samples/texts/2274792/page_9.md

C.3 Disparate Benefits with Transmission Errors

This model is identical to the disparate benefits model in the main text except that teaching has an error rate, $\epsilon$, which is the frequency with which an individual who pays a learning cost fails to learn the behavior. Equation 7 is replaced by Equation C.8 and Equation 8 is replaced by Equation C.9.

$$y_{sk1}^{\mathrm{\prime }\mathrm{\prime }}=\frac{1}{2}(r_{sk}-ϵ)\underset{\begin{array}{c}\text{Allele}\\ \text{from mother}\end{array}}{\underbrace{(y_{fk1}^{\mathrm{\prime }}+(y_{mk0}^{\mathrm{\prime }}+y_{mk1}^{\mathrm{\prime }})\sum _h{y}_{fh1}^{\mathrm{\prime }})}}(C.8)y\text{'}\text{'}\_\{sk1\}\; =\; \backslash frac\{1\}\{2\}(r\_\{sk\}\; -\; \backslash epsilon)\; \backslash underbrace\{\backslash left(\; y\text{'}\_\{fk1\}\; +\; (y\text{'}\_\{mk0\}\; +\; y\text{'}\_\{mk1\})\; \backslash sum\_h\; y\text{'}\_\{fh1\}\; \backslash right)\}\_\{\backslash substack\{\backslash text\{Allele\}\; \backslash \backslash \; \backslash text\{from\; mother\}\}\}\; \backslash quad\; (C.8)$$

$$y_{sk0}^{\mathrm{\prime }\mathrm{\prime }}=\frac{1}{2}\underset{\begin{array}{c}\text{Allele from mother}\\ \text{Allele from father}\end{array}}{\underbrace{(y_{fk0}^{\mathrm{\prime }}+(1-r_{sk}+ϵ)y_{fk1}^{\mathrm{\prime }}+(y_{mk0}^{\mathrm{\prime }}+y_{mk1}^{\mathrm{\prime }})\sum _h(y_{fh0}^{\mathrm{\prime }}+(1-r_{sk}+ϵ)y_{fh1}^{\mathrm{\prime }}))}}(C.9)y\text{'}\text{'}\_\{sk0\}\; =\; \backslash frac\{1\}\{2\}\; \backslash underbrace\{\backslash left(\; y\text{'}\_\{fk0\}\; +\; (1-r\_\{sk\}+\backslash epsilon)y\text{'}\_\{fk1\}\; +\; (y\text{'}\_\{mk0\}+y\text{'}\_\{mk1\})\; \backslash sum\_h\; (y\text{'}\_\{fh0\}\; +\; (1-r\_\{sk\}+\backslash epsilon)y\text{'}\_\{fh1\})\; \backslash right)\}\_\{\backslash substack\{\backslash text\{Allele\; from\; mother\}\; \backslash \backslash \; \backslash text\{Allele\; from\; father\}\}\}\; \backslash quad\; (C.9)$$

As in the main text, I ran numeric simulations of this system for various combinations of female net benefits, $b_f - \mu$ and male net benefits $b_m - \mu$, and an innovation rate of $r = 0.005$. Since more than 90% of females learn the trait from their mother I conservatively set $\epsilon$ to 0.1 as the error rate for the simulation. 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 converged to a shared equilibrium or ran for $10^7$ generations.