The dynamics and consequences of host–parasite coevolution depend on the nature of host genotype-by-parasite genotype interactions (G × G) for host and parasite fitness. G × G with crossing reaction norms can yield cyclic dynamics of allele frequencies (“Red Queen” dynamics) while G × G where the variance among host genotypes differs between parasite genotypes results in selective sweeps (“arms race” dynamics). Here, we investigate the relative potential for arms race and Red Queen coevolution...