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

We introduce a Cannings model with directional selection via a paintbox construction and establish a strong duality with the line counting process of a new \emph{Cannings ancestral selection graph} in discrete time. This duality also yields a formula for the fixation probability of the beneficial type. Haldane’s formula states that for a single selectively advantageous individual in a population of haploid individuals of size $N$ the prob-ability of fixation is asymptotically (as $N\to \mathrm{\infty }$) equal to the selective advantage of haploids $s_N$ divided by half of the offspring variance. For a class of offspring distributions within Kingman attraction we prove this asymptotics for sequences $s_N$ obeying $N^{-1}\ll s_N\ll N^{-1\mathrm{/}2}$, which is a regime of “moderately weak selection”. It turns out that for $s_N\ll N^{-2\mathrm{/}3}$ the Cannings ancestral selection graph is so close to the ancestral selection graph of a Moran model that a suitable coupling argument allows to play the problem back asymptotically to the fixation probability in the Moran model, which can be computed explicitly.