Here is my proposal for an ansatz for P(Y(n+1)|Y(n)). That is, given that at least n people already believe X, how likely it is that at least one more person also believes X. Let N be the total population of the world. If n/N is close to zero, then I expect P(Y(n+1)|Y(n)) is also close to zero, and if n/N is close to 1, then P(Y(n+1)|Y(n)) is also close to 1. That is, if I know that a tiny proportion of people believe something, that's very weak evidence that a slightly larger proportion believe it also, and if I know that almost everyone believes it, that's very strong evidence that even more people believe it.

One family of functions that have this property are the functions f(n) = (n/N)^C, where C is some fixed positive number. Actually it's convenient to set C = c/N where c is some other fixed positive number. I don't have a story to tell about why P(Y(n+1)|Y(n)) should behave this way, I bring it up only because f(n) does the right thing near 1 and N, and is pretty simple.

To evaluate P(Y(n)), we take the integral of

(c/N)log(t/N)dt

from 1 to n, and exponentiate it. The result is, up to a multiplicative constant

exp(c times (x log x - x)) = (x/e)^(cx)

where x = n/N. I think it's a good idea to leave this as a function of x. Write K for the multiplicative constant. We have P(Proportion x of the population believes X) = K(x/e)^(cx). A graph of this function for K = 1, c = 1 can be found here and a graph of its reciprocal (whose relevance is explained in the parent) can be found here