A > 2EB, which is the case for sufficiently small E (E < 1⁄4 suffices). The expected payoff is now robust (continuous) to small perturbations in our choice of Po.
Your solution does have Omega maximize right answers. My solution works if Omega wants the “correct” result summed over all Everett branches: for every you that 2-boxes, there exists an empty box A, even if it doesn’t usually go to the 2-boxer.
Both answers are correct, but for different problems. The “classical” Newcomb’s problem is unphysical, just as byrnema initially described. A “Quantum Newcomb’s problem” requires specifying how Omega deals with quantum uncertainty.
Interesting. Since the spirit of Newcomb’s problem depends on 1-boxing have a higher payoff, I think it makes sense to additionally postulate your solution to quantum uncertainty, as it maintains the same maximizer. That’s so even if the Everett interpretation of QM is wrong.
The math is actually quite straight-forward, if anyone cares to see it. Consider a generalized Newcomb’s problem. Box A either contains $A or nothing, while box B contains $B (obviously A>B, or there is no actual problem). Let Pb the probability that you 1-box. Let Po be the probability that Omega fills box A (note that only quantum randomness counts, here. If you decide by a “random” but deterministic process, Omega knows how it turns out, even if you don’t, so Pb=0 or 1). Let F be your expected return.
Regardless of what Omega does, you collect the contents of box A, and have a (1-Pb) probability of collecting the contents of box B. F(Po=1)= A + (1-Pb)B
F(Po=0)=(1-Pb)B
For the non-degenerate cases, these add together as expected. F(Po, Pb) = Po(A + (1-Pb)B) + (1-Po)[(1-Pb)B]
Suppose Po = Pb := P
F(P) = P(A + (1-P)B) + [(1-P)^2] B
=P(A + B—PB) + (1-2P+P^2) B
=PA + PB - (P^2)B + B − 2PB + (P^2)B
=PA + PB + B − 2PB
=B + P(A-B)
If A > B, F(P) is monotonically increasing, so P = 1 is the gives maximum return. If A<B, P=0 is the maximum (I hope it’s obvious to everyone that if box B has MORE money than a full box A, 2-boxing is ideal).
I’m not sure why you take Po = Pb. If Omega is trying to maximize his chance of predicting correctly then he’ll take Po = 1 if Pb > 1⁄2 and Pb = 0 if Pb < 1⁄2. Then, assuming A > B / 2, the optimal choice is Po = 1⁄2.
Actually, if Omega behaves this way there is a jump discontinuity in expected value at Po=1/2. We can move the optimum away from the discontinuity by postulating there is some degree of imprecision in our ability to choose a quantum coin with the desired characteristic. Maybe when we try to pick a coin with bias Po we end up with a coin with bias Po+e, where e is an error chosen from a uniform distribution over [Po-E, Po+E]. The optimal choice of Po is now 1⁄2 + 2E, assuming A > 2EB, which is the case for sufficiently small E (E < 1⁄4 suffices). The expected payoff is now robust (continuous) to small perturbations in our choice of Po.
A good point.
Your solution does have Omega maximize right answers. My solution works if Omega wants the “correct” result summed over all Everett branches: for every you that 2-boxes, there exists an empty box A, even if it doesn’t usually go to the 2-boxer.
Both answers are correct, but for different problems. The “classical” Newcomb’s problem is unphysical, just as byrnema initially described. A “Quantum Newcomb’s problem” requires specifying how Omega deals with quantum uncertainty.
Interesting. Since the spirit of Newcomb’s problem depends on 1-boxing have a higher payoff, I think it makes sense to additionally postulate your solution to quantum uncertainty, as it maintains the same maximizer. That’s so even if the Everett interpretation of QM is wrong.