Ah; now we begin to compare different things. To compare the effort of pulling the lever, against the utility of the additional lives. At this point, yes, the actual magnitude and not just the sign of the difference between f(A+B-C+D) and f(A+D) becomes important; yet D is unknown and unknowable. This means that the magnitude of the difference can only be known with certainty if f(x) is linear; in the case of a nonlinear f(x), the magnitude cannot be known with certainty. I can easily pick out a nonlinear, monotonically increasing function such that the difference between f(A+B-C+D) and f(A+D) can be made arbitrarily small for any positive integer A, B and C (where A+B>C) by simply selecting a suitable positive integer D. A simple example would be f(x)=sqrt(x).

Now, the hypothetical moral agent is in a quandary. Using effort to pick a solution costs utilions. The cost is a simple, straightforward constant; he known how much that costs. But, with f(x)=sqrt(x), without knowing D, he cannot tell whether the utilions of saving the people is greater or lesser than the utilion cost of picking a solution. (For the purpose of simplicity, I will assume that no-one will ever know that he was in a position to make the choice - that is, his reputation is safe, no matter what he selects). Therefore, he has to make an estimate. He has to guess a value of D. There are multiple strategies that can be followed here:

• Try to estimate the most probable value of D. This would require something along the lines of the Drake equation - picking the most likely numbers for the different elements, picking the most likely size of an extraterrestrial civilisation, and doing some multiplication.

• Take the most pessimistic possible value of D; D=0. That is, plan as though I am in the worst possible universe; if I am correct, and D=0, then I take the correct action, while if I am incorrect and D later proves greater than zero, then that is a pleasant surprise. This guards against getting an extremely unpleasant surprise if it later turns out that D is substantially lower than the most likely estimate; utilions in the future are more likely to go up than down.

• Ignore the cost, and simply take the option that saves the most lives, regardless of effort. This strategy actually reduces the cost slightly (as one does not need to expend the very slight cost of calculating the cost), and has the benefit of allowing immediate action. It is the option that I would prefer that everyone who is not me should take (because if other people take it, then I have a greater chance of getting my life saved at the cost of no effort on my part). I might choose this option out of a sense of fairness (if I wish other people to take this option, it is only reasonable to consider that other people may wish me to take it) or out of a sense of duty (saving lives is important).