Thanks, Derek. What do you think about what I proposed here?
If one puts weight w on the welfare range (WR) of humans relative to that of chickens being N, and 1 - w on it being n, the expected welfare range of:
Humans relative to that of chickens is E(“WR of humans”/”WR of chickens”) = w*N + (1 - w)*n.
Chickens relative to that of humans is E(“WR of chickens”/”WR of humans”) = w/N + (1 - w)/n.
You [Carl Shulman] are arguing that N can plausibly be much larger than n. For the sake of illustration, we can say N = 389 (ratio between the 86 billion neurons of a humans and 221 M of a chicken), n = 3.01 (reciprocal of RP’s [Rethink Priorities’] median welfare range of chickens relative to humans of 0.332), and w = 1⁄12 (since the neuron count model was one of the 12 RP considered, and all of them were weighted equally [I think RP only used 7 or 8 models for the final welfare ranges, and not neuron counts, but my point does not depend on the weight]). Having the welfare range of:
Chickens as the reference, E(“WR of humans”/”WR of chickens”) = 35.2. So 1/E(“WR of humans”/”WR of chickens”) = 0.0284.
Humans as the reference (as RP did), E(“WR of chickens”/”WR of humans”) = 0.305.
So, as you said, determining welfare ranges relative to humans results in animals being weighted more heavily. However, I think the difference is much smaller than the suggested above. Since N and n are quite different, I guess we should combine them using a weighted geometric mean, not the weighted mean as I did above. If so, both approaches output exactly the same result:
E(“WR of humans”/”WR of chickens”) = N^w*n^(1 - w) = 4.49. So 1/E(“WR of humans”/”WR of chickens”) = (N^w*n^(1 - w))^-1 = 0.223.
E(“WR of chickens”/”WR of humans”) = (1/N)^w*(1/n)^(1 - w) = 0.223.
The reciprocal of the expected value is not the expected value of the reciprocal, so using the mean leads to different results. However, I think we should be using the geometric mean, and the reciprocal of the geometric mean is the geometric mean of the reciprocal. So the 2 approaches (using humans or chickens as the reference) will output the same ratios regardless of N, n and w as long as we aggregate N and n with the geometric mean. If N and n are similar, it no longer makes sense to use the geometric mean, but then both approaches will output similar results anyway, so RP’s approach looks fine to me as a 1st pass. Does this make any sense?
It is an intriguing use of a geometric mean, but I don’t think it is right because I think there is no right way to do it given just the information you have specified. (The geometric mean may be better as a heuristic than the naive approach—I’d have to look at it in a range of cases—but I don’t think it is right.)
The section on Ratio Incorporation goes into more detail on this. The basic issue is that we could arrive at a given ratio either by raising or lowering the measure of each of the related quantities and the way you get to a given ratio matters for how it should be included in expected values. In order to know how to find the expected ratio, at least in the sense you want for consequentialist theorizing, you need to look at the details behind the ratios.
Thanks, Derek. What do you think about what I proposed here?
It is an intriguing use of a geometric mean, but I don’t think it is right because I think there is no right way to do it given just the information you have specified. (The geometric mean may be better as a heuristic than the naive approach—I’d have to look at it in a range of cases—but I don’t think it is right.)
The section on Ratio Incorporation goes into more detail on this. The basic issue is that we could arrive at a given ratio either by raising or lowering the measure of each of the related quantities and the way you get to a given ratio matters for how it should be included in expected values. In order to know how to find the expected ratio, at least in the sense you want for consequentialist theorizing, you need to look at the details behind the ratios.