Consider the following discrete-time population model. We start with a single "good" individual who reproduces asexually into $k$ children and dies in the process. At generation $t=2$, those children themselves each asexually reproduce into $k$ children of their own and die in the process. That is, at generation $t=n$, there are $k^n$ individuals.


Good individuals' children are good with probability $1-q$ and evil with probability $q$.


Evil individuals' children are evil with probability $1-q$ and good with probability $q$.


In all of this, $q<1/2$, with the idea that good individuals are more likely to beget good individuals and evil individuals are more likely to beget evil individuals.




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I am most interested in the asymptotic behavior of the probability that the good individuals outnumber the evil individuals. That is, even if we start with a good individual with a moderately high probability of begetting good individuals, might this asymptotic probability tend to $1/2$, with good and evil equally matched?


Calling the number of good individuals $G\_n$ and the number of evil individuals $E\_n$, I am curious about $\lim\_{n \to \infty} Pr(G\_n > E\_n) = f(q)$. In particular, one might expect that if $q$ is close enough to $1/2$ that the asymptotic probability $f(q)=1/2$, while if $q$ is close enough to $0$ that $f(q) > 1/2$.


Indeed, $f(q)$ can be shown through Chebyshev's inequality to be strictly greater than $1/2$ for $q$ small enough, and $f(1/2) = 1/2$.


My question is, is there some $0<q\_c<1/2$ for which


$$f(q)=\begin{cases}
>1/2 & q<q\_c \\
1/2 & q\_c< q \leq 1/2
\end{cases}? $$


Note that the $k$ dependence is implicit in the above; I expect $q\_c$ is a function of $k$. An explicit form for $f(q)$ would also be very interesting.



Population growth with good and evil children - probability good outnumbers evil
The process you describe has been studied extensively as a mutation model [5], as a model of broadcasting on trees [2], as a representation of the Ising model on the Bethe lattice [1]. A very general relevant analysis in the context of mutitype branching processes is in [3], some of the results there were refined in [4].


Your intuition was correct. There is a phase transition, and the critical value $q\_c$ on the $k$-ary tree is given by the equation
$$(1-2q\_c)^2=1/k \,,$$
see [1] or Theorem 1.1 in [2]. For $q \ge q\_c$, one has (with your notation) $f(q)=1/2$, while for $q<q\_c$ one has $f(q)>1/2$.


[1] BLEHER, P. M., Ruiz, J. and ZAGREBNOV, V. A. (1995). On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice. J Statist. Phys. 79 473-482.


[2] Evans, William, Claire Kenyon, Yuval Peres, and Leonard J. Schulman. "Broadcasting on trees and the Ising model." Annals of Applied Probability (2000): 410-433. <https://www.jstor.org/stable/2667156>


[3] KESTEN, H. and STIGUM, B. P. (1966). Additional limit theorems for indecomposable multi-dimensional Galton-Watson processes. Ann. Math. Statist. 37 1463-1481.


[4] Mossel, Elchanan, and Yuval Peres. "Information flow on trees." The Annals of Applied Probability 13, no. 3 (2003): 817-844.


[5] STEEL, M. (1989). Distribution in bicolored evolutionary trees. Ph.D. thesis, Massey Univ., Palmerston North, New Zealand.


