Hey, in hindsight I realize that the paper + summarization don’t make clear that this does depend on model assumptions/empirical points, sorry. I’ve edited the post to make this clearer (here is the previous version without the edits, in case it’s of interest.)
tl;dr: This comes from model assumptions which seem reasonable, but empirical investigations + historical case studies, or alternatively sci-fi scenarios could flip the conclusion.
In particular, let L′=−r⋅L+f(a⋅L,b⋅K), i.e. roughly L(t)=L(t−1)⋅(1−r)+f(a⋅L,b⋅K), so each year you lose r% of people, but you also do some movement building, for which you spend a⋅L labor and b⋅K capital.
Then for some functions f which determine movement building, this already implies that the movement has a maximum size. So for instance, if you havef(a⋅L,b⋅K)=log(11a⋅L+1b⋅K), then with infinite capital this reduces to f(a⋅L,b⋅K)=log(11a⋅L+1∞)=log(11a⋅L)=log(a⋅L)
But then even if you allocate all labor to movement building (so that a=1, or something), you’d have something like L′=−r⋅L+log(L) , and this eventually converges to the point where log(L)=r⋅L no matter where you start.
Now, above I’ve omitted some constants, and our function isn’t quite the same, but that’s essentially what’s going on (see ρR<0,λ<1 in equation 6 in page 4.) I.e., if you lose movement participants as a percentage but have a recruitment function that eventually has “brutal” diminishing returns (sub-linear diminishing returns to labor + throwing money at movement building doesn’t solve it), you get a similar result (movement converges to a constant.)
But you could also imagine a scenario where the returns are less brutal—e.g., you’re always able to recruit an additional participant by throwing money at the problem, or every movement builder can sort of eternally always recruit a person every year, etc. You could also imagine more sci-fi like scenario, where humanity is expanding exponentially (cubically) in space, and a social movement is a constant fraction of humanity.
More realistically, if f instead looks like √(a∗L)⋅(b∗K), which has diminishing returns but not brutally so, movement size can increase forever because you can always throw more money at the problem until √(a∗L)⋅(b∗K)>r⋅L
Note that if you have a less brutal recruitment function, this increases the appeal of movement building, not of earn to giving.
Also, I’m not sure whether “brutal” is the right way to be talking about this. “Brutal” is the term I use when I think about this but if I recall correctly the function we use is standard in the literature, and it seems plausible when you start to think about groups which reach a large size. But there is definitely an empirical question here about how movement results actually look like.
Hey, in hindsight I realize that the paper + summarization don’t make clear that this does depend on model assumptions/empirical points
FWIW this was clear to me, I was using “conclusions” to mean “conclusions, given the model assumptions”, not “conclusions, which the authors definitely think are true”.
Re: Labor grows to a constant size
Hey, in hindsight I realize that the paper + summarization don’t make clear that this does depend on model assumptions/empirical points, sorry. I’ve edited the post to make this clearer (here is the previous version without the edits, in case it’s of interest.)
tl;dr: This comes from model assumptions which seem reasonable, but empirical investigations + historical case studies, or alternatively sci-fi scenarios could flip the conclusion.
In particular, let L′=−r⋅L+f(a⋅L,b⋅K), i.e. roughly L(t)=L(t−1)⋅(1−r)+f(a⋅L,b⋅K), so each year you lose r% of people, but you also do some movement building, for which you spend a⋅L labor and b⋅K capital.
Then for some functions f which determine movement building, this already implies that the movement has a maximum size. So for instance, if you havef(a⋅L,b⋅K)=log(11a⋅L+1b⋅K), then with infinite capital this reduces to f(a⋅L,b⋅K)=log(11a⋅L+1∞)=log(11a⋅L)=log(a⋅L)
But then even if you allocate all labor to movement building (so that a=1, or something), you’d have something like L′=−r⋅L+log(L) , and this eventually converges to the point where log(L)=r⋅L no matter where you start.
Now, above I’ve omitted some constants, and our function isn’t quite the same, but that’s essentially what’s going on (see ρR<0,λ<1 in equation 6 in page 4.) I.e., if you lose movement participants as a percentage but have a recruitment function that eventually has “brutal” diminishing returns (sub-linear diminishing returns to labor + throwing money at movement building doesn’t solve it), you get a similar result (movement converges to a constant.)
But you could also imagine a scenario where the returns are less brutal—e.g., you’re always able to recruit an additional participant by throwing money at the problem, or every movement builder can sort of eternally always recruit a person every year, etc. You could also imagine more sci-fi like scenario, where humanity is expanding exponentially (cubically) in space, and a social movement is a constant fraction of humanity.
More realistically, if f instead looks like √(a∗L)⋅(b∗K), which has diminishing returns but not brutally so, movement size can increase forever because you can always throw more money at the problem until √(a∗L)⋅(b∗K)>r⋅L
Note that if you have a less brutal recruitment function, this increases the appeal of movement building, not of earn to giving.
Also, I’m not sure whether “brutal” is the right way to be talking about this. “Brutal” is the term I use when I think about this but if I recall correctly the function we use is standard in the literature, and it seems plausible when you start to think about groups which reach a large size. But there is definitely an empirical question here about how movement results actually look like.
FWIW this was clear to me, I was using “conclusions” to mean “conclusions, given the model assumptions”, not “conclusions, which the authors definitely think are true”.
Right, thanks, it seemed better to be too paranoid than to be too little.