Note that killing people is not the only way to raise the median. Another technique is taking resources and redistributing them. The optimal first-level strategy is to only allow minimum-necessary-for-survival to those below the median (which, depending on what it thinks “survival” means, might include just freezing them, or cutting off all unnecessary body parts and feeding them barely nutritious glop while storing them in the dark), and distribute everything else equally between the rest.
Also, given this strategy, the median of human consumption is 2×R/(N-1), where R is the total amount of resources and N is the total amount of humans. The utility function then becomes sqrt(2×R/(N-1)) × N × T. Which means that for the same resources, its utility is maximized if the maximum number of people use them. Thus, the AI will spend its time finding the smallest possible increment above “minimum necessary for survival”, and maximize the number of people it can sustain, keeping (N-1)/2 people at the minimum and (N-1)/2+1 just a tiny bit above it, and making sure it does this for the longest possible time.
Note that killing people is not the only way to raise the median. Another technique is taking resources and redistributing them. The optimal first-level strategy is to only allow minimum-necessary-for-survival to those below the median (which, depending on what it thinks “survival” means, might include just freezing them, or cutting off all unnecessary body parts and feeding them barely nutritious glop while storing them in the dark), and distribute everything else equally between the rest.
Also, given this strategy, the median of human consumption is 2×R/(N-1), where R is the total amount of resources and N is the total amount of humans. The utility function then becomes sqrt(2×R/(N-1)) × N × T. Which means that for the same resources, its utility is maximized if the maximum number of people use them. Thus, the AI will spend its time finding the smallest possible increment above “minimum necessary for survival”, and maximize the number of people it can sustain, keeping (N-1)/2 people at the minimum and (N-1)/2+1 just a tiny bit above it, and making sure it does this for the longest possible time.