Here is an illustration of how one can easily be much more confident than that. If welfare per animal-year was proportional to f(x) = 2^x, where x is the number of neurons, its elasticity would be x*f’(x)/f(x) = x*ln(2)*2^x/2^x = ln(2)*x. Even for the 302 neurons of adult nematodes, which are the animals with the fewest neurons, the elasticity would be 209 (= ln(2)*302). For my assumption that welfare per animal-year is proportional to g(x) = x^a, its elasticity is x*a*x^(a − 1)/x^a = a. So, for a number of neurons close to that of adult nematodes, I think welfare per animal-year being proportional to 2^x is roughly as plausible as it being proportional to x^302. For a number of neurons close to that of humans, I believe welfare per animal-year being proportional to 2^x is roughly as plausible as it being proportional to x^(86*10^9). If the elasticity y follows a normal distribution with mean m, and standard deviation s, the probability p(y) of a given elasticity is proportional to e^(-(y—m)^2/(2*s^2))/s. Given 2 values for the elasticity y1 and y2, the ratio between their probabilities is p(y2)/p(y1) = e^((-(y2 - m)^2 + (y1 - m)^2)/(2*s^2)). For m = 0.5, s = 0.25, y1 = 0.5 (equal the expected elasticity m), and y2 = 302 (the value I am arguing is very unlikely), p(y2)/p(y1) = e^((-(302 − 0.5)^2 + (0.5 − 0.5)^2)/(2*0.25^2)) = e^(-727*10^3) = 10^(-log10(e)*727*10^3) = 10^(-316*10^3).
The above does not show that the welfare of animals with the fewest neurons dominates. However, it illustrate one can not only get astronomical stakes, but also astronomically low probabilities of such stakes holding. For any probability distribution describing real world phenomena, the probability and stakes are not independent. So one cannot just come up with a function implying astronomical stakes, and then independently guess a probability of such stakes holding.
Production functions usually have elasticities from 0 to 1, which is part of why my speculative best guess is that welfare per animal-year is proportional to “number of neurons”^0.5 (0.5 is the mean between 0 and 1). Doubling inputs (which could be energy) usually results in doubling the output (which could be welfare) at most. As far as I know, it would be unthinkable to have a function supposed to model the production of something in the real world with an elasticity of 302 or 86 billion. Here is a related chat I had with Gemini.
Here is an illustration of how one can easily be much more confident than that. If welfare per animal-year was proportional to f(x) = 2^x, where x is the number of neurons, its elasticity would be x*f’(x)/f(x) = x*ln(2)*2^x/2^x = ln(2)*x. Even for the 302 neurons of adult nematodes, which are the animals with the fewest neurons, the elasticity would be 209 (= ln(2)*302). For my assumption that welfare per animal-year is proportional to g(x) = x^a, its elasticity is x*a*x^(a − 1)/x^a = a. So, for a number of neurons close to that of adult nematodes, I think welfare per animal-year being proportional to 2^x is roughly as plausible as it being proportional to x^302. For a number of neurons close to that of humans, I believe welfare per animal-year being proportional to 2^x is roughly as plausible as it being proportional to x^(86*10^9). If the elasticity y follows a normal distribution with mean m, and standard deviation s, the probability p(y) of a given elasticity is proportional to e^(-(y—m)^2/(2*s^2))/s. Given 2 values for the elasticity y1 and y2, the ratio between their probabilities is p(y2)/p(y1) = e^((-(y2 - m)^2 + (y1 - m)^2)/(2*s^2)). For m = 0.5, s = 0.25, y1 = 0.5 (equal the expected elasticity m), and y2 = 302 (the value I am arguing is very unlikely), p(y2)/p(y1) = e^((-(302 − 0.5)^2 + (0.5 − 0.5)^2)/(2*0.25^2)) = e^(-727*10^3) = 10^(-log10(e)*727*10^3) = 10^(-316*10^3).
The above does not show that the welfare of animals with the fewest neurons dominates. However, it illustrate one can not only get astronomical stakes, but also astronomically low probabilities of such stakes holding. For any probability distribution describing real world phenomena, the probability and stakes are not independent. So one cannot just come up with a function implying astronomical stakes, and then independently guess a probability of such stakes holding.
Production functions usually have elasticities from 0 to 1, which is part of why my speculative best guess is that welfare per animal-year is proportional to “number of neurons”^0.5 (0.5 is the mean between 0 and 1). Doubling inputs (which could be energy) usually results in doubling the output (which could be welfare) at most. As far as I know, it would be unthinkable to have a function supposed to model the production of something in the real world with an elasticity of 302 or 86 billion. Here is a related chat I had with Gemini.