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CCC comments on The $125,000 Summer Singularity Challenge - Less Wrong
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A billion is nine orders of magnitude. As a very rough estimate, then, adding an order of magnitude to the number of lives in existence divides the motivation to extend an arbitrary stranger's life by an order of magnitude. And the same for any other multiplier.
That is, if G is chosen such that f(x)-f(x-1)=G, then f(Mx)-f(Mx-1)=G/M for any given x and any multiplier M. If I then define my hedons such that f(0)=0 and f(1)=1...
...then I get that f(x) is the harmonic series.
For 10,000 people, on this entirely arbitrary (and extremely large) scale, I get a value f(x) between 9 and 10; for seven billion, f(x) lies between 23 and 24 (source)
That's pretty much the natural logarithm of x (plus a constant, plus a term O(1/n)).
Hm. Yes, to the level of approximation I'm using here, I could as easily have used a log function. And would have, if I'd thought of it; the log function is used enough that I'd expect its properties to be easier for whoever reads my post to imagine.