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

## TL;DR

This paper extends Haldane's formula to a class of Cannings models under moderately weak selection, establishing duality with a new ancestral selection graph and deriving fixation probabilities in this regime.

## Contribution

It introduces a Cannings ancestral selection graph and proves Haldane's asymptotics for a range of weak selection intensities, connecting to Moran model results.

## Key findings

- Established duality between Cannings model and ancestral selection graph.
- Proved Haldane's asymptotics for selection strengths satisfying N^{-1} << s_N << N^{-1/2}.
- Linked Cannings model fixation probabilities to Moran model calculations.

## Abstract

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 $N\to \infty$) 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^{-1/2} $, which is a regime of "moderately weak selection". It turns out that for $ s_N \ll N^{-2/3} $ 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.