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 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 beneficial ancestor the probability of fixation is asmptotically equal to the selective advantage $s_N$ divided by half of the offspring variance. For a class of offspring distributions within Kingman attraction we prove this asymptotics for sequences $s_N$ with $N^{-2/3} \gg s_N \gg N^{-1}$. It turns out that in this regime of "moderately weak selection" the Cannings ancestral selection graph is so close to the ancestral selection graph of a Moran model that a suitable coupling argument works. In a companion paper we treat the case of moderately strong selection, $N^{-2/3} \ll s_N\ll 1$, which, other than the case considered in the present paper, admits a more classical approach to Haldane's formula via branching process approximations.