  We study ancestral structures for the two-type Moran model with mutation and
frequency-dependent selection under the nonlinear dominance or
fittest-type-wins scheme. Under appropriate conditions, both lead, in
distribution, to the same type-frequency process. Reasoning through the
mutations on the ancestral selection graph (ASG), we develop the corresponding
killed and pruned lookdown ASG and use them to determine the present and
ancestral type distributions. To this end, we establish factorial moment
dualities to the Moran model and a relative. We extend the results to the
diffusion limit and present applications for finite population size as well as
moderate and weak selection.
