$$x''_{sjk1} = \frac{1}{2} \left( \tau_{sjk} x'_{fjk1} + \sum_a \sum_b x'_{makb} \sum_d \tau_{sjd} x'_{fjd1} + \sum_g \sum_h x'_{mjgh} \sum_m \tau_{smk} x'_{fmk1} \right. \\ \left. + \sum_n x'_{njkn} \sum_p \sum_q \tau_{spq} x'_{fpq1} \right) \quad (C.2)$$

$$\begin{aligned} x''_{sjk0} = \frac{1}{2} & (x'_{fjk0} + (1-\tau_{sjk})x'_{fjk1} + \sum_a \sum_b x'_{makb} \sum_d ((1-\tau_{sjd})x'_{fjd1} + x'_{fjd0}) \\ & + \sum_g \sum_h x'_{mjgh} \sum_m ((1-\tau_{smk})x'_{fmk1} + x'_{fmk0}) \\ & + \sum_n x'_{njk} \sum_p \sum_q ((1-\tau_{spq})x'_{fpq1} + x'_{fpq0})) \end{aligned} \quad (C.3)$$

Individuals innovate a trait with a probability of $\nu$, increasing the fraction of encultured individuals, Equation C.4, and decreasing the fraction of unencultured individuals, Equation C.5.

$$x'''_{sk1} = x''_{sjk1} + \nu x''_{sjk0} \quad (C.4)$$

$$x'''_{sjk0} = x''_{sjk0}(1 - \nu) \quad (C.5)$$

As in the main text, I ran numeric simulations of this system for various combinations of teaching costs, $t$, and reproductive benefits of the cultural trait, $b$ and an innovation rate of $r = 0.005$. As in the main text, for each parameter combination, I ran the simulation starting with four initial allele frequencies. In each frequency one allele started at 5% of the population and the rest were evenly distributed among the rest of the population. I started the frequency of the cultural trait at zero in all four initial conditions and ran the simulations until they either converged to a shared equilibrium or ran for $10^7$ generations.

The results of the simulation are shown in Figure C.1