Haldane's formula in Cannings models: The case of moderately strong selection

For a class of Cannings models we prove Haldane’s formula, $\pi(s_N) \sim \frac{2s_N}{\rho^2}$, for the fixation probability of a single beneficial mutant in the limit of large population size $N$ and in the regime of moderately strong selection, i.e. for $s_N\sim N^{-b}$ and $0<b<1\mathrm{/}2$. Here, $s_N$ is the selective advantage of an individual carrying the beneficial type, and ${\rho }^2$ is the (asymptotic) offspring variance. Our assumptions on the reproduction mechanism allow for a coupling of the beneficial allele’s frequency process with slightly supercritical Galton-Watson processes in the early phase of fixation.