I believe that the proposed function does not follow the rule that adding positive value members is positive value. You can double the population to get any average utility that is greater than half the original utility, while not increasing the other part of the equation by doubling it (starts at, say, 0.98 and can be at most 1)
The correct answer is to factor out "more good can be done with more resources" from "more good can be done by using resources better". With this factorization, arguments for the repugnant conclusion only show that you want more resources, not that you're better off using resources by spamming minimally valuable lives.
I believe that the proposed function does not follow the rule that adding positive value members is positive value.
Right- the point is that the original repugnant conclusion is avoided if you replace "adding any number of people with positive happiness leads to a superior aggregation" with "there is some number of people with below-average utility who can be added which leads to a superior aggregation."
Consider the following facts:
This sounds a lot like the mere addition paradox, illustrated by the following diagram:
This is seems to lead directly to the repugnant conclusion - that there is a huge population of people who's lives are barely worth living, but that this outcome is better because of the large number of them (in practice this conclusion may have a little less bite than feared, at least for non-total utilitarians).
But that conclusion doesn't follow at all! Consider the following aggregation formula, where au is the average utility of the population and n is the total number of people in the population:
au(1-(1/2)n)
This obeys the two properties above, and yet does not lead to a repugnant conclusion. How so? Well, property 2 is immediate - since only the average utility appears, the reallocating utility in a more egalitarian way does not decrease the aggregation. For property 1, define f(n)=1-(1/2)n. This function f is strictly increasing, so if we add more members of the population, the product goes up - this allows us to diminish the average utility slightly (by decreasing the utility of the people we've added, say), and still end up with a higher aggregation.
How do we know that there is no repugnant conclusion? Well, f(n) is bounded above by 1. So let au and n be the average utility and size of a given population, and au' and n' those of a population better than this one. Hence au(f(n)) < au'(f(n')) < au'. So the average utility can never sink below au(f(n)): the average utility is bounded.
So some weaker versions of the mere addition argument do not imply the repugnant conclusion.