Lines of descent in a Moran model with frequency-dependent selection and mutation

We study ancestral structures for the two-type Moran model with two-way mutation and frequency-dependent selection that follows the nonlinear dominance or fittest-type-wins scheme. Both schemes lead, in distribution, to the same type-frequency process. Reasoning through the mutation structure on the ancestral selection graph (ASG), we derive processes suitable to determine the type distribution of the present and ancestral population, leading to, respectively, the killed and pruned lookdown ASG. To this end, we establish factorial moment dualities to the Moran model and a relative thereof, respectively. Finally, we extend the results to the diffusion limit.