Minor comment on one small paragraph:
Price’s Law says that half of the contributions in a field come from the square root of the number of contributors. In other words, productivity increases linearly as the number of contributors increases exponentially. Therefore, as the number of AI safety researchers increases exponentially, we might expect the total productivity of the AI safety community to increase linearly.
I think Price’s law is false, but I don’t know what law it should be instead. I’ll look at the literature on the rate of scientific progress (eg. Cowen & Southwood (2019)) to see if I could find any relationship between number of researchers and research productivity.
Price’s law is a poor fit; Lotka’s law is a better fit
The most prominent citation for Price’s law, Nicholls (1988), says that Price’s law is a poor fit (section 4: Validity of the Price Law):
Little empirical investigation of the Price law has been carried out to date [4,14]. Glänzel and Schubert [12] have reported some empirical results. They analyzed Lotka’s Chemical Abstracts data and found that the most prolific authors contributed less that 20% of the total number of papers. They also refer, but without details, to the examination of “several dozens” of other empirical data sets and conclude that “in the usually studied populations of scientists, even the most productive authors are not productive enough to fulfill the requirements of Price’s conjecture” [12]. Some incidental results of scientometric studies suggest that about 15% of the authors will be necessary to generate 50% of the papers [16,17].
To further examine the empirical validity of Price’s hypothesis, 50 data sets were collected and analyzed here. … the contribution of the most prolific group of authors fell considerably short of the [50% of the papers] predicted by Price. … The actual proportion of all authors necessary to generate at least 50% of the papers was found to be much larger that . Table 2 summarizes these results. In some cases, …, more than half of the total number of papers is generated by those authors contributing only a single paper each. The absolute and relative size of for various population sizes t is given in Table 3. All the empirical results referred to here are consistent; and, unfortunately, there seems little reason to suppose that further empirical results would offer any support for the Price law.
Nicholls (1988) continues, saying that Lotka’s law (number of authors with publications is proportional to ) has good empirical support, and finds to be a good fit for sciences and humanities, and to be a good fit in social sciences.
A different paper, Chung & Cox (1990), also finds that Price’s Law is a poor fit while Lotka’s law with between 1.95 to 3.26 to be a good fit in finance.
(Allison, Price, Griffith, Moravcsik & Stewart (1976) discusses the mathematical relationship between Price’s Law and Lotka’s Law: neither implies the other; nor are they contradictory.)
Later edits:
Porby, in his post Why I think strong general AI is coming soon, mentions a tangentially related idea: core researchers contribute much more insight than newer researchers. New researchers need a lot of time to become core researchers.
In Porby’s model, the research productivity at year may be proportional to the number of researchers at year .
I think the dramatic impact would be stronger without the “The end”, and instead adding more blank space.
Idea copied from a comment on the final chapter of Three Worlds Collide.