Wright-Fisher diffusions for evolutionary games with death-birth updating

We investigate spatial evolutionary games with death-birth updating in large finite populations. Within growing spatial structures subject to appropriate conditions, the density processes of a fixed type are proven to converge to the Wright-Fisher diffusions with drift. In addition, convergence in the Wasserstein distance of the laws of their occupation measures holds. The proofs of these results develop along an equivalence between the laws of the evolutionary games and certain voter models and rely on the analogous results of voter models on large finite sets by convergences of the Radon-Nikodym derivative processes. As another application of this equivalence of laws, we show that in a general, large population of size $N$, for which the stationary probabilities of the corresponding voting kernel are comparable to uniform probabilities, a first-derivative test among the major methods for these evolutionary games is applicable at least up to weak selection strengths in the usual biological sense (that is, selection strengths of the order $\mathcal O(1/N)$).