Craster’s Incestual Recursion

He marries his daughters, and they give him more daughters, and on and on it goes.
Edison Tollett

During one of my usual afternoon discussions with fellow graduate students, we strayed on the precarious path of incest and argued as to where incest would fit on the nature vs nurture spectrum. I preferred to attribute the cultural embargo on incest to nurture, as incest had been widespread in antiquity, whereas my friend was of the opinion that the incest taboo is hard-wired in human beings, citing the near-universal forbiddance of sexual relations within close family.

This made me wonder about the sustainability of a population where the only means of population growth (for whatever reason) is incest. Those of you who watch/read Game of Thrones will be able to think of just such a population, the Craster’s keep. Craster (who the above quote refers to) is an old coot living beyond The Wall. He lives there with his daughters who are also his wives, i.e., he incestuously fathers new children with them. All new-born boys are given away (to the White Walkers) so that there is only one male in the house.

To model this population, I came up with the following recursion relation:

$T_n = T_{n-1} + \frac{1}{2}T_{n-16}$

The notation in this equation is as follows: $T$ represents the population of females in the house and $n$ represents the number of years since the population started (so $T_n$ means the population of females in the $n{\textrm{th}}$ year after start). I assume that the probability of a girl being born, the only gender contributing to population growth, is exactly a half (thus the $\frac{1}{2}$). I also assume that the Craster family moved to this house without any young ones and that a daughter turns into a wife and begins to contribute to population growth at the age of 15. Every eligible (for procreation) female in the house generates one healthy offspring per year under my assumption, thus the $T_{n-16}$. I ignore some factors like fertility, menopause, etc. to keep the calculation simple.

The next thing I wanted was to get $T_n$ as a function of $n$ to see how this population changed over time. The standard method of solving such a recurrence relation involves finding the roots of the characteristic polynomial, so I did. And the roots turned out, not surprisingly, to be complex. If the roots $r_1, r_2, \ldots$ are all distinct, the solution to the recurrence relation is

$T_n = k_1r_1^n + k_2r_2^n + \ldots + k_{16}r_{16}^n$

where the coefficients $k$‘s are determined by initial conditions. Here I made the assumption that Craster came with only one wife to the house. Then

$T_n = \left \{ \begin{array}{l l} 0 & \quad n<0 \\ \frac{1}{2} n+1 & \quad 0\leq n<16\end{array} \right .$

The solution to recurrence relation written above can be cast into a matrix form:

$\mathbf{Vk} = \mathbf{T}$

where $\mathbf{k}$ is the vector of coefficients $k_i$, $\mathbf{T}$ is the vector of female populations $T_n$ and $\mathbf{V}$ is the Vandermonde matrix:

$\mathbf{V} = \begin{pmatrix} 1 & 1 & 1 & \cdots & 1 \\ r_1 & r_2 & r_3 & \cdots & r_{16} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ r_1^{15} & r_2^{15} & r_3^{15} & \cdots & r_{16}^{15} \\ \end{pmatrix}$

Since $\mathbf{T}$ is known, I could calculate $\mathbf{k}$ by inverting $\mathbf{V}$. It turns out that there is a neat algorithm that inverts a Vandermonde matrix faster than a standard Gaussian elimination.

Armed with $\mathbf{k}$, I could now get the population at any time. (for $T_n = \mathbf{r}^n\mathbf{k}$, where $\mathbf{r} = (r_1 \, r_2 \, r_3 \, \cdots \, r_{16})$.) This is what the population looks like:

Fascinating! There is a nearly exponential increase in population after 15 years, around the time when there are more than one wives available to Craster. Moreover, It is mentioned that he had nineteen wives during the War of the Five Kings which, using this graph, suggests that it had been 38 years since Craster’s keep started. Assuming he moved there at the age of 20, he should have been 58 during the war. Frankly though, he looks rather old.

PS – 1. I haven’t read the books (I should!), so if any of my assumptions are factually wrong or if there are more details about Craster and his keep that can be explored with this model, let me know!

2. This is my first blog post, so if you spot any rookie mistakes, let me know!

EDIT: (11/23/14) An earlier version of this post erroneously assumed $T_n = n + 1$ for $0 \leq n \leq 16$. Thanks to Darkhan and Liliana for pointed out the error!