As a function of M, |P| is very likely to be exponential and so it will take O(M) symbols to specify a member of P.
O-ops, I didn’t think about it, thanks! Maybe it would be better to change it so input is “a=b” or “a!=b”, and a always gets “a=b”.
That aside, why are you assuming that program b “wants” anything? Essentially all of P won’t be programs that have any sort of “want”. If it is a precondition of the problem that b is such a program, what selection procedure is assumed between those that do “want” money from this scenario? Note that being selected for running is also a precondition for getting any money at all, so this selection procedure is critically important—far more so than anything the program might output!
Programmer who wrote b decided that it should be consequentialist agent who wants to get money. (Or, if this program is actually, a, it wants to maximize the payment for b just because such a program was chosen by Omega by pure luck)
Suppose b “knows” that Omega runs this experiment for all programs b. Then the optimal behaviour for a competent b (by a ridiculously small margin) is to 1-box.
Suppose b suspects that box-choosing programs are slightly less likely to be run if they 1-box on equal inputs. Then the optimal behaviour for b is to 2-box, because the average extra payoff for 1-boxing on equal inputs is utterly insignificant while the average penalty for not being chosen to run is very much greater. Anything that affects probability of being run as box-chooser with probability greater than 1000/|P| (which is on the order of 1/10^10^10^10^100) matters far more than what the program actually does.
In the original Newcombe problem, you know that you are going to get money based on your decision. In this problem, a running program does not know this. It doesn’t know whether it’s a or b or both, and every method for selecting a box-chooser is a different problem with different optimal strategies.
O-ops, I didn’t think about it, thanks! Maybe it would be better to change it so input is “a=b” or “a!=b”, and a always gets “a=b”.
Programmer who wrote b decided that it should be consequentialist agent who wants to get money. (Or, if this program is actually, a, it wants to maximize the payment for b just because such a program was chosen by Omega by pure luck)
I’ll try to make it clearer:
Suppose b “knows” that Omega runs this experiment for all programs b. Then the optimal behaviour for a competent b (by a ridiculously small margin) is to 1-box.
Suppose b suspects that box-choosing programs are slightly less likely to be run if they 1-box on equal inputs. Then the optimal behaviour for b is to 2-box, because the average extra payoff for 1-boxing on equal inputs is utterly insignificant while the average penalty for not being chosen to run is very much greater. Anything that affects probability of being run as box-chooser with probability greater than 1000/|P| (which is on the order of 1/10^10^10^10^100) matters far more than what the program actually does.
In the original Newcombe problem, you know that you are going to get money based on your decision. In this problem, a running program does not know this. It doesn’t know whether it’s a or b or both, and every method for selecting a box-chooser is a different problem with different optimal strategies.